Research ArticleCongestion Behavior under Uncertainty on Morning Commutewith Preferred Arrival Time Interval

LingLing Xiao1 Ronghui Liu2 and HaiJun Huang1

1 School of Economics and Management Beihang University Beijing 100191 China2 Institute for Transport Studies University of Leeds Leeds LS2 9JT UK

Correspondence should be addressed to Ronghui Liu rliuitsleedsacuk

Received 5 November 2013 Accepted 18 December 2013 Published 17 February 2014

Academic Editor Huimin Niu

Copyright copy 2014 LingLing Xiao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper extends the bottleneck model to study congestion behavior of morning commute with flexible work schedule Theproposed model assumes a stochastic bottleneck capacity which follows a uniform distribution and homogeneous commuters whohave the same preferred arrival time intervalThe commuters are fully aware of the stochastic properties of travel time and scheduledelay distributions at all departure times that emerge from day-to-day capacity variations The commutersrsquo departure time choicefollows user equilibrium (UE) principle in terms of the expected trip cost Analytical and numerical solutions of this model areprovided The equilibrium departure time patterns are examined which show that the stochastic capacity increases the mean tripcost and lengthens the rush hour The adoption of flexitime results in less congestion and more efficient use of bottleneck capacitythan fixed-time work schedule The longer the flexi-time interval is the more uniformly distributed the departure times are

1 Introduction

The bottleneck model was first proposed by Vickrey [1] andhas subsequently inspiredmany to developmore realistic ext-ensions to gain qualitative and theoretical insights into tran-sport policy measures on commuter travel behavior (egSmith [2] Daganzo [3] Braid [4] Arnott et al [5] Yangand Huang [6] Ramadurai et al [7]) In these analysescommuters must choose their departure times to minimizethe sum of travel delays (depending on the bottleneckcapacity) and schedule delays (depending on the differencebetween actual and desired arrival times) At user equilib-rium no commuter could reduce the total commute cost byunilaterally changing hisher departure time

The study of decision making under risk (and uncer-tainty) has a long history in the fields of economics psychol-ogy transport and beyond (Machina [8] Cubitt and Sugden[9] Gollier and Treich [10] Birnbaum and Schmidt [11])Manymathematical tools and analytical frameworks are usedtomodel peoplersquos behavior and predict likely choice outcomesin varying settingsThe theory has also evolved fromexpectedutility to nonexpected utility and been used to describe choicebehavior in risky situations (Starmer [12] de Palma et al [13])

This paper applies the expected utility theories to cap-ture the departure time choice behavior in morning com-mute problem under uncertainty Morning commute playsan important role in a monocentric city and the trafficcongestion in such network is caused by concentration oftravel demand around the work start time The introductionof flexible work schedule is one of the transport demandmanagement measures for alleviating peak congestion Thepaper provides useful insight into travelerrsquos decision makingin contrast to fixed-time schedule pattern Henderson [14]incorporated the productivity effect to analyze the equilib-rium and optimum solution with staggered work hours Munand Yonekawa [15] extended bottleneck congestion to studythe case that part of the firms in city adopts flexitime andincorporated the effects on urban productivity

Most literatures onmorning commute have assumed thatthe capacity of the bottleneck is deterministic and the trafficdemand is also deterministic or governed by a predeterminedelastic demand function In reality not only does traveltime increase as traffic volume increases towards capacitybut also the travel time becomes increasingly random andunpredictable due to the chaotic behavior of traffic at themicro level The source of variation in road capacity may

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 767851 9 pageshttpdxdoiorg1011552014767851

2 Discrete Dynamics in Nature and Society

occur due to physical and operational factors such as roadrepairs construction accidents and bad weather The varia-tions in road capacity from physical and operational reasonsare what make the analysis of travel behavior so complex andyet interesting As such understanding travelersrsquo attitudesand their behavior in varying settings is key to developingsustainable transport policesThere has been recent attentionto the stochastic nature of the bottleneck models (Siu andLo [16] Li et al [17] Xiao et al [18]) A reasonable way tocapture these variations and their impact on the networkperformance is to formulate the problem using probabilitydistributions (Chen et al [19])

The focus of this paper is to analyze the departure timechoice behavior under uncertainty in the morning commut-ing problem with flexi-time work schedule It is expectedthat the stochastic capacity leads to uncertainty in queuingtravel time and trip cost which in turn influences thecommutersrsquo travel choice behavior We assume that travelersare fully aware of the stochastic properties of the travel timeand schedule delay distributions throughout the morningpeak period which emerges from their day-to-day travelexperience Furthermore we consider homogenous travelershave the same preferred arrival time interval (PATI) Weformulate a stochastic bottleneck model for this flexi-timecommute problem and derive its analytical solution Theproperties of the model are investigated

The solution of the proposedmodel shows that the capac-ity variability of the bottleneck leads to significant changes indeparture time patterns which are different to those derivedunder deterministic conditions In a deterministic bottleneckmodel with flexi-time work schedule an individual canchoose either to depart in the tails of the rush hour whentravel time is low and pay the penalty of arriving at workearly or late or to depart close to the PATI when traveltime is high but schedule delay cost is low In other wordsunder the deterministic equilibrium schedule delay earlylateand arrival on time cannot occur simultaneously for a givendeparture time (Henderson [14] Mun and Yonekawa [15])We demonstrate that with day-to-day stochastic capacitycommuters departing at the same time may endure scheduledelay earlylate or not and may experience queuing delay ornot on different days

The rest of this paper is organized as follows Section 2formulates separately a deterministic and a stochastic bottle-neck model with homogeneous PATI The equilibrium solu-tions for the departure time pattern are derived for each caseThe theoretical properties of the proposed stochastic bottle-neck model with PATI are investigated and compared withthe deterministic case Numerical examples are presented inSection 3 to illustrate further the equilibrium properties ofthe model Section 4 provides conclusion remarks

2 Departure Patterns of Morning Commutewith Flexible Arrival Time

21 The Deterministic Case We formulate the peak periodcongestion based on the bottleneck model developed byVickrey [1] Suppose a single road connecting a residential

area and the Commercial Business District (CBD) which hasa bottleneck just before the CBD It is assumed that vehiclesdrive at constant speed from home to the bottleneck pointtravel time for this portion of trip is constant and representedas 119879free Queue develops when traffic flow rate exceeds thebottleneck capacity 119904 Travel time for a vehicle departing attime instant 119905 is represented as follows

119879 (119905) = 119879free +119876 (119905)

119904 (1)

where 119879(119905) is travel time and 119876(119905) is length of queue Thesecond term represents the waiting time within the queuebehind the bottleneck We set 119879free = 0 hereafterThis settingwill not affect the qualitative property

The queue length that a trip maker departing at time 119905encounters is calculated as follows

119876 (119905) = maxint119905

1199050

[119903 (119909) minus 119904] d119909 0 (2)

and the cumulative departures as

119877 (119905) = int

119905

1199050

119903 (119909) d119909 (3)

where 119903(119909) is the departure rate at time instant 119905 and 1199050is the

earliest time with positive departure rateSuppose that every morning a fixed number of 119873

individuals commute from home to office located in theCBD driving along the road stated above All workers haveidentical skills and preferences Unlike Vickrey [1] whichassumes that there is only one preferred arrive time 119905lowast herefirms in the CBD adopt a flexi-time work schedule such thatemployees arriving at office earlier or later 120575(120575 ge 0) than 119905lowastincur no scheduling cost Hereafter we call this time period[119905lowastminus 120575 119905lowast+ 120575] as PATI

Some commutersmay still arrive at the destination earlieror later than PATI in order to avoid a long queue at thebottleneck The cost for commuters traveling from home tothe CBD consists of three components the cost of traveltime and the cost of schedule delay early or late It can beformulated as follows

119862 (119905) = 120572119879 (119905) + SDE (119905) + SDL (119905) (4)

where 120572 is the value of travel timeThe cost of schedule delayearly (SDE) and schedule delay late (SDL) for a commuterwho leaves home at time 119905 can be expressed as

SDE (119905) = 120573 (119905lowastminus 120575 minus (119905 + 119879 (119905)))

SDL (119905) = 120574 (119905 + 119879 (119905) minus (119905lowast+ 120575))

(5)

where 120573 and 120574 denote the value of schedule delay early andthe value of schedule delay late respectively

Discrete Dynamics in Nature and Society 3

0

Num

bers

of t

rave

lers

Cumulative departures

Cumulative arrivals

Departure times

120579 = 1N

T(t2)

T(t1)

r(t)

t0 t1 t2 120575 teminus120575

Figure 1 Departure time distributions in the deterministic case

Substituting (1) and (5) into (4) and applying the equi-librium condition 120597119862(119905)120597119905 = 0 we have the equilibriumdeparture rate as follows

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 lt 119905

1

119904 if 1199051le 119905 le 119905

2

120572119904

(120572 + 120574) if 119905

2lt 119905 le 119905

119890

(6)

where 1199050and 119905119890are the earliest time and the latest time with

positive departure rate respectively And 1199051and 1199052are the

watershed times for which an individual arrives at work ontime Then

1199050= 119905lowast+120574 minus 120573

120573 + 120574120575 minus

120574

120573 + 120574sdot119873

119904

1199051=120572 minus 120573

120572119905lowastminus120572 minus 120573

120572120575 minus

120573120574

120572 (120573 + 120574)sdot119873

119904

1199052=120572 minus 120573

120572119905lowast+120572 + 120573

120572120575 minus

120573120574

120572 (120573 + 120574)sdot119873

119904

119905119890= 119905lowast+120574 minus 120573

120573 + 120574120575 +

120573

120573 + 120574sdot119873

119904

(7)

Figure 1 depicts the cumulative departures (from home)and arrivals (at CBD) in equilibrium under the deterministiccapacity case For simplicity we set 119905lowast to be zero and then alltravelers have the same PATI [minus120575 120575] The horizontal distancebetween the departure and arrival curves gives the travel timeSeveral critical time points are indicated Travelers departingat time interval [119905

1 1199052] have the longest travel time but will

arrive on time that is within PATI Travelers departing before1199051will arrive earlier than desiredwhile the travelers departing

after 1199052will arrive late at the destination

22 The Stochastic Case The deterministic case models asingle-day departure time equilibrium In the real world road

capacitiesmay vary from day to day due to unexpected eventssuch as incidents and weather conditions Because of thecapacity fluctuations both commutersrsquo travel time and theirschedule delays are stochastic In this section we hypothesizethat a constant long-term departure time patternmay emergegiven the responses of the travelers to the day-to-day capacityvariation Each commuter chooses an optimal departure timewhich minimizes hisher long-term expected trip cost Wecall this pattern if it exists a long-term equilibrium pattern

221 Assumptions and Travelersrsquo Cost Function The follow-ing assumptions are made in the model formulation

(A1) Commuters are homogeneouswith the same120572120573 and120574 values and the same PATI

(A2) The capacity of the bottleneck is constant within aday but fluctuates from day to day The uncertaintyof capacity is completely exogenous and independentof departures

(A3) The capacity is a nonnegative stochastic variablechanging around a certain mean capacity FollowingLi et al [17] we assume that stochastic capacityfollows a uniform distribution within interval [120579119904 119904]where 119904 is the design capacity and 120579(lt1) is a positiveparameter which denotes the lowest rate of availablecapacity

(A4) Commuters are aware of the capacity degenerationprobability and their departure time choice followsthe user equilibrium (UE) principle in terms of meantrip cost

We assume that the capacity of the single bottleneckis stochastic but the commutersrsquo departure time choice isdeterministic The calculation of the mean trip cost relies onthe calculations of the mean travel time the mean scheduledelay early and late For simplicity we set the 119905lowast to be zeroand then the PATI becomes [minus120575 120575] Under the stochasticcondition the mean trip cost with respect to departure time119905 can be formulated as follows

119864 [119888 (119905)] = 119864 [120572119879 (119905) + SDE (119905) + SDL (119905)] (8)

The equilibriumcondition for commutersrsquo departure timechoice in a single bottleneckwith stochastic capacity is that nocommuter can reduce hisher mean trip cost by unilaterallyaltering hisher departure time This condition implies thatthe commutersrsquomean trip cost is fixedwith respect to the timeinstant with positive departure rate That is

120597119864 [119888 (119905)]

120597119905= 0 if 119903 (119905) gt 0 (9)

222 Mathematical Formulations and Derivations Due tothe stochastic capacity over days the travel time experiencedby a traveler departing at the same time 119905 varies from day today This is equivalent to saying that commuters departinghome to work at the same time may endure schedule delayearlylate or not and may experience queuing delay or not ondifferent days Consequently there are six situations which

4 Discrete Dynamics in Nature and Society

may occur (I) always arrive early (II) possibly arrive earlyor on time (III) always arrive on time (IV) possibly arriveon time or late (V) always arrive late and (VI) always arrivelate but queue may exist We propose the following simpleextension of trip cost function under the six situations tomodel user departure time choice under degradable capac-ities and we use 119905

1 1199052 1199053 1199054 and 119905

5to denote the watershed

lines separating the six casesAs we assume that stochastic capacity is completely

exogenous and independent of departure flows the expec-tation of travel time and schedule delay cost with respect todifferent situations can be derived respectively as follows

119864 [119879 (119905)]

=

int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 119905)119891 (119904) 119889119904 119905

0le 119905 le 119905

5

int

119877(119905)(119905minus1199050)

120579119904

(119877 (119905)

119904minus 119905 + 119905

0)119891 (119904) 119889119904 119905

5lt 119905 le 119905

119890

119864 [SDE (119905)]

=

120573int

119904

120579119904

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

0le 119905 le 119905

1

120573int

119904

119877(119905)(minus120575minus1199050)

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

1lt 119905 le 119905

2

119864 [SDL (119905)]

=

120574int

119877(119905)(120575minus1199050)

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

3lt 119905 le 119905

4

120574int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

4le 119905 le 119905

5

(10)

where 119891(119904) = 1(119904 minus 120579119904) Substituting (10) into (8) we get theexpected trip cost with respect to each situation Accordingto (9) the equilibrium departure rates for the six situationscan be expressed as follows

Situation 1 No commuters experience schedule delay latersubject to all possible values of the bottleneck capacity Weget the departure rate

119903 (119905) =120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1 1199050le 119905 lt 119905

1 (11)

The boundary condition for this situation is SDL(1199051) = 0

when 119904 = 120579119904 and hence we have 119877(1199051) = minus(119905

0+ 120575)120579119904

Situation 2 If the capacity of the bottleneck is large enoughonly schedule delay early will occur On the contrary noschedule delay occurs when the capacity is small The water-shed capacity satisfies 119879(119905) + 119905 = minus120575 Equivalently we have119904 = 119877(119905)(minus120575 minus 119905

0) and the departure rate

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

1199051lt 119905 le 119905

2

(12)

The boundary condition for this case is SDL(1199052) = 0when

119904 = 119904 and hence we have 119877(1199052) = minus(119905

0+ 120575)119904

Situation 3 No commuters experience schedule delay subjectto all possible values of the bottleneck capacityTherefore thedeparture rate is

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1 1199052lt 119905 le 119905

3 (13)

The boundary condition for this case is SDE(1199053) =

SDL(1199053) = 0 when 119904 = 120579119904 Hence we have 119877(119905

3) = (120575 minus 119905

0)120579119904

Situation 4 If the capacity of the bottleneck is large enoughindividuals arrive on time On the contrary schedule delaylate occurs when the capacity is smallThewatershed capacitysatisfies 119879(119905) + 119905 = 120575 Equivalently we have 119904 = 119877(119905)(120575 minus 119905

0)

Therefore

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

1199053lt 119905 le 119905

4

(14)

The boundary condition for this case is SDE(1199054) = 0when

119904 = 119904 and then we have 119877(1199054) = (120575 minus 119905

0)119904

Situation 5 Similar to Situation 1 we have

119903 (119905) =120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1 1199054lt 119905 le 119905

5 (15)

The boundary condition for this case is 119877(1199053) = 119904(119905

5minus 1199050)

that is the queue length at time 1199055equals to zero when 119904 = 119904

Situation 6 Similar to Situation 2 we can find a watershedcapacity of the bottleneck such that the queue length equalszero that is119877(119905) = 119904(119905minus119905

0) and hence the watershed capacity

is 119877(119905)(119905 minus 1199050) We have

119903 (119905) =d119877 (119905)d119905

=(120572 + 120574) 119877 (119905) (119905 minus 119905

0) minus (120572120579 + 120574) 119904

(120572 + 120574) (ln119877 (119905) minus ln 120579119904 (119905 minus 1199050))

1199053lt 119905 le 119905

119890

(16)

The boundary condition for this case is 119903(119905119890) = 0

Equivalently we have 119877(119905119890) = 119904(119905

119890minus 1199050) where 119904 = 119904(120572120579 +

120574)(120572 + 120574) is the mean capacity under the stochastic case

223 Determination of the Watershed Time Instants Sincethe departure rate 119903(119905) = 0 if 119905 gt 119905

119890 the cumulative departures

at time 119905119890are equal to the traffic demand that is 119877(119905

119890) =

119873 = 119904(119905119890minus 1199050) Therefore we have 119905

119890= 1199050+ 119873119904 Moreover

the equilibrium condition of the stochastic bottleneck impliesthat 119864[119888(119905

0)] = 119864[119888(119905

119890)] and hence we have

1199050=119873

119904sdot

1

1205960minus 1

+1205920

1 minus 1205960

120575 119905119890=119873

119904sdot

1205960

1205960minus 1

+1205920

1 minus 1205960

120575

(17)

where

1205960= 1 minus

(1 minus 120579) (120573 + 120574)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

1205920=

(1 minus 120579) (120574 minus 120573)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

(18)

Discrete Dynamics in Nature and Society 5

Using the boundary conditions of Situations 1ndash6 we canobtain the watershed lines as follows

1199051= 12059611199050+ 1205921120575 119905

2= 12059621199050+ 1205922120575

1199053= 12059631199050+ 1205923120575 119905

4= 12059641199050+ 1205924120575

1199055= 12059651199050+ 1205925120575

(19)

where

1205961= 1 minus

(120572 minus 120573) 120579120585

120572 120596

2=(120572 + 120573)

120572minus 120585

1205963=120573

120572minus 120579120585 + 1 120596

4=(120572 + 120574) (1 minus 120585)

120572+120573

120572

1205965= 1 +

(120574 + 120573)

(120572 minus (120572 + 120574) 120585) 120592

1=(120573 minus 120572) 120579120585

120572

1205922=120573

120572minus 120585 120592

3=120573

120572+ 120579120585

1205924=((120572 + 120574) 120585 + (120573 minus 120574))

120572 120592

5=

(120573 minus 120574)

(120572 minus (120572 + 120574) 120585)

120585 =ln 120579minus1

(1 minus 120579)

(20)

With the resulting stochastic departure pattern at long-term equilibrium the experienced day-to-day travel timesand number of travelers experiencing queues change accord-ing to varied capacities over days Given the boundary condi-tions of Situations 1ndash6 the cumulative departures and arrivalsof a stochastic bottleneck are given in Figure 2 The two solidcurves denote the cumulative departures and cumulativearrivals as in Figure 1 The dotted lines are the maximumand the minimum capacity of the bottleneck that is 119904 and120579119904 The earliest departure time from home to workplace is 119905

0

and if the bottleneck capacity equals 120579119904 commuters departingbetween 119905

1and 1199053will arrive at workplace on time If the

capacity equals 119904 commuters departing between 1199052and 1199054will

arrive at workplace on time Commuters departing withintime interval [119905

2 1199053] will always arrive at workplace on time

whatever the variability of capacity is At the beginningcommuters depart from home with a constant departurerate until the watershed time 119905

1 Subsequently the departure

rate will gradually drop down After time instant 1199054 another

constant departure rate will last until the watershed timeinstant 119905

5and the queuing length at this time will be zero

Later the departure rate continues to drop down to zero attime 119905

119890

224 Properties of the Stochastic Bottleneck Model In thissubsection we investigate the theoretical properties of theequilibrium solution of the proposed stochastic bottleneckmodel with PATI

Theorem 1 At equilibrium the expected trip cost for allcommuters is a monotonically increasing function of trafficdemand and a monotonically decreasing function of 120575-valuethat is 120597119864[119888(119905

0)]120597119873 gt 0 and 120597119864[119888(119905

0)]120597120575 lt 0 hold

Proof Since 119864[119888(1199050)] = (minus120575 minus 119905

0)120573 and 119905

0= 119873(119904(120596

0minus 1)) +

1205920120575(1 minus 120596

0) we have 120597119864[119888(119905

0)]120597120575 = minus2120573120574(120573 + 120574) lt 0 and

0Departure times

Cumulative departures

Cumulative arrivals

PATI

120579 = 09

0

Num

bers

of t

rave

lers

N

T(t2)

T(t3)T(t4)

T(t1)

t0 t1 t2 t3 t4 t5 te

120579s

s

Figure 2 Departure time distributions in the stochastic case

120597119864[119888(1199050)]120597119873 = 120573((1minus120596

0)119904)The definitions of120573 and 119904 imply

that they are positive hence to prove 120597119864[119888(1199050)]120597119873 gt 0 we

only need to prove 1 minus1205960gt 0 Since 0 lt 120579 lt 1 and 119904 = 119904(120572120579 +

120574)(120572 + 120574) gt 119904(120572120579 + 120574120579)(120572 + 120574) = 120579119904 we have 1 minus 120579 gt 0 Andln 119904 minus ln 120579119904 gt 0 clearly holds Therefore both the numeratorand denominator of the second term in the right-hand sideof the first equation of (18) are positive therefore we have1 minus 1205960gt 0 This completes the proof

Theorem 2 At equilibrium the expected trip cost for allcommuters is amonotonically decreasing function of parameter120579-value that is 120597119864[119888(119905

0)]120597120579 lt 0 holds

Proof Submitting 1199050= 119873(119904(120596

0minus 1)) + 120592

0120575(1 minus 120596

0) into

119864[119888(1199050)] = (minus120575 minus 119905

0)120573 then the first-order derivative can be

given as follows

120597119864 [119888]

120597120579= 120573

119873

119904sdot120572 + 120574

120573 + 1205741199011015840(120579) (21)

where

119901 (120579) =ln 119904 minus ln (120579119904)

1 minus 120579

1199011015840(120579) =

1

1 minus 120579(119901 (120579) minus

120574

120579 (120572120579 + 120574))

(22)

Since ln(119904(120579119904)) lt 1 minus 120579119904119904 = 120574(1 minus 120579)(120572120579 + 120574) we then have

119901 (120579) minus120574

120579 (120572120579 + 120574)

lt120574

120572120579 + 120574minus

120574

120579 (120572120579 + 120574)=

120574

120572120579 + 120574(1 minus

1

120579) lt 0

(23)

It is clear that 1199011015840(120579) lt 0 and therefore we get 120597119864[119888]120597120579 lt 0This completes the proof

Theorem 3 At equilibrium the departure rate is a monoton-ically decreasing function of the departure time throughout thewhole peak period that is d119903(119905)d119905 le 0 119905 isin [119905

0 119905119890]

6 Discrete Dynamics in Nature and Society

Proof According to (11)ndash(16) the departure rate 119903(119905)

is continuous within each of intervals [1199050 1199051) (1199051 1199052) (1199052 1199053)

(1199053 1199054) (1199054 1199055) and (119905

5 119905119890] To prove the departure rate

is continuous during the peak period we calculate thefollowing limitations

lim119905rarr 119905+

1

119903 (119905) = lim119905rarr 119905+

1

120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

1

119903 (119905)

lim119905rarr 119905minus

2

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

2

119903 (119905)

lim119905rarr 119905minus

3

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1

=120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

= lim119905rarr 119905+

3

119903 (119905)

lim119905rarr 119905minus

4

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

4

119903 (119905)

lim119905rarr 119905+

5

119903 (119905) =(120572 + 120574) 119877 (119905

3) (1199053minus 1199050) minus (120572120579 + 120574) 119904

(120572 + 120574) ln (119877 (1199053) (120579119904 (119905

3minus 1199050)))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

5

119903 (119905)

(24)

This proves 119903(119905) is continuous indeed within the interval[1199050 119905119890]

Equations (11) (13) and (15) state that the departure rate119903(119905) is constant for 119905

0le 119905 le 119905

1 1199052le 119905 le 119905

3 and 119905

4le 119905 le

1199055 hence it is monotonically decreasing within these three

intervals By definition the cumulative departure flow 119877(119905) isnondecreasing with respect to time 119905Thus the denominatorsof the right-hand sides in (12) and (14) are nondecreasingwith respect to time 119905 Therefore the right-hand sides of (12)and (14) are nonincreasing with respect to time 119905 that is thedeparture rate 119903(119905) is monotonically decreasing within [119905

1 1199052]

and [1199053 1199054] The proof of d119903(119905)d119905 le 0 for all 119905 isin (119905

5 119905119890] can be

found in Xiao et al [18]In summary the departure rate 119903(119905) is monotonically dec-

reasing within all four intervals and at their boundaries Con-sidering the continuity of 119903(119905) for all 119905 isin [119905

0 119905119890] we conclude

that 119903(119905) is monotonically decreasing within [1199050 119905119890] This

completes the proof

Proposition 4 When parameter 120579 approaches one thestochastic bottleneck model immediately follows the determin-istic model

Proof According to the LrsquoHopitalrsquos rule we have lim120579rarr1

(1 minus

120579)(ln 120579minus1) = 1 We then have

lim120579rarr1

1205960= lim120579rarr1

1205965=minus120573

120574

lim120579rarr1

1205961= lim120579rarr1

1205962= lim120579rarr1

1205963= lim120579rarr1

1205964=120573

120572

lim120579rarr1

1205920= lim120579rarr1

1205925=(120574 minus 120573)

120574

lim120579rarr1

1205921= lim120579rarr1

1205922=(120573 minus 120572)

120572 lim

120579rarr1

1205923= lim120579rarr1

1205924=(120573 + 120572)

120572

(25)

lim120579rarr1

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 le 119905

1

119904 if 1199051le 119905 le 119905

4

120572119904

(120572 + 120574) if 119905

4lt 119905 le 119905

119890

(26)

Substituting (26) into (19) we have

1199050= minus

120574

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575 119905

5= 119905119890=

120573

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575

1199051= 1199052= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120573 minus 120572

120572120575

1199053= 1199054= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120572 + 120573

120572120575

(27)

Therefore we get the same traffic flow pattern with that froma deterministic bottleneck model

Proposition 5 When the number of commuters is givenincreasing the value of parameter 120579 will result in a decrease inthe length of peak period

Proof According to (16) we have d119904d120579 = 120572(120572 + 120574) gt 0This implies that 119904 is a monotonic increasing function withrespect to 120579 According to (17) we can obtain the length ofpeak period as follows

119905119890minus 1199050=119873

119904

1205960

1205960minus 1

+1205920

1 minus 1205960

120575 minus119873

119904

1

1205960minus 1

minus1205920

1 minus 1205960

120575 =119873

119904

(28)

Since119873 is constant and 119904 is a monotonic increasing functionwith respect to 120579 119905

119890minus 1199050is also a monotonic increasing

function with respect to 120579 This completes the proof

The above proof also shows that the length of peak periodis not affected by the 120575-value

3 Numerical Examples

The input parameters of our numerical example are 120572 =

64 $h 120573 = 39 $h 120574 = 1521 $h 119873 = 6000 veh 119904 =

4000 vehh 120575 = 10 min and 120579 = 09 By solving the proposed

Discrete Dynamics in Nature and Society 7

Table 1 The influence of parameter 120579 on the mean trip cost and the watershed time instant

120579 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

100 362 minus110 minus074 minus074 minus040 minus040 040 040 150095 378 minus114 minus077 minus073 minus046 minus031 036 039 152090 395 minus118 minus080 minus073 minus052 minus021 031 037 155085 413 minus123 minus085 minus072 minus059 minus009 026 035 157080 433 minus128 minus089 minus071 minus066 005 021 032 160

minus15 minus1 minus05 0 050

1

2

3

4

Departure time (hour)

Expe

cted

cost

($)

Total travel costTravel time cost

SDE costSDL cost

120579 = 09 120575 = 10min

Figure 3 The mean equilibrium trip cost and other components

model we obtained the time period with positive departurerate during [minus118 037] (see Table 1) The mean trip cost themean travel time cost and themean schedule delay cost (SDEand SDL) are depicted in Figure 3 We find that the mean tripcosts of all commuters are the same and are equal to 395 $but endure a trade-off between cost of travel time and cost ofschedule delay It is interesting that the waiting time cost isnot zero at the end of the peak period indicating that queuestill exists

It is interesting to investigate the impact of the fractionparameter 120579 on the solution of the stochastic bottleneckmodel We changed the parameter 120579 from 08 to 10 andcomputed themean trip cost and the watershed time instantsThe results are shown in Table 1 the row highlighted is withthe default 120579-value It can be seen that 119905

1= 1199052 1199053= 1199054 and 119905

5=

119905119890when 120579 = 10This confirmsProposition 4We can also find

that the length of time period with positive departure rateincreases with the 120579-value This confirms Proposition 5 Inaddition the second column of Table 1 shows that the meantrip cost decreases with the increase of 120579-valueThis confirmsTheorem 2 Since decreasing the 120579-value is equivalent toincreasing the travel time uncertainty commuters will leavehome earlier than before for avoiding the potential losscaused by uncertainty risk

The departure rates against different 120579-values are dis-played in Figure 4 It can be clearly seen that the resultsconfirm Propositions 4 and 5 that the stochastic bottleneckmodel immediately follows the deterministic model when

120579 = 10

120579 = 09

120579 = 08

0

2

4

6

8

10

Dep

artu

re ra

te (v

ehh

)

minus15 minus1 minus05 0 05Departure time (hour)

times103

120575 = 10min

Figure 4 The influence of parameter 120579 on departure rate

120579 approaches one and that enlarging the parameter 120579 willresult in a decrease in the length of peak period Figure 4 alsoshows that in equilibrium the departure rate during the peakperiod is monotonically decreasing which is consistent withTheorem 3

Table 2 lists the mean trip costs and the watershed timeinstants when 120579 = 09 and 120575-value from 0 minutes to 20minutes It is shown that in equilibrium the mean trip cost ismonotonically decreasing with increasing 120575 whilst the lengthof the peak period remains unchangedThis is consistent withTheorem 1

Figure 5 depicts the departure rates against different 120575-values at 120579 = 09 It can be seen that with larger 120575-value (orlonger PATI) the peak periods shift to later and the amountof earlier departures (at a rate higher than capacity) reducesThe overall departure time patterns become more flat andapproach designed capacity of the bottleneck with increasing120575-valueThis suggests that traffic congestion can be alleviatedby adopting flexi-timework schedule similar to that achievedthrough congestion pricing policy (see Arnott et al [5])

Figure 6 depicts the mean trip times with different pre-ferred arrival time intervals when 120579 = 09 One can observefrom this figure that adopting flexi-time work schedule canreduce the commutersrsquo travel time or the queue behind thebottleneck The areas below three curves are 06208 05940and 05131 hours respectively It shows further that the

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

occur due to physical and operational factors such as roadrepairs construction accidents and bad weather The varia-tions in road capacity from physical and operational reasonsare what make the analysis of travel behavior so complex andyet interesting As such understanding travelersrsquo attitudesand their behavior in varying settings is key to developingsustainable transport policesThere has been recent attentionto the stochastic nature of the bottleneck models (Siu andLo [16] Li et al [17] Xiao et al [18]) A reasonable way tocapture these variations and their impact on the networkperformance is to formulate the problem using probabilitydistributions (Chen et al [19])

The focus of this paper is to analyze the departure timechoice behavior under uncertainty in the morning commut-ing problem with flexi-time work schedule It is expectedthat the stochastic capacity leads to uncertainty in queuingtravel time and trip cost which in turn influences thecommutersrsquo travel choice behavior We assume that travelersare fully aware of the stochastic properties of the travel timeand schedule delay distributions throughout the morningpeak period which emerges from their day-to-day travelexperience Furthermore we consider homogenous travelershave the same preferred arrival time interval (PATI) Weformulate a stochastic bottleneck model for this flexi-timecommute problem and derive its analytical solution Theproperties of the model are investigated

The solution of the proposedmodel shows that the capac-ity variability of the bottleneck leads to significant changes indeparture time patterns which are different to those derivedunder deterministic conditions In a deterministic bottleneckmodel with flexi-time work schedule an individual canchoose either to depart in the tails of the rush hour whentravel time is low and pay the penalty of arriving at workearly or late or to depart close to the PATI when traveltime is high but schedule delay cost is low In other wordsunder the deterministic equilibrium schedule delay earlylateand arrival on time cannot occur simultaneously for a givendeparture time (Henderson [14] Mun and Yonekawa [15])We demonstrate that with day-to-day stochastic capacitycommuters departing at the same time may endure scheduledelay earlylate or not and may experience queuing delay ornot on different days

The rest of this paper is organized as follows Section 2formulates separately a deterministic and a stochastic bottle-neck model with homogeneous PATI The equilibrium solu-tions for the departure time pattern are derived for each caseThe theoretical properties of the proposed stochastic bottle-neck model with PATI are investigated and compared withthe deterministic case Numerical examples are presented inSection 3 to illustrate further the equilibrium properties ofthe model Section 4 provides conclusion remarks

2 Departure Patterns of Morning Commutewith Flexible Arrival Time

21 The Deterministic Case We formulate the peak periodcongestion based on the bottleneck model developed byVickrey [1] Suppose a single road connecting a residential

area and the Commercial Business District (CBD) which hasa bottleneck just before the CBD It is assumed that vehiclesdrive at constant speed from home to the bottleneck pointtravel time for this portion of trip is constant and representedas 119879free Queue develops when traffic flow rate exceeds thebottleneck capacity 119904 Travel time for a vehicle departing attime instant 119905 is represented as follows

119879 (119905) = 119879free +119876 (119905)

119904 (1)

where 119879(119905) is travel time and 119876(119905) is length of queue Thesecond term represents the waiting time within the queuebehind the bottleneck We set 119879free = 0 hereafterThis settingwill not affect the qualitative property

The queue length that a trip maker departing at time 119905encounters is calculated as follows

119876 (119905) = maxint119905

1199050

[119903 (119909) minus 119904] d119909 0 (2)

and the cumulative departures as

119877 (119905) = int

119905

1199050

119903 (119909) d119909 (3)

where 119903(119909) is the departure rate at time instant 119905 and 1199050is the

earliest time with positive departure rateSuppose that every morning a fixed number of 119873

individuals commute from home to office located in theCBD driving along the road stated above All workers haveidentical skills and preferences Unlike Vickrey [1] whichassumes that there is only one preferred arrive time 119905lowast herefirms in the CBD adopt a flexi-time work schedule such thatemployees arriving at office earlier or later 120575(120575 ge 0) than 119905lowastincur no scheduling cost Hereafter we call this time period[119905lowastminus 120575 119905lowast+ 120575] as PATI

Some commutersmay still arrive at the destination earlieror later than PATI in order to avoid a long queue at thebottleneck The cost for commuters traveling from home tothe CBD consists of three components the cost of traveltime and the cost of schedule delay early or late It can beformulated as follows

119862 (119905) = 120572119879 (119905) + SDE (119905) + SDL (119905) (4)

where 120572 is the value of travel timeThe cost of schedule delayearly (SDE) and schedule delay late (SDL) for a commuterwho leaves home at time 119905 can be expressed as

SDE (119905) = 120573 (119905lowastminus 120575 minus (119905 + 119879 (119905)))

SDL (119905) = 120574 (119905 + 119879 (119905) minus (119905lowast+ 120575))

(5)

where 120573 and 120574 denote the value of schedule delay early andthe value of schedule delay late respectively

Discrete Dynamics in Nature and Society 3

0

Num

bers

of t

rave

lers

Cumulative departures

Cumulative arrivals

Departure times

120579 = 1N

T(t2)

T(t1)

r(t)

t0 t1 t2 120575 teminus120575

Figure 1 Departure time distributions in the deterministic case

Substituting (1) and (5) into (4) and applying the equi-librium condition 120597119862(119905)120597119905 = 0 we have the equilibriumdeparture rate as follows

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 lt 119905

1

119904 if 1199051le 119905 le 119905

2

120572119904

(120572 + 120574) if 119905

2lt 119905 le 119905

119890

(6)

where 1199050and 119905119890are the earliest time and the latest time with

positive departure rate respectively And 1199051and 1199052are the

watershed times for which an individual arrives at work ontime Then

1199050= 119905lowast+120574 minus 120573

120573 + 120574120575 minus

120574

120573 + 120574sdot119873

119904

1199051=120572 minus 120573

120572119905lowastminus120572 minus 120573

120572120575 minus

120573120574

120572 (120573 + 120574)sdot119873

119904

1199052=120572 minus 120573

120572119905lowast+120572 + 120573

120572120575 minus

120573120574

120572 (120573 + 120574)sdot119873

119904

119905119890= 119905lowast+120574 minus 120573

120573 + 120574120575 +

120573

120573 + 120574sdot119873

119904

(7)

Figure 1 depicts the cumulative departures (from home)and arrivals (at CBD) in equilibrium under the deterministiccapacity case For simplicity we set 119905lowast to be zero and then alltravelers have the same PATI [minus120575 120575] The horizontal distancebetween the departure and arrival curves gives the travel timeSeveral critical time points are indicated Travelers departingat time interval [119905

1 1199052] have the longest travel time but will

arrive on time that is within PATI Travelers departing before1199051will arrive earlier than desiredwhile the travelers departing

after 1199052will arrive late at the destination

22 The Stochastic Case The deterministic case models asingle-day departure time equilibrium In the real world road

capacitiesmay vary from day to day due to unexpected eventssuch as incidents and weather conditions Because of thecapacity fluctuations both commutersrsquo travel time and theirschedule delays are stochastic In this section we hypothesizethat a constant long-term departure time patternmay emergegiven the responses of the travelers to the day-to-day capacityvariation Each commuter chooses an optimal departure timewhich minimizes hisher long-term expected trip cost Wecall this pattern if it exists a long-term equilibrium pattern

221 Assumptions and Travelersrsquo Cost Function The follow-ing assumptions are made in the model formulation

(A1) Commuters are homogeneouswith the same120572120573 and120574 values and the same PATI

(A2) The capacity of the bottleneck is constant within aday but fluctuates from day to day The uncertaintyof capacity is completely exogenous and independentof departures

(A3) The capacity is a nonnegative stochastic variablechanging around a certain mean capacity FollowingLi et al [17] we assume that stochastic capacityfollows a uniform distribution within interval [120579119904 119904]where 119904 is the design capacity and 120579(lt1) is a positiveparameter which denotes the lowest rate of availablecapacity

(A4) Commuters are aware of the capacity degenerationprobability and their departure time choice followsthe user equilibrium (UE) principle in terms of meantrip cost

We assume that the capacity of the single bottleneckis stochastic but the commutersrsquo departure time choice isdeterministic The calculation of the mean trip cost relies onthe calculations of the mean travel time the mean scheduledelay early and late For simplicity we set the 119905lowast to be zeroand then the PATI becomes [minus120575 120575] Under the stochasticcondition the mean trip cost with respect to departure time119905 can be formulated as follows

119864 [119888 (119905)] = 119864 [120572119879 (119905) + SDE (119905) + SDL (119905)] (8)

The equilibriumcondition for commutersrsquo departure timechoice in a single bottleneckwith stochastic capacity is that nocommuter can reduce hisher mean trip cost by unilaterallyaltering hisher departure time This condition implies thatthe commutersrsquomean trip cost is fixedwith respect to the timeinstant with positive departure rate That is

120597119864 [119888 (119905)]

120597119905= 0 if 119903 (119905) gt 0 (9)

222 Mathematical Formulations and Derivations Due tothe stochastic capacity over days the travel time experiencedby a traveler departing at the same time 119905 varies from day today This is equivalent to saying that commuters departinghome to work at the same time may endure schedule delayearlylate or not and may experience queuing delay or not ondifferent days Consequently there are six situations which

4 Discrete Dynamics in Nature and Society

may occur (I) always arrive early (II) possibly arrive earlyor on time (III) always arrive on time (IV) possibly arriveon time or late (V) always arrive late and (VI) always arrivelate but queue may exist We propose the following simpleextension of trip cost function under the six situations tomodel user departure time choice under degradable capac-ities and we use 119905

1 1199052 1199053 1199054 and 119905

5to denote the watershed

lines separating the six casesAs we assume that stochastic capacity is completely

exogenous and independent of departure flows the expec-tation of travel time and schedule delay cost with respect todifferent situations can be derived respectively as follows

119864 [119879 (119905)]

=

int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 119905)119891 (119904) 119889119904 119905

0le 119905 le 119905

5

int

119877(119905)(119905minus1199050)

120579119904

(119877 (119905)

119904minus 119905 + 119905

0)119891 (119904) 119889119904 119905

5lt 119905 le 119905

119890

119864 [SDE (119905)]

=

120573int

119904

120579119904

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

0le 119905 le 119905

1

120573int

119904

119877(119905)(minus120575minus1199050)

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

1lt 119905 le 119905

2

119864 [SDL (119905)]

=

120574int

119877(119905)(120575minus1199050)

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

3lt 119905 le 119905

4

120574int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

4le 119905 le 119905

5

(10)

where 119891(119904) = 1(119904 minus 120579119904) Substituting (10) into (8) we get theexpected trip cost with respect to each situation Accordingto (9) the equilibrium departure rates for the six situationscan be expressed as follows

Situation 1 No commuters experience schedule delay latersubject to all possible values of the bottleneck capacity Weget the departure rate

119903 (119905) =120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1 1199050le 119905 lt 119905

1 (11)

The boundary condition for this situation is SDL(1199051) = 0

when 119904 = 120579119904 and hence we have 119877(1199051) = minus(119905

0+ 120575)120579119904

Situation 2 If the capacity of the bottleneck is large enoughonly schedule delay early will occur On the contrary noschedule delay occurs when the capacity is small The water-shed capacity satisfies 119879(119905) + 119905 = minus120575 Equivalently we have119904 = 119877(119905)(minus120575 minus 119905

0) and the departure rate

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

1199051lt 119905 le 119905

2

(12)

The boundary condition for this case is SDL(1199052) = 0when

119904 = 119904 and hence we have 119877(1199052) = minus(119905

0+ 120575)119904

Situation 3 No commuters experience schedule delay subjectto all possible values of the bottleneck capacityTherefore thedeparture rate is

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1 1199052lt 119905 le 119905

3 (13)

The boundary condition for this case is SDE(1199053) =

SDL(1199053) = 0 when 119904 = 120579119904 Hence we have 119877(119905

3) = (120575 minus 119905

0)120579119904

Situation 4 If the capacity of the bottleneck is large enoughindividuals arrive on time On the contrary schedule delaylate occurs when the capacity is smallThewatershed capacitysatisfies 119879(119905) + 119905 = 120575 Equivalently we have 119904 = 119877(119905)(120575 minus 119905

0)

Therefore

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

1199053lt 119905 le 119905

4

(14)

The boundary condition for this case is SDE(1199054) = 0when

119904 = 119904 and then we have 119877(1199054) = (120575 minus 119905

0)119904

Situation 5 Similar to Situation 1 we have

119903 (119905) =120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1 1199054lt 119905 le 119905

5 (15)

The boundary condition for this case is 119877(1199053) = 119904(119905

5minus 1199050)

that is the queue length at time 1199055equals to zero when 119904 = 119904

Situation 6 Similar to Situation 2 we can find a watershedcapacity of the bottleneck such that the queue length equalszero that is119877(119905) = 119904(119905minus119905

0) and hence the watershed capacity

is 119877(119905)(119905 minus 1199050) We have

119903 (119905) =d119877 (119905)d119905

=(120572 + 120574) 119877 (119905) (119905 minus 119905

0) minus (120572120579 + 120574) 119904

(120572 + 120574) (ln119877 (119905) minus ln 120579119904 (119905 minus 1199050))

1199053lt 119905 le 119905

119890

(16)

The boundary condition for this case is 119903(119905119890) = 0

Equivalently we have 119877(119905119890) = 119904(119905

119890minus 1199050) where 119904 = 119904(120572120579 +

120574)(120572 + 120574) is the mean capacity under the stochastic case

223 Determination of the Watershed Time Instants Sincethe departure rate 119903(119905) = 0 if 119905 gt 119905

119890 the cumulative departures

at time 119905119890are equal to the traffic demand that is 119877(119905

119890) =

119873 = 119904(119905119890minus 1199050) Therefore we have 119905

119890= 1199050+ 119873119904 Moreover

the equilibrium condition of the stochastic bottleneck impliesthat 119864[119888(119905

0)] = 119864[119888(119905

119890)] and hence we have

1199050=119873

119904sdot

1

1205960minus 1

+1205920

1 minus 1205960

120575 119905119890=119873

119904sdot

1205960

1205960minus 1

+1205920

1 minus 1205960

120575

(17)

where

1205960= 1 minus

(1 minus 120579) (120573 + 120574)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

1205920=

(1 minus 120579) (120574 minus 120573)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

(18)

Discrete Dynamics in Nature and Society 5

Using the boundary conditions of Situations 1ndash6 we canobtain the watershed lines as follows

1199051= 12059611199050+ 1205921120575 119905

2= 12059621199050+ 1205922120575

1199053= 12059631199050+ 1205923120575 119905

4= 12059641199050+ 1205924120575

1199055= 12059651199050+ 1205925120575

(19)

where

1205961= 1 minus

(120572 minus 120573) 120579120585

120572 120596

2=(120572 + 120573)

120572minus 120585

1205963=120573

120572minus 120579120585 + 1 120596

4=(120572 + 120574) (1 minus 120585)

120572+120573

120572

1205965= 1 +

(120574 + 120573)

(120572 minus (120572 + 120574) 120585) 120592

1=(120573 minus 120572) 120579120585

120572

1205922=120573

120572minus 120585 120592

3=120573

120572+ 120579120585

1205924=((120572 + 120574) 120585 + (120573 minus 120574))

120572 120592

5=

(120573 minus 120574)

(120572 minus (120572 + 120574) 120585)

120585 =ln 120579minus1

(1 minus 120579)

(20)

With the resulting stochastic departure pattern at long-term equilibrium the experienced day-to-day travel timesand number of travelers experiencing queues change accord-ing to varied capacities over days Given the boundary condi-tions of Situations 1ndash6 the cumulative departures and arrivalsof a stochastic bottleneck are given in Figure 2 The two solidcurves denote the cumulative departures and cumulativearrivals as in Figure 1 The dotted lines are the maximumand the minimum capacity of the bottleneck that is 119904 and120579119904 The earliest departure time from home to workplace is 119905

0

and if the bottleneck capacity equals 120579119904 commuters departingbetween 119905

1and 1199053will arrive at workplace on time If the

capacity equals 119904 commuters departing between 1199052and 1199054will

arrive at workplace on time Commuters departing withintime interval [119905

2 1199053] will always arrive at workplace on time

whatever the variability of capacity is At the beginningcommuters depart from home with a constant departurerate until the watershed time 119905

1 Subsequently the departure

rate will gradually drop down After time instant 1199054 another

constant departure rate will last until the watershed timeinstant 119905

5and the queuing length at this time will be zero

Later the departure rate continues to drop down to zero attime 119905

119890

224 Properties of the Stochastic Bottleneck Model In thissubsection we investigate the theoretical properties of theequilibrium solution of the proposed stochastic bottleneckmodel with PATI

Theorem 1 At equilibrium the expected trip cost for allcommuters is a monotonically increasing function of trafficdemand and a monotonically decreasing function of 120575-valuethat is 120597119864[119888(119905

0)]120597119873 gt 0 and 120597119864[119888(119905

0)]120597120575 lt 0 hold

Proof Since 119864[119888(1199050)] = (minus120575 minus 119905

0)120573 and 119905

0= 119873(119904(120596

0minus 1)) +

1205920120575(1 minus 120596

0) we have 120597119864[119888(119905

0)]120597120575 = minus2120573120574(120573 + 120574) lt 0 and

0Departure times

Cumulative departures

Cumulative arrivals

PATI

120579 = 09

0

Num

bers

of t

rave

lers

N

T(t2)

T(t3)T(t4)

T(t1)

t0 t1 t2 t3 t4 t5 te

120579s

s

Figure 2 Departure time distributions in the stochastic case

120597119864[119888(1199050)]120597119873 = 120573((1minus120596

0)119904)The definitions of120573 and 119904 imply

that they are positive hence to prove 120597119864[119888(1199050)]120597119873 gt 0 we

only need to prove 1 minus1205960gt 0 Since 0 lt 120579 lt 1 and 119904 = 119904(120572120579 +

120574)(120572 + 120574) gt 119904(120572120579 + 120574120579)(120572 + 120574) = 120579119904 we have 1 minus 120579 gt 0 Andln 119904 minus ln 120579119904 gt 0 clearly holds Therefore both the numeratorand denominator of the second term in the right-hand sideof the first equation of (18) are positive therefore we have1 minus 1205960gt 0 This completes the proof

Theorem 2 At equilibrium the expected trip cost for allcommuters is amonotonically decreasing function of parameter120579-value that is 120597119864[119888(119905

0)]120597120579 lt 0 holds

Proof Submitting 1199050= 119873(119904(120596

0minus 1)) + 120592

0120575(1 minus 120596

0) into

119864[119888(1199050)] = (minus120575 minus 119905

0)120573 then the first-order derivative can be

given as follows

120597119864 [119888]

120597120579= 120573

119873

119904sdot120572 + 120574

120573 + 1205741199011015840(120579) (21)

where

119901 (120579) =ln 119904 minus ln (120579119904)

1 minus 120579

1199011015840(120579) =

1

1 minus 120579(119901 (120579) minus

120574

120579 (120572120579 + 120574))

(22)

Since ln(119904(120579119904)) lt 1 minus 120579119904119904 = 120574(1 minus 120579)(120572120579 + 120574) we then have

119901 (120579) minus120574

120579 (120572120579 + 120574)

lt120574

120572120579 + 120574minus

120574

120579 (120572120579 + 120574)=

120574

120572120579 + 120574(1 minus

1

120579) lt 0

(23)

It is clear that 1199011015840(120579) lt 0 and therefore we get 120597119864[119888]120597120579 lt 0This completes the proof

Theorem 3 At equilibrium the departure rate is a monoton-ically decreasing function of the departure time throughout thewhole peak period that is d119903(119905)d119905 le 0 119905 isin [119905

0 119905119890]

6 Discrete Dynamics in Nature and Society

Proof According to (11)ndash(16) the departure rate 119903(119905)

is continuous within each of intervals [1199050 1199051) (1199051 1199052) (1199052 1199053)

(1199053 1199054) (1199054 1199055) and (119905

5 119905119890] To prove the departure rate

is continuous during the peak period we calculate thefollowing limitations

lim119905rarr 119905+

1

119903 (119905) = lim119905rarr 119905+

1

120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

1

119903 (119905)

lim119905rarr 119905minus

2

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

2

119903 (119905)

lim119905rarr 119905minus

3

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1

=120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

= lim119905rarr 119905+

3

119903 (119905)

lim119905rarr 119905minus

4

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

4

119903 (119905)

lim119905rarr 119905+

5

119903 (119905) =(120572 + 120574) 119877 (119905

3) (1199053minus 1199050) minus (120572120579 + 120574) 119904

(120572 + 120574) ln (119877 (1199053) (120579119904 (119905

3minus 1199050)))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

5

119903 (119905)

(24)

This proves 119903(119905) is continuous indeed within the interval[1199050 119905119890]

Equations (11) (13) and (15) state that the departure rate119903(119905) is constant for 119905

0le 119905 le 119905

1 1199052le 119905 le 119905

3 and 119905

4le 119905 le

1199055 hence it is monotonically decreasing within these three

intervals By definition the cumulative departure flow 119877(119905) isnondecreasing with respect to time 119905Thus the denominatorsof the right-hand sides in (12) and (14) are nondecreasingwith respect to time 119905 Therefore the right-hand sides of (12)and (14) are nonincreasing with respect to time 119905 that is thedeparture rate 119903(119905) is monotonically decreasing within [119905

1 1199052]

and [1199053 1199054] The proof of d119903(119905)d119905 le 0 for all 119905 isin (119905

5 119905119890] can be

found in Xiao et al [18]In summary the departure rate 119903(119905) is monotonically dec-

reasing within all four intervals and at their boundaries Con-sidering the continuity of 119903(119905) for all 119905 isin [119905

0 119905119890] we conclude

that 119903(119905) is monotonically decreasing within [1199050 119905119890] This

completes the proof

Proposition 4 When parameter 120579 approaches one thestochastic bottleneck model immediately follows the determin-istic model

Proof According to the LrsquoHopitalrsquos rule we have lim120579rarr1

(1 minus

120579)(ln 120579minus1) = 1 We then have

lim120579rarr1

1205960= lim120579rarr1

1205965=minus120573

120574

lim120579rarr1

1205961= lim120579rarr1

1205962= lim120579rarr1

1205963= lim120579rarr1

1205964=120573

120572

lim120579rarr1

1205920= lim120579rarr1

1205925=(120574 minus 120573)

120574

lim120579rarr1

1205921= lim120579rarr1

1205922=(120573 minus 120572)

120572 lim

120579rarr1

1205923= lim120579rarr1

1205924=(120573 + 120572)

120572

(25)

lim120579rarr1

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 le 119905

1

119904 if 1199051le 119905 le 119905

4

120572119904

(120572 + 120574) if 119905

4lt 119905 le 119905

119890

(26)

Substituting (26) into (19) we have

1199050= minus

120574

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575 119905

5= 119905119890=

120573

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575

1199051= 1199052= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120573 minus 120572

120572120575

1199053= 1199054= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120572 + 120573

120572120575

(27)

Therefore we get the same traffic flow pattern with that froma deterministic bottleneck model

Proposition 5 When the number of commuters is givenincreasing the value of parameter 120579 will result in a decrease inthe length of peak period

Proof According to (16) we have d119904d120579 = 120572(120572 + 120574) gt 0This implies that 119904 is a monotonic increasing function withrespect to 120579 According to (17) we can obtain the length ofpeak period as follows

119905119890minus 1199050=119873

119904

1205960

1205960minus 1

+1205920

1 minus 1205960

120575 minus119873

119904

1

1205960minus 1

minus1205920

1 minus 1205960

120575 =119873

119904

(28)

Since119873 is constant and 119904 is a monotonic increasing functionwith respect to 120579 119905

119890minus 1199050is also a monotonic increasing

function with respect to 120579 This completes the proof

The above proof also shows that the length of peak periodis not affected by the 120575-value

3 Numerical Examples

The input parameters of our numerical example are 120572 =

64 $h 120573 = 39 $h 120574 = 1521 $h 119873 = 6000 veh 119904 =

4000 vehh 120575 = 10 min and 120579 = 09 By solving the proposed

Discrete Dynamics in Nature and Society 7

Table 1 The influence of parameter 120579 on the mean trip cost and the watershed time instant

120579 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

100 362 minus110 minus074 minus074 minus040 minus040 040 040 150095 378 minus114 minus077 minus073 minus046 minus031 036 039 152090 395 minus118 minus080 minus073 minus052 minus021 031 037 155085 413 minus123 minus085 minus072 minus059 minus009 026 035 157080 433 minus128 minus089 minus071 minus066 005 021 032 160

minus15 minus1 minus05 0 050

1

2

3

4

Departure time (hour)

Expe

cted

cost

($)

Total travel costTravel time cost

SDE costSDL cost

120579 = 09 120575 = 10min

Figure 3 The mean equilibrium trip cost and other components

model we obtained the time period with positive departurerate during [minus118 037] (see Table 1) The mean trip cost themean travel time cost and themean schedule delay cost (SDEand SDL) are depicted in Figure 3 We find that the mean tripcosts of all commuters are the same and are equal to 395 $but endure a trade-off between cost of travel time and cost ofschedule delay It is interesting that the waiting time cost isnot zero at the end of the peak period indicating that queuestill exists

It is interesting to investigate the impact of the fractionparameter 120579 on the solution of the stochastic bottleneckmodel We changed the parameter 120579 from 08 to 10 andcomputed themean trip cost and the watershed time instantsThe results are shown in Table 1 the row highlighted is withthe default 120579-value It can be seen that 119905

1= 1199052 1199053= 1199054 and 119905

5=

119905119890when 120579 = 10This confirmsProposition 4We can also find

that the length of time period with positive departure rateincreases with the 120579-value This confirms Proposition 5 Inaddition the second column of Table 1 shows that the meantrip cost decreases with the increase of 120579-valueThis confirmsTheorem 2 Since decreasing the 120579-value is equivalent toincreasing the travel time uncertainty commuters will leavehome earlier than before for avoiding the potential losscaused by uncertainty risk

The departure rates against different 120579-values are dis-played in Figure 4 It can be clearly seen that the resultsconfirm Propositions 4 and 5 that the stochastic bottleneckmodel immediately follows the deterministic model when

120579 = 10

120579 = 09

120579 = 08

0

2

4

6

8

10

Dep

artu

re ra

te (v

ehh

)

minus15 minus1 minus05 0 05Departure time (hour)

times103

120575 = 10min

Figure 4 The influence of parameter 120579 on departure rate

120579 approaches one and that enlarging the parameter 120579 willresult in a decrease in the length of peak period Figure 4 alsoshows that in equilibrium the departure rate during the peakperiod is monotonically decreasing which is consistent withTheorem 3

Table 2 lists the mean trip costs and the watershed timeinstants when 120579 = 09 and 120575-value from 0 minutes to 20minutes It is shown that in equilibrium the mean trip cost ismonotonically decreasing with increasing 120575 whilst the lengthof the peak period remains unchangedThis is consistent withTheorem 1

Figure 5 depicts the departure rates against different 120575-values at 120579 = 09 It can be seen that with larger 120575-value (orlonger PATI) the peak periods shift to later and the amountof earlier departures (at a rate higher than capacity) reducesThe overall departure time patterns become more flat andapproach designed capacity of the bottleneck with increasing120575-valueThis suggests that traffic congestion can be alleviatedby adopting flexi-timework schedule similar to that achievedthrough congestion pricing policy (see Arnott et al [5])

Figure 6 depicts the mean trip times with different pre-ferred arrival time intervals when 120579 = 09 One can observefrom this figure that adopting flexi-time work schedule canreduce the commutersrsquo travel time or the queue behind thebottleneck The areas below three curves are 06208 05940and 05131 hours respectively It shows further that the

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

0

Num

bers

of t

rave

lers

Cumulative departures

Cumulative arrivals

Departure times

120579 = 1N

T(t2)

T(t1)

r(t)

t0 t1 t2 120575 teminus120575

Figure 1 Departure time distributions in the deterministic case

Substituting (1) and (5) into (4) and applying the equi-librium condition 120597119862(119905)120597119905 = 0 we have the equilibriumdeparture rate as follows

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 lt 119905

1

119904 if 1199051le 119905 le 119905

2

120572119904

(120572 + 120574) if 119905

2lt 119905 le 119905

119890

(6)

where 1199050and 119905119890are the earliest time and the latest time with

positive departure rate respectively And 1199051and 1199052are the

watershed times for which an individual arrives at work ontime Then

1199050= 119905lowast+120574 minus 120573

120573 + 120574120575 minus

120574

120573 + 120574sdot119873

119904

1199051=120572 minus 120573

120572119905lowastminus120572 minus 120573

120572120575 minus

120573120574

120572 (120573 + 120574)sdot119873

119904

1199052=120572 minus 120573

120572119905lowast+120572 + 120573

120572120575 minus

120573120574

120572 (120573 + 120574)sdot119873

119904

119905119890= 119905lowast+120574 minus 120573

120573 + 120574120575 +

120573

120573 + 120574sdot119873

119904

(7)

Figure 1 depicts the cumulative departures (from home)and arrivals (at CBD) in equilibrium under the deterministiccapacity case For simplicity we set 119905lowast to be zero and then alltravelers have the same PATI [minus120575 120575] The horizontal distancebetween the departure and arrival curves gives the travel timeSeveral critical time points are indicated Travelers departingat time interval [119905

1 1199052] have the longest travel time but will

arrive on time that is within PATI Travelers departing before1199051will arrive earlier than desiredwhile the travelers departing

after 1199052will arrive late at the destination

22 The Stochastic Case The deterministic case models asingle-day departure time equilibrium In the real world road

capacitiesmay vary from day to day due to unexpected eventssuch as incidents and weather conditions Because of thecapacity fluctuations both commutersrsquo travel time and theirschedule delays are stochastic In this section we hypothesizethat a constant long-term departure time patternmay emergegiven the responses of the travelers to the day-to-day capacityvariation Each commuter chooses an optimal departure timewhich minimizes hisher long-term expected trip cost Wecall this pattern if it exists a long-term equilibrium pattern

221 Assumptions and Travelersrsquo Cost Function The follow-ing assumptions are made in the model formulation

(A1) Commuters are homogeneouswith the same120572120573 and120574 values and the same PATI

(A2) The capacity of the bottleneck is constant within aday but fluctuates from day to day The uncertaintyof capacity is completely exogenous and independentof departures

(A3) The capacity is a nonnegative stochastic variablechanging around a certain mean capacity FollowingLi et al [17] we assume that stochastic capacityfollows a uniform distribution within interval [120579119904 119904]where 119904 is the design capacity and 120579(lt1) is a positiveparameter which denotes the lowest rate of availablecapacity

(A4) Commuters are aware of the capacity degenerationprobability and their departure time choice followsthe user equilibrium (UE) principle in terms of meantrip cost

We assume that the capacity of the single bottleneckis stochastic but the commutersrsquo departure time choice isdeterministic The calculation of the mean trip cost relies onthe calculations of the mean travel time the mean scheduledelay early and late For simplicity we set the 119905lowast to be zeroand then the PATI becomes [minus120575 120575] Under the stochasticcondition the mean trip cost with respect to departure time119905 can be formulated as follows

119864 [119888 (119905)] = 119864 [120572119879 (119905) + SDE (119905) + SDL (119905)] (8)

The equilibriumcondition for commutersrsquo departure timechoice in a single bottleneckwith stochastic capacity is that nocommuter can reduce hisher mean trip cost by unilaterallyaltering hisher departure time This condition implies thatthe commutersrsquomean trip cost is fixedwith respect to the timeinstant with positive departure rate That is

120597119864 [119888 (119905)]

120597119905= 0 if 119903 (119905) gt 0 (9)

222 Mathematical Formulations and Derivations Due tothe stochastic capacity over days the travel time experiencedby a traveler departing at the same time 119905 varies from day today This is equivalent to saying that commuters departinghome to work at the same time may endure schedule delayearlylate or not and may experience queuing delay or not ondifferent days Consequently there are six situations which

4 Discrete Dynamics in Nature and Society

may occur (I) always arrive early (II) possibly arrive earlyor on time (III) always arrive on time (IV) possibly arriveon time or late (V) always arrive late and (VI) always arrivelate but queue may exist We propose the following simpleextension of trip cost function under the six situations tomodel user departure time choice under degradable capac-ities and we use 119905

1 1199052 1199053 1199054 and 119905

5to denote the watershed

lines separating the six casesAs we assume that stochastic capacity is completely

exogenous and independent of departure flows the expec-tation of travel time and schedule delay cost with respect todifferent situations can be derived respectively as follows

119864 [119879 (119905)]

=

int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 119905)119891 (119904) 119889119904 119905

0le 119905 le 119905

5

int

119877(119905)(119905minus1199050)

120579119904

(119877 (119905)

119904minus 119905 + 119905

0)119891 (119904) 119889119904 119905

5lt 119905 le 119905

119890

119864 [SDE (119905)]

=

120573int

119904

120579119904

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

0le 119905 le 119905

1

120573int

119904

119877(119905)(minus120575minus1199050)

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

1lt 119905 le 119905

2

119864 [SDL (119905)]

=

120574int

119877(119905)(120575minus1199050)

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

3lt 119905 le 119905

4

120574int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

4le 119905 le 119905

5

(10)

where 119891(119904) = 1(119904 minus 120579119904) Substituting (10) into (8) we get theexpected trip cost with respect to each situation Accordingto (9) the equilibrium departure rates for the six situationscan be expressed as follows

Situation 1 No commuters experience schedule delay latersubject to all possible values of the bottleneck capacity Weget the departure rate

119903 (119905) =120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1 1199050le 119905 lt 119905

1 (11)

The boundary condition for this situation is SDL(1199051) = 0

when 119904 = 120579119904 and hence we have 119877(1199051) = minus(119905

0+ 120575)120579119904

Situation 2 If the capacity of the bottleneck is large enoughonly schedule delay early will occur On the contrary noschedule delay occurs when the capacity is small The water-shed capacity satisfies 119879(119905) + 119905 = minus120575 Equivalently we have119904 = 119877(119905)(minus120575 minus 119905

0) and the departure rate

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

1199051lt 119905 le 119905

2

(12)

The boundary condition for this case is SDL(1199052) = 0when

119904 = 119904 and hence we have 119877(1199052) = minus(119905

0+ 120575)119904

Situation 3 No commuters experience schedule delay subjectto all possible values of the bottleneck capacityTherefore thedeparture rate is

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1 1199052lt 119905 le 119905

3 (13)

The boundary condition for this case is SDE(1199053) =

SDL(1199053) = 0 when 119904 = 120579119904 Hence we have 119877(119905

3) = (120575 minus 119905

0)120579119904

Situation 4 If the capacity of the bottleneck is large enoughindividuals arrive on time On the contrary schedule delaylate occurs when the capacity is smallThewatershed capacitysatisfies 119879(119905) + 119905 = 120575 Equivalently we have 119904 = 119877(119905)(120575 minus 119905

0)

Therefore

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

1199053lt 119905 le 119905

4

(14)

The boundary condition for this case is SDE(1199054) = 0when

119904 = 119904 and then we have 119877(1199054) = (120575 minus 119905

0)119904

Situation 5 Similar to Situation 1 we have

119903 (119905) =120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1 1199054lt 119905 le 119905

5 (15)

The boundary condition for this case is 119877(1199053) = 119904(119905

5minus 1199050)

that is the queue length at time 1199055equals to zero when 119904 = 119904

Situation 6 Similar to Situation 2 we can find a watershedcapacity of the bottleneck such that the queue length equalszero that is119877(119905) = 119904(119905minus119905

0) and hence the watershed capacity

is 119877(119905)(119905 minus 1199050) We have

119903 (119905) =d119877 (119905)d119905

=(120572 + 120574) 119877 (119905) (119905 minus 119905

0) minus (120572120579 + 120574) 119904

(120572 + 120574) (ln119877 (119905) minus ln 120579119904 (119905 minus 1199050))

1199053lt 119905 le 119905

119890

(16)

The boundary condition for this case is 119903(119905119890) = 0

Equivalently we have 119877(119905119890) = 119904(119905

119890minus 1199050) where 119904 = 119904(120572120579 +

120574)(120572 + 120574) is the mean capacity under the stochastic case

223 Determination of the Watershed Time Instants Sincethe departure rate 119903(119905) = 0 if 119905 gt 119905

119890 the cumulative departures

at time 119905119890are equal to the traffic demand that is 119877(119905

119890) =

119873 = 119904(119905119890minus 1199050) Therefore we have 119905

119890= 1199050+ 119873119904 Moreover

the equilibrium condition of the stochastic bottleneck impliesthat 119864[119888(119905

0)] = 119864[119888(119905

119890)] and hence we have

1199050=119873

119904sdot

1

1205960minus 1

+1205920

1 minus 1205960

120575 119905119890=119873

119904sdot

1205960

1205960minus 1

+1205920

1 minus 1205960

120575

(17)

where

1205960= 1 minus

(1 minus 120579) (120573 + 120574)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

1205920=

(1 minus 120579) (120574 minus 120573)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

(18)

Discrete Dynamics in Nature and Society 5

Using the boundary conditions of Situations 1ndash6 we canobtain the watershed lines as follows

1199051= 12059611199050+ 1205921120575 119905

2= 12059621199050+ 1205922120575

1199053= 12059631199050+ 1205923120575 119905

4= 12059641199050+ 1205924120575

1199055= 12059651199050+ 1205925120575

(19)

where

1205961= 1 minus

(120572 minus 120573) 120579120585

120572 120596

2=(120572 + 120573)

120572minus 120585

1205963=120573

120572minus 120579120585 + 1 120596

4=(120572 + 120574) (1 minus 120585)

120572+120573

120572

1205965= 1 +

(120574 + 120573)

(120572 minus (120572 + 120574) 120585) 120592

1=(120573 minus 120572) 120579120585

120572

1205922=120573

120572minus 120585 120592

3=120573

120572+ 120579120585

1205924=((120572 + 120574) 120585 + (120573 minus 120574))

120572 120592

5=

(120573 minus 120574)

(120572 minus (120572 + 120574) 120585)

120585 =ln 120579minus1

(1 minus 120579)

(20)

With the resulting stochastic departure pattern at long-term equilibrium the experienced day-to-day travel timesand number of travelers experiencing queues change accord-ing to varied capacities over days Given the boundary condi-tions of Situations 1ndash6 the cumulative departures and arrivalsof a stochastic bottleneck are given in Figure 2 The two solidcurves denote the cumulative departures and cumulativearrivals as in Figure 1 The dotted lines are the maximumand the minimum capacity of the bottleneck that is 119904 and120579119904 The earliest departure time from home to workplace is 119905

0

and if the bottleneck capacity equals 120579119904 commuters departingbetween 119905

1and 1199053will arrive at workplace on time If the

capacity equals 119904 commuters departing between 1199052and 1199054will

arrive at workplace on time Commuters departing withintime interval [119905

2 1199053] will always arrive at workplace on time

whatever the variability of capacity is At the beginningcommuters depart from home with a constant departurerate until the watershed time 119905

1 Subsequently the departure

rate will gradually drop down After time instant 1199054 another

constant departure rate will last until the watershed timeinstant 119905

5and the queuing length at this time will be zero

Later the departure rate continues to drop down to zero attime 119905

119890

224 Properties of the Stochastic Bottleneck Model In thissubsection we investigate the theoretical properties of theequilibrium solution of the proposed stochastic bottleneckmodel with PATI

Theorem 1 At equilibrium the expected trip cost for allcommuters is a monotonically increasing function of trafficdemand and a monotonically decreasing function of 120575-valuethat is 120597119864[119888(119905

0)]120597119873 gt 0 and 120597119864[119888(119905

0)]120597120575 lt 0 hold

Proof Since 119864[119888(1199050)] = (minus120575 minus 119905

0)120573 and 119905

0= 119873(119904(120596

0minus 1)) +

1205920120575(1 minus 120596

0) we have 120597119864[119888(119905

0)]120597120575 = minus2120573120574(120573 + 120574) lt 0 and

0Departure times

Cumulative departures

Cumulative arrivals

PATI

120579 = 09

0

Num

bers

of t

rave

lers

N

T(t2)

T(t3)T(t4)

T(t1)

t0 t1 t2 t3 t4 t5 te

120579s

s

Figure 2 Departure time distributions in the stochastic case

120597119864[119888(1199050)]120597119873 = 120573((1minus120596

0)119904)The definitions of120573 and 119904 imply

that they are positive hence to prove 120597119864[119888(1199050)]120597119873 gt 0 we

only need to prove 1 minus1205960gt 0 Since 0 lt 120579 lt 1 and 119904 = 119904(120572120579 +

120574)(120572 + 120574) gt 119904(120572120579 + 120574120579)(120572 + 120574) = 120579119904 we have 1 minus 120579 gt 0 Andln 119904 minus ln 120579119904 gt 0 clearly holds Therefore both the numeratorand denominator of the second term in the right-hand sideof the first equation of (18) are positive therefore we have1 minus 1205960gt 0 This completes the proof

Theorem 2 At equilibrium the expected trip cost for allcommuters is amonotonically decreasing function of parameter120579-value that is 120597119864[119888(119905

0)]120597120579 lt 0 holds

Proof Submitting 1199050= 119873(119904(120596

0minus 1)) + 120592

0120575(1 minus 120596

0) into

119864[119888(1199050)] = (minus120575 minus 119905

0)120573 then the first-order derivative can be

given as follows

120597119864 [119888]

120597120579= 120573

119873

119904sdot120572 + 120574

120573 + 1205741199011015840(120579) (21)

where

119901 (120579) =ln 119904 minus ln (120579119904)

1 minus 120579

1199011015840(120579) =

1

1 minus 120579(119901 (120579) minus

120574

120579 (120572120579 + 120574))

(22)

Since ln(119904(120579119904)) lt 1 minus 120579119904119904 = 120574(1 minus 120579)(120572120579 + 120574) we then have

119901 (120579) minus120574

120579 (120572120579 + 120574)

lt120574

120572120579 + 120574minus

120574

120579 (120572120579 + 120574)=

120574

120572120579 + 120574(1 minus

1

120579) lt 0

(23)

It is clear that 1199011015840(120579) lt 0 and therefore we get 120597119864[119888]120597120579 lt 0This completes the proof

Theorem 3 At equilibrium the departure rate is a monoton-ically decreasing function of the departure time throughout thewhole peak period that is d119903(119905)d119905 le 0 119905 isin [119905

0 119905119890]

6 Discrete Dynamics in Nature and Society

Proof According to (11)ndash(16) the departure rate 119903(119905)

is continuous within each of intervals [1199050 1199051) (1199051 1199052) (1199052 1199053)

(1199053 1199054) (1199054 1199055) and (119905

5 119905119890] To prove the departure rate

is continuous during the peak period we calculate thefollowing limitations

lim119905rarr 119905+

1

119903 (119905) = lim119905rarr 119905+

1

120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

1

119903 (119905)

lim119905rarr 119905minus

2

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

2

119903 (119905)

lim119905rarr 119905minus

3

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1

=120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

= lim119905rarr 119905+

3

119903 (119905)

lim119905rarr 119905minus

4

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

4

119903 (119905)

lim119905rarr 119905+

5

119903 (119905) =(120572 + 120574) 119877 (119905

3) (1199053minus 1199050) minus (120572120579 + 120574) 119904

(120572 + 120574) ln (119877 (1199053) (120579119904 (119905

3minus 1199050)))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

5

119903 (119905)

(24)

This proves 119903(119905) is continuous indeed within the interval[1199050 119905119890]

Equations (11) (13) and (15) state that the departure rate119903(119905) is constant for 119905

0le 119905 le 119905

1 1199052le 119905 le 119905

3 and 119905

4le 119905 le

1199055 hence it is monotonically decreasing within these three

intervals By definition the cumulative departure flow 119877(119905) isnondecreasing with respect to time 119905Thus the denominatorsof the right-hand sides in (12) and (14) are nondecreasingwith respect to time 119905 Therefore the right-hand sides of (12)and (14) are nonincreasing with respect to time 119905 that is thedeparture rate 119903(119905) is monotonically decreasing within [119905

1 1199052]

and [1199053 1199054] The proof of d119903(119905)d119905 le 0 for all 119905 isin (119905

5 119905119890] can be

found in Xiao et al [18]In summary the departure rate 119903(119905) is monotonically dec-

reasing within all four intervals and at their boundaries Con-sidering the continuity of 119903(119905) for all 119905 isin [119905

0 119905119890] we conclude

that 119903(119905) is monotonically decreasing within [1199050 119905119890] This

completes the proof

Proposition 4 When parameter 120579 approaches one thestochastic bottleneck model immediately follows the determin-istic model

Proof According to the LrsquoHopitalrsquos rule we have lim120579rarr1

(1 minus

120579)(ln 120579minus1) = 1 We then have

lim120579rarr1

1205960= lim120579rarr1

1205965=minus120573

120574

lim120579rarr1

1205961= lim120579rarr1

1205962= lim120579rarr1

1205963= lim120579rarr1

1205964=120573

120572

lim120579rarr1

1205920= lim120579rarr1

1205925=(120574 minus 120573)

120574

lim120579rarr1

1205921= lim120579rarr1

1205922=(120573 minus 120572)

120572 lim

120579rarr1

1205923= lim120579rarr1

1205924=(120573 + 120572)

120572

(25)

lim120579rarr1

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 le 119905

1

119904 if 1199051le 119905 le 119905

4

120572119904

(120572 + 120574) if 119905

4lt 119905 le 119905

119890

(26)

Substituting (26) into (19) we have

1199050= minus

120574

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575 119905

5= 119905119890=

120573

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575

1199051= 1199052= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120573 minus 120572

120572120575

1199053= 1199054= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120572 + 120573

120572120575

(27)

Therefore we get the same traffic flow pattern with that froma deterministic bottleneck model

Proposition 5 When the number of commuters is givenincreasing the value of parameter 120579 will result in a decrease inthe length of peak period

Proof According to (16) we have d119904d120579 = 120572(120572 + 120574) gt 0This implies that 119904 is a monotonic increasing function withrespect to 120579 According to (17) we can obtain the length ofpeak period as follows

119905119890minus 1199050=119873

119904

1205960

1205960minus 1

+1205920

1 minus 1205960

120575 minus119873

119904

1

1205960minus 1

minus1205920

1 minus 1205960

120575 =119873

119904

(28)

Since119873 is constant and 119904 is a monotonic increasing functionwith respect to 120579 119905

119890minus 1199050is also a monotonic increasing

function with respect to 120579 This completes the proof

The above proof also shows that the length of peak periodis not affected by the 120575-value

3 Numerical Examples

The input parameters of our numerical example are 120572 =

64 $h 120573 = 39 $h 120574 = 1521 $h 119873 = 6000 veh 119904 =

4000 vehh 120575 = 10 min and 120579 = 09 By solving the proposed

Discrete Dynamics in Nature and Society 7

Table 1 The influence of parameter 120579 on the mean trip cost and the watershed time instant

120579 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

100 362 minus110 minus074 minus074 minus040 minus040 040 040 150095 378 minus114 minus077 minus073 minus046 minus031 036 039 152090 395 minus118 minus080 minus073 minus052 minus021 031 037 155085 413 minus123 minus085 minus072 minus059 minus009 026 035 157080 433 minus128 minus089 minus071 minus066 005 021 032 160

minus15 minus1 minus05 0 050

1

2

3

4

Departure time (hour)

Expe

cted

cost

($)

Total travel costTravel time cost

SDE costSDL cost

120579 = 09 120575 = 10min

Figure 3 The mean equilibrium trip cost and other components

model we obtained the time period with positive departurerate during [minus118 037] (see Table 1) The mean trip cost themean travel time cost and themean schedule delay cost (SDEand SDL) are depicted in Figure 3 We find that the mean tripcosts of all commuters are the same and are equal to 395 $but endure a trade-off between cost of travel time and cost ofschedule delay It is interesting that the waiting time cost isnot zero at the end of the peak period indicating that queuestill exists

It is interesting to investigate the impact of the fractionparameter 120579 on the solution of the stochastic bottleneckmodel We changed the parameter 120579 from 08 to 10 andcomputed themean trip cost and the watershed time instantsThe results are shown in Table 1 the row highlighted is withthe default 120579-value It can be seen that 119905

1= 1199052 1199053= 1199054 and 119905

5=

119905119890when 120579 = 10This confirmsProposition 4We can also find

that the length of time period with positive departure rateincreases with the 120579-value This confirms Proposition 5 Inaddition the second column of Table 1 shows that the meantrip cost decreases with the increase of 120579-valueThis confirmsTheorem 2 Since decreasing the 120579-value is equivalent toincreasing the travel time uncertainty commuters will leavehome earlier than before for avoiding the potential losscaused by uncertainty risk

The departure rates against different 120579-values are dis-played in Figure 4 It can be clearly seen that the resultsconfirm Propositions 4 and 5 that the stochastic bottleneckmodel immediately follows the deterministic model when

120579 = 10

120579 = 09

120579 = 08

0

2

4

6

8

10

Dep

artu

re ra

te (v

ehh

)

minus15 minus1 minus05 0 05Departure time (hour)

times103

120575 = 10min

Figure 4 The influence of parameter 120579 on departure rate

120579 approaches one and that enlarging the parameter 120579 willresult in a decrease in the length of peak period Figure 4 alsoshows that in equilibrium the departure rate during the peakperiod is monotonically decreasing which is consistent withTheorem 3

Table 2 lists the mean trip costs and the watershed timeinstants when 120579 = 09 and 120575-value from 0 minutes to 20minutes It is shown that in equilibrium the mean trip cost ismonotonically decreasing with increasing 120575 whilst the lengthof the peak period remains unchangedThis is consistent withTheorem 1

Figure 5 depicts the departure rates against different 120575-values at 120579 = 09 It can be seen that with larger 120575-value (orlonger PATI) the peak periods shift to later and the amountof earlier departures (at a rate higher than capacity) reducesThe overall departure time patterns become more flat andapproach designed capacity of the bottleneck with increasing120575-valueThis suggests that traffic congestion can be alleviatedby adopting flexi-timework schedule similar to that achievedthrough congestion pricing policy (see Arnott et al [5])

Figure 6 depicts the mean trip times with different pre-ferred arrival time intervals when 120579 = 09 One can observefrom this figure that adopting flexi-time work schedule canreduce the commutersrsquo travel time or the queue behind thebottleneck The areas below three curves are 06208 05940and 05131 hours respectively It shows further that the

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

may occur (I) always arrive early (II) possibly arrive earlyor on time (III) always arrive on time (IV) possibly arriveon time or late (V) always arrive late and (VI) always arrivelate but queue may exist We propose the following simpleextension of trip cost function under the six situations tomodel user departure time choice under degradable capac-ities and we use 119905

1 1199052 1199053 1199054 and 119905

5to denote the watershed

lines separating the six casesAs we assume that stochastic capacity is completely

exogenous and independent of departure flows the expec-tation of travel time and schedule delay cost with respect todifferent situations can be derived respectively as follows

119864 [119879 (119905)]

=

int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 119905)119891 (119904) 119889119904 119905

0le 119905 le 119905

5

int

119877(119905)(119905minus1199050)

120579119904

(119877 (119905)

119904minus 119905 + 119905

0)119891 (119904) 119889119904 119905

5lt 119905 le 119905

119890

119864 [SDE (119905)]

=

120573int

119904

120579119904

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

0le 119905 le 119905

1

120573int

119904

119877(119905)(minus120575minus1199050)

(minus120575 minus119877 (119905)

119904minus 1199050)119891 (119904) 119889119904 119905

1lt 119905 le 119905

2

119864 [SDL (119905)]

=

120574int

119877(119905)(120575minus1199050)

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

3lt 119905 le 119905

4

120574int

119904

120579119904

(119877 (119905)

119904+ 1199050minus 120575)119891 (119904) 119889119904 119905

4le 119905 le 119905

5

(10)

where 119891(119904) = 1(119904 minus 120579119904) Substituting (10) into (8) we get theexpected trip cost with respect to each situation Accordingto (9) the equilibrium departure rates for the six situationscan be expressed as follows

Situation 1 No commuters experience schedule delay latersubject to all possible values of the bottleneck capacity Weget the departure rate

119903 (119905) =120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1 1199050le 119905 lt 119905

1 (11)

The boundary condition for this situation is SDL(1199051) = 0

when 119904 = 120579119904 and hence we have 119877(1199051) = minus(119905

0+ 120575)120579119904

Situation 2 If the capacity of the bottleneck is large enoughonly schedule delay early will occur On the contrary noschedule delay occurs when the capacity is small The water-shed capacity satisfies 119879(119905) + 119905 = minus120575 Equivalently we have119904 = 119877(119905)(minus120575 minus 119905

0) and the departure rate

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

1199051lt 119905 le 119905

2

(12)

The boundary condition for this case is SDL(1199052) = 0when

119904 = 119904 and hence we have 119877(1199052) = minus(119905

0+ 120575)119904

Situation 3 No commuters experience schedule delay subjectto all possible values of the bottleneck capacityTherefore thedeparture rate is

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1 1199052lt 119905 le 119905

3 (13)

The boundary condition for this case is SDE(1199053) =

SDL(1199053) = 0 when 119904 = 120579119904 Hence we have 119877(119905

3) = (120575 minus 119905

0)120579119904

Situation 4 If the capacity of the bottleneck is large enoughindividuals arrive on time On the contrary schedule delaylate occurs when the capacity is smallThewatershed capacitysatisfies 119879(119905) + 119905 = 120575 Equivalently we have 119904 = 119877(119905)(120575 minus 119905

0)

Therefore

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

1199053lt 119905 le 119905

4

(14)

The boundary condition for this case is SDE(1199054) = 0when

119904 = 119904 and then we have 119877(1199054) = (120575 minus 119905

0)119904

Situation 5 Similar to Situation 1 we have

119903 (119905) =120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1 1199054lt 119905 le 119905

5 (15)

The boundary condition for this case is 119877(1199053) = 119904(119905

5minus 1199050)

that is the queue length at time 1199055equals to zero when 119904 = 119904

Situation 6 Similar to Situation 2 we can find a watershedcapacity of the bottleneck such that the queue length equalszero that is119877(119905) = 119904(119905minus119905

0) and hence the watershed capacity

is 119877(119905)(119905 minus 1199050) We have

119903 (119905) =d119877 (119905)d119905

=(120572 + 120574) 119877 (119905) (119905 minus 119905

0) minus (120572120579 + 120574) 119904

(120572 + 120574) (ln119877 (119905) minus ln 120579119904 (119905 minus 1199050))

1199053lt 119905 le 119905

119890

(16)

The boundary condition for this case is 119903(119905119890) = 0

Equivalently we have 119877(119905119890) = 119904(119905

119890minus 1199050) where 119904 = 119904(120572120579 +

120574)(120572 + 120574) is the mean capacity under the stochastic case

223 Determination of the Watershed Time Instants Sincethe departure rate 119903(119905) = 0 if 119905 gt 119905

119890 the cumulative departures

at time 119905119890are equal to the traffic demand that is 119877(119905

119890) =

119873 = 119904(119905119890minus 1199050) Therefore we have 119905

119890= 1199050+ 119873119904 Moreover

the equilibrium condition of the stochastic bottleneck impliesthat 119864[119888(119905

0)] = 119864[119888(119905

119890)] and hence we have

1199050=119873

119904sdot

1

1205960minus 1

+1205920

1 minus 1205960

120575 119905119890=119873

119904sdot

1205960

1205960minus 1

+1205920

1 minus 1205960

120575

(17)

where

1205960= 1 minus

(1 minus 120579) (120573 + 120574)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

1205920=

(1 minus 120579) (120574 minus 120573)

(120572120579 + 120574) (ln 119904 minus ln 120579119904)

(18)

Discrete Dynamics in Nature and Society 5

Using the boundary conditions of Situations 1ndash6 we canobtain the watershed lines as follows

1199051= 12059611199050+ 1205921120575 119905

2= 12059621199050+ 1205922120575

1199053= 12059631199050+ 1205923120575 119905

4= 12059641199050+ 1205924120575

1199055= 12059651199050+ 1205925120575

(19)

where

1205961= 1 minus

(120572 minus 120573) 120579120585

120572 120596

2=(120572 + 120573)

120572minus 120585

1205963=120573

120572minus 120579120585 + 1 120596

4=(120572 + 120574) (1 minus 120585)

120572+120573

120572

1205965= 1 +

(120574 + 120573)

(120572 minus (120572 + 120574) 120585) 120592

1=(120573 minus 120572) 120579120585

120572

1205922=120573

120572minus 120585 120592

3=120573

120572+ 120579120585

1205924=((120572 + 120574) 120585 + (120573 minus 120574))

120572 120592

5=

(120573 minus 120574)

(120572 minus (120572 + 120574) 120585)

120585 =ln 120579minus1

(1 minus 120579)

(20)

With the resulting stochastic departure pattern at long-term equilibrium the experienced day-to-day travel timesand number of travelers experiencing queues change accord-ing to varied capacities over days Given the boundary condi-tions of Situations 1ndash6 the cumulative departures and arrivalsof a stochastic bottleneck are given in Figure 2 The two solidcurves denote the cumulative departures and cumulativearrivals as in Figure 1 The dotted lines are the maximumand the minimum capacity of the bottleneck that is 119904 and120579119904 The earliest departure time from home to workplace is 119905

0

and if the bottleneck capacity equals 120579119904 commuters departingbetween 119905

1and 1199053will arrive at workplace on time If the

capacity equals 119904 commuters departing between 1199052and 1199054will

arrive at workplace on time Commuters departing withintime interval [119905

2 1199053] will always arrive at workplace on time

whatever the variability of capacity is At the beginningcommuters depart from home with a constant departurerate until the watershed time 119905

1 Subsequently the departure

rate will gradually drop down After time instant 1199054 another

constant departure rate will last until the watershed timeinstant 119905

5and the queuing length at this time will be zero

Later the departure rate continues to drop down to zero attime 119905

119890

224 Properties of the Stochastic Bottleneck Model In thissubsection we investigate the theoretical properties of theequilibrium solution of the proposed stochastic bottleneckmodel with PATI

Theorem 1 At equilibrium the expected trip cost for allcommuters is a monotonically increasing function of trafficdemand and a monotonically decreasing function of 120575-valuethat is 120597119864[119888(119905

0)]120597119873 gt 0 and 120597119864[119888(119905

0)]120597120575 lt 0 hold

Proof Since 119864[119888(1199050)] = (minus120575 minus 119905

0)120573 and 119905

0= 119873(119904(120596

0minus 1)) +

1205920120575(1 minus 120596

0) we have 120597119864[119888(119905

0)]120597120575 = minus2120573120574(120573 + 120574) lt 0 and

0Departure times

Cumulative departures

Cumulative arrivals

PATI

120579 = 09

0

Num

bers

of t

rave

lers

N

T(t2)

T(t3)T(t4)

T(t1)

t0 t1 t2 t3 t4 t5 te

120579s

s

Figure 2 Departure time distributions in the stochastic case

120597119864[119888(1199050)]120597119873 = 120573((1minus120596

0)119904)The definitions of120573 and 119904 imply

that they are positive hence to prove 120597119864[119888(1199050)]120597119873 gt 0 we

only need to prove 1 minus1205960gt 0 Since 0 lt 120579 lt 1 and 119904 = 119904(120572120579 +

120574)(120572 + 120574) gt 119904(120572120579 + 120574120579)(120572 + 120574) = 120579119904 we have 1 minus 120579 gt 0 Andln 119904 minus ln 120579119904 gt 0 clearly holds Therefore both the numeratorand denominator of the second term in the right-hand sideof the first equation of (18) are positive therefore we have1 minus 1205960gt 0 This completes the proof

Theorem 2 At equilibrium the expected trip cost for allcommuters is amonotonically decreasing function of parameter120579-value that is 120597119864[119888(119905

0)]120597120579 lt 0 holds

Proof Submitting 1199050= 119873(119904(120596

0minus 1)) + 120592

0120575(1 minus 120596

0) into

119864[119888(1199050)] = (minus120575 minus 119905

0)120573 then the first-order derivative can be

given as follows

120597119864 [119888]

120597120579= 120573

119873

119904sdot120572 + 120574

120573 + 1205741199011015840(120579) (21)

where

119901 (120579) =ln 119904 minus ln (120579119904)

1 minus 120579

1199011015840(120579) =

1

1 minus 120579(119901 (120579) minus

120574

120579 (120572120579 + 120574))

(22)

Since ln(119904(120579119904)) lt 1 minus 120579119904119904 = 120574(1 minus 120579)(120572120579 + 120574) we then have

119901 (120579) minus120574

120579 (120572120579 + 120574)

lt120574

120572120579 + 120574minus

120574

120579 (120572120579 + 120574)=

120574

120572120579 + 120574(1 minus

1

120579) lt 0

(23)

It is clear that 1199011015840(120579) lt 0 and therefore we get 120597119864[119888]120597120579 lt 0This completes the proof

Theorem 3 At equilibrium the departure rate is a monoton-ically decreasing function of the departure time throughout thewhole peak period that is d119903(119905)d119905 le 0 119905 isin [119905

0 119905119890]

6 Discrete Dynamics in Nature and Society

Proof According to (11)ndash(16) the departure rate 119903(119905)

is continuous within each of intervals [1199050 1199051) (1199051 1199052) (1199052 1199053)

(1199053 1199054) (1199054 1199055) and (119905

5 119905119890] To prove the departure rate

is continuous during the peak period we calculate thefollowing limitations

lim119905rarr 119905+

1

119903 (119905) = lim119905rarr 119905+

1

120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

1

119903 (119905)

lim119905rarr 119905minus

2

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

2

119903 (119905)

lim119905rarr 119905minus

3

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1

=120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

= lim119905rarr 119905+

3

119903 (119905)

lim119905rarr 119905minus

4

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

4

119903 (119905)

lim119905rarr 119905+

5

119903 (119905) =(120572 + 120574) 119877 (119905

3) (1199053minus 1199050) minus (120572120579 + 120574) 119904

(120572 + 120574) ln (119877 (1199053) (120579119904 (119905

3minus 1199050)))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

5

119903 (119905)

(24)

This proves 119903(119905) is continuous indeed within the interval[1199050 119905119890]

Equations (11) (13) and (15) state that the departure rate119903(119905) is constant for 119905

0le 119905 le 119905

1 1199052le 119905 le 119905

3 and 119905

4le 119905 le

1199055 hence it is monotonically decreasing within these three

intervals By definition the cumulative departure flow 119877(119905) isnondecreasing with respect to time 119905Thus the denominatorsof the right-hand sides in (12) and (14) are nondecreasingwith respect to time 119905 Therefore the right-hand sides of (12)and (14) are nonincreasing with respect to time 119905 that is thedeparture rate 119903(119905) is monotonically decreasing within [119905

1 1199052]

and [1199053 1199054] The proof of d119903(119905)d119905 le 0 for all 119905 isin (119905

5 119905119890] can be

found in Xiao et al [18]In summary the departure rate 119903(119905) is monotonically dec-

reasing within all four intervals and at their boundaries Con-sidering the continuity of 119903(119905) for all 119905 isin [119905

0 119905119890] we conclude

that 119903(119905) is monotonically decreasing within [1199050 119905119890] This

completes the proof

Proposition 4 When parameter 120579 approaches one thestochastic bottleneck model immediately follows the determin-istic model

Proof According to the LrsquoHopitalrsquos rule we have lim120579rarr1

(1 minus

120579)(ln 120579minus1) = 1 We then have

lim120579rarr1

1205960= lim120579rarr1

1205965=minus120573

120574

lim120579rarr1

1205961= lim120579rarr1

1205962= lim120579rarr1

1205963= lim120579rarr1

1205964=120573

120572

lim120579rarr1

1205920= lim120579rarr1

1205925=(120574 minus 120573)

120574

lim120579rarr1

1205921= lim120579rarr1

1205922=(120573 minus 120572)

120572 lim

120579rarr1

1205923= lim120579rarr1

1205924=(120573 + 120572)

120572

(25)

lim120579rarr1

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 le 119905

1

119904 if 1199051le 119905 le 119905

4

120572119904

(120572 + 120574) if 119905

4lt 119905 le 119905

119890

(26)

Substituting (26) into (19) we have

1199050= minus

120574

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575 119905

5= 119905119890=

120573

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575

1199051= 1199052= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120573 minus 120572

120572120575

1199053= 1199054= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120572 + 120573

120572120575

(27)

Therefore we get the same traffic flow pattern with that froma deterministic bottleneck model

Proposition 5 When the number of commuters is givenincreasing the value of parameter 120579 will result in a decrease inthe length of peak period

Proof According to (16) we have d119904d120579 = 120572(120572 + 120574) gt 0This implies that 119904 is a monotonic increasing function withrespect to 120579 According to (17) we can obtain the length ofpeak period as follows

119905119890minus 1199050=119873

119904

1205960

1205960minus 1

+1205920

1 minus 1205960

120575 minus119873

119904

1

1205960minus 1

minus1205920

1 minus 1205960

120575 =119873

119904

(28)

Since119873 is constant and 119904 is a monotonic increasing functionwith respect to 120579 119905

119890minus 1199050is also a monotonic increasing

function with respect to 120579 This completes the proof

The above proof also shows that the length of peak periodis not affected by the 120575-value

3 Numerical Examples

The input parameters of our numerical example are 120572 =

64 $h 120573 = 39 $h 120574 = 1521 $h 119873 = 6000 veh 119904 =

4000 vehh 120575 = 10 min and 120579 = 09 By solving the proposed

Discrete Dynamics in Nature and Society 7

Table 1 The influence of parameter 120579 on the mean trip cost and the watershed time instant

120579 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

100 362 minus110 minus074 minus074 minus040 minus040 040 040 150095 378 minus114 minus077 minus073 minus046 minus031 036 039 152090 395 minus118 minus080 minus073 minus052 minus021 031 037 155085 413 minus123 minus085 minus072 minus059 minus009 026 035 157080 433 minus128 minus089 minus071 minus066 005 021 032 160

minus15 minus1 minus05 0 050

1

2

3

4

Departure time (hour)

Expe

cted

cost

($)

Total travel costTravel time cost

SDE costSDL cost

120579 = 09 120575 = 10min

Figure 3 The mean equilibrium trip cost and other components

model we obtained the time period with positive departurerate during [minus118 037] (see Table 1) The mean trip cost themean travel time cost and themean schedule delay cost (SDEand SDL) are depicted in Figure 3 We find that the mean tripcosts of all commuters are the same and are equal to 395 $but endure a trade-off between cost of travel time and cost ofschedule delay It is interesting that the waiting time cost isnot zero at the end of the peak period indicating that queuestill exists

It is interesting to investigate the impact of the fractionparameter 120579 on the solution of the stochastic bottleneckmodel We changed the parameter 120579 from 08 to 10 andcomputed themean trip cost and the watershed time instantsThe results are shown in Table 1 the row highlighted is withthe default 120579-value It can be seen that 119905

1= 1199052 1199053= 1199054 and 119905

5=

119905119890when 120579 = 10This confirmsProposition 4We can also find

that the length of time period with positive departure rateincreases with the 120579-value This confirms Proposition 5 Inaddition the second column of Table 1 shows that the meantrip cost decreases with the increase of 120579-valueThis confirmsTheorem 2 Since decreasing the 120579-value is equivalent toincreasing the travel time uncertainty commuters will leavehome earlier than before for avoiding the potential losscaused by uncertainty risk

The departure rates against different 120579-values are dis-played in Figure 4 It can be clearly seen that the resultsconfirm Propositions 4 and 5 that the stochastic bottleneckmodel immediately follows the deterministic model when

120579 = 10

120579 = 09

120579 = 08

0

2

4

6

8

10

Dep

artu

re ra

te (v

ehh

)

minus15 minus1 minus05 0 05Departure time (hour)

times103

120575 = 10min

Figure 4 The influence of parameter 120579 on departure rate

120579 approaches one and that enlarging the parameter 120579 willresult in a decrease in the length of peak period Figure 4 alsoshows that in equilibrium the departure rate during the peakperiod is monotonically decreasing which is consistent withTheorem 3

Table 2 lists the mean trip costs and the watershed timeinstants when 120579 = 09 and 120575-value from 0 minutes to 20minutes It is shown that in equilibrium the mean trip cost ismonotonically decreasing with increasing 120575 whilst the lengthof the peak period remains unchangedThis is consistent withTheorem 1

Figure 5 depicts the departure rates against different 120575-values at 120579 = 09 It can be seen that with larger 120575-value (orlonger PATI) the peak periods shift to later and the amountof earlier departures (at a rate higher than capacity) reducesThe overall departure time patterns become more flat andapproach designed capacity of the bottleneck with increasing120575-valueThis suggests that traffic congestion can be alleviatedby adopting flexi-timework schedule similar to that achievedthrough congestion pricing policy (see Arnott et al [5])

Figure 6 depicts the mean trip times with different pre-ferred arrival time intervals when 120579 = 09 One can observefrom this figure that adopting flexi-time work schedule canreduce the commutersrsquo travel time or the queue behind thebottleneck The areas below three curves are 06208 05940and 05131 hours respectively It shows further that the

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Using the boundary conditions of Situations 1ndash6 we canobtain the watershed lines as follows

1199051= 12059611199050+ 1205921120575 119905

2= 12059621199050+ 1205922120575

1199053= 12059631199050+ 1205923120575 119905

4= 12059641199050+ 1205924120575

1199055= 12059651199050+ 1205925120575

(19)

where

1205961= 1 minus

(120572 minus 120573) 120579120585

120572 120596

2=(120572 + 120573)

120572minus 120585

1205963=120573

120572minus 120579120585 + 1 120596

4=(120572 + 120574) (1 minus 120585)

120572+120573

120572

1205965= 1 +

(120574 + 120573)

(120572 minus (120572 + 120574) 120585) 120592

1=(120573 minus 120572) 120579120585

120572

1205922=120573

120572minus 120585 120592

3=120573

120572+ 120579120585

1205924=((120572 + 120574) 120585 + (120573 minus 120574))

120572 120592

5=

(120573 minus 120574)

(120572 minus (120572 + 120574) 120585)

120585 =ln 120579minus1

(1 minus 120579)

(20)

With the resulting stochastic departure pattern at long-term equilibrium the experienced day-to-day travel timesand number of travelers experiencing queues change accord-ing to varied capacities over days Given the boundary condi-tions of Situations 1ndash6 the cumulative departures and arrivalsof a stochastic bottleneck are given in Figure 2 The two solidcurves denote the cumulative departures and cumulativearrivals as in Figure 1 The dotted lines are the maximumand the minimum capacity of the bottleneck that is 119904 and120579119904 The earliest departure time from home to workplace is 119905

0

and if the bottleneck capacity equals 120579119904 commuters departingbetween 119905

1and 1199053will arrive at workplace on time If the

capacity equals 119904 commuters departing between 1199052and 1199054will

arrive at workplace on time Commuters departing withintime interval [119905

2 1199053] will always arrive at workplace on time

whatever the variability of capacity is At the beginningcommuters depart from home with a constant departurerate until the watershed time 119905

1 Subsequently the departure

rate will gradually drop down After time instant 1199054 another

constant departure rate will last until the watershed timeinstant 119905

5and the queuing length at this time will be zero

Later the departure rate continues to drop down to zero attime 119905

119890

224 Properties of the Stochastic Bottleneck Model In thissubsection we investigate the theoretical properties of theequilibrium solution of the proposed stochastic bottleneckmodel with PATI

Theorem 1 At equilibrium the expected trip cost for allcommuters is a monotonically increasing function of trafficdemand and a monotonically decreasing function of 120575-valuethat is 120597119864[119888(119905

0)]120597119873 gt 0 and 120597119864[119888(119905

0)]120597120575 lt 0 hold

Proof Since 119864[119888(1199050)] = (minus120575 minus 119905

0)120573 and 119905

0= 119873(119904(120596

0minus 1)) +

1205920120575(1 minus 120596

0) we have 120597119864[119888(119905

0)]120597120575 = minus2120573120574(120573 + 120574) lt 0 and

0Departure times

Cumulative departures

Cumulative arrivals

PATI

120579 = 09

0

Num

bers

of t

rave

lers

N

T(t2)

T(t3)T(t4)

T(t1)

t0 t1 t2 t3 t4 t5 te

120579s

s

Figure 2 Departure time distributions in the stochastic case

120597119864[119888(1199050)]120597119873 = 120573((1minus120596

0)119904)The definitions of120573 and 119904 imply

that they are positive hence to prove 120597119864[119888(1199050)]120597119873 gt 0 we

only need to prove 1 minus1205960gt 0 Since 0 lt 120579 lt 1 and 119904 = 119904(120572120579 +

120574)(120572 + 120574) gt 119904(120572120579 + 120574120579)(120572 + 120574) = 120579119904 we have 1 minus 120579 gt 0 Andln 119904 minus ln 120579119904 gt 0 clearly holds Therefore both the numeratorand denominator of the second term in the right-hand sideof the first equation of (18) are positive therefore we have1 minus 1205960gt 0 This completes the proof

Theorem 2 At equilibrium the expected trip cost for allcommuters is amonotonically decreasing function of parameter120579-value that is 120597119864[119888(119905

0)]120597120579 lt 0 holds

Proof Submitting 1199050= 119873(119904(120596

0minus 1)) + 120592

0120575(1 minus 120596

0) into

119864[119888(1199050)] = (minus120575 minus 119905

0)120573 then the first-order derivative can be

given as follows

120597119864 [119888]

120597120579= 120573

119873

119904sdot120572 + 120574

120573 + 1205741199011015840(120579) (21)

where

119901 (120579) =ln 119904 minus ln (120579119904)

1 minus 120579

1199011015840(120579) =

1

1 minus 120579(119901 (120579) minus

120574

120579 (120572120579 + 120574))

(22)

Since ln(119904(120579119904)) lt 1 minus 120579119904119904 = 120574(1 minus 120579)(120572120579 + 120574) we then have

119901 (120579) minus120574

120579 (120572120579 + 120574)

lt120574

120572120579 + 120574minus

120574

120579 (120572120579 + 120574)=

120574

120572120579 + 120574(1 minus

1

120579) lt 0

(23)

It is clear that 1199011015840(120579) lt 0 and therefore we get 120597119864[119888]120597120579 lt 0This completes the proof

Theorem 3 At equilibrium the departure rate is a monoton-ically decreasing function of the departure time throughout thewhole peak period that is d119903(119905)d119905 le 0 119905 isin [119905

0 119905119890]

6 Discrete Dynamics in Nature and Society

Proof According to (11)ndash(16) the departure rate 119903(119905)

is continuous within each of intervals [1199050 1199051) (1199051 1199052) (1199052 1199053)

(1199053 1199054) (1199054 1199055) and (119905

5 119905119890] To prove the departure rate

is continuous during the peak period we calculate thefollowing limitations

lim119905rarr 119905+

1

119903 (119905) = lim119905rarr 119905+

1

120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

1

119903 (119905)

lim119905rarr 119905minus

2

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

2

119903 (119905)

lim119905rarr 119905minus

3

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1

=120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

= lim119905rarr 119905+

3

119903 (119905)

lim119905rarr 119905minus

4

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

4

119903 (119905)

lim119905rarr 119905+

5

119903 (119905) =(120572 + 120574) 119877 (119905

3) (1199053minus 1199050) minus (120572120579 + 120574) 119904

(120572 + 120574) ln (119877 (1199053) (120579119904 (119905

3minus 1199050)))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

5

119903 (119905)

(24)

This proves 119903(119905) is continuous indeed within the interval[1199050 119905119890]

Equations (11) (13) and (15) state that the departure rate119903(119905) is constant for 119905

0le 119905 le 119905

1 1199052le 119905 le 119905

3 and 119905

4le 119905 le

1199055 hence it is monotonically decreasing within these three

intervals By definition the cumulative departure flow 119877(119905) isnondecreasing with respect to time 119905Thus the denominatorsof the right-hand sides in (12) and (14) are nondecreasingwith respect to time 119905 Therefore the right-hand sides of (12)and (14) are nonincreasing with respect to time 119905 that is thedeparture rate 119903(119905) is monotonically decreasing within [119905

1 1199052]

and [1199053 1199054] The proof of d119903(119905)d119905 le 0 for all 119905 isin (119905

5 119905119890] can be

found in Xiao et al [18]In summary the departure rate 119903(119905) is monotonically dec-

reasing within all four intervals and at their boundaries Con-sidering the continuity of 119903(119905) for all 119905 isin [119905

0 119905119890] we conclude

that 119903(119905) is monotonically decreasing within [1199050 119905119890] This

completes the proof

Proposition 4 When parameter 120579 approaches one thestochastic bottleneck model immediately follows the determin-istic model

Proof According to the LrsquoHopitalrsquos rule we have lim120579rarr1

(1 minus

120579)(ln 120579minus1) = 1 We then have

lim120579rarr1

1205960= lim120579rarr1

1205965=minus120573

120574

lim120579rarr1

1205961= lim120579rarr1

1205962= lim120579rarr1

1205963= lim120579rarr1

1205964=120573

120572

lim120579rarr1

1205920= lim120579rarr1

1205925=(120574 minus 120573)

120574

lim120579rarr1

1205921= lim120579rarr1

1205922=(120573 minus 120572)

120572 lim

120579rarr1

1205923= lim120579rarr1

1205924=(120573 + 120572)

120572

(25)

lim120579rarr1

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 le 119905

1

119904 if 1199051le 119905 le 119905

4

120572119904

(120572 + 120574) if 119905

4lt 119905 le 119905

119890

(26)

Substituting (26) into (19) we have

1199050= minus

120574

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575 119905

5= 119905119890=

120573

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575

1199051= 1199052= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120573 minus 120572

120572120575

1199053= 1199054= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120572 + 120573

120572120575

(27)

Therefore we get the same traffic flow pattern with that froma deterministic bottleneck model

Proposition 5 When the number of commuters is givenincreasing the value of parameter 120579 will result in a decrease inthe length of peak period

Proof According to (16) we have d119904d120579 = 120572(120572 + 120574) gt 0This implies that 119904 is a monotonic increasing function withrespect to 120579 According to (17) we can obtain the length ofpeak period as follows

119905119890minus 1199050=119873

119904

1205960

1205960minus 1

+1205920

1 minus 1205960

120575 minus119873

119904

1

1205960minus 1

minus1205920

1 minus 1205960

120575 =119873

119904

(28)

Since119873 is constant and 119904 is a monotonic increasing functionwith respect to 120579 119905

119890minus 1199050is also a monotonic increasing

function with respect to 120579 This completes the proof

The above proof also shows that the length of peak periodis not affected by the 120575-value

3 Numerical Examples

The input parameters of our numerical example are 120572 =

64 $h 120573 = 39 $h 120574 = 1521 $h 119873 = 6000 veh 119904 =

4000 vehh 120575 = 10 min and 120579 = 09 By solving the proposed

Discrete Dynamics in Nature and Society 7

Table 1 The influence of parameter 120579 on the mean trip cost and the watershed time instant

120579 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

100 362 minus110 minus074 minus074 minus040 minus040 040 040 150095 378 minus114 minus077 minus073 minus046 minus031 036 039 152090 395 minus118 minus080 minus073 minus052 minus021 031 037 155085 413 minus123 minus085 minus072 minus059 minus009 026 035 157080 433 minus128 minus089 minus071 minus066 005 021 032 160

minus15 minus1 minus05 0 050

1

2

3

4

Departure time (hour)

Expe

cted

cost

($)

Total travel costTravel time cost

SDE costSDL cost

120579 = 09 120575 = 10min

Figure 3 The mean equilibrium trip cost and other components

model we obtained the time period with positive departurerate during [minus118 037] (see Table 1) The mean trip cost themean travel time cost and themean schedule delay cost (SDEand SDL) are depicted in Figure 3 We find that the mean tripcosts of all commuters are the same and are equal to 395 $but endure a trade-off between cost of travel time and cost ofschedule delay It is interesting that the waiting time cost isnot zero at the end of the peak period indicating that queuestill exists

It is interesting to investigate the impact of the fractionparameter 120579 on the solution of the stochastic bottleneckmodel We changed the parameter 120579 from 08 to 10 andcomputed themean trip cost and the watershed time instantsThe results are shown in Table 1 the row highlighted is withthe default 120579-value It can be seen that 119905

1= 1199052 1199053= 1199054 and 119905

5=

119905119890when 120579 = 10This confirmsProposition 4We can also find

that the length of time period with positive departure rateincreases with the 120579-value This confirms Proposition 5 Inaddition the second column of Table 1 shows that the meantrip cost decreases with the increase of 120579-valueThis confirmsTheorem 2 Since decreasing the 120579-value is equivalent toincreasing the travel time uncertainty commuters will leavehome earlier than before for avoiding the potential losscaused by uncertainty risk

The departure rates against different 120579-values are dis-played in Figure 4 It can be clearly seen that the resultsconfirm Propositions 4 and 5 that the stochastic bottleneckmodel immediately follows the deterministic model when

120579 = 10

120579 = 09

120579 = 08

0

2

4

6

8

10

Dep

artu

re ra

te (v

ehh

)

minus15 minus1 minus05 0 05Departure time (hour)

times103

120575 = 10min

Figure 4 The influence of parameter 120579 on departure rate

120579 approaches one and that enlarging the parameter 120579 willresult in a decrease in the length of peak period Figure 4 alsoshows that in equilibrium the departure rate during the peakperiod is monotonically decreasing which is consistent withTheorem 3

Table 2 lists the mean trip costs and the watershed timeinstants when 120579 = 09 and 120575-value from 0 minutes to 20minutes It is shown that in equilibrium the mean trip cost ismonotonically decreasing with increasing 120575 whilst the lengthof the peak period remains unchangedThis is consistent withTheorem 1

Figure 5 depicts the departure rates against different 120575-values at 120579 = 09 It can be seen that with larger 120575-value (orlonger PATI) the peak periods shift to later and the amountof earlier departures (at a rate higher than capacity) reducesThe overall departure time patterns become more flat andapproach designed capacity of the bottleneck with increasing120575-valueThis suggests that traffic congestion can be alleviatedby adopting flexi-timework schedule similar to that achievedthrough congestion pricing policy (see Arnott et al [5])

Figure 6 depicts the mean trip times with different pre-ferred arrival time intervals when 120579 = 09 One can observefrom this figure that adopting flexi-time work schedule canreduce the commutersrsquo travel time or the queue behind thebottleneck The areas below three curves are 06208 05940and 05131 hours respectively It shows further that the

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Proof According to (11)ndash(16) the departure rate 119903(119905)

is continuous within each of intervals [1199050 1199051) (1199051 1199052) (1199052 1199053)

(1199053 1199054) (1199054 1199055) and (119905

5 119905119890] To prove the departure rate

is continuous during the peak period we calculate thefollowing limitations

lim119905rarr 119905+

1

119903 (119905) = lim119905rarr 119905+

1

120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=120572

120572 minus 120573sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

1

119903 (119905)

lim119905rarr 119905minus

2

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 minus 120573 (ln ((minus120575 minus 1199050) 119904) minus ln119877 (119905))

=119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

2

119903 (119905)

lim119905rarr 119905minus

3

119903 (119905) =119904 (1 minus 120579)

ln 120579minus1

=120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

= lim119905rarr 119905+

3

119903 (119905)

lim119905rarr 119905minus

4

119903 (119905) =120572119904 (1 minus 120579)

120572 ln 120579minus1 + 120574 (ln119877 (119905) minus ln ((120575 minus 1199050) 120579119904))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905+

4

119903 (119905)

lim119905rarr 119905+

5

119903 (119905) =(120572 + 120574) 119877 (119905

3) (1199053minus 1199050) minus (120572120579 + 120574) 119904

(120572 + 120574) ln (119877 (1199053) (120579119904 (119905

3minus 1199050)))

=120572

120572 + 120574sdot119904 (1 minus 120579)

ln 120579minus1= lim119905rarr 119905minus

5

119903 (119905)

(24)

This proves 119903(119905) is continuous indeed within the interval[1199050 119905119890]

Equations (11) (13) and (15) state that the departure rate119903(119905) is constant for 119905

0le 119905 le 119905

1 1199052le 119905 le 119905

3 and 119905

4le 119905 le

1199055 hence it is monotonically decreasing within these three

intervals By definition the cumulative departure flow 119877(119905) isnondecreasing with respect to time 119905Thus the denominatorsof the right-hand sides in (12) and (14) are nondecreasingwith respect to time 119905 Therefore the right-hand sides of (12)and (14) are nonincreasing with respect to time 119905 that is thedeparture rate 119903(119905) is monotonically decreasing within [119905

1 1199052]

and [1199053 1199054] The proof of d119903(119905)d119905 le 0 for all 119905 isin (119905

5 119905119890] can be

found in Xiao et al [18]In summary the departure rate 119903(119905) is monotonically dec-

reasing within all four intervals and at their boundaries Con-sidering the continuity of 119903(119905) for all 119905 isin [119905

0 119905119890] we conclude

that 119903(119905) is monotonically decreasing within [1199050 119905119890] This

completes the proof

Proposition 4 When parameter 120579 approaches one thestochastic bottleneck model immediately follows the determin-istic model

Proof According to the LrsquoHopitalrsquos rule we have lim120579rarr1

(1 minus

120579)(ln 120579minus1) = 1 We then have

lim120579rarr1

1205960= lim120579rarr1

1205965=minus120573

120574

lim120579rarr1

1205961= lim120579rarr1

1205962= lim120579rarr1

1205963= lim120579rarr1

1205964=120573

120572

lim120579rarr1

1205920= lim120579rarr1

1205925=(120574 minus 120573)

120574

lim120579rarr1

1205921= lim120579rarr1

1205922=(120573 minus 120572)

120572 lim

120579rarr1

1205923= lim120579rarr1

1205924=(120573 + 120572)

120572

(25)

lim120579rarr1

119903 (119905) =

120572119904

(120572 minus 120573) if 119905

0le 119905 le 119905

1

119904 if 1199051le 119905 le 119905

4

120572119904

(120572 + 120574) if 119905

4lt 119905 le 119905

119890

(26)

Substituting (26) into (19) we have

1199050= minus

120574

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575 119905

5= 119905119890=

120573

120573 + 120574sdot119873

119904+120574 minus 120573

120573 + 120574120575

1199051= 1199052= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120573 minus 120572

120572120575

1199053= 1199054= minus

120573120574

120572 (120573 + 120574)sdot119873

119904+120572 + 120573

120572120575

(27)

Therefore we get the same traffic flow pattern with that froma deterministic bottleneck model

Proposition 5 When the number of commuters is givenincreasing the value of parameter 120579 will result in a decrease inthe length of peak period

Proof According to (16) we have d119904d120579 = 120572(120572 + 120574) gt 0This implies that 119904 is a monotonic increasing function withrespect to 120579 According to (17) we can obtain the length ofpeak period as follows

119905119890minus 1199050=119873

119904

1205960

1205960minus 1

+1205920

1 minus 1205960

120575 minus119873

119904

1

1205960minus 1

minus1205920

1 minus 1205960

120575 =119873

119904

(28)

Since119873 is constant and 119904 is a monotonic increasing functionwith respect to 120579 119905

119890minus 1199050is also a monotonic increasing

function with respect to 120579 This completes the proof

The above proof also shows that the length of peak periodis not affected by the 120575-value

3 Numerical Examples

The input parameters of our numerical example are 120572 =

64 $h 120573 = 39 $h 120574 = 1521 $h 119873 = 6000 veh 119904 =

4000 vehh 120575 = 10 min and 120579 = 09 By solving the proposed

Discrete Dynamics in Nature and Society 7

Table 1 The influence of parameter 120579 on the mean trip cost and the watershed time instant

120579 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

100 362 minus110 minus074 minus074 minus040 minus040 040 040 150095 378 minus114 minus077 minus073 minus046 minus031 036 039 152090 395 minus118 minus080 minus073 minus052 minus021 031 037 155085 413 minus123 minus085 minus072 minus059 minus009 026 035 157080 433 minus128 minus089 minus071 minus066 005 021 032 160

minus15 minus1 minus05 0 050

1

2

3

4

Departure time (hour)

Expe

cted

cost

($)

Total travel costTravel time cost

SDE costSDL cost

120579 = 09 120575 = 10min

Figure 3 The mean equilibrium trip cost and other components

model we obtained the time period with positive departurerate during [minus118 037] (see Table 1) The mean trip cost themean travel time cost and themean schedule delay cost (SDEand SDL) are depicted in Figure 3 We find that the mean tripcosts of all commuters are the same and are equal to 395 $but endure a trade-off between cost of travel time and cost ofschedule delay It is interesting that the waiting time cost isnot zero at the end of the peak period indicating that queuestill exists

It is interesting to investigate the impact of the fractionparameter 120579 on the solution of the stochastic bottleneckmodel We changed the parameter 120579 from 08 to 10 andcomputed themean trip cost and the watershed time instantsThe results are shown in Table 1 the row highlighted is withthe default 120579-value It can be seen that 119905

1= 1199052 1199053= 1199054 and 119905

5=

119905119890when 120579 = 10This confirmsProposition 4We can also find

that the length of time period with positive departure rateincreases with the 120579-value This confirms Proposition 5 Inaddition the second column of Table 1 shows that the meantrip cost decreases with the increase of 120579-valueThis confirmsTheorem 2 Since decreasing the 120579-value is equivalent toincreasing the travel time uncertainty commuters will leavehome earlier than before for avoiding the potential losscaused by uncertainty risk

The departure rates against different 120579-values are dis-played in Figure 4 It can be clearly seen that the resultsconfirm Propositions 4 and 5 that the stochastic bottleneckmodel immediately follows the deterministic model when

120579 = 10

120579 = 09

120579 = 08

0

2

4

6

8

10

Dep

artu

re ra

te (v

ehh

)

minus15 minus1 minus05 0 05Departure time (hour)

times103

120575 = 10min

Figure 4 The influence of parameter 120579 on departure rate

120579 approaches one and that enlarging the parameter 120579 willresult in a decrease in the length of peak period Figure 4 alsoshows that in equilibrium the departure rate during the peakperiod is monotonically decreasing which is consistent withTheorem 3

Table 2 lists the mean trip costs and the watershed timeinstants when 120579 = 09 and 120575-value from 0 minutes to 20minutes It is shown that in equilibrium the mean trip cost ismonotonically decreasing with increasing 120575 whilst the lengthof the peak period remains unchangedThis is consistent withTheorem 1

Figure 5 depicts the departure rates against different 120575-values at 120579 = 09 It can be seen that with larger 120575-value (orlonger PATI) the peak periods shift to later and the amountof earlier departures (at a rate higher than capacity) reducesThe overall departure time patterns become more flat andapproach designed capacity of the bottleneck with increasing120575-valueThis suggests that traffic congestion can be alleviatedby adopting flexi-timework schedule similar to that achievedthrough congestion pricing policy (see Arnott et al [5])

Figure 6 depicts the mean trip times with different pre-ferred arrival time intervals when 120579 = 09 One can observefrom this figure that adopting flexi-time work schedule canreduce the commutersrsquo travel time or the queue behind thebottleneck The areas below three curves are 06208 05940and 05131 hours respectively It shows further that the

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Table 1 The influence of parameter 120579 on the mean trip cost and the watershed time instant

120579 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

100 362 minus110 minus074 minus074 minus040 minus040 040 040 150095 378 minus114 minus077 minus073 minus046 minus031 036 039 152090 395 minus118 minus080 minus073 minus052 minus021 031 037 155085 413 minus123 minus085 minus072 minus059 minus009 026 035 157080 433 minus128 minus089 minus071 minus066 005 021 032 160

minus15 minus1 minus05 0 050

1

2

3

4

Departure time (hour)

Expe

cted

cost

($)

Total travel costTravel time cost

SDE costSDL cost

120579 = 09 120575 = 10min

Figure 3 The mean equilibrium trip cost and other components

model we obtained the time period with positive departurerate during [minus118 037] (see Table 1) The mean trip cost themean travel time cost and themean schedule delay cost (SDEand SDL) are depicted in Figure 3 We find that the mean tripcosts of all commuters are the same and are equal to 395 $but endure a trade-off between cost of travel time and cost ofschedule delay It is interesting that the waiting time cost isnot zero at the end of the peak period indicating that queuestill exists

It is interesting to investigate the impact of the fractionparameter 120579 on the solution of the stochastic bottleneckmodel We changed the parameter 120579 from 08 to 10 andcomputed themean trip cost and the watershed time instantsThe results are shown in Table 1 the row highlighted is withthe default 120579-value It can be seen that 119905

1= 1199052 1199053= 1199054 and 119905

5=

119905119890when 120579 = 10This confirmsProposition 4We can also find

that the length of time period with positive departure rateincreases with the 120579-value This confirms Proposition 5 Inaddition the second column of Table 1 shows that the meantrip cost decreases with the increase of 120579-valueThis confirmsTheorem 2 Since decreasing the 120579-value is equivalent toincreasing the travel time uncertainty commuters will leavehome earlier than before for avoiding the potential losscaused by uncertainty risk

The departure rates against different 120579-values are dis-played in Figure 4 It can be clearly seen that the resultsconfirm Propositions 4 and 5 that the stochastic bottleneckmodel immediately follows the deterministic model when

120579 = 10

120579 = 09

120579 = 08

0

2

4

6

8

10

Dep

artu

re ra

te (v

ehh

)

minus15 minus1 minus05 0 05Departure time (hour)

times103

120575 = 10min

Figure 4 The influence of parameter 120579 on departure rate

120579 approaches one and that enlarging the parameter 120579 willresult in a decrease in the length of peak period Figure 4 alsoshows that in equilibrium the departure rate during the peakperiod is monotonically decreasing which is consistent withTheorem 3

Table 2 lists the mean trip costs and the watershed timeinstants when 120579 = 09 and 120575-value from 0 minutes to 20minutes It is shown that in equilibrium the mean trip cost ismonotonically decreasing with increasing 120575 whilst the lengthof the peak period remains unchangedThis is consistent withTheorem 1

Figure 5 depicts the departure rates against different 120575-values at 120579 = 09 It can be seen that with larger 120575-value (orlonger PATI) the peak periods shift to later and the amountof earlier departures (at a rate higher than capacity) reducesThe overall departure time patterns become more flat andapproach designed capacity of the bottleneck with increasing120575-valueThis suggests that traffic congestion can be alleviatedby adopting flexi-timework schedule similar to that achievedthrough congestion pricing policy (see Arnott et al [5])

Figure 6 depicts the mean trip times with different pre-ferred arrival time intervals when 120579 = 09 One can observefrom this figure that adopting flexi-time work schedule canreduce the commutersrsquo travel time or the queue behind thebottleneck The areas below three curves are 06208 05940and 05131 hours respectively It shows further that the

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Table 2 The influence of parameter 120575 on the mean trip cost and the watershed time instants

120575min 119864[119888(119905)] 1199050

1199051

1199052

1199053

1199054

1199055

119905119890

119905119890minus 1199050

0 498 minus128 minus080 minus080 minus080 minus055 021 027 1555 446 minus123 minus080 minus072 minus068 minus038 026 032 15510 395 minus118 minus080 minus073 minus052 minus021 031 037 15515 343 minus113 minus080 minus074 minus036 minus004 036 042 15520 291 minus108 minus080 minus075 minus020 013 041 047 155

minus15 minus1 minus05 0 050

25

5

75

10

Departure time (hour)

Dep

artu

re ra

te (v

ehh

)

120579 = 09

120575 = 0min

120575 = 10min

120575 = 20min

times103

Figure 5 The influence of parameter 120575 on departure rate

adoption of flexi-time in equilibrium leads to less congestionthan under the fixed work schedule

Figure 7 shows the joint effect of the 120579-value and the120575-value on the equilibrium trip cost For a fixed 120579 theexpected trip cost declines with an increase of the preferredarrival time interval Since the schedule delay costs enduredby commuters are reduced with an increase of the lengthPATI without changing the departure pattern this meansthat transport policies to encourage firms in CBD to adoptflexi-time can ease overall system traffic congestion On theother hand for a fixed 120575-value the expected trip cost declineswith the decrease of 120579-value which confirms that improvingsystem reliability and reducing uncertainty will increase thesystemrsquos effectiveness

4 Conclusion

This paper investigated the travel choice behavior underuncertainty on morning commute problem by consideringthe capacity variability of a highway bottleneck The bot-tleneck model was applied to analyze the departure timepattern of a group of homogeneous commuters with thesame preferred arrival time interval The capacity of thebottleneck is assumed to follow a uniform distribution and

0

02

04

06

08

Mea

n tr

avel

tim

e (ho

ur)

minus15 minus1 minus05 0 05Departure time (hour)

120575 = 0min

120575 = 10min

120575 = 20min

120579 = 09

Figure 6 Travel time with different 120575-value

3

3

35

35

35

4

4

4

45

45

45

5

5

08 085 09 095 100

5

10

15

20

120575(m

in)

120579

N = 6000 veh

Figure 7 The equilibrium trip cost 119864[119862] with different 120579 and 120575

the commutersrsquo departure time choice to follow the UEprinciple in terms of the mean trip cost The analyticalsolution of the stochastic bottleneckmodel was derived Bothanalytical and numerical results show that increasing thecapacity variation results in longer peak period and highercommutersrsquo mean trip cost In addition it is shown that withlonger flexi-time interval the departure time distributions

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

become flatter This suggests that flexi-time is an effectivedemand management measure for alleviating peak conges-tion For future research we will further improve the modelwith consideration of heterogeneous commuters and travelrisk and apply the model in analyzing such policy measuresas congestion pricing metering and flexible work scheme

Conflict of Interests

The authors declare that there is no conflict of interests regar-ding the publication of this paper

Acknowledgments

The authors acknowledge thank the financial supportfrom the National Basic Research Program of China(2012CB725401) the PhD Student Innovation Fund of Bei-hang University (302976) and the China Scholarship Coun-cil

References

[1] W S Vickrey ldquoCongestion theory and transport investmentrdquoAmerican Economic Review vol 59 pp 251ndash261 1969

[2] M J Smith ldquoThe existence of a time-dependent equilibrium di-stribution of arrivals at a single bottleneckrdquo Transportation Sci-ence vol 18 no 4 pp 385ndash394 1984

[3] C F Daganzo ldquoUniqueness of a time-dependent equilibriumdistribution of arrivals at a single bottleneckrdquo TransportationScience vol 19 no 1 pp 29ndash37 1985

[4] R M Braid ldquoUniform versus peak-load pricing of a bottleneckwith elastic demandrdquo Journal of Urban Economics vol 26 no 3pp 320ndash327 1989

[5] R Arnott A de Palma and R Lindsey ldquoEconomics of a bot-tleneckrdquo Journal of Urban Economics vol 27 no 1 pp 111ndash1301990

[6] H Yang and H-J Huang ldquoThe multi-class multi-criteria traf-fic network equilibrium and systems optimumproblemrdquoTrans-portation Research Part B vol 38 no 1 pp 1ndash15 2004

[7] G Ramadurai S V Ukkusuri J Zhao and J-S Pang ldquoLinearcomplementarity formulation for single bottleneck model withheterogeneous commutersrdquoTransportation Research Part B vol44 no 2 pp 193ndash214 2010

[8] M J Machina ldquoChoice under uncertainty problem solved andunsolvedrdquo Journal of Economic Perspectives vol 1 no 1 pp 121ndash154 1987

[9] R P Cubitt and R Sugden ldquoDynamic decision-making underuncertainty an experimental investigation of choice betweenaccumulator gamblesrdquo Journal of Risk and Uncertainty vol 22no 2 pp 103ndash128 2001

[10] C Gollier andN Treich ldquoDecision-making under scientific un-certainty the economics of the precautionary principlerdquo Journalof Risk and Uncertainty vol 27 no 1 pp 77ndash103 2003

[11] M H Birnbaum and U Schmidt ldquoAn experimental investiga-tion of violations of transitivity in choice under uncertaintyrdquoJournal of Risk and Uncertainty vol 37 no 1 pp 77ndash91 2008

[12] C Starmer ldquoDevelopments in non-expected utility theory thehunt for a descriptive theory of choice under riskrdquo Journal ofEconomic Literature vol 38 no 2 pp 332ndash382 2000

[13] A de Palma M Ben-Akiva D Brownstone et al ldquoRisk unce-rtainty and discrete choice modelsrdquo Marketing Letters vol 19no 3-4 pp 269ndash285 2008

[14] J V Henderson ldquoThe economics of staggered work hoursrdquoJournal of Urban Economics vol 9 no 3 pp 349ndash364 1981

[15] S-I Mun and M Yonekawa ldquoFlextime traffic congestion andurban productivityrdquo Journal of Transport Economics and Policyvol 40 no 3 pp 329ndash358 2006

[16] B Siu and H K Lo ldquoEquilibrium trip scheduling in congestedtraffic under uncertaintyrdquo in Proceedings of the 18th Interna-tional SymposiumonTransportation andTrafficTheoryWHKLam S CWong andHK Lo Eds pp 19ndash38 Elsevier OxfordUK 2009

[17] H Li M Bliemer and P Bovy ldquoDeparture time distribution inthe stochastic bottleneck modelrdquo International Journal of ITSResearch vol 6 no 2 pp 79ndash86 2008

[18] L L Xiao H J Huang and R Liu ldquoCongestion behavior andtolls in a bottleneckmodel with stochastic capacityrdquoTransporta-tion Science 2013

[19] A Chen Z Ji and W Recker ldquoTravel time reliability with risk-sensitive travelersrdquoTransportationResearchRecord no 1783 pp27ndash33 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of