Regional Science and Urban Economics 43 (2013) 764–782

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Regional Science and Urban Economics

j ourna l homepage: www.e lsev ie r .com/ locate / regec

Trip-timing decisions with traffic incidents☆

Mogens Fosgerau a,b,⁎, Robin Lindsey c

a Technical University of Denmark, Denmarkb Centre for Transport Studies, Swedenc Sauder School of Business, University of British Columbia, Canada

☆ We are grateful to the editor and two referees for veFosgerau would also like to thank Stefan Mabit and Lars Peearlier draft. Both authors thank participants at the KuhmoJune30, 2011.Mogens Fosgerauhas been supportedby theD⁎ Corresponding author. Tel.: +45 54 25 65 21.

E-mail addresses: mf@transport.dtu.dk (M. Fosgerau),(R. Lindsey).

1 The 2012 Report does not repeat this estimate. As Haltion of nonrecurrent congestion is difficult to determine betude and timing of recurrent congestion, and vice versa.prevalence of nonrecurrent congestion because incident-iafter the incidents are cleared away.

0166-0462/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.regsciurbeco.2013.07.002

a b s t r a c t

a r t i c l e i n f o

Article history:Received 28 June 2012Received in revised form 30 June 2013Accepted 8 July 2013Available online 19 July 2013

JEL classification:C61D62R41

Keywords:Departure-time decisionsBottleneck modelTraffic incidentsCongestionScheduling utilityMorning commuteEvening commute

This paper analyzes traffic bottleneck congestion when drivers randomly cause incidents that temporarily blockthe bottleneck. Drivers have general scheduling preferences for time spent at home and at work. They indepen-dently choose morning departure times from home to maximize expected utility without knowing whether anincident has occurred. The resulting departure timepatternmay be compressed or dispersed according towheth-er or not the bottleneck is fully utilized throughout the departure period on days without incidents. For both theuser equilibrium (UE) and the social optimum (SO) the departure pattern changes from compressed to dispersedwhen the probability of an incident becomes sufficiently high. The SO can be decentralized with a time-varyingtoll, but drivers are likely to be strictly worse off than in the UE unless they benefit from the toll revenues in someway. A numerical example is presented for illustration. Finally, the model is extended to encompass minorincidents in which the bottleneck retains some capacity during an incident.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Traffic congestion imposes a heavy burden in urban areas. The TexasTransportation Institute conducts an annual survey of traffic congestionin the US. According to its 2012 report, in 2011 congestion caused an es-timated 5.5 billion hours of travel delay and 2.9 billion gallons of extrafuel consumption with a total cost of $121 billion (Schrank et al., 2012).The average cost per automobile commuter in the urban areas studiedwas $818. Nonrecurring traffic congestion due to accidents, bad weather,special events, and other shocks accounts for a large fraction of the totaldelays. According to Schrank et al. (2011, Appendix B, p. B-27) incident-related delays alone contribute 52–58% of total delay in US urban areas.1

ry helpful comments. Mogenster Østerdal for comments on an-Nectar Conference, Stockholm,anish Strategic Research Council.

robin.lindsey@sauder.ubc.ca

l (1993) observes, the contribu-cause it depends on themagni-Drivers may underestimate thenduced queues can persist long

ghts reserved.

Unanticipated travel delays upset peoples' travel plans, and maycause them to arrive late with serious consequences for commuting,business, and other types of trips. Travelers can sometimes adjust tothe threat of delays by changing their transport mode or destination,or even canceling trips, but a more common response is to adjustdeparture times. Researchers have long been interested in studyingthe adjustment process, and they have adopted various modelingapproaches. In an early and insightful study, Gaver (1968) derived theoptimal departure time for a driver faced with stochastic travel timewho incurs costs from both travel time and schedule delay. The optimalpolicy, which Gaver called a headstart strategy, entails a probabilistictrade-off between arriving early and arriving late. Gaver assumedthat travel time has a constant and exogenous variance, and he didnot attempt to derive an endogenous travel time distribution as adynamic equilibrium. His approach was adopted and extended byKnight (1974), Hall (1983), Noland and Small (1995), and Noland(1997).

All these studies usemodelswith flow congestion. An alternative ap-proach is to use the Vickrey (1969) bottleneck model in which conges-tion delay takes the form of queuing. A series of studies by Arnott et al.(1991, 1999) and Lindsey (1994, 1999) introduced stochasticity into thebottleneck model by assuming that capacity and/or demand fluctuaterandomly from day to day, but are constant during the period of use

4 Holding spare capacity is broadly consistent with policies of reserving shoulder lanesfor use during accidents and other disruptions.

5 A notational glossary is provided at the end of the paper.6 Throughout the paperwewill refer to “drivers” even though individuals are treated as

a continuum in the model so that there are no discrete or atomic agents. Reference to“drivers”, “users”, “commuters” and so on is common in the bottleneck model literature,

765M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

on a given day. For want of a better term, we will call this the “daily-shocks” model.2

Our paper differs from these earlier bottleneck-model studies inthree ways. First, they adopted the traditional specification of trip-timing preferences used by Vickrey (1969) in which individuals havea preferred time to arrive at their destination and incur a scheduledelay cost proportional to the amount of time they arrive earlier orlater. Following Börjesson et al. (2012) we will call this the “step”model. Here we adopt a more general scheduling utility functionapproach that incorporates preferences for time spent at different activ-ities. We apply the model to commuting trips by specifying preferencesfor time spent at home and at work.3

Second, and more fundamentally, we assume that capacity canfluctuate while trips are being made rather than being determined be-fore travel begins. Third, we assume that capacity reductions are dueto incidents caused by drivers during their trip. The timing of shocks istherefore endogenous to the model rather than exogenous as in earlierstudies. Since drivers are responsible for most incidents, this within-day, endogenous specification of capacity fluctuations accounts for asignificant portion of nonrecurring congestion that occurs. It also pro-vides the basis for assessing tolling and other policies to reduce thecosts of congestion by altering peoples' travel decisions. For most ofthe paper we assume that capacity is reduced to zero by an incident al-though in a final section we examine a variant of the model in whichloss of capacity is partial.

Two unpublished studies cover part of the same ground as we do.Schrage (2006) derives the unregulated and socially optimal departurerates for a single road link when the accident rate is a function of theinflow rate and therefore endogenous. Her model differs from ours inthree main respects. First, she uses the Henderson (1974) flow conges-tion model in which a driver's travel time is determined by the aggre-gate departure rate when he starts his trip. This model has no statevariable analogous to queue length in the bottleneck model. Second,capacity is reduced only partially in an incident and it subsequentlyrecovers slowly, and deterministically, rather than all at once. Third,drivers are assumed to know whether and when an accident has oc-curred before they depart. Schrage derives the optimal time-varyingand state-dependent toll that decentralizes the social optimum, butshe does not solve for the timing of departures in either the unregulateduser equilibrium or the social optimum. In independentwork, Peer et al.(2010) use the bottleneck model to analyze incidents in which, likeSchrage (2006), capacity loss is partial. They treat incident timing asexogenous and assume that an incident persists until all drivers havecompleted their trips. They also adopt the “step” model of trip-timingpreferences. Finally, they limit attention to the unregulated user equilib-rium and do not examine the social optimum or tolling.

In our paper we undertake a systematic analysis of both user(i.e., Nash) equilibrium and socially optimal trip-timing decisionswhen drivers do not know whether an incident has occurred beforethey decide when to depart. We solve for the optimal time-varying(but state-independent) toll that decentralizes the social optimum.One of the questions we address is whether the bottleneck operates atcapacity throughout the travel period on days when no incident occurs,

2 Arnott et al. (1991, 1999) and Li et al. (2008) analyze user equilibrium in the daily-shocks model, whereas Lindsey (1994, 1999) focuses on the social optimum. Other recentpapers have also studied random travel times using the bottleneck model. Xin andLevinson (2007) assume that travel times are exogenous and independently distributedover time, and their model does not feature incidents per se. Fosgerau (2010) showshow the dynamics of random congestion induce characteristic loops in the relationshipbetween the mean and the variance of travel time over different times of day. de Palmaand Fosgerau (2011) analyze random queue sorting whereby travel time is random fromthe perspective of individual travelers, but capacity and demand are fixed.

3 Jenelius et al. (2011) use a similar scheduling utility function approach to study the ef-fects of unpredictable travel time shocks on trip-timing decisions. They apply themodel toa full day of activity includingmorning and evening commutes. Their model differs in fea-turing shocks that are exogenous and independent of time of day. There is also no trafficcongestion in their model.

or whether some capacity goes “unused”. We show that for both theuser equilibrium and social optimum, spare capacity does exist forpart, or all, of the travel period if incidents are sufficiently probable.4

In contrast to the daily-shocks model, departures can be more spreadout in the user equilibrium than in the social optimum. Another differ-ence is that the socially-optimal departure rate can decrease, ratherthan increase, over time.

The paper is organized as follows. Section 2 describes the model.Section 3 summarizes the main features of user equilibrium and socialoptimum for the deterministic variant of the model with no incidents.Section 4 derives properties of the user equilibrium with incidents.Section 5 conducts a parallel analysis of the social optimum. Section 6presents a numerical example calibrated for morning commutes, andthen considers a variant for evening commutes. Section 7 undertakes apartial analysis of an extension of the model in which the bottleneckretains some capacity during an incident. Finally, Section 8 concludeswith a summary and ideas for extension.

2. The model

A continuum of N identical individuals drive alone from a commonorigin through a bottleneck to a common destination.5 To be concrete,in most of the paper the trip is assumed to be a morning commutefrom home (H) to work (W). (However, an evening commute is alsoexamined in the example section.) Departure time from home is denot-ed by t. Drivers6 depart at a rate ρ (t) during a set of times T; cumulativedepartures are thus R(t) = ∫ {v ∈ T|v ≤ t}ρ(v)dv.7 Free-flow travel timebefore and after reaching the bottleneck is normalized to zero. A driverdeparting at t encounters a queuing delay of q (t) at the bottleneck andreaches work at time a = t + q (t). Drivers have scheduling prefer-ences8 described by the utility function

u t; að Þ ¼Z t

tH

β vð ÞdvþZ tW

aγ vð Þdv: ð1Þ

The limits of integration, tH and tW, are chosen such that all travel takesplace within the interval [tH, tW]. Function β(∙) N 0 denotes the flow ofutility from being at home, and function γ(∙) N 0 denotes utility frombeing at work. Functions β(∙) and γ(∙) are assumed to be continuouslydifferentiable with derivatives β′ b 0 and γ′ N 0 and to intersect attime t⁎.9 Utility from time spent driving is normalized to zero. Theseassumptions ensure that, for any fixed trip duration, there is a uniquedeparture time t, t b t⁎, that maximizes scheduling utility. They also as-sure that u (t, a) is strictly increasing in t, strictly decreasing in a, andglobally strictly concave. Two final assumptions, Lim

v→tHβ vð Þ ¼ ∞ and Lim

v→tW

γ vð Þ ¼ ∞ , will ensure existence of a Nash equilibrium in departuretimes.10

and it facilitates exposition.7 All statements about ρ in the paper will be “almost surely”, since ρ can take arbitrary

values on sets of Lebesgue measure zero without affecting aggregate behavior or welfare.To ease exposition this detail will be ignored.

8 This formulation of scheduling preferences originates from Vickrey (1969, 1973) andhas been used by Tseng and Verhoef (2008), Fosgerau and Engelson (2011), Fosgerau andde Palma (2012), Jenelius et al. (2011), and Börjesson et al. (2012).

9 The notation differs from that in the stepmodel where β denotes the cost per minuteof arriving before t⁎, and γ denotes the cost per minute of arriving after t⁎. The assump-tions β′ b 0 and γ′ N 0 rule out the step model because the (implicit) β(∙) and γ(∙) func-tions in that model are constants except at t⁎ where γ(∙) steps up. This is notparticularly restrictive since the step-model preferences can be approximated arbitrarilyclosely by differentiable functions. Nevertheless, the assumptions could be generalizedas in Fosgerau and Engelson (2011).10 These assumptions are relaxed in the example of Section 6where β(∙) and γ(∙) are lin-ear functions.

766 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

If no incident is in progress, the bottleneck has a flow capacity of s.Incidents are caused by a randomly determined driver and block thebottleneck for a deterministic period Δ N 0. The incident occurs whenthe driver reaches the head of the queue (if any) and is about to crossthe bottleneck.11 At most one driver causes an incident on a given day.Let ξ ∈ [0,N] be the random variable that indicates the position of theculpable driver in the departure schedule if an incident occurs. Variableξ has a continuously differentiable density f(ξ) and a cumulativefunction F(ξ), where F (0) = 0 and F (N) b 1. Function f(∙) will be called“incident risk” and F (N) “incident probability”.12 A baseline assumptionis that incident risk is constant, and this will be assumed in the numer-ical example of Section 6. A day with an incident is called a “Bad day”,and a daywithout an incident is called a “Good day”. Any costs associat-ed with incidents other than delay are ignored.13 For the analysis of thesocial optimum, we require f(∙) to be differentiable.

Drivers independently choose their departure times to maximizeexpected scheduling utility while taking the departure rate as givenandwithout knowingwhether an incident has occurred. The first driverdeparts at time t0. If the bottleneck operates at capacity from t0 on, anddriver ξ causes an incident, the incident occurs at time t0 þ R ξð Þ

s andcapacity is restored at t0 þ R ξð Þ

s þ Δ . A driver who is delayed by anincident will be said to incur “queuing delay” even if the driver causesthe incident and there is no queue of drivers ahead. The duration of anincident, Δ, is assumed to be long enough that the queue does not dissi-pate until after the last driver departs. For future reference this is calledthe “persistent-queue” assumption. If a queue develops on Good days, itmay or may not dissipate before the last driver departs.

3. User equilibrium and system optimumwithout incidents

As a first step in analyzing themodel, and also for later reference, webriefly describe the user equilibrium and social optimum for a setting inwhich incidents do not occur.

3.1. User equilibrium

Fosgerau and de Palma (2012) analyze user equilibrium (UE) in themodel without incidents and their treatment is briefly summarizedhere. Let superscript e denote UE and 0 the setting without incidents.It is easy to show that departures take place during an interval Te0 =[t0e0,tNe0]. A queue exists throughout the interior of Te0, but disappears attime tN

e0 so that te0N ¼ te00 þ Ns . de Palma and Fosgerau (2011) refer to

elimination of the queue at tNe0 as the “no residual queue” property of UE.Since a queue exists in the interior of Te0, a driver departing at time t

exits the bottleneck at te00 þ Re0 tð Þs . Scheduling utility is constant on Te0

and equal to

u t; te00 þ Re0 tð Þs

!¼Z t

v¼tH

β vð ÞdvþZ tW

v¼te00 þRe0 tð Þs

γ vð Þdv:

11 The mechanics of the model are the same if an incident occurs anywhere betweenhome and the exit point from the bottleneck. All drivers ahead of the culprit are unaffectedby the incident.12 Incident risk depends on ξ, but not on the identity of the individual driver. The order inwhich drivers depart therefore does not affect aggregate variables of interest. Incident riskalso does not depend on the rate at which drivers arrive at the bottleneck or on whetherthere is a queue. (However, the probability that an incident occurs within a short intervalof time is proportional to the arrival rate at the bottleneck.) Relaxing these assumptionswould complicate the analysis.13 Results of interest are unaffected if incidents create additional costs (e.g., related toemergency response, vehicle repair, filing of insurance claims, etc.) as long as the costsare independent of t and t0

w0.

The UE departure rate is derived by differentiating u(∙) with respect to t,setting the derivative to zero, and rearranging terms:

ρe0 tð Þs

¼ β tð Þγ te00 þ Re0 tð Þ

s

� � : ð2Þ

Since the first and last drivers receive the same expected utility, ute00 ; te00� �

¼ u te00 þ Ns ; t

e00 þ N

s

� �. This condition can be written as

Z te00 þNs

v¼te00

β vð Þ−γ vð Þð Þdv ¼ 0: ð3Þ

Eq. (3) gives an implicit formula for t0e0. It states that a driver who shiftsfrom departing first to departing last gains additional utility at homethat just offsets foregone utility at work.

A representative user equilibrium is shown in Fig. 1. The first driverdeparting at t0e0 has a scheduling utility equal to the area under curveabgjdef. This area is smaller by area bgd than the ideal of leaving homeat t⁎, arriving immediately at work, and gaining utility of abcdef. Thelast driver departing at tN

e0 has a utility equal to the area under thecurve abcdkefwhich is less than the ideal by area dke. Condition (3) as-sures that areas dke and bgd are equal. Now consider a driver departingat time t, and call this ‘driver t’. From time t0e0 to t, driver t gains area bghc

more utility than the first driver. From time t to te00 þ Re0 tð Þs , driver t is

caught in the queue and gains less utility than the first driver by areahlmj. In equilibrium, queuing time is such that area hlmj matches areabghc. This is equivalent to the condition that driver t's utility fromtime spent at home from t0

e0 to t equals the first driver's utility from

time spent at work from t to te00 þ Re0 tð Þs .

Note that driver t's queuing time is shorter the higher his utility fromwork because the driver foregoes more utility while traveling ratherthan being at work. Thus, if utility functions β(·) and γ(·) were shiftedupwards an equal amount, area bghc would not change, but area hlmjwould become taller and narrower. This observation helps to explain adifference between the morning commute and evening commute ex-amples in Section 6.

3.2. The social optimum

The social optimum (SO), denoted by superscript w, is derived bychoosing ρ(t) to maximize aggregate scheduling utility:

U ¼Z

t∈Tw0ρ tð Þu t; að Þdt;

where Tw0 is the set of SO departure times. The departure rate ismaintained at capacity throughout Tw0 so that no queue is allowed toform, a = t for each driver, and tw0

N ¼ tw00 þ N

s . Aggregate schedulingutility is therefore

U ¼Z tw0

0 þNs

t¼tw00

su t; tð Þdt ¼ sZ tw0

0 þNs

t¼tw00

Z t

v¼tH

β vð ÞdvþZ tW

v¼tγ vð Þdv

" #dt:

The first-order condition for t0w0 is

Z tw00 þN

s

v¼tw00

β vð Þ−γ vð Þð Þdv ¼ 0: ð4Þ

Eq. (4) for t0w0 is identical to Eq. (3) for t0e0. Departures therefore occurover the same time interval in UE and SO: Tw0 = Te0.

It is straightforward to show that the SO can be decentralized bylevying a time-varying toll, τw0(t), such that u(t,t) − τw0(t) is constantfor t ∈ Tw0 and lower for t ∉ Tw0. If demand were elastic, the toll mustbe zero at the beginning and end of the travel period (Arnott et al.,

15 Thefirst expression for expectedutility in Eq. (9) is explained as follows. Thefirst termpertains to utility when there is no incident which occurs with probability 1 − F(Re(t)).

767M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

1993). Here the number of drivers is fixed, and a constant can be addedto the toll schedule without affecting trip timing. Nevertheless, it isnatural to set the toll to zero at the beginning and end of the departureperiod so that τw0(t0w0) = τw0(tNw0) = 0.

4. User equilibrium with incidents

Consider now the case of interest in which incidents can occur. Webegin by establishing some general characteristics of UE. This is follow-ed by separate analyses of the compressed departures and disperseddepartures configurations.

4.1. General characteristics of user equilibrium

Lemmas 1 and 2 below summarize properties of a UE.

Lemma 1. (a): The UE departure rate, ρe (t), is strictly positive on aninterval Te = (t0e,tNe ). (b): t∗ ∈ (t0e,tNe ). (c): Re(tNe ) = N ≤ s(tNe − t0

e): onGood days the no-residual queue property holds.

Proof. Part (a): ρe (t) cannot drop to zero in the interior of Te since oth-erwise some driver could increase expected utility by departing duringthe gap. Part (b): Clearly t0e b t⁎; otherwise any driver departing after t⁎

could increase utility by departing at t⁎ instead. Suppose tNe ≤ t∗ so

that β(tNe ) ≥ γ(tNe ). If the last driver departed dt later, his expectedutility would change by dE(u|tNe ) = (β(tNe ) − (1 − F(N))γ(tNe ))dt.Givenβ(tNe ) ≥ γ(tNe ) and F(N) N 0, dE(u|tNe ) N 0 and tN

e cannot be an indi-vidually optimal departure time. Part (c): If Re(tNe ) N s(tNe − t0

e), therewould be a queue at tN

e . This would violate the no residual queueproperty, and the last driver could leave home later without arrivingat work later. □

Lemma 2. In UE the last departure time, tNe , is such that

F Nð Þ≤1−β teN� �

γ teN� � : ð5Þ

Proof. By Lemma 1, on Good days there is no residual queue at tNe . Andby the persistent-queue assumption, if an incident occurs the queuepersists until after tNe . A driver departing just after tNe therefore encoun-ters a queue with probability F (N), and the driver's expected utilitychanges at a rate

∂Eðu tj Þ∂t ¼ β tð Þ− 1−F Nð Þð Þγ tð Þ: ð6Þ

Expression (6)must be non-positive for t ≥ tNe ; otherwise the last driver

could increase utility by departing later. Since Eq. (6) is largest fort =tN

e , inequality (5) must hold.14 □

Expression (6) is readily interpreted. β(t) is the marginal benefit attime t from staying longer at home, and γ(t) is the marginal cost ofdelaying arrival at work. Departing later implies arriving later if noincident has occurred which is the case with probability 1 − F (N). Ifan incident has occurred, there is no cost of delaying departure sincethe driver merely spends less time queuing and reaches work at thesame time. If F (N) is sufficiently large, and Δ is sufficiently small, thepersistent queue assumption will be violated since any tN

e that satisfiesEq. (5) will occur after the queue dissipates. The persistent queueassumption therefore imposes bounds on F (N) and Δ.

User equilibrium follows one of two patterns. In one, queuing onGood days persists until the last driver has departed. Similar to UE

14 There is no additional equilibrium condition analogous to Eq. (5) that applies to t0e be-

cause an incident cannot occur before t0e. Given t0

e b t*, departing before t0e would clearly

not be optimal.

with no incidents, all drivers complete their trips within the minimumfeasible time interval ofN/s. This patternwill be called “compressed” de-partures. In the second pattern, called “dispersed” departures, queuingends on Good days before the last driver departs and departures extendfor a time interval longer than N/s. The compressed-departures anddispersed-departures patterns are examined separately in the nexttwo subsections.

4.2. User equilibrium with compressed departures

Themain characteristics of a compressed-departuresUE are summa-rized in the following theorem.

Theorem 1. Assume departures are compressed. Then a unique Nashequilibrium exists. The equilibrium departure rate is

ρe tð Þs

¼β tð Þ

1−F Re tð Þ� �� �γ te0 þ Re tð Þ

s

� �þ F Re tð Þ� �

γ te0 þ Re tð Þs

þ Δ� �

þ f Re tð Þ� �Z te0þRe tð Þs þΔ

v¼te0þRe tð Þs

γ vð Þdv

0BB@1CCA

: ð7Þ

The departure time set Te is determined by the conditions tNe =

t0e + N/s and

Z teN

v¼te0

β vð Þ−γ vð Þð Þdv ¼ F Nð ÞZ teNþΔ

v¼teN

γ vð Þdv: ð8Þ

Proof. Expected utility from departing at t is

Eðu tj Þ ¼ 1−F Re tð Þ� �� �u t; te0 þ

Re tð Þs

� �

þ F Re tð Þ� �u t; te0 þ

Re tð Þs

þ Δ� �

¼Z t

v¼tH

β vð ÞdvþZ tW

v¼te0þRe tð Þs

γ vð Þdv

−F Re tð Þ� �Z v¼te0þRe tð Þs þΔ

v¼te0þRe tð Þs

γ vð Þdv;

ð9Þ

which is constant during Te.15 Differentiate and set to zero to obtain

β tð Þ−γ te0 þRe tð Þs

� �ρe tð Þs

− f Re tð Þ� �ρe tð Þ

Z te0þRe tð Þs þΔ

v¼te0þRe tð Þs

γ vð Þdv

−F Re tð Þ� �ρe tð Þs

γ te0 þRe tð Þs

þ Δ� �

−γ te0 þRe tð Þs

� �� �¼ 0:

Collecting terms in ρe (t) yields Eq. (7) which simplifies to Eq. (2) if f(n) = 0, ∀n. Utility from departing at t0e is given by Eq. (9) with t = t0

e:

Eðu te0�� � ¼ u te0; t

e0

� � ¼ Z te0

v¼tH

β vð ÞdvþZ tW

v¼te0

γ vð Þdv: ð10Þ

The driver departs at time t and arrives at te0 þ Re tð Þs when the Re (t) preceding drivers have

passed the bottleneck. The second term pertains to utility when an incident has occurred,with probability F (Re (t)). Since the bottleneck is shut for a period Δ, the driver arrives atte0 þ Re tð Þ

s þ Δ.

16 This equation is explained as follows.When an incident occurs at time v, Re (v) drivershave passed the bottleneck. During the interval [v, v + Δ], no further drivers can pass. Af-ter v + Δ, drivers pass the bottleneck again at rate s as long as a queue persists. Cumula-tive passages through the bottleneck by time a N v + Δ are therefore Re (v) + s(a − v − Δ). A driverwho departs at time t is preceded by Re (t) other drivers. This drivertherefore arrives when the bottleneck has processed this number of drivers: Re (v) + s(a − v − Δ) = Re (t).17 The first term in Eq. (14) covers instances inwhich no incident occurs. The probabilityof no incident is 1 − F(Re(t)). The driver encounters no queue, and therefore arrives im-mediately at t and receives a utility of u(t,t). The second term covers instances in which

an incident occurs before et which happens with probability F Re et� �� �. The number of

drivers who have departed by t is Re (t). Were there no incident, the driver departing att would pass the bottleneck at te0 þ Re tð Þ

s . With an incident, arrival is delayed by Δ and the

driver passes the bottleneck at te0 þ Re tð Þs þ Δ. The last term covers instances inwhich an in-

cident occurs after et but before the driver departs at t. If an incident occurs at time v, thedriver arrives atvþ Δþ Re tð Þ−Re vð Þ

s as explained above Eq. (14). Since there is no queue beforev, the driver responsible for an incident at time v leaves home at time v. The probabilitythat this driver causes an accident is f(Re(v)), and the rate at which drivers are departingat time v is ρ(v). This explains the integrand of the last term.

768 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

Expected utility from departing at tNe is given by Eq. (9) with t = tNe :

Eðu teN�� � ¼ Z teN

v¼tH

β vð ÞdvþZ tW

v¼teN

γ vð Þdv−F Nð ÞZ teNþΔ

v¼teN

γ vð Þdv: ð11Þ

Equating Eqs. (10) and (11) yields Condition (8). The left-hand side ofEq. (8) is decreasing in t0

e, while (given tNe = t0

e + N/s) the right-handside is increasing. Any solution is thus unique. Existence is guaranteedby the assumptions Lim

v→tΗβ vð Þ ¼ ∞ and Lim

v→tWγ vð Þ ¼ ∞. □

The candidate compressed UE described in Theorem 1 can be testedby solving for tNe using Eq. (8), and substituting the result into Condi-tion (5). If Eq. (5) is satisfied, the UE is indeed compressed. If Eq. (5) isviolated, the UE is dispersed. Condition (5) is assumed to be satisfiedin the balance of this subsection.

The right-hand side of Eq. (8) is an increasing function of F (N) and Δ.Thus, the greater the incident probability and the longer an incident lasts,the earlier departures begin. This result is consistent with the Gaver(1968) analysis of headstart strategies, mentioned in the introduction.

Corollary 1. Assume departures are compressed, and f (n) is independentof n. Then ρe is decreasing so that Re is concave.

Proof. Concavity of Re follows from Eq. (7) since β(t) in the nu-merator is a decreasing function of t, whereas in the denominator:

γ te0 þ Re tð Þs

� �;γ te0 þ Re tð Þ

s þ Δ� �

;∫te0þRe tð Þ

s þΔ

v¼te0þRe tð Þs

γ vð Þdv and F (Re (t)) are all

increasing; γ te0 þ Re tð Þs þ Δ

� �Nγ te0 þ Re tð Þ

s

� �; and f (Re (t)) is constant. □

Concavity of the cumulative departure schedule implies that thedeparture rate decreases monotonically over time, and also that ongood days queuing can occur only during one connected time period.Concavity is also a property of user equilibrium in the daily-shocksmodel (Arnott et al., 1999, Proposition 1).

From Eq. (7) the initial departure rate is

ρe te0� �s

¼ β te0� �

γ te0� �þ f 0ð Þ

Z te0þΔ

v¼te0

γ vð Þdv: ð12Þ

With no incidents, the initial departure rate is given by Eq. (2):

ρe0 te00� �s

¼β te00� �γ te00� � : ð13Þ

Compared to Eq. (13), Eq. (12) includes an additional term in the denom-inator, but since t0

e b t0e0, β(t0e) N β(t0e0) and γ(t0e) b γ(t0e0). It is therefore

unclear whether incident risk induces drivers to depart at a faster orslower initial rate. (In the example presented in Section 6 the departurerate is faster.) However, if there is no incident risk for the first drivers(i.e., f (0) = 0), then it follows from Eq. (13), Eq. (12), and t0

e b t0e0 that

the initial departure rate is higher when incidents can occur. This is alsoa property of the model with exogenous incidents in Peer et al. (2010).

4.3. User equilibrium with dispersed departures

If a candidate UE with compressed departures is computed usingEq. (8), and inequality (5) is violated, the UE is dispersed. Dependingon the time path of f(∙), the departure rate can be nonmonotonic, andon Good days queuing can occur in disjoint time intervals. This is con-ceivable if f(∙) has a pronounced double peak, but it seems unlikelysince the natural baseline is for incident risk to be constant. To keepthe analysis tractable it is assumed that any queuing on Good days

occurs during a single interval beginning at t0e. Thus, suppose that on

Good days there is a queue for t∈ te0;et� �and no queue for t∈ et; teNh i

.

For t∈ te0;et� �, the departure rate is given by Eq. (7). For t∈ et; teNh i

,

Eq. (9) does not apply and expected utility must be computed afresh.If an incident occurs at time v≤et , a driver departing at t arrives attime a ¼ te0 þ Re tð Þ

s þ Δ . If v≥et , arrival time is defined by the conditionRe(v) + s(a − v − Δ) = Re(t), or a ¼ vþ Δþ Re tð Þ−Re vð Þ

s .16 Expectedutility is therefore17

E uð jtÞ ¼ 1−F Re tð Þ� �� �u t; tð Þ þ F Re et� �� �

u t; te0 þRe tð Þs

þ Δ� �

þZ t

v¼et ρ vð Þ f Re vð Þ� �u t; vþ Δþ Re tð Þ−Re vð Þ

s

� �dv:

ð14Þ

Differentiating Eq. (14) with respect to t, and setting the derivativeto zero, one obtains

β tð Þ− 1−F Re tð Þ� �� �γ tð Þ− f Re tð Þ� �

ρe tð ÞZ tþΔ

v¼tγ vð Þdv

−ρe tð Þs

F Re et� �� �γ te0 þ

Re tð Þs

þ Δ� �

−ρe tð Þs

Z t

v¼etρ vð Þ f Re vð Þ� �γ vþ Δþ Re tð Þ−Re vð Þ

s

� �dv ¼ 0:

Collecting terms in ρe(t) yields

ρe tð Þs

¼ β tð Þ− 1−F Re tð Þ� �� �γ tð Þ

sf Re tð Þ� �Z tþΔ

v¼tγ vð Þdvþ F Re et� �� �

γ te0 þ Re tð Þs

þ Δ� �

þZ t

v¼etρ vð Þ f Re vð Þ� �γ vþ Δþ Re tð Þ−Re vð Þ

s

� �dv

26643775: ð15Þ

The denominator of Eq. (15) is strictly positive. The numerator must bepositive until t = tN

e , and nonpositive thereafter. Thus, the departurerate drops to zero at tNe which is defined by the condition

β teN� � ¼ 1−F Nð Þð Þγ teN

� �: ð16Þ

Condition (5) therefore holds as an equality when departures aredispersed. The UE with dispersed departures can be solved numericallyusing the following iterative procedure:

1 Guess t0e.2 Integrate Eq. (7) from t = t0

e to t ¼ et, whereet is defined by the condi-tion R et� � ¼ s et−te0

� �.

3 Integrate Eq. (15) from t ¼ et to t = tNe where tN

e is defined byCondition (16).

18 Optimal control methods are described in Kamien and Schwartz (1981, Part II) andLeonard and Van Long (1992, Chap. 6).19 Lindsey (1994) also uses this approach.

769M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

4 If Re(tNe ) matches N within a tolerance limit, then stop. Otherwise re-turn to step 1.

5. The social optimum with incidents

5.1. Preliminary results

The SO with incidents maximizes total expected utility:

E Uð Þ ¼Z

t∈Twρ tð ÞE u tj Þdt;ð ð17Þ

where Tw is the set of SO departure times. The SO departure rate, ρw(t),maximizes E(U) subject to the feasibility constraints ρw(t) ≥ 0 andRw(tNw) = N. Before tackling this optimal control problem some generalproperties of ρw(t) will be deduced.

Lemma 3. (a): The SO departure rate never exceeds capacity: ρw(t) ≤ s.(b): t∗ ∈ int(Tw).

Lemma 3 is proved in Appendix B.1. Part (a) is obvious: exceedingcapacity would cause queuing on Good days without giving any driverextra time at home or work. Part (b) is also intuitive: to maximizetotal scheduling utility at home and work, the first driver must departwhen home time is more valuable, and the last driver must departwhen work time has greater value.

Lemma 4. In the SO, the last departure time, tNw, is such that

F Nð Þ≤1−β twN� �

γ twN� � : ð18Þ

Proof. If the last driver is rescheduled to depart slightly after tNw, otherdrivers are unaffected and the change in total expected utility is limitedto the last driver. The proof of Lemma 2 therefore applies to Lemma 4. □

Condition (18) on tNw has the same functional form as Condition (5)

on tNe . This congruence will be used later to compare the timing of

departures in the SO and UE.Lemmas 3 and 4 establish some bounds on the rate and timing

of departures in the SO, but a number of questions remain.Should ρw(t) ever be reduced below capacity? If so, how doesρw(t) vary over time thereafter? Is ρw(t) a continuous function?Is it ever optimal to reduce ρw(t) low enough and long enoughto eliminate queuing for at least some incident states? Variousbehaviors are possible. For example, suppose an early incident islikely, the incident risk then declines, and Δ is large. The modelis then similar to the daily-shocks model for which ρw(t) is weak-ly increasing over time (Lindsey, 1994). If, alternatively, an earlyincident is likely, but Δ is small, it is optimal to hold ρw(t) belowcapacity long enough to clear any queue from the probable, butshort-lived incident. A third possibility is that incident risk isincreasing with n. In this case it may be prudent to accelerate de-partures in order to induce incidents earlier, and allow them to becleared away sooner so that most drivers do not arrive inordi-nately late.

Given the wide range of possible solutions, the SO is difficult to ana-lyze in full generality. Technical obstacles also arise. These can becircumvented by reformulating the optimization problem (see below),but the solution is tedious. Consequently, attention will be focused onthe optimal timing of departures and on whether the departure rateshould ever be reduced below capacity.

5.2. Optimal control formulation

Three technical difficulties arise if optimal control theory18 is appliedto maximize Eq. (17) with respect to ρw(t). First, the Hamiltonian de-pends on lagged values of ρw. This problem is partly overcome byusing the index or position of drivers in the departure schedule, n, asthe running variable rather than t. The control variable becomes thetime headway between successive drivers rather than the departurerate. Second, theHamiltonian depends on lagged values of the state var-iable R. This problem is circumvented by replacing Rwith a set of state-contingent queuing times.19 Third, the equation of motion for queuingtime q (t) is not differentiable at q = 0. This problem is addressed byimposing a nonnegativity constraint on queuing time that binds duringtime intervals when ρw(t) = s.

The optimal solution of the reformulated problem is assumed tocomprise two stages. In Stage 1, which includes drivers n∈ 0; bn

, head-way – denoted by h – is maintained at h(n) = 1/s. This is equivalent toholding the departure rate at capacity. In Stage 2, which encompassesthe remaining driversn∈ bn;N�

, headway is increased above 1/s. The op-timal value of bn is solved as described below. If bn ¼ N, Stage 2 is degen-erate and the departure rate is held at capacity throughout the travelperiod. On Good days, all drivers travel within a time span of N/s, andthe departure schedule is compressed in the same sense as it is com-pressed in the UE. If bn ¼ 0, Stage 1 is degenerate and the departurerate is held below capacity throughout.

Total expected utility for the two stages combined is:

E Uð Þ ¼Z N

n¼0

1−F nð Þð ÞU t nð Þ; t nð Þ þ qþ nð Þ� �þZ n

ξ¼0f ξð ÞU t nð Þ; t nð Þ þ qξ nð Þ

� �dξ

24 35dn; ð19Þ

where t(n) is departure time for driver n, q+(n) is queuing time experi-enced by driver n if no incident has occurred, and qξ(n) is queuing timeexperienced by driver n if driver ξ ≤ n has caused an incident. Theequations of motion for q+(n) and qξ(n) are:

dqþ nð Þdn

¼1s−h nð Þ if qþ nð ÞN0; or qþ nð Þ ¼ 0 and h nð Þb1

s0 otherwise

( );

dqξ nð Þdn

¼1s−h nð Þ if qξ nð ÞN0; or qξ nð Þ ¼ 0 and h nð Þb1

s0 otherwise

( ); ξbn:

In Stage 1, the constraint q+(n) ≥ 0 is binding. Given the persistent-queue assumption, the nonnegativity constraint qξ(n) ≥ 0 can beignored in both stages.

5.2.1. Stage 1: Departure rate held at capacityThe equations of motion and constraints for Stage 1 are:

dt nð Þdn

¼ h nð Þ costate variable μ1 nð Þ≤0ð Þ; ð20Þ

qþ nð Þ≥0 multiplier Ψ nð Þ≥0ð Þ; ð21Þ

dqþ nð Þdn

¼ 1s−h nð Þ costate variable λþ nð Þ≤0

� �; ð22Þ

dqξ nð Þdn

¼ 1s−h nð Þ; ξbn costate variable λξ nð Þ≤0

� �: ð23Þ

Eq. (20) stipulates that departure time, a state variable, increases at arate equal to the headway between successive drivers. Costate variableμ1 reflects the benefit of occupying or “using up” departure time slots.

770 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

Because departure time slots are valuable in the interior of Tw, μ1 b 0.Initial conditions are:

t 0ð Þ free; ð24Þ

μ1 0ð Þ ¼ 0; ð25Þqþ 0ð Þ ¼ 0; ð26Þ

qξ ξð Þ ¼ Δ; ξ∈ 0;N½ �: ð27Þ

The departure time for the first driver is chosen freely as per Condi-tion (24). Costate variable μ1 is therefore zero for the first driver as perCondition (25). Queuing time on Good days is initially zero as perCondition (26). If an incident occurs, queuing time jumps from 0 to Δas per Condition (27).

The Hamiltonian is20

Ω ¼ 1−F nð Þð ÞU t nð Þ; t nð Þ þ qþ nð Þ� �þ Z n

ξ¼0f ξð ÞU t nð Þ; t nð Þ þ qξ nð Þ

� �dξ

þμ1 nð Þh nð Þ þ 1−F nð Þð ÞΨ nð Þqþ nð Þ þ 1−F nð Þð Þλþ nð Þ 1s−h nð Þ

� �þZ n

ξ¼0f ξð Þλξ nð Þdξ 1

s−h nð Þ

� �:

ð28Þ

Optimality conditions are21

∂Ω∂h ¼ μ1 nð Þ|ffl{zffl}

1ð Þ

− 1−F nð Þð Þλþ nð Þ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}2ð Þ

−Z n

ξ¼0f ξð Þλξ nð Þdξ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

3ð Þ

¼ 0; ð29Þ

∂μ1 nð Þ∂n ¼ − ∂H

∂t nð Þ ¼ −

β t nð Þð Þ|fflfflfflffl{zfflfflfflffl}1ð Þ

− 1−F nð Þð Þγ t nð Þ þ qþ nð Þ� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}2ð Þ

−Z n

ξ¼0f ξð Þγ t nð Þ þ qξ nð Þ

� �dξ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

3ð Þ

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;; ð30Þ

∂λþ nð Þ∂n ¼ − ∂H

∂qþ nð Þ ¼ γ t nð Þ þ qþ nð Þ� �−Ψ nð Þ; ð31Þ

∂λξ nð Þ∂n ¼ − ∂H

∂qξ nð Þ ¼ γ t nð Þ þ qξ nð Þ� �

; ξbn: ð32Þ

Eq. (29) identifies the net benefit from marginally increasing the head-way for driver n. Term (1) is the opportunity cost of allocating moredeparture time to driver n. Term (2) is the expected benefit from reduc-ing queuing time when no incident has occurred. Similarly, term (3) isthe expected benefit from reducing queuing time when an incidenthas occurred. At the optimum, the opportunity cost matches theexpected benefits. Eq. (30) describes the evolution of μ1(n), whereμ1(n) b 0 corresponds to the disbenefit of using up departure timeslots. The term in braces is the rate of change in driver n's expectedutility as the driver's departure time increases. Term (1) is driver n'sutility from staying longer at home. Term (2) is driver n's expectedloss of work-time utility if no incident has occurred, and term (3) isthe corresponding loss if an incident has occurred. Finally, Eqs. (31)and (32) describe the evolution of the costate variables, λ+(n) andλξ(n), which specify the shadow benefit of queuing time, and thus arenegative. Eq. (31) governs the disbenefit of queuing time when there

20 In Eq. (28) the multiplier Ψ(n) and the costate variables λ+(n) and λξ(n) are multi-plied by their respective probability and probability densities. This facilitates interpreta-tion of the optimality conditions.21 Second-order conditions for an optimum are assumed to hold.

is no incident. This disbenefit declines over time as fewer drivers remainwho will arrive late. Eq. (32) is interpreted similarly.

5.2.2. Solution with compressed departuresStage 1 can prevail for none, some, or all of the travel period. If it

prevails for all of it, the SO is compressed. The departure rate is held atcapacity, andfirst-order Condition (29) is not needed to derive the solu-tion. Total expected utility is given by Eq. (19) with q+(n) = 0, t(n) =t0w + n/s, and t(n) + qξ(n) = t(ξ) + Δ + (n − ξ)/s = t0

w +n/s + Δ.Using Eq. (1), this yields

E Uð Þ ¼Z N

n¼0

Z tw0 þn=s

v¼tH

β vð ÞdvþZ tW

v¼tw0 þn=sγ vð Þdv

−F nð ÞZ tw0 þn=sþΔ

v¼tw0 þn=sγ vð Þdv

2666437775dn:

The first-order condition for t0w is

∂E Uð Þ∂tw0

¼Z N

n¼0

β tw0 þ n=s� �

−γ tw0 þ n=s� �

−F nð Þ γ tw0 þ n=sþ Δ� �

−γ tw0 þ n=s� �� �" #

dn ¼ 0;

or Z N

n¼0β tw0 þ n=s� �

−γ tw0 þ n=s� �� �

dn ¼Z N

n¼0F nð Þ γ tw0 þ n=sþ Δ

� �−γ tw0 þ n=s

� �� �dn:

ð33Þ

Using integration byparts, the right-hand side of Eq. (33) can bewritten

F nð ÞZ tw0 þn=sþΔ

t¼tw0 þn=sγ vð Þdvj

N

n¼0

−Z N

n¼0f nð Þ

Z tw0 þn=sþΔ

t¼tw0 þn=sγ vð Þdtdn

¼ F Nð ÞZ twNþΔ

t¼twN

γ tð Þdt−Z N

n¼0f nð Þ

Z tw0 þn=sþΔ

t¼tw0 þn=sγ tð Þdtdn:

Changing the variable of integration from n to t = t0w + n/s yields, finally,

Z twN

t¼tw0

β tð Þ−γ tð Þð Þdt ¼

F Nð ÞZ twNþΔ

t¼twN

γ tð Þdt−sZ twN

t¼tw0

f s t−tw0� �� � Z tþΔ

v¼tγ vð Þdv

� �dt:

ð34Þ

Holding F (N) fixed, the right-hand side of Eq. (34) is a decreasing func-tion of f(∙) on the interval (0,N). Departures in the SO therefore beginlater the higher the incident risk for any given incident probability. Theright-hand side of Eq. (34) is an increasing function of Δ so that, similarto the UE, departures begin earlier the longer incidents last.

To determinewhether the SO is indeed compressed, tNw can be solvedwith Eq. (34) and substituted into Condition (18).22 If the SO and UE areboth compressed it is possible to compare their trip timing and welfareas is done in the following theorem.

Theorem 2. Assume departures are compressed in both UE and SO. Then(a): Departures begin later in the SO than in the UE, but earlier than inthe model without incidents: t0

e b t0w b t0

e0 = t0w0. (b): The SO can be

implemented using a time-dependent toll. (c): If the toll is constrained tobe non-negative, drivers are strictly worse off than in the UE if they donot benefit from the toll revenues.

22 Condition (18) can be derived using the optimal control formulation in this subsectionby substituting μ1(N) = 0,Ψ(N)q+(N) = 0,h Nð Þ ¼ 1

s, and t(N) + qξ(N) = t0w + N/s + Δ

into the Hamiltonian (28), differentiating it with respect to t(N), and evaluating the deriv-ative at t(N) = tN

w.

771M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

Theorem2 is proved in Appendix B.2. Part (a) of Theorem2 indicatesthat departures begin and end later in the SO than UE if both are com-pressed.23 To see why, note that in the UE the first and last driversgain the same expected utility or, equivalently, incur the same expectedprivate travel cost. The first driver is therefore indifferent between con-tinuing to depart first, and switching to depart last. However, byswitching to last the driver no longer imposes an accident risk on otherdrivers. Starting from the UE departure schedule this switch is thereforesocially desirable at the margin, and the SO therefore has a later depar-ture schedule than the user equilibrium. Looked at another way, thelast driver does not impose an accident externality because there areno subsequent drivers to delay. However, the last driver is concernedabout being delayed by an incident caused by one of the earlier drivers.By contrast, the first driver is not concerned about being delayed, butdoes impose an externality on all the other drivers. Individuals are there-fore biased toward departing too early. In this respect, the UE is moresensitive than the SO to incident risks.

Part (b) of Theorem 2 asserts that, as in themodel without incidents,the SO departure schedule can bedecentralizedwith a time-varying toll.Part (c) states that the toll makes drivers worse off. This is because thetoll must be higher at the beginning of the departure period than atthe end so that drivers delay departure.24 Since negative tolls areruled out by assumption, the first driver has to pay a positive toll.

As noted above, Conditions (5) and (18) have the same functionalform. Since tN

w N tNe if the UE and SO are both compressed, Condition (5)

is more stringent than Condition (18). It is therefore possible for Condi-tion (5) to fail for a candidate compressedUE so that theUE is dispersed,but for Condition (18) to hold so that the SO is compressed.25 The resultsof Theorem 2 still apply in this case as formalized in the followingtheorem.

Theorem 3. Assume departures are dispersed in the UE but compressed inthe SO. Then (a): Departures begin and end later in the SO than in the UE,but earlier than in the model without incidents: t0

e b t0w b t0

e0 = t0w0. (b):

The SO can be implemented using a time-dependent toll. (c): If the toll isconstrained to be non-negative, drivers are strictly worse off than in theUE if they do not benefit from the toll revenues.

Theorem 3 is proved in Appendix B.3. The intuition underlyingTheorem 3 is the same as for Theorem 2.

5.2.3. Stage 2: Departure rate held below capacityThe equations of motion and constraints for Stage 2 of the SO are

similar to Stage 1:

dt nð Þdn

¼ h nð Þ costate variable μ2 nð Þ≤0ð Þ; ð35Þ

dqξ nð Þdn

¼ 1s−h nð Þ; ξbn costate variable λξ nð Þ≤0

� �: ð36Þ

In Stage 2 the departure rate is held below capacity so that q+ (n) = 0.This variable and its associated multiplier, Ψ(n), are thus omitted.Terminal conditions are:

μ2 Nð Þ ¼ 0; ð37Þ

23 This contrasts with the daily-shocksmodel inwhich the SO can begin earlier (Lindsey,1994).24 This is unlike either the model without incidents or the daily-shocks model (seeLindsey (1994, Proposition 10)).25 This possibility contrasts with a property of (deterministic) flow-congestion modelsthat departures aremore spread out in the system optimum than user equilibrium as longas flow is not hypercongested. See Chu (1995), Small and Verhoef (2007, Section 4.1.2),and DePalma and Arnott (2012).

λξ Nð Þ ¼ 0; ξ∈ 0;Nð �: ð38Þ

Costate variable μ2 for departure time is zero for the last driver, as perCondition (37), because departure time slots after t(N) are available,but undesirable. The shadow value of queuing time is also zero for thelast driver as per Condition (38) because there are no further driversto be delayed by a queue.

The Hamiltonian for Stage 2 is

Ω ¼ 1−F nð Þð ÞU t nð Þ; t nð Þð Þ þZ n

ξ¼0f ξð ÞU t nð Þ; t nð Þ þ qξ nð Þ

� �dξ

þ μ2 nð Þh nð Þ þZ n

ξ¼0f ξð Þλξ nð Þdξ 1

s−h nð Þ

� �:

ð39Þ

Optimality conditions are:

∂Ω∂h ¼ μ2 nð Þ−

Z n

ξ¼0f ξð Þλξ nð Þdξ ¼ 0; ð40Þ

∂μ2 nð Þ∂n ¼ − ∂H

∂t nð Þ ¼ −β t nð Þð Þ− 1−F nð Þð Þγ t nð Þð Þ

−Z n

ξ¼0f ξð Þγ t nð Þ þ qξ nð Þ

� �dξ

8<:9=;; ð41Þ

∂λξ nð Þ∂n ¼ − ∂H

∂qξ nð Þ ¼ γ t nð Þ þ qξ nð Þ� �

; ξbn: ð42Þ

Eqs. (40) and (41) have similar interpretations to Eqs. (29) and (30) forStage 1. Integrating Eq. (42), andusing terminal Condition (38), gives anequation for λξ(n):

λξ nð Þ ¼ −Z N

v¼nγ t ξð Þ þ Δþ s−1 v−ξð Þ� �

dv: ð43Þ

Differentiating Eq. (40)with respect to n, andmatching the resultingexpression for ∂μ2 nð Þ

∂n with Eq. (41), one obtains:

β t nð Þð Þ ¼ 1−F nð Þð Þγ t nð Þð Þ− f nð Þλn nð Þ; n∈ bn;N �: ð44Þ

Eq. (44) characterizes the optimal departure time for driver n. Ex-cept for the last term, it has a similar interpretation to Eq. (6) forthe UE. The left-hand side is the marginal benefit to driver n fromdelaying departure. The first term on the right-hand side is theexpected marginal cost to driver n of delaying arrival at workwhen there is no incident. The second term on the right-hand sideis the expected additional marginal external cost that driver n im-poses on subsequent drivers. This cost arises because any incidentwill end later and hence cause subsequent drivers to arrive laterand lose more time at work.

Given terminal Condition (38), λN(N) = 0 and Eq. (44) simplifieswith n = N to

β tw Nð Þ� � ¼ 1−F Nð Þð Þγ tw Nð Þ� �; ð45Þ

where superscriptw is added to clarify comparison with the UE. Condi-tion (18) therefore holds as an equality if departures are dispersed inthe SO. Eq. (45) for tw(N) is identical to Eq. (16) for tNe with dispersed de-partures inUE. Hence tNw = tN

e : if departures are dispersed in both the SOand UE, then departures end at the same time. This contrasts with thecase where both UE and SO are compressed in which SO departuresbegin and end later.

If driver ξ causes an incident, driver n N ξ is delayed at the bottleneckuntil time

t nð Þ þ qξ nð Þ ¼ t ξð Þ þ Δþ s−1 n−ξð Þ: ð46Þ

Fig. 1. Scheduling utility and timing of departures.

26 For “rear-end and sideswipe collisions” themean is 58 min, and for “hit-object, broad-side, and ‘other’ types of collisions” the mean is 62 min (Table 7 of their paper).

772 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

Using Eq. (46), and differentiating Eq. (44)with respect to n, one obtains(see Appendix B.4) an expression for the SO headway:

h nð Þ ¼f nð Þ

γ t nð Þð Þ þ γ t nð Þ þ Δð Þþs−1

Z N

v¼nγ′ t nð Þ þ Δþ s−1 v−nð Þ� �

dv

0@ 1Aþ λn nð Þ∂ f nð Þ=∂n

1−F nð Þð Þγ′ t nð Þð Þ−β′t nð Þð Þ

þ f nð ÞZ N

v¼nγ′ t nð Þ þ Δþ s−1 v−nð Þ� �

dv

: ð47Þ

The denominator of Eq. (47) is strictly positive. If f(n) = ∂f(n)/∂n = 0,the numerator is zero which is inconsistent with the requirement thath(n) N 1/s in Stage 2. Thus, Stage 2 can prevail for a given n only if inci-dent risk f (n) is sufficiently high and/or increasing. Optimal headway isan increasing function of f (n) because queuing becomes more likely inthe future, and increasing the headway reduces queuing time. In the lastterm of the numerator, λn(n) b 0 for n b N. Optimal headway istherefore smaller if incident risk is increasing. This is because accelerat-ing departures induces incidents to occur and end sooner, therebyallowing later drivers to arrive with shorter delays.

As noted above, if departures are dispersed in the SO, then they arealso dispersed in the UE. The SO and UE with dispersed departures aredifficult to compare because the UE departure rate in Eq. (15) is verydifferent in functional form from the departure rate implied by the SOheadway in Eq. (47). It is easy to show that the SO can still be supportedby a time-varying toll, but results analogous to those in parts (a) and (c)of Theorems 2 and 3 have eluded us.

5.2.4. Numerical solution methodThe SO with dispersed departures cannot, in general, be solved

analytically. To solve it numerically, Eq. (43) with ξ = n can besubstituted into Eq. (44) to obtain an equation for departure time duringStage 2, t2(n). The remaining unknowns, t0w and bn, are solved using twoconditions. First, the departure time for driver bn must be the same inStage 1 and Stage 2:

t1 bn� � ¼ tw0 þ bn=s ¼ t2 bn� �: ð48Þ

Second, the costate variables μ1 and μ2 must match at bn:μ1 bn� � ¼ μ2 bn� �: ð49Þ

μ1 bn� � is computed by integrating Eq. (30) forward with respect to n,starting at n = 0 with μ1(0) = 0. μ2 bn� � is computed by integratingEq. (41) backward with respect to n, starting at n = N withμ2(N) =0. (If Stage 2 is optimal throughout the departure period,then bn ¼ 0 and Eqs. (48) and (49) do not have an interior solution.) Anotable feature of the solution with bnN0 is that h2 bn� �N1=s . Optimalheadway is therefore discontinuous, and takes an upward jump at thetransition point from Stage 1 to Stage 2.

6. A numerical example

An example is now presented to obtain further insights into thecharacteristics of the UE and SO, as well as to get a sense of the quanti-tative importance of incidents. Following Fosgerau and Engelson(2011), the scheduling utility functions for home andwork are assumedto be linear: β(t) = β0 − β1t and γ(t) = γ0 + γ1t, with β0, β1, γ0 andγ1 all strictly positive. Following Börjesson et al. (2012), this will becalled the “slope” model.

As demonstrated in the previous two sections, both the UE and SOare sensitive to the shape of the incident risk function. A reasonablebaseline assumption is that incident risk is constant and this is assumedfor the example: f (n) = f, where f is a constant. Appendix B.5 givespartial analytical solutions for the UE and SO.

6.1. Calibration

Tseng and Verhoef (2008) derive non-parametric estimates of func-tions β(t) and γ(t) for morning commuting trips by car or public trans-port. Their estimates are approximated fairly well by the slope model.Linear regressions of β(t) and γ(t) were therefore performed usingthe seven observations for their mixed-logit estimates (see Table 3 oftheir paper). These produced estimates of β1 = 8.86 €/h2 and γ1 =25.42 €/h2. A complication with the estimates of β0 and γ0 is thattime spent traveling is preferred to time spent at work for workingmore than about 10 min before t⁎. Time spent traveling is also preferredto time spent at home shortly before t⁎. Such preferences would inducevery high or infinite (i.e., mass) departures in UE (see Arnott et al.,1990). To avoid this, parameters β0 and γ0 were set by trial and errorto generate reasonable departure rates over the full travel period. Thevalues chosen were β0 = γ0 = 40; with β0 = γ0 this implies t⁎ = 0.The parameterized utility functions used are therefore β(t) = 40 −8.86t and γ(t) = 40 + 25.42t.

There are relatively few studies of incident duration. For accidentsthat involve one lane closure, Golob et al. (1987) find a mean incidentduration of about one hour.26 Jones et al. (1991) obtain a similar meanof 55 min. Nam and Mannering (2000) obtain a much higher mean of162.5 min for incidents in an Incident Response Team database thattend to be more severe than average. For the example here, a smallervalue of 30 min was selected for parameter Δ. This compensates in arough way for the assumption that incidents completely block thebottleneck which is often not the case for multi-lane highways. It is inline with Hall (1993) who assumes that durations are uniformlydistributed between 5 and 60 min with a mean of 32.5 min. It is alsoconsistent with Koster and Rietveld (2011) who assume that incidentslast for 35 min and reduce capacity by 80%.

Parameters N and s were set to N = 8,000 and s = 4,000, and inci-dent risk was set to f = 0.2/N so that incident probability is fN = 0.2.This implies that an incident occurs once per work week. On average,half the commuters would pass the bottleneck before the incidentoccurred so that individual commuters would experience an incident-related delay on one day out of ten. The results of the example arereported in the next subsection. The following subsection examines amodified version of the example that is descriptive of an eveningcommute.

Table 1Comparison of user equilibrium and social optimum (morning).

No incidentsfN = 0

IncidentsfN = 0.2

Effect ofincident risk

UE SO UE SO UE SO

1 First departure time [h] −1.00 −1.00 −1.101 −1.037 −0.101 −0.0372 Last departure time [h] 1.00 1.00 0.899 0.936 −0.101 −0.0373 Expected average social

trip cost [€]17.14 5.71 20.78 8.43 3.64 2.72

4 Average social trip coston Good days [€]

17.14 5.71 18.16 5.74 1.02 0.03

5 Average social trip coston Bad days [€]

– – 31.23 19.21 – –

6 Row 5 − Row 4 13.07 13.47

Fig. 2. Total cost of incidents in user equilibrium and social optimum (morning).

Fig. 3. Individual cost of incident in user equilibrium (morning).

773M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

6.2. Results for the morning commute

In the example, departures are compressed in both the UE and SO.27

The two solutions are compared in Table 1.TheUE begins about 6 min earlier thanwith no incident risk. The ini-

tial departure rate is 15,389 veh/h: appreciably higher than the rate of13,405 veh/h without incidents.28 Consistent with Theorem 2, the SOalso begins earlier than without incidents but the time shift is muchsmaller than for the UE. Expected trip costs are measured by the lossof expected utility relative to an ideal in which bottleneck capacity iseffectively infinite, incidents never occur, and all drivers can thereforetravel from home to work simultaneously at t∗. Without incidents, thecost of a trip is €17.14 in the UE and €5.71 in the SO. SO cost is onlyone third as large as UE cost because, in the slope model, two thirds oftrip costs in the UE are due to queuing time which is avoided in theSO. With incidents, expected trip cost increases by €3.64 to €20.78 inthe UE, and by €2.72 to €8.43 in the SO. The proportional increase inexpected cost is larger for the SO, and in this respect the SO is less effec-tive than the UE at adapting to incidents. One reason is that departuresbegin later in the SO than the UE, so that incident-related delays imposea higher cost from late arrivals. Another is that the SO with compresseddepartures is designed to avoid queuing on Good days whilemaintaining full capacity utilization. The SO has no “margin of reserve”,and it is therefore vulnerable to capacity breakdowns.

The increase in expected cost can be decomposed into an increase onGood days and an increase on Bad days. Row 4 of Table 1 shows that thecost increase on Good days is negligible for the SO (€0.03) but apprecia-ble for theUE (€1.02). This is because the shift toward earlier departuresis more pronounced in the UE. On Bad days, expected trip costs arehigher in the UE (€31.23) than the SO (€19.21). But the difference incosts between Good days and Bad days is actually slightly higher inthe SO (€13.47) than the UE (€13.07). In the UE, drivers adjust theirdeparture times in response to incident risk in order to reduce thecosts of lateness on Bad days while sacrificing some utility on Gooddays. As explained above, the SO is less flexible because the rate ofdepartures is fixed, and only the timing of the travel period is adjusted.

Fig. 2 plots the total cost of an incident (measured relative to Gooddays) as a function ofwhen the incident occurs.29 The SO curve lies slight-ly above the UE curve at all times. Both curves decline monotonicallybecause later incidents affect fewer drivers. However, both curvesare concave because drivers who depart later are more adverselyaffected by incidents so that even late incidents impose a significant loss.

Figs. 3 and 4 plot individual driver's trip costs in UE and SO (includ-ing toll) according towhether or not they encounter an incident. In each

27 Condition (5) for the UE is satisfied for fN b 0.4482, and Condition (18) for the SO issatisfied for fN b 0.4936. The UE and SO are therefore both compressed unless incidentprobability is nearly 1/2.28 Introduction of a small incident risk invariably leads to an increase in the initial depar-ture rate if β1 b 2γ1; see Appendix B.5.29 Since incidents occur as drivers pass the bottleneck, the probability density of inci-dents is uniform with respect to timing.

case the cost with no incident decreases with departure time while thecost with an incident increases. The increase in cost due to an incidentgrows from about €10 for the first driver to €35 for the last driver.Departing later is therefore riskier, and if drivers were risk-averse theywould tend to depart in order of decreasing risk aversion.30

Oneway to reduce the costs of incidents is to reduce their frequency.Another is to reduce their duration. The effects of incident frequencyand duration are easily determined in themodel by varying parametersf andΔ. Fig. 5 shows how incident probability affects expected costs pertrip for the UE, the SO, and the SO including toll costs. In all three casesthe relationship is nearly linear. The elasticities are respectively 1.009for the UE, 0.9928 for the SO, and 0.9899 for the SO including toll. Inthe case of the SO the elasticity cannot exceed one; it would equal oneif the departure schedule were held fixed independent of the incidentprobability, but this is generally not optimal. This reasoning does notapply to the UE, and the elasticity is slightly above one. Thus, drivers'uncoordinated response to incident risks is collectively inefficient.

Fig. 6 presents analogous results for incident duration. In each case,expected cost increases more than in proportion to duration. The elastic-ities are 1.109 for the UE, 1.111 for the SO, and 1.100 for the SO includingtoll. This suggests that highest priority should be given to road linkswhere major incidents are common. The duration of incidents that re-quire assistance is equal to the sum of detection time, time required to

30 de Palma et al. (2012) develop a model of route choice with drivers that differ in riskaversion, and show how the most risk-averse drivers select the safer route.

Fig. 4. Individual cost of incident in social optimumwith toll (morning).

Fig. 5. Effect of incident probability on incident costs (morning).

774 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

dispatch emergency vehicles, travel time to the incident location, and ser-vice time (Hall, 2002). Incident duration can be reduced by expeditingany of these stages. Carson et al. (1999) determine that even short reduc-tions in incident response and clearance times can yield high benefit-to-cost ratios.

A notable feature of Fig. 6 is how much the toll boosts the privatecosts of incidents borne by drivers in the SO. Fig. 7 displays the tollschedule for three levels of incident duration as well as a case withoutincidents (effectively, incidents of zero duration). As incident durationincreases, the toll schedule increases rapidly and it also advances slowlyas departures begin earlier. With Δ = 1.5 h, the toll begins at €16.87and rises to a maximum of €30.24 before declining smoothly to zero.Such a high toll is attributable to the combined effects of inelasticdemand, relatively strong trip-timing preferences, and long-lastingincidents that shut capacity down completely. In practice, tolls are likelyto be lower than this.

6.3. Results for the evening commute

To this pointwe have used themodel to describe incidents that occurduring the morning commute. Yet themodel can be applied to other sit-uations in which people wish to move from one location to another atthe same time. An obvious instance is the evening commute. The eveningcommute has not been studied as extensively as the morning commute,either theoretically or empirically, but it is generally considered thatwork time constraints are a major – if not dominating – factor in deter-mining when people leave work. In theoretical studies that have usedthe step model to describe the evening commute it is typically assumedthat scheduling costs are determined by departure time (from work)rather than arrival time (at home), and that the unit cost of departingfrom work early is greater than the unit cost of departing late.31 In theslope model the corresponding assumption is that β1 N γ1. This is whatBörjesson et al. (2012) find using a stated preference data set for a sam-ple of people traveling for a variety of purposes.32 In contrast to Tsengand Verhoef (2008), their estimate of β1 is more than three times aslarge as their estimate of γ1 (see their Table 6).

We now briefly investigate how incident risk affects commutingbehavior in the evening commute, and compare it with the morning. Tosimplify the comparison, instead of adopting the Börjesson et al. (2012)estimates we simply interchange the estimates of β1 and γ1, and useβ1 = 25.42 €/h2 and γ1 = 8.86 €/h2.33 Other parameter values are keptthe same as for the morning.

31 See, for example, Fargier (1983), de Palma and Lindsey (2002), and Zhang et al. (2008).32 Rather surprisingly, they find no significant differences in scheduling parameters foreither morning and afternoon trips, or for different trip purposes.33 The ratioβ1/γ1 = 2.86 is similar to the ratio of 3.26 that Börjesson et al. (2012) obtain.

In the absence of incident risk, the evening commute is a mirrorimage of the morning commute for both UE and SO in terms of the de-parture period and trip costs. However, the initial departure rate islower in the evening than the morning. Substituting parameter valuesinto the formula ρ te00

� � ¼ 2β0þβ1N=s2β0−γ1N=s

s one obtains an initial departurerate of 8,403 veh/h for the evening compared to 13,405 veh/h for themorning. Fig. 1 offers an explanation for the difference. The schedulingutility functions drawn there depict a morning commute with γ1(∙)steeper than β(∙) in the neighborhood of t⁎ (in the slope model thetwo functions are linear). For the evening commute, γ1(∙) is flatterthan β(∙). Area hlmj is therefore taller than the area shown, andcorrespondingly narrower. Queuing time therefore grows more slowly.The intuition for this is that utility from time spent at home is relativelyinsensitive to time of day, and workers therefore have less to gain bydelaying departure from work. Another difference from the morning isthat introduction of a small incident risk leads to a reduction in theinitial departure rate.34

Table 2 displays other properties of the evening commute corre-sponding to those for the morning commute shown in Table 1. Depar-tures are again compressed in both the UE and SO.35 Compared to themorning, departures in the UE and SO begin later. The UE beginsabout 4 min earlier than without incident risk compared to 6 minearlier for the morning. Incident risk has a smaller effect because, witha smaller value of γ1, arriving late is less costly. For the same reason, av-erage trip costs for UE and SO are lower on bothGood days and Bad daysthan in the morning.

For the UE, expected trip cost increases by only €2.61 compare to€3.64 for the morning. For the SO, expected trip costs increases by€2.26 compared to €2.72 for the morning. Again, the proportional in-crease in expected cost ismuch larger for the SO than theUE for reasonssimilar to those for the morning.

Fig. 8 plots the total cost of an incident as a function of when the in-cident occurs. Unlike for the morning shown in Fig. 2, the SO curvecrosses the UE curve twice rather than lying wholly above it. Both curvesare still concave, but less so than for the morning because late incidentsare not as costlywhen the penalty for arriving late at home is less severe.

Figs. 9 and 10 plot individual driver's trip costs in UE and SOaccording to whether or not they encounter an incident. The two setsof curves are much flatter than in Figs. 3 and 4 for the morning. Forboth UE and SO, the cost incurred when encountering an incidentgrows from about €16 for the first driver to €25 for the last driver.Departing later is therefore riskier, but much less so than for the morn-ing commute.

34 This happens if 2β0 2γ1−β1ð Þ þ β1γ1Ns −Δ� �

b0, which is satisfied in the example; seeAppendix B.5.35 Condition (5) for the UE is satisfied for fN b 0.5778, and Condition (18) for the SO issatisfied for fN b 0.6763.

Fig. 7. Effect of incident duration on socially optimal toll (morning).

Fig. 6. Effect of incident duration on incident costs (morning).

Table 2Comparison of user equilibrium and social optimum (evening).

No incidentsfN = 0

IncidentsfN = 0.2

Effect of incidentrisk

UE SO UE SO UE SO

1 First departure time [h] −1.00 −1.00 −1.074 −1.013 −0.074 −0.0132 Last departure time [h] 1.00 1.00 0.936 0.987 −0.074 −0.0133 Expected average social

trip cost [€]17.14 5.71 19.75 7.97 2.61 2.26

4 Average social trip coston Good days [€]

17.14 5.71 17.53 5.72 0.41 0.01

5 Average social trip coston Bad days [€]

– – 28.66 16.98 – –

6 Row 5 − Row 4 11.13 11.26

775M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

7. Incidents with partial reductions in bottleneck capacity

Most incidents on multi-lane highways do not reduce capacity tozero. 36 Suppose an incident reduces bottleneck capacity from s tok ∈ [0,s). Call an incident “major” if k = 0 (as in the basic model), and“minor” if k N 0. Minor incidents are more complicated to analyzethanmajor incidents, and some properties ofmajor incidents do not ex-tend to minor incidents. Consequently, the analysis of minor incidentshere is brief.

7.1. User equilibrium

Lemmas 1 and 2 describing UE for major incidents also apply tominor incidents.37 However, the delay due to a minor incident dependson when it occurs and it is necessary to consider the amount of timeelapsed between an incident and when a driver departs. Fig. 11depicts the various possibilities. A driver departing in the intervalt ∈ [t0, R−1(kΔ)] encounters one of two cases: (1) no incident has yetoccurred, or (2) an incident has occurred that ends after the driver ar-rives at work. A driver departing in the interval t ∈ [R−1(kΔ), t0 + Δ]can encounter a third case: (3) an incident has occurred that ends be-fore the driver arrives at work. Finally, a driver departing after t0 + Δcan encounter a fourth case: (4) an incident has ended before the driverdeparts. Case 2 can temporarily “disappear” either before or aftert0 + Δ. (Fig. 11 shows an example in which Case 2 disappears aftert0 + Δ.) However, Case 2 necessarily “reappears” before the last driver

36 See Highway Capacity Manual 2000 (2000, Chapter 22, Table 22-6).37 The proof of Lemma 2 for major incidents is slightly modified; see Appendix B.6.

departs at tN. The delay due to an incident differs between the cases; for-mulas are derived in Appendix B.6.

The first departure in UE is determined (see Appendix B.6) by thecondition

Z teN

v¼te0

β vð Þ−γ vð Þð Þdv ¼ F Re teN−ksΔ

� �� �Z teNþs−ks Δ

v¼teN

γ vð Þdv

þZ teN

v¼teN−ksΔf Re vð Þ� �

ρe vð ÞZ teNþs−k

s teN−vð Þt¼teN

γ tð Þdtdv:ð50Þ

If k = 0, Eq. (50) simplifies to Eq. (8) formajor incidents. Eq. (50) in-volves lagged values of the departure rate, ρe(v), for v∈ teN−k

sΔ; teN

, and

has to be solved numerically. The initial departure rate turns out tobe as in Eq. (2): the same formula as without incidents. Since t0

e b t0e0,

ρe(t0e) N ρe0(t0e0). Thus, unlike for major incidents, the risk of minor inci-dents unambiguously raises the initial departure rate. This is becauseminor incidents do not eliminate capacity, and thus impose only ashort delay on early drivers. The risk taken by postponing departure istherefore smaller, and the queue required to deter drivers from alldeparting later is correspondingly longer.

7.2. The social optimum

Lemma 3 for major incidents carries over to minor incidents and theproof is the same. However, the SO for minor incidents is tedious to de-rive since the SO departure rate depends on both lagged and leadingvalues of itself. Attention is limited here to compressed departures.The SO departure period is derived in Appendix B.6 where it is shown

Fig. 8. Total cost of incidents in user equilibrium and social optimum (evening).

Fig. 9. Individual cost of incident in user equilibrium (evening).Fig. 11. Timing of minor incidents (morning).

776 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

that ∂t0w/∂k N 0. The SO therefore begins earlier when incidents aremore severe (i.e., k is smaller). We have been unable to rank t0

w and t0e.

The UE and SO with compressed departures can be derived for theexample in Section 6. Fig. 12 plots for UE and SO of the morning com-mute the increase in expected trip cost due to incidents as the fractionof capacity lost in an incident (i.e., 1 − k/s) increases from 0 to 1. Thetwo curves rise smoothly, and at k = 0 they reach the values of €3.64and €2.72 that apply for major incidents. Both curves are convex sothat the expected cost of an incident is more than proportional to thecapacity lost. This is broadly consistent with the finding in Section 6that the cost of incidents is more than proportional to their duration.

8. Concluding remarks

Incidents are amajor cause of traffic congestion in large urban areas.This paper uses the bottleneck model to analyze the effects of incidentson trip-timing decisions and trip costs. Incidents are assumed to becaused by individual drivers while they travel. Unlike in previous stud-ies except for Schrage (2006), the timing of incidents is endogenous totraveler behavior. This enriches the realism and lessons derived fromthe model, but also adds complexity.

Several general results deserve highlighting. First, three biases areidentified in the timing of departures in the UE relative to the SO. Oneis the familiar bias toward departing too quickly in the early stages ofthe travel period which leads to queuing on Good days in the UE, butnot in the SO. The other two biases are driven by incident risk andthey act in opposite directions. Drivers are biased toward departing

Fig. 10. Individual cost of incident in social optimumwith toll (evening).

too early because they do not want to be delayed by an incident andarrive seriously late. But they are also biased toward departing too latebecause they ignore the delays they will impose on subsequent driversif they cause an incident. This last bias is missing frommodels in whichincidents occur exogenously.

Second, if the probability of an incident is sufficiently high, then inboth the UE and the SO bottleneck capacity is not fully used for theentire travel period on Good days when no incident occurs, and the de-parture pattern is “dispersed”. For the SO this can be interpreted as a pol-icy of maintaining reserve capacity in order to moderate the adverseeffects of incidents on Bad days. A third result is that if departures arecompressed in the SO, then drivers are worse off than in the UE if theSO is decentralized using a non-negative time-varying toll. Thismight ag-gravate resistance to congestion pricing and correspondingly strengthenarguments for using toll revenues in a way that benefits drivers.

Further lessons concerning how the probability, duration, andmagnitude of incidents affect expected travel costs can be drawn fromnumerical examples. For the parameterized slope-model example weuse, incidents that occur early in the travel period are more costlythan later incidents because early incidents affect more travelers. Theexpected costs of incidents in both the UE and SO are approximatelylinear in the probability of incidents, but convex in their magnitudeand duration. This suggests that, not surprisingly, priority for incidentmanagement should be given to shortening major and long-livedtypes of incidents.

There are many ways in which the paper could be extended. Theanalysis could be refined by relaxing the persistent-queue assumption.Heterogeneity in drivers' scheduling preferences could be introduced.

Fig. 12. Increase in expected trip cost fromminor incidents (morning).

777M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

The duration of incidents could be treated as a random variable. Morethan one incident per day could be allowed. The assumption thatmultipleincidents never occur is reasonable if incidents are rare. For the base caseof the numerical example it is assumed that the probability of no incidentis 0.8. Given the underlying hazard rate, more than one incident wouldoccur with probability 0.0215, or about once every 45 days. Moreover, ifa second incident occurs while the first is ongoing, the second incidenthas no effect until the first is cleared up. In the case of minor incidents,the effect of two concurrent incidents may be approximately equal tothe impact of the one that is more severe (Hall, 2002).

Another extension is to treat travel demand as price sensitive. Thenumber of trips taken would then differ for the user equilibrium andsocial optimum. It would also depend on the frequency, severity, andduration of incidents. By reducing the expected costs of travel, incidentmanagement schemeswould stimulatemore trips, and correspondinglymore incidents, that would partially offset the policy benefits (Hall,1993; Koster and Rietveld, 2011).

Another interesting extension is to assume that individuals are riskaverse. Börjesson et al. (2012) find that scheduling models severelyunderestimate the disutility that individuals incur from travel timevariability.38 They offer two plausible explanations for this. One is that,contrary to what is assumed in scheduling models, people do not haveafixed preferred arrival time (t⁎ in themodel here) because they can re-schedule many activities given advance information about travel timedurations. The other explanation is that people may dislike uncertaintyitself because of the anxiety it creates, the costs incurred when decidedwhen to depart, or the costs of formulating contingency plans. AsBörjesson et al. (2012) put it, being ‘delayed’ can be worse than justbeing ‘late’. Risk aversion could be introduced into the model in acrudeway by adding to the expected utility function a term proportion-al to the standard deviation of travel time.

Incident risk could be assumed to depend on time of day or othercircumstances. For instance, incident risk is higher at night Vargheseand Shankar, 2007 and in rain (FHWA Road Weather ManagementProgram, 2009). A more radical step would be to reformulate themodel using a flow congestion model as in Schrage (2006). One reasonfor doing so is that queuing congestion militates against maintainingspare capacity since it does not offer a smooth trade-off between highercapacity utilization on Good days and extra delay on Bad days. Such atrade-off does exist with flow congestion. Another reason is that boththe frequency and severity of incidents per vehicle-km driven coulddepend on speed and/or the density of vehicles on the road (Sheferand Rietveld, 1997). The empirical evidence on this is limited and incon-sistent.39 A few studies have found that severe accidents aremore com-mon in light traffic. However, Wang et al. (2009) conclude that trafficcongestion has no statistically significant effect on accident frequencyon the M25 London orbital motorway. To the extent that incidentrates do vary with traffic conditions, the introduction of congestiontolls will affect not only congestion externalities but also incident/accident externalities (Dickerson et al., 2000).

A final extension worth noting is information about incidents. Wehave assumed that drivers know the risk function for incidents, f(∙),but not whether an incident has occurred. It is true that pre-trip anden-route information about travel conditions are now available fromvarious media. Nevertheless, incidents often occur after people havemade their travel decisions and can alter them only with difficulty — ifat all. Changes of route may be possible in urban areas with denseroad networks, but drivers may be reluctant to switch route if alterna-tives are either circuitous or unfamiliar. For example, the shortestroute from metro Vancouver to the US border is via the George MasseyTunnel on Highway 99. Long delays are frequently encountered at thetunnel. The closest alternative crossing is via the Alex Fraser Bridge on

38 This finding shows up in both the step model and the slope model they estimate.39 See, for example, Dickerson et al. (2000), Noland and Quddus (2005), and Small andVerhoef (2007, Section 3.4.6).

Highway 91 which is about 10 km to the east. In settings such asthese, travelers have little alternative to queuing once they are en route.

In other settings where travelers can respond to information, the ef-fectiveness of the responsewill dependon a number of factors includingthe timeliness and precision of information, the fraction of travelerswho have access to it, the proportion of them who choose to respondand how, whether alternative routes are congested, and so on. Theseand other design considerations for Advanced Traveler Information Sys-tems have been extensively studied since the late 1980s.40 The effects ofinformation provision in an extended version of the current model arenot easy to envisage.

Appendix A. Notational glossary

A.1. Latin characters

a arrival time at work [clock time]e superscript denoting user equilibriumE () expectations operatorf probability density of an incident (“incident risk”)F cumulative probability of an incident (“incident probability”)h headway between successive departures [h]H subscript denoting homek bottleneck capacity during incident (k = 0 for major inci-

dents) [veh./h]n index of drivers in order of departurebn value of n at which socially optimal departure rate drops

below capacityN number of driversq (t) queuing time [h]q+ (n) queuing time in social optimum if no incident has occurred

[h]qξ (n) queuing time in social optimum if driver ξ causes an incident

[h]R cumulative departures [veh]s bottleneck capacity with no incident [veh./h]t departure time from home [clock time]t0 time when first driver departs from home [clock time]tH time at which utility accounting begins (at home) [clock

time]tN time when last driver departs from home [clock time]tW time at which utility accounting ends (at work) [clock time]t⁎ preferred time to transfer instantaneously from home to

work [clock time]T set of departure timesu scheduling utilityU aggregate utilityv clock time (used as variable for integration)w superscript denoting social optimumW subscript denoting work

A.2. Greek characters

β rate of utility derived from time spent at homeβ0, β1 parameters of linear β () functionγ rate of utility derived from time spent at workγ0, γ1 parameters of linear γ() functionΔ duration of incident [h]λ+ (n) costate variable for q+ (n)λξ (n) costate variable for qξ (n)μ1 costate variable for departure time in Stage 1 of social optimumμ2 costate variable for departure time in Stage 2 of social

optimum

40 For reviews see Rietveld (2011) and Chorus and Timmermans (2011).

778 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

ξ index of driver who causes an incidentρ aggregate departure rate from origin [veh./h]τ toll [€]Ψ multiplier on nonnegativity constraint q+ (n) ≥ 0Ω Hamiltonian for social optimum

A.3. Other characters

0 superscript denoting model without incidents.

Appendix B. Mathematical appendixes

B.1. Proof of Lemma 3

Part (a): Suppose ρw(t) N s on some interval. Then Rw(t) intersects afamily of lines s(t − bt) parameterized bybt. For any suchbt, a queue existsfor all t∈bT ≡ t : Rw tð ÞNs t−bt� �n o

. Reducing Rw(·) to s(t − bt) on bT post-pones departures from home for a cohort of drivers without delayingtheir arrivals at work. The cohort's utility increases strictly withoutaffecting the utility of other drivers. Hence ρw(t) ≤ s on Tw.

Part (b), proof that t0w b t⁎: Following the approach of Section 5.2,write total expected utility as:

E Uð Þ ¼Z N

n¼0

1−F nð Þð ÞU t nð Þ; t nð Þð ÞþZ n

ξ¼0f ξð ÞU t nð Þ; t nð Þ þ qξ nð Þ

� �dξ

24 35dn;where t(n) is the departure time of driver n, and qξ (n) is queuing timefor driver n if driver ξ b n causes an incident. Consider slightly advanc-ing the departure times of all drivers by dt N 0. Total expected utilitychanges by

dE Uð Þ ¼ dtZ N

n¼0

1−F nð Þð Þ γ t nð Þð Þ−β t nð Þð Þð ÞþZ n

ξ¼0f ξð Þ γ t nð Þ þ qξ nð Þ

� �−β t nð Þð Þ

� �dξ

24 35dn¼ dt

Z N

n¼0

−β t nð Þð Þ þ 1−F nð Þð Þγ t nð Þð ÞþZ n

ξ¼0f ξð Þγ t nð Þ þ qξ nð Þ

� �dξ

24 35dn:If t0

w ≥ t*, then β(t(n)) ≤ γ(t(n)) ≤ γ(t(n) + qξ(n)), ∀ n, withβ(t(n)) b γ(t(n)) for n N 0. Hence dE(U) N 0, and t0

w ≥ t* cannot beoptimal.

Part (b), proof that tNw N t*: Suppose tNw ≤ t*, and let the last driver de-part dt later. The last driver's expected utility changes by dE(u|tNw) =(β(tNw) − (1 − F(N))γ(tNw))dt. Given β(tNw) ≥ γ(tNw) and, F(N) N 0,dE(u|tNw) N 0, and tN

w ≤ t* cannot be optimal. □

B.2. Proof of Theorem 2

Part (a): The right-hand side of Eq. (34) can be bounded above andbelow:

0bE Nð ÞZ twNþΔ

t¼twN

γ tð Þdt−sZ twN

t¼tw0

f s t−tw0� �� � Z tþΔ

v¼tγ vð Þdv

� �dt

bF Nð ÞZ twNþΔ

t¼twN

γ tð Þdt:

Comparison with Eq. (8) for t0e shows that t0w ∈ (t0e,t0e,0).Part (b): If a toll implements the SO, expected utility net of the

toll must be constant over the departure interval Tw and lower outsideTw. Such a toll has the form τw tð Þ ¼ E u tj Þ þ c ¼ ∫

t

v¼tHβ vð Þdvþ

�∫tW

v¼tγ vð Þdv− F nð Þ∫tþΔ

v¼tγ vð Þdvþ c where c is a constant.

Part (c): Expected utility when departing in the UE at tNe is given by

Eq. (11): E ujteN� � ¼ ∫

teNv¼tH

β vð Þdvþ 1−F Nð Þð Þ∫tW

v¼teNγ vð Þdvþ F Nð Þ

∫tW

v¼teNþΔγ vð Þdv. The same equation applies for E(u|tNw) in the SO since,

as in the UE, there is no queue on Good days, and the delay due to an in-

cident does not dependonwhen it occurs. ThereforeEðu twN�� �

−Eðu teN�� � ¼

∫v¼teN

twN β vð Þ−γ vð Þð Þdvþ F Nð Þ ∫twNv¼teN

γ vð Þdv−�

∫twNþΔ

v¼teNþΔγ vð ÞdvÞ. The first

term in this expression is negative since tNw N tNe N t*. The second expres-

sion is also negative since tNw N tN

e , Δ N 0, and γ(·) is an increasing func-tion. Expected utility gross of the toll is therefore lower in the SO thanthe UE. If the toll cannot be negative, drivers are therefore worse off inthe SO if toll revenues are not used in a way that benefits them.

B.3. Proof of Theorem 3

Part (a): If UE departures are dispersed, then β(tNe ) = (1 −F(N))γ(tNe ) and t0

e b tNe − N/s. If SO departures are compressed, then

β(tNw) b (1 − F(N))γ(tNw) and t0w = tN

w − N/s. Therefore, t0w = tN

w −N/s b tN

e − N/s N t0e. The proof that t0w b t0

w0 is the same as for Theorem 2.Part (b): The proof that the SO can be implemented using a time-

dependent toll is the same as for Theorem 2.Part (c): The proof is similar to the proof for Theorem 2. Expected

utility when departing in the UE at tNe is given by Eq. (14):

Eðue teN�� � ¼ 1−F Nð Þð Þu teN ; t

eN

� �þ F Re et� �� �u teN ; t

e0 þ

Nsþ Δ

� �þZ teN

v¼etρ vð Þ f Re vð Þ� �u teN ; vþ Δþ N−Re vð Þ

s

� �dv:

ð51Þ

where a superscript e has been added to u since formulas for expectedutility differ for the UE and SO. Integrating the last term of Eq. (51) byparts, and rearranging terms, one obtains

Eðue teN�� � ¼ u teN ; t

eN

� �−F Nð Þ

Z teNþΔ

v¼teN

γ vð Þdv

þZ teN

v¼et F Re vð Þ� �1þ ρe vð Þ

s

� �γ vþ Δþ N−Re vð Þ

s

� �dv:

ð52Þ

Expected utility when departing in the SO at tNw is

Eðuw twN�� � ¼ u twN ; t

wN

� �−F Nð Þ

Z twNþΔ

v¼twN

γ vð Þdv: ð53Þ

Subtracting Eq. (52) from Eq. (53):

Eðuw twN�� �

−Eðue teN�� � ¼ Z twN

v¼teN

β vð Þ−γ vð Þð Þdv

þF Nð ÞZ twN

v¼teN

γ vð Þdv−Z twNþΔ

v¼teNþΔγ vð Þdv

!

−Z teN

v¼et F Re vð Þ� �1−ρe vð Þ

s

� �γ vþ Δþ N−Re vð Þ

s

� �dv:

The first two terms in this expression are negative (see proof ofTheorem 2). The third term is also negative since ρe(v) b s for v∈et; teNh i

. Expected utility net of the toll is therefore lower in the SO than

UE. If the toll cannot be negative, and drivers do not benefit from tollrevenues, drivers are worse off.

779M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

B.4. Optimal headway in Stage 2

Differentiating Eq. (44) totally with respect to n one obtains:

β′ t nð Þð Þh nð Þ ¼ 1−F nð Þð Þγ′ t nð Þð Þh nð Þ− f nð Þγ t nð Þð Þ

− f nð Þ ∂λn nð Þ∂n −λn nð Þ ∂ f nð Þ

∂n :ð54Þ

Now

∂λn nð Þ∂n ¼ ∂λξ nð Þ

∂n���ξ¼n

þ ∂λξ nð Þ∂ξ

���ξ¼n

:

Given Eq. (43),

∂λξ nð Þ∂n

���ξ¼n

¼ γ t nð Þ þ Δð Þ; ð55Þ

and

∂λξ nð Þ∂ξ ¼ −

Z N

v¼nγ′ t ξð Þ þ Δþ s−1 v−ξð Þ� �

dv h ξð Þ−1s

� �;

so that

∂λξ nð Þ∂ξ j

ξ¼n¼ −

Z N

v¼nγ′ t nð Þ þ Δþ s−1 v−nð Þ� �

dv h nð Þ−1s

� �¼ 1

s−h nð Þ

� �Z N

v¼nγ′ t nð Þ þ Δþ s−1 v−nð Þ� �

dv:ð56Þ

Adding Eqs. (55) and (56) yields

∂λn nð Þ∂n ¼ λ t nð ÞΔð Þ

þ 1s−h nð Þ

� �Z N

v¼nγ′

t nð Þ þ Δþ s−1 v−nð Þ� �

dv:ð57Þ

Substituting Eq. (57) into Eq. (54), and rearranging terms, yieldsEq. (47) in the text for the optimal headway.

B.5. Analytical solution for linear example

B.5.1. User equilibriumGiven Eq. (8), the first departure in UE is at time

te0 ¼β0−γ0− β1 þ γ1ð Þ N

2s−fsΔ γ0 þ γ1

Nsþ Δ

2

� �� �β1 þ γ1 þ fγ1Δs

:

From Eq. (9), cumulative arrivals, Re(t), are given by the positive root ofthe quadratic equation

γ1

2s1þ 2fsΔð Þ Re tð Þ� �2 1þ fsΔð Þ γ0 þ γ1t

e0

� �þ fsγ1

2Δ2

� �Re tð Þ

−β0s t−te0� �þ β1s

2t2− te0

� �2� �¼ 0:

Assuming without loss of generality that β0 = γ0, it can be shown that

Sgn∂ρ te0� �∂ f j

f¼0

!¼ 2β0 2γ1−β1ð Þ þ β1γ1

Ns−Δ

� �:

A small incident risk leads to an increase in the initial departure rate un-less parameter γ1 is small compared to β1.

B.5.2. Social optimum

Stage 1. If the SO is compressed, Eq. (34) applies and the first departureoccurs at time

tw0 ¼ β0−γ0

β1 þ γ1− N

2s− γ1

β1 þ γ1

fNΔ2

:

It is easy to check that t0w N t0e if f N 0. Condition (18) is satisfied for the

candidate compressed SO if the following quadratic equation inZ ≡ fNγ1 is negative:

−ΔZ2

þ 2 β0 þβ1γ0

γ1

� �þ β1 þ γ1ð Þ N

sþ Δ

� �� �Z− β1 þ γ1ð Þ2 N

sb0: ð58Þ

Condition (58) is clearly satisfied for small values of f.The costate variable for departure time during Stage 1 is a negative,

convex, and quadratic function of n:

μ1 nð Þ ¼ γ0−β0 þ β1 þ γ1ð Þtw0� �

nþ fΔγ1 þβ1 þ γ1

s

� �n2

2;n∈ 0; bn

:

Stage 2. Departure time in Stage 2 is given by Eq. (44) which has an an-alytical solution:

t2 nð Þ ¼β0−γ0− fγ0 N−2nð Þ− fγ1 Δþ N−n

2s

� �N−nð Þ

β1 þ γ1 þ fγ1 N−2nð Þ ;n∈ bn;N :

The last driver departs at

twN ¼ t2 Nð Þ ¼ β0−γ0 1−fNð Þβ1−γ1 1−fNð Þ :

If the SO entails only Stage 2, bn ¼ 0 and the first driver departs at

tw0 ¼ t2 0ð Þ ¼β0−γ0−fN γ0 þ γ1 Δþ N

2s

� �� �β1 þ γ1 1þ fNð Þ :

The costate variable for departure time in Stage 2, μ2(n), is solved nu-merically by integrating Eq. (41) backwards with respect to n fromn = N.

B.6. Minor incidents

B.6.1. User equilibrium: Proof of Lemma 2 for minor incidentsA driver who deviates from a candidate compressed UE and departs

at t∈ teN ; teN þ s−k

s Δ� �

has an expected utility

Eðu tj Þ ¼ 1−F Nð Þð Þu t; tð Þ þZtN

v¼tN−ksΔ

f R vð Þð Þρ vð Þu t;sktN−

s−kk

v� �

dv

þF R tN−ksΔ

� �� �u t; tN þ s−k

sΔ

� �:

The derivative of expected utility is

∂Eðu tj Þ∂t ¼ β tð Þ− 1−F Nð Þð Þγ tð Þ;

which is the same as for major incidents. Hence Condition (5) stillapplies. □

780 M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

B.6.2. Timing of departures in user equilibriumAs noted in the text, the delay imposed by an incident on a driver de-

pends on when it occurs. Four cases are possible.

Case 1. No incident has occurredA driver departing at t arrives at a = t0 + R(t)/s. The contribution to

the driver's expected utility is:

Δue1 ¼ 1−F R tð Þð Þð Þu t; t0 þ R tð Þ=sð Þ: ð59Þ

Case 2. An incident has occurred that persists throughout the driver'strip

If the incident occurs at time v, a driver departing at t arrives whenthe number of drivers who have passed the bottleneck catchesup with cumulative departures at t: s(v − t0) + k(a − v) = R(t).Hence a ¼ R tð Þþst0

k − s−kk v. Since the driver must arrive before the inci-

dent ends, a ≤ v + Δ. This condition translates to v≥t0 þ R tð Þ−kΔs . The

incident must also occur before the driver departs: v ≤ t. Accountingfor these conditions, the contribution to the driver's expected utility is:

Δue2

Ztv¼Max t0 ;t0þR tð Þ−kΔ

s

f R vð Þð Þρ vð Þ t;R tð Þ þ st0

k− s−k

kv

� �dv: ð60Þ

Case 3. An incident has occurred that ends before the driver arrivesArrival time is determined by the condition s(v − t0) + kΔ +

s(a − v − Δ) = R(t) which yields a ¼ t0 þ s−kk Δþ R tð Þ

s . The incident can-not end before the driver departs: v N t − Δ. But it must end before thedriver arrives: v≤ t0 þ R tð Þ−kΔ

s . The contribution to the driver's expectedutility is:

Δue3 ¼

ZMin t;t0þR tð Þ−kΔs

v¼Max t0 ;t−Δ½ �

f R vð Þð Þρ vð Þv t; t0 þs−kk

Δþ R tð Þs

� �dv

¼ F R Min t; t0 þR tð Þ−kΔ

s

� � �� �−F R Max t0; t−Δ½ �ð Þð Þ

� �u t; t0 þ

s−kk

Δþ R tð Þs

� �:

ð61Þ

Case 4. An incident has ended before the driver departsArrival time is determined by the same condition as for Case 3 so

thata ¼ t0 þ s−kk Δþ R tð Þ

s . The incidentmust end before the driver departs:v b t − Δ. The contribution to the driver's expected utility is:

Δue4 ¼ F R Max t0; t−Δ½ �ð Þð Þu t; t0 þ

s−ks

Δþ R tð Þs

� �: ð62Þ

For the last driver who departs at tW, Cases 1–4 are all appli-cable. Using Eqs. (59)–(62) and R(tN) = N, the last driver's expectedutility is

Eðu tNj Þ ¼ 1−F Nð Þð Þu tN ; tNð ÞþZtNv¼tN−k

sΔ

f R vð Þð Þρ vð Þu tN ;sktN−

s−kk

v� �

dv

þ F R tN−ksΔ

� �� �u tN ; tN þ s−k

sΔ

� �:

Equilibrium departure period. The timing of the departure period [t0e,tNe ] isdetermined by the condition tN

e = t0e + N/s and the condition that E(u|

tN) = u(t0,t0). Using Eq. (1), the latter conditionworks out to Eq. (50) inthe text:

Z teN

v¼te0

β vð Þ−γ vð Þð Þdv ¼ F R teN−ksΔ

� �� �Z teNþs−ks

v¼teN

γ vð Þdv

þZ teN

v¼teN−ks

f Re vð Þ� �ρe vð Þ

Z teNþs−ks teN−vð Þ

t¼teN

γ tð Þdtdv:ð63Þ

Eq. (63) can be solved numerically using the following iterativeprocedure:

1. Guess t0e and set t = t0e.

2. Increment departure time by a small time step dt from t to t + dt.3. Determine which of Cases 1–4 are applicable. Set the derivative of

E(u|t) to zero, and solve for ρe (t).4. Update Re(t + dt) = Re(t) + ρe(t)dt.5. Repeat steps 2–4 until t = t0

e + N/s.6. Evaluate Condition (63) using the values computed for ρe tð Þ; t∈

teN− ksΔ; t

eN

. If Condition (63) is satisfied to within a tolerance

limit, then stop. Otherwise return to step 1.

Initial departure rate. For the first few drivers, only Cases 1 and 2 apply.Using Eqs. (59) and (60), expected utility is

1−F R tð Þð Þð Þu t; t0 þ R tð Þ=sð Þ

þZtv¼t0

f R vð Þð Þρ vð Þu t;R tð Þ þ st0

k− s−k

kv

� �dv:

Differentiating this expressionwith respect to t, and evaluating it at t =t0, one obtains

ρe te0� �s

¼ β te0� �

γ te0� � :

This is the same as Eq. (2) with t = t0e0 for the model without incidents.

Since t0e b t0

e0, ρe(t0e) N ρe0(t0e0).

B.6.3. Social optimumExpected utility of drivers for the SO is derived in a similar way to

UE, but using the formula that cumulative departures equal R(t) =s(t − t0).

Case 1. No incident has occurredA driver departing at t arrives at t. The contribution to the driver's

expected utility is:

Δuw1 ¼ 1−F s t−t0ð Þð Þð Þu t; tð Þ: ð64Þ

Case 2. An incident has occurred that persists throughout the driver'strip

If the incident occurs at time v, a driver departing at t arrives whenthe number of drivers who have passed through the bottleneck catchesupwith cumulative departures at t: s (v − t0) + k(a − v) = s(t − t0).The driver therefore arrives ata ¼ s

k t− s−kk v. Since the drivermust arrive

before the incident ends, a ≤ v + Δ, which translates to v≥t– ksΔ. The

incident must also occur before the driver departs: v ≤ t + Δ. Account-ing for these conditions, the contribution to the driver's expected utilityis:

Δuw2 ¼

Ztv¼Max t0 ;t−k

sΔ½ �f s v−t0ð Þð Þsu t;

skt− s−k

kv

� �dv: ð65Þ

781M. Fosgerau, R. Lindsey / Regional Science and Urban Economics 43 (2013) 764–782

Case 3. An incident has occurred that ends before the driver arrivesArrival time is determined by the condition s(v − t0) + kΔ +

(s − v − Δ) = s(t − t0) which yields a ¼ t þ s−ks Δ . Since the driver

must arrive after the incident ends, v≤t− s−ks Δ . The contribution to

the driver's expected utility is:

Δuw3 ¼

Zt−ksΔ

v¼Ma t0 ;t−Δ½ �

f s v−t0ð Þð Þsu t; t þ s−ks

Δ� �

dv;

which can be broken out into three intervals:

Δuw3 ¼

0 if tbt0 þksΔ

F s t−t0ð Þ−kΔð Þu t; t þ s−ks

Δ� �

if t∈ t0 þksΔ; t0 þ Δ

� �F s t−t0ð Þ−kΔð Þ−F s t−Δ−t0ð Þð Þð Þu t; t þ s−k

sΔ

� �if tNt0 þ Δ:

8>>>>>>>><>>>>>>>>:ð66Þ

Case 4. An incident has ended before the driver departsArrival time is determined by the same condition as for Case 3 so

that a ¼ t s−ks Δ. The contribution to expected utility is:

Δuw4 ¼ F s t−Δ−t0ð Þð Þu t; t þ s−k

sΔ

� �: ð67Þ

Summing terms in Eqs. (64)–(67) one obtains

E u tj Þ ¼

1−F s t−t0ð Þð Þð Þu t; tð Þ

þsZtv¼t0

f s v−t0ð Þð Þu t;skt− s−k

kv

� �dv if tbt0 þ

ksΔ

1−F s t−t0ð Þð Þð Þu t; tð Þ þ F s t−t0ð Þ−kΔð Þu t; t þ s−ks

Δ� �

þsZt

v¼t−ksΔ

f s v−t0ð Þð Þu t;skt− s−k

kv

� �dv if tNt0 þ

ksΔ

:

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

0BBBBBBBBBBBBBB@Changing the variable of integration from t to n, total expected utilitybecomes:

E Uð Þ ¼Z kΔ

n¼0

1−F nð Þð ÞZ t0þn=s

v¼tH

β vð ÞdvþZ tW

v¼t0þn=sγ vð Þdv

þZ n

ξ¼0f ξð Þ

Z t0þn=s

v¼tH

β vð ÞdvþZ tW

v¼t0þnk−s−k

sk ξγ vð Þdv

!dv

2666437775dn

þZ N

n¼kΔ

1−F nð Þð ÞZ t0þn=s

v¼tH

β vð ÞdvþZ tW

v¼t0þn=sγ vð Þdv

þF n−kΔð ÞZ t0þn=s

v¼tH

β vð ÞdvþZ tW

v¼t0þnsþ s−kð ÞΔ

γ vð Þdv !

dv

þZ n

ξ¼n−kΔf ξð Þ

Z t0þn=s

v¼tH

β vð ÞdvþZ tW

v¼t0þnk−s−k

sk ξγ vð Þdv

!

26666666664

37777777775dn:

ð68Þ

Differentiating Eq. (68) with respect to t0, and collecting terms, theequation defining the optimal t0w can be writtenZ N

n¼0β t nð Þð Þ−γ t nð Þð Þð Þdnþ

Z N

n¼0F nð Þγ t nð Þð Þdn

−Z N

n¼kΔF n−kΔð Þγ tΔ nð Þð Þdn

−Z kΔ

n¼0

Z n

ξ¼0f ξð Þγ tξ nð Þ

� �dξdn−

Z N

n¼kΔ

Z n

ξ¼n−kΔf ξð Þγ tξ nð Þ

� �dξdn ¼ 0;

ð69Þ

where t(n) = t0w + n/s, tΔ nð Þ ¼ tw0 þ n=sþ s−k

s Δ, and tξ nð Þ ¼ tw0 þ n=sþs−ksk n−ξð Þ. Condition (69) has the form

Z N

n¼0β t nð Þð Þ−γ t nð Þð Þð Þdnþ Z ¼ 0:

Now

∂Z∂k ¼ Δ

s

Z N

n¼kΔF n−kΔð Þγ

0

tΔ nð Þð Þ

þ k−2Z kΔ

n¼0

Z n

ξ¼0f ξð Þγ

0

tξ nð Þ� �

n−ξð Þdξdn

þk−2Z N

n¼kΔ

Z n

ξ¼n−kΔf ξð Þγ

0

tξ nð Þ� �

n−ξð Þdξdn:

ð70Þ

All components of Eq. (70) are strictly positive. Given Eq. (69) andt(n) = t0

w + n/s, this implies that ∂t0w/∂k N 0.

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