1 Time-Space Diagram

8/9/2019 1 Time-Space Diagram

1/14

1.2251.225J (ESD 205) Transportation Flow SystemsJ (ESD 205) Transportation Flow Systems

Lecture 1Lecture 1Cumulative Plots & TimeCumulative Plots & Time--Space DiagramsSpace Diagrams

Prof.Prof. IsmailIsmail Chabini and Prof.Chabini and Prof. Amedeo OdoniAmedeo Odoni


2/14

Cumulative PlotsCumulative PlotsObserver: count the total number of vehicles,N(t), that passed in

front of him/her during time interval [0,t].

Lane 2

Lane 1

Link entry Link exit

Observer

1

2

3

4

5

time

N(t)

0

N(t)

)(~

q(t)

tN

1.225, 10/29/02t1 t2 t t3 t4 t5 Lecture 1, Page 2


3/14

Observations onObservations on N(t)N(t)

N(t) is a step function. (not smooth)~~ (

( tN ) is a smooth approximation ofN(t) tNd ) exists.dt~

((

tq ) = tNd ) : instantaneous flow at time t.dt

~N(T) N(0) N(T) N(0)Average flow =

T T

1.225, 10/29/02 Lecture 1, Page 3


4/14

ArrivalArrival--Departure Cumulative PlotsDeparture Cumulative Plots

Single lane

Observer A:A(t) Observer D:D(t)

1.225, 10/29/02 Lecture 1, Page 4

D(t)

A(t)

t0 t1A-1(n)

n1

D-1(n)time

N(t)

00

t

?

n?


5/14

Accumulated Items:Accumulated Items: Q(t) = A(t)Q(t) = A(t)--D(t)D(t)??

Q(t): number of items (cars, planes) accumulated between the twoobservers.

Q(t) = Q(0) + [A(t) - A(0)] - [D(t) - D(0)]= (Q(0) + D(0) - A(0)) + A(t) - D(t)

= A(t) - D(t) ( if Q(0)+D(0)-A(0) = 0 )

1.225, 10/29/02 Lecture 1, Page 5


6/14

Waiting Under FIFO OrderWaiting Under FIFO Order

Vehicles depart in the same order as they entered a link (i.e. segmentof road) (First-In-First-Out) FIFO

Item n is observed at the entrance of a link at timeA-1(n).Item n is observed at the exit of a link at timeD-1(n).Waiting time of the item n: w(n) = D-1(n) - A-1(n)

1.225, 10/29/02 Lecture 1, Page 6


7/14

Q(t)Q(t),, w(n)w(n), and Elemental Waiting, and Elemental Waiting

D(t)

A(t)

t0

elemental waitings

t+dt

N(t)

time0 A-1(n) D-1(n)

n w(n)n+dn

t1

n1

t

Q(t)

1.225, 10/29/02 Lecture 1, Page 7


8/14

Total Waiting TimeTotal Waiting Time

Elemental waiting during [t, t+dt]: Q(t)dt

( )

t1

( (

Total waiting during [t0, t1]: Area =

t1

dttQ = t0 ( tA ) tD ))dtt0

Elemental waiting during [n, n+dn]: w(n)dnTotal waiting during [0, n1]: Area = n1 dnnw( )0Average wait suffered by vehicle arriving between t0 and t1

Area =Area t1 t0 =Q 1W =( ( ( (tA 1) tA ) t1 t0 tA 1) tA ) 0 0

Q = W (Queuing formula)

1.225, 10/29/02 Lecture 1, Page 8


9/14

TimeTime--Space Diagram: Analysis at a Fixed PositionSpace Diagram: Analysis at a Fixed Position

position

h1 h2 h4h3

0

L

x

timet0

T

1.225, 10/29/02 Lecture 1, Page 9


10/14

Flows and HeadwaysFlows and Headways

m(x): number of vehicles that passed in front of an observer atpositionx during time interval [0,T]. (ex. m(x)=5)

( (Flow rate: xq ) = xm )T

Headway hj(x): time separation of consecutive vehiclesxm )(

hj (x)Average headway: h (x) = j=1

(xm ) 1What is the relationship between q(x) and h (x) ?

1.225, 10/29/02 Lecture 1, Page 10


11/14

Flow Rate vs. Average HeadwayFlow Rate vs. Average Headway

m(x)

IfTis large, T hj (x)j=1

m(x)

hj (x)1 = T j=1 =h (x)Then,xq ) xm ) xm )( ( (

1(xq ) This is intuitively correct.

h (x)

q(x) is also called volume in traffic flow systems circles (i.e. 1.225)

q(x) is also called frequency in scheduled systems circles (i.e.1.224)

1.225, 10/29/02 Lecture 1, Page 11


12/14

TimeTime--Space Diagram: Analysis at Fixed TimeSpace Diagram: Analysis at Fixed Time

position

t

s1

s2

L

0 timet0

1.225, 10/29/02 Lecture 1, Page 12


13/14

Density and SpacingDensity and Spacing

n(t):number of vehicles in a stretch of lengthL at time t.

(

(

Density tk )= tn )

L

si(t):spacing between vehicle i and vehicle i+1.tn )(

L si (t)i=1 tn )(si (t)

1( = L i=1 = ts )

( ( (tk ) tn ) tn )1

( tk )ts )

(Is this intuitive?)(

1.225, 10/29/02 Lecture 1, Page 13


14/14

Lecture 1 SummaryLecture 1 Summary

Cumulative plots:A(t), D(t), Q(t), w(n) Q = W

Time-Space Diagram: Analysis at a fixed position1

xq ) (h (x)

Time-Space Diagram: Analysis at a fixed time1

tk ) (ts )(

1.225, 10/29/02 Lecture 1, Page 14

Documents

1 Time-Space Diagram