Dynamic model of peak period congestion

Tmnrpn. &S.-B Vol. 18B, No. 4/5, pp. 339-355, Printed m the U.S.A.

DYNAMIC

0191-2615/84 %3X0+ .W Pergamon Press Ltd.

1984

MODEL

Departmentof Civil Engineering,

OF PEAK PERIOD

CONGESTION

MOSHE BEN-AKIVA Massachusetts Institute of Technology, U.S.A.

Centre d’Etudes des Transports

Cambridge,

MICHBLE CYNA Urbains, Ministere des Transports,

Bagneux,

MA 02139,

France

and

Department

of Geography,

ANDRE DE PALMA McMaster University, Hamilton,

(Received 14 July 1982; in revisedform

Ontario,

Canada

13 November 1983)

Abstract-This paper examines the problem of peak period traffic congestion and the analysis of alternative congestion relief methods. It presents a dynamic model of the queues and delays at a single point of traflic congestion because there is ample evidence to suggest that the major delays to users occur at bottlenecks. The model consists of a deterministic queueing model and a model of arrival rate as a function of travel time and schedule delay. A dynamic simulation model also describes the evolution of queues from day to day. The model is used to study the impacts of changes in capacity, total demand, flexibility of work start time and traffic control. Among the numerical results is a demonstration that additional capacity always significantly reduces the duration of the congestion period, but may result in a less significant improvement in maximum delays.

1. INTRODUCTION

a recent paper, de Palma et al. (1983) developed a theoretical stochastic equilibrium model of peak period traffic congestion. This paper presents a dynamic extension and a simulation procedure of this model. Simulation experiments are performed to predict the impacts on peak period congestion of various policy measures. We consider travellers who go through bottlenecks of limited capacity and who may be able to adjust their departure times to avoid slow traffic. Consider, for example, a commuter who has to be at his/her work place at an official work start time. This desired arrival time may be during the peak period and the road that the commuter takes is congested at that time. This commuter may have a choice between on time arrival with a long travel time, and a late or an early arrival with a shorter travel time. The difference between actual and desired arrival time is the schedule deZayincurred by the traveller who trades it off against travel time (Kraft and Wohl, 1967). Cosslett (1977), Small (1982) and Abkowitz (1980) developed econometric demand models of work trip scheduling that are based on this tradeoff between travel time and schedule delay. They estimated logit models of the choice of departure or arrival time with different specifications of the utility functions. Wohl(l970) analyzed a system where workers commute to a single destination with all congestion occurring at a single bottleneck facility. More recently, Hendrickson and Kocur (198 1) developed a model for the same problem. They assume a deterministic utility function of departure time choice and at equilibrium this utility is equal for all individuals in the system. The present model which was developed independently also addresses the same problem, but with a more general model that employs a probabilistic demand model of the type estimated by Cosslett (1977), Small (1982) and Abkowitz (1980) and a dynamic model of day to day changes of the type that was studied by de Palma and Lefkvre (1983). In

2.

THE

STOCHASTIC

EQUILIBRIUM

MODEL

This section summarizes the essential features of the stochastic equilibrium model presented in de Palma et al. (1983). Consider the system depicted in Fig. 1 where N drivers go every day from point A to point F. This can represent a home to work trip with three segments: AC with a constant travel time t,; DF with a constant travel time tz; and CD which 339

M. BEN-AKIVA et al.

340

\

(the

boltleneck)

/F’work’

C

D

(,)

t

I

I

‘I

I

‘2

'0

i

Fig. 1, The setting of the model

is a bottleneck of fixed capacity S. The travel time from C to D is a constant r,; but at point C a queue may develop. Denote the waiting time for a driver arriving at C at time t as t,(t). Thus, the total travel time from A to F for a driver arriving at time t at C is:

tt(t> = t, + t,(r) + t, + t,

(1)

Let D(t)be the number of cars in the queue at time t. The waiting time for an arrival at C at time t is derived from a model of a deterministic queue as follows:

D(t)

t,(t) = -

(2)

s

with no congestion o(t) = 0 and t,(t) = 0. Let r(t) be the arrival rate at point C. In a no congestion case o(t) = 0 and the outflow is equal to the arrival rate. With congestion D(t)> 0 and the outflow is equal to the capacity. It was shown in de Palma et al. (1983) that there can exist at most one congestion period. Denote by t, and t; the beginning and the end times of a congestion period, respectively. Then, the rate of change of the queue length is given by d(t)

= 0, for t I t, and

t 2 t;

(3)

6(t)

= r(t) - s, for t, < t < ti.

(4)

and

The beginning

of the congestion

period

is given by the first occurrence

(54

r(t,) = s and the end of the congestion

period is determined D(tJ

= 0

of

from

(5b)

A driver’s choice of a departure time from A at time t-t1 is the same as a choice of an arrival time t at C. Let V(t) be an observable utility for an arrival at C at time f and p be a scale parameter. The probability of time t being chosen is given by the following continuous logit model (see Ben-Akiva and Watanatada, 1977 and Litinas and Ben-Akiva, 1979):

Dynamic

model

of peak period

congestion

341

where E is the logit denominator. The N cars arrive at C between t, and t, + T and the arrival rate at point C at time t, as will be discussed later, can be written as

r(t)=Np(t)=gexp

[

tV(t)

(7)

1

The two key variables that vary among alternative departure times are the travel time and the schedule delay. The marginal disutility of an additional unit of travel time is a > 0. Thus, the utility associated with travel time is - att(t). Let the time interval [t* - A, t * + A], where A 2 0, be the desired time period for arrival at point F, where t * denotes the centre of the period and A is a measure of work start time flexibility. Let t”and Fbe the arrival times at C that correspond to r * - A and I * + A, respectively. In other words, arrivals at the bottleneck during the interval [t,, fi are early arrivals with a schedule delay disutility of - fl(t* - A - t - t,(t) - t, - t2) and arivals during [[ tl + T] are late arrivals with a disutility of - j?y(t + t,(t) + t,, + t, - t * - A) where /I and y are constant parameters. Note that y is the ratio of the marginal disutilities of late and early arrivals, which in most applications is expected to be greater than one. Thus, the utility associated with both travel time and schedule delay can be shown to be V(t) = /Wt + (PO - a)&(t) + /3A 101- #?O(t* - t,, - t2)

(8)

where

0(t)=

1 for t, It

It”

0 for t”

i --y

(9)

for t=ItIt,+T

and t”=t*-to-t2-A-t,(i) tZ=t*-tO-t2+A-t,(fj.

(10)

The basic model above assumes that cars in the queue are stacked up at point C-the entrance to the bottleneck. The extended model described below considers the density of cars in the road section before the bottleneck (see Fig. 2). As before, the speeds on AB, CD and DF are fixed and the constant travel times are t,, r,,, and t2, respectively. However, the speed on BC depends on the number of vehicles between B and C. Thus, in this model the queue is assumed to be uniformly distributed within the segment BC. Denote now by D(t) the number of vehicles between B and C at time t. Note that in this model, t is the arrival time at point B. With no congestion, the BC travel time is a constant t,. (In the basic model t, = 0.) In the extended model, t,(t), the travel time on BC, is

(11) Therefore, congestion appears whenever D(r) > SC,,(and D(t) can be greater than 0 without congestion), and the definition of the beginning and end of the congestion period is given

342

M. BEN-AKIVA ef al. A

F

Fig. 2. The setting of the extended model

D(t,) = D(tJ The queue

length

equation

for this model

d(t)

and for the assumed

waiting

(12)

D(t) t,(t)

(13)

is

= r(t) - ~

time model b(t)

= s . t,

in (11) it is

= r(t) -

Min

(14)

It is shown in de Palma et al. (1983) that for t,,-+O this extended model reduces to the basic model. Note that in this formulation the effective queue length at time t is equal to Max [D(t) - St,, 0] and the effective

waiting

time at the bottleneck

at time t is

t,(t) - L The theoretical properties of this model were analyzed in detail in de Palma et al. (1983). It was shown that there can be at most one congestion period. However, when congestion occurs, the model cannot be completely solved analytically and a numerical solution technique is necessary. Moreover, a simulation technique is also necessary to analyze a variety of extensions of this model. In the following section, we present a dynamical extension of this model that describes the evolution of the departure time distribution and of the queueing delays from day to day. Thus, it also predicts the transient impacts from the implementation of a new policy. The steady state solution of this dynamic model is the stochastic equilibrium model that was presented in this section. The dynamical model also provides a natural algorithm for a simulation approach.

3.

THE

DYNAMIC

MODEL

The dynamic extension is based on the approach that was developed by de Palma and Lefevre (1983). The setting of the system is the same as before, with additional notation to indicate day-to-day variability. Thus, V, r, D and t, are functions of t, the arrival time as C (or B) and o, the day for which they are defined. Let r(t,o) be the arrival rate at the bottleneck at time t on day w. The number of individuals arriving during an interval of time [t, t + 61 on day o is

Dynamic

model

of peak period

343

congestion

s lf6

x(t, 0) =

r(u, co) du

(15)

f

Denote by R(t, t’, w) do the fraction of individuals who shift from an arrival during [t, t + 61 to an arrival during [t’, t’ + 61 during the time interval [w, o + do]. The rate of change of the number of individuals arriving during the interval [t, t + 61 can be expressed as the difference per unit of time between the number of individuals shifting to [t, t + 61 and the number of individuals shifting from [t, r + 61, as follows,

~

am

=

,;rwT oW( t’,t,w) -

Assume that the transition rates among different the following simple dynamical logit model

x(t, 0) 1 R(t,t’,w). I’fl

departure

times can be modelled

with

(17)

where r(t, o) is the observable utility of an arrival time during the interval [t, t + 61 on day o and R is a constant transition rate out of the current state. The structure of this dynamical model is a special case of a more general dynamical nested logit model developed by de Palma and Lefevre (1983), de Palma and Ben-Akiva (1981) and Ben-Akiva and de Palma (1983). The simplifying assumption made here is that the utility of a shift to a new state is not dependent on the attributes of the current state. Substitute (17) in (16) to get

eY(‘.w)ip

ax(t,w) = R N am

[ Let 6+0

to obtain

the following

F e

W”, N/P

continuous

-x(4 0)

(18) 1

time limit of this dynamical

system

(19)

where the continuous logit model presented earlier replaces the discrete logit model. This equation describes the evolution of the arrival rate at the bottleneck from day to day. Note the R is the only new parameter, and all the other variables and parameters are the same as in the equilibrium model with an additional argument o where it is appropriate. For w -+ co this dynamical system reaches a stationary state where lim V(t, 0) = V(t) Cl-z.2 lim r(t,o) w-=

(20)

= r(t)

(21)

and r(t) is given by eqn (7). Thus, the stationary state is the equilibrium model described in a previous section of this paper. 4.

THE

STRUCTURE

OF

THE

solution

of the

SIMULATION

The dynamic simulation is the numerical solution of eqn (18). The time o is transformed into a discrete variable representing a day or an iteration; at the beginning of the day there

M. BEN-AKIVA et al.

344

is no congestion. For a day, compute r and D, and update t, which is now the starting point for the next day. The process stops when the system reaches a stationary state; that is, the relative difference between one day and the next is smaller than a given tolerance. See Fig. 3 and Appendix 1. Values for the unknown parameters of the demand model were derived from the estimates obtained by Small (1982). The following values for the utility function defined in eqns (8) and (9) were used in the simulation program for time measured in hours: LX= 6.4, /II = 3.9, and By = 15.2. These values imply that a minute of late schedule delay is valued the same as 3.9 min of early schedule delay and an additional minute of an early schedule delay is valued the same as .6 minutes of extra travel time.

5.

NUMERICAL

EXAMPLES

OF

STATIONARY

TIME

A numerical This simulation

AND

TRANSIENT

DEPARTURE

DISTRIBUTIONS

example of a stationary distribution of departure times is given in Fig. 4. was performed with the Basic Model (i.e. t, = 0) for the following variables: Total Number of Cars (N) = 10,000 Capacity of the Bottleneck (s) = 4000 cars/hr Range of On-Time Arrivals (d) = fU2 hr Scale Parameter of the Choice Model (p) = 1

Note that in the Basic Model the congestion

initial

v

(81,

t

condi

(IO),

;

period begins when the departure rate exceeds the

tians

(10)

> E (6)

Extended

Model

legend: variable

Fig. 3. The structure

of the simulation.

(calculated

from

equation)

345

Dynamic model of peak period congestion DEPARTURE (CA’IS/HR )

/

t max,ckl1,‘_ h

tq30

j fuY2

fxm

Pm0 DSPERTURE

TIME

Fig. 4. Stationary distribution of departure times.

capacity of the bottleneck at time t,. After the departure rate attains its maximum value it begins to decrease and the point in time when it is again equal to the capacity of the bottleneck is denoted at tmax,del.At this point in time, the queue length attains its maximum value which is equal to the area indicated by A,. The queue v_anjshes at time ti when the area indicated as A, is equal to A,. Note that the time interval [t, ?j of departures with on-time arrivals is not equal to one hour (i.e. 2 times A) because of the delays at the bottleneck. Figure 5 demonstrates the dynamical evolution of the departure time distribution towards the stationary state. In this example, the initial condition of the dynamic simulation was a situation without congestion which is reflected in a flat departure rate distribution between t”and r A more relevant dynamical evolution is depicted in Fig. 6. In this case the initial condition is the stationary distribution given in Fig. 4. The bottleneck capacity is increased by 50% from 4000 cars/hr to 6000 cars/hr. Initially, the system still experiences a significant level of congestion during a shorter interval of time. However, as drivers modify their departure times the system evolves towards the stationary distribution shown in Fig. 7, where the remaining level of congestion is very small. A comparison of Figs. 4 and 7 demonstrates the effects of increased capacity on the stationary departure rate distribution. A large fraction of the drivers who previously arrived early to avoid long delays are now at work on-time. The same effect holds for late arrivals. A fraction of the drivers who avoided the long delays by departing late are now leaving home earlier and are able to arrive on-time at work. This example demonstrates the significant effect that changing the capacity of a bottleneck may have on the temporal pattern of the arrival rate. Thus, an analysis of peak period traffic congestion that assumes a fixed distribution of departure times may result in erroneous conclusions.

M. BEN-AKIVAet al. DEPARTURE ORS/IiR

RATE 1

LAY 10 \

3A’t 1

DEPARTURE

Fig.

5. Transient

distributions

DEPARTmE

of departure

times starting congestion.

from

TIME

the initial

condition

RATE

CARS

bXl

DEPARTURE

Fig. 6. Transient

distributions

of departure

TIMF

times for a 50% increase

in capacity.

of no

347

Dynamic model of peak period congestion DEPARTURE

RATE

t mdtq 00

mo

I

14’ 800

10

so0 DEPARTURE

TI,E

Fig. 7. Stationary distribution of departure times for a 50% increase in capacity.

6. SIMULATION

EXPERIMENTS

Simulation experiments were conducted to analyze the effects of a number of policy related variables on the stationary state: Capacity and number of cars in the system (N/sFIt can be shown that the distribution of departure times and delays are only functions of the ratio of the total volume of traffic over the capacity of the bottleneck (N/s). Thus, a change in the bottleneck capacity can be determined either by changing s directly or by changing N. Increasing N because of urban growth or decreasing N through carpooling incentives are other instances of change in the ratio N/s, Thus, the effect of a change in this ratio is of practical importance and will be studied in the simulation experiments. Work start time flexibility (A+A represents the extent to which one can choose his arrival time. It can also represent a formal flexitime policy. Thus, the effectiveness of increasing the flexibility in arrival time can be predicted by the model. Trafic control measures (t,Ft, represents the uniform spreading of the arriving traffic on the approach to the bottleneck. It can occur without external intervention by the natural tendency of some drivers to speed up and some to slow down to form a more evenly distributed traffic stream when a bottleneck is observed ahead. But it can also result from a traffic control device such as a regulatory system of traffic lights with real time control which can have this spreading effect. Randomness and Information (pFp measures the randomness of the traffic behaviour. By increasing information, randomness can be affected. Thus, the value of p could be changed by policy, but the means of changing p are difficult to define. In the simulation experiments, the values of the p$cy variables are changed one at a time. For the steady state solutions, the outputs are: t,, t$ t, t, t,,(t), r(t), D(t) and V(t). The results are presented with graphs that describe summary stationary state outputs as functions of the

348

M. BEN-AKIVA et al.

policy variables.

The most important

(i) the average

1 f, + T time 6 = r(t)&(t) dt N s possible delay tvmax=i’Max,Elc,,lrt,,(t) and end times of the congestion period (t3 -

waiting

(ii) the maximum (iii) the beginning The following experiments:

values

N s A t, p

graphs are for

were

= = = = =

selected

to

represent

the

base

tJ.

case

for

the

simulation

1500(Total number of cars) lOOO(Capacity of the bottleneck in cars/hr) O(Flexitime parameter in hr) O.Z(Traffic control parameter in hr) l(Randomness parameter of the demand model)

Note that this base case is different from the numerical example given in the previous section. It is based on the same demand model parameters but uses the extended model with t, > 0. The policy analysis graphs are drawn for the following ranges of values: 0
< 2

(Figs.

S-lo),

0 < A < 42 min (Figs. ll-13), 0 < t, < 0.5 hr with t, + t, = 0.5 hr and are discussed below.

(Figs 14-16)

.l

0

t (HR) .5

Fig. 8. Average

1

1.5

2

waiting time as a function of :. s

Dynamic

model

of peak period

congestion

349

.8 -

.6 -

.4 i

0

1 .5

I 1

Fig. 9. Maximum

I 1.5

I 2

waiting time as a function of E. s

8:u)

7:M

7:oo

6:M

6:oO

I

.5

,

1

Fig. 10. t,, r;

1.5 and

i

2

as a function E. s

!WR)

350

M.

BEN-AKIVA

et

al

.1

A (HR)

0

.2

.1

11. Average

Fig.

t

“w

.4

.3

waiting

.5

time as a function

.6

.7

of d.

(HR .6

.5

.4

.3

.2

0

1

I

I

I

I

I

.1

.2

.3

.4

.5

.6

Fig. 12. Maximum

waiting

time as a function

of d.

A

UJR

Dynamic

model

of peak period

congestion

351

t’ q 8:00

-

7:30

-

t

I

7:ooqI I I

6:30

I I

-

I I I I

6:OO

I .1

I .2

I .3

I .4

I .5

Fig. 13. t,, “;, 6 t”as a function

t”-t”

I .6 AC

of A.

(RR) .4

.3

.2

0 .1

Fig.

14. Average

.2

delay

.3

as a function

.4

.5

of t,(r, + t, = 0.5 hr).

tu

(HR)

M. BEN-AKIVAet al.

352

0

I

’

I .1

Fig. 15. Maximum

I

I

.2

delay

!3

as a function

.4

of

!5

tU (HR)

f,(t, + to= 0.5 hr).

8:00-

t’ q 7:30

7:oo

6:30

6:OC .1

.2

.3

.4

.5

Fig. 16. t,, t;, i as a function of r.(t. + t0 = 0.5 hr).

tu (Hx

Dynamic model of peak period congestion

353

Varia_tionof (N/s) (Figs. S-10)4 as well as tVmXare increasing functions of (N/s). t, and r( = 9 are decreasing functions of (N/s) and ti is an increasing function of (N/s). In the three cases, the relationships are roughly linear. For higher levels of congestion the relationship between the ratio of (N/s) and the average or maximum delay is convex. These results show that increasing the capacity of a congested bottleneck leads to a shorter congestion period, as expected. Less obvious is the result that the relative decrease of the average delay is substantially greater than that of the maximum delay; thus, while a capacity addition can result in great global time savings, the reduction of congestion during the height of the peak period is likely to be relatively small. In a recent empirical study of traffic delays on the expressways in the Boston downtown area Lisco (1983) found a stable relationship between the ratio of daily traffic volume over capacity and the maximum delay at bottlenecks. The form of Lisco’s empirical relationship is the same as that of Fig. 9 which depicts the relationship between (N/s) and to,, for the stationary states predicted by the model. Variation of A (Figs 1 l-13+< and t,,_ are both decreasing functions of A up to a critical value A’ when congestion vanishes. They are both concave except for large values of A approaching A’. Thus, the incremental efficiency of an additional minute of flexibility i,s an increasing function of A for values not approaching A’. The graphs for t,, ti, t”and t as functions of A, indicate that from A = 0 to A = 30 min, the length of the congestion period, t; - t,, is constant, although the level of congestion measured by < is decreasing. Note that as A increases the congestion period converges towards the end of the pe$od of on-time arrivals. It was shown in de Palma et al (1983) that for this mod$ ti 2 t. It can also be shown that for value A’ at which congestion disappears t, = ti > t where the strict inequality holds in this case because t,, > 0. A more intuitive explanation of this phenomenon is the fact that the time of maximum delay always occurs when the departure rate is decreasing and therefore the congestion period converges toward the end of the period of on-time arrivals. A comparison of Figs 8 and 11 shows, for example, that the effectiveness of increasing A from 0 to 30 min may be approximately the same as that of doubling the bottleneck capacity. Variation of t, (Figs. 14-16)-The sum t, + to is kept constant so that comparisons could be made directly between the various values of t,,t$ and i: < and turn, are decreasing functions of t,, which shows that it is possible to decrease the level of congestion with traffic control measures. t; is roughly constant and tq is linearly increasing. Thus, the duration of the queue is approximately linearly decreasing with t,. The effect of increasing t, is primarily to delay the beginning of the congestion state; that is, to increase t,. Mathematically, this is simply a result of the condition from eqn (12) that D(t,) = s . t,. It states chat for a greater value of t,, more traffic must be accumulated in the roadway section BC before a queue begins to form at point C. In other words, the model assumes that as t,, increases the traffic before the bottleneck is spread uniformly over a longer section of road. As a consequence the beginning of the congestion state is delayed and the average and maximum delays decrease with increasing t,. Variation of p (Fig. 17)_The changes of p cannot be defined precisely in physical terms. Thus, instead of drawing a graph, the exploration of the effect of ~1 is limited to a comparison of two values to indicate the level of sensitivity and direction of change. When p increases, < and t,,_ decrease; the duration of the queue decreases very slowly. Thus, it appears that increasing p has a qualitatively similar effect to that from increasing A. Increasing the randomness parameter is equivalent to reducing the sensitivity by which drivers react to changes in travel time and schedule delay. Thus, the departure time distribution is more uniform which tends to reduce the level of congestion. Another use of these results can be to define the policy objectives and search for appropriate policy measures. For example, if an acceptable level of congestion can be defined, the corresponding values of the policy variables that are required can be obtained from the graphs. For this numerical example, consider a maximum acceptable level of congestion of < - t, = 5 min which is exceeded in the base case. Three policies are available: (i) increasing the effective capacity by 27% so that (N/s) = 1.18, (ii) having flexible work hours with A = 25 min, (iii) regulating the arrival rate so that t, = 30 min.

M. BEN-AKIVA et al.

354

bin) 1

2

1

8.93

4.45

t

t”

“”

u

t’ q

4

t,t

~~, 1

36.7

1

26.7

7.

1

7:21

Fig. 17. Variations

Each of these policies changes the value be obtained by employing the simulation

7:03

8:05

8:lO

(

7~25

7:39

of the system with p.

of only one policy variable; to test composite policies.

further

results

can

CONCLUSION

The simulation model developed in this paper was used to demonstrate the phenomenon of shifting peaks and the relative effectiveness of measures to alleviate traffic congestion. These results pertain to a simple situation of a single bottleneck, a constant total demand and a homogenous traffic stream. In further research the dynamic simulation model will be applied to more complex traffic patterns and queueing systems. In particular, the basic model structure is being extended to incorporate elastic total demand, route and mode choice. A model that explicitly treats multiple traffic streams can be used to analyze traffic control policies such as priority treatment of high occupancy vehicles and public transport improvements. The simulation can also be used to analyze policies of staggered work hours and alternative schemes of peak period pricing. The parameters of the departure choice models should also be further investigated by classifying the population of drivers into market segments with different tradeoffs and desired arrival times and testing the sensitivity of the results to variations in these parameters.

Acknowledgements-Partially supported by the Office of Planning Methods & Support, Urban Mass Transportation Administration, U.S. Dept. of Transportation. We are indebted to Bernard Francois for many useful suggestions and for his assistance in developing the simulation program.

REFERENCES Abkowitz M. (1980) The Impact of Service Reliability on Work Travel Behavior. Ph. D. Thesis, Department of Civil Engineering, Massachusetts Institute of Technology. Ben-Akiva M. and de Palma A. (1983) Modelling and Analysis of Dynamic Residential Location Choice. Working Paper No. 83-19. Department of Economics, McMaster University, Hamilton, Ontario. Ben-Akiva M. and Watanatada T. (1977) Applications of a continuous spatial choice logit model. Prepared for NBERNSF Conference on Decision Rules Under Uncertainty. (Structural Analysis of Discrete Data: with Econometric Applications. Edited by C. F. Manski and D. McFadden 1981) M.I.T. Press; Cambridge, Mass. Cosslett S. (1977) Demand Model Estimation and VaGdation, (Edited by D. McFadden et al.) Urban Travel Demand Forecasting Project, lnstitute of Transportation Studies, University of California, Berkeley. Cyna M. (1981) Congestion and Schedule Delay. S. M. Thesis. Department of Civil Engineering, Massachusetts Institute of Technology. de Palma A. and Ben-Akiva M. (1981) An interactive dynamic model of residential location choice. Paper prepared for presentation at the Structural Economic Analysis and Planning in Time and Space Conference, Umea, Sweden. de Palma A. and Lefevre C. (1983) Individual decision making in dynamic collective systems. J. Math. Socio. 9, 103-124. de Palma A. Ben-Akiva M., Lef&vre C. and Litinas N. (1983) Stochastic equilibrium model of peak period traffic congestion. Transport Sci. 17(4). Hendrickson C. and Kocur G. (1981) Schedule delay and departure time decisions in a deterministic model. Transpn Sci. 15, 62-11. Kraft G. and Wohl M. (1967) New directions for passenger demand analysis and forecasting. Transpn Research l(3) 205-230.

Dynamic

model

of peak period

congestion

355

Lisco T. E. (1983) A procedure for Predicting Queues and Delays on Expressways in Urban Core Areas, Central Transportation Planning Staff, Tech. Rep. 36, Central Transportation Planning Staff, Boston, Mass. Litinas N. and Ben-Akiva M. (1979) Behavioral modelling of continuous spatial distributions of trips, residential locations and workplaces. Unpublished paper, Massachusetts Institute of Technology, Cambridge, Mass. Small K. (1982) The scheduling of consumer activities: work trips. The Am. Econ. Rev. 72(3), 467479. Wohl M. (1970) A methodology for forecasting peak and off-peak travel volumes. Highway Res. Record 322, 183-219.

APPENDIX

1

Relationships used in the simulation The interval It,. t, + Tl is divided into M interval of equal length T/MC = 6). Compute t,, V, r, and D for the extremities of each interval. These points are indexed by r- = 0, .I , M. Use eqn (1 I)-to compute t,(f, w). Use eqn (8) to obtain V(t, 0). The denominator of the logit model is approximated by

(A.11 Equation

(19) is used in the following

discrete

N r(t, w) = RF exp

form:

‘V(t,co) 1 +(l-R)r(t,o-l),

r=O ,...,

M.

(A.2)

[u

A test on the value of t, leads to two distinct and one for the Extended Mode1 (t, > 0). The Basic Model-t, and t; are defined by

branches

r(t,, o) = s and a test on r(t, o) gives t,

Thus,

D(t, o) is computed

of the simulation:

one for the Basic Model

(t. = 0)

D(t,, o) = D(t;, u) = 0.

(A.3)

by -s

D(C +l,w)=D(t,o)+~

(A.4)

> The first value of t such that D(t + 1, o) I 0 corresponds to I;. D(t, o) is computed between equal to zero outside that interval. The extended mode-The following linear approximation of eqn (13) is used to compute

fq and t; and set

D(f

D(t + 1, o)

(A.9

t, and ri are obtained from D(t,w) = St.. The final step is the same for both models. A convergence value, equal to the maximum relative difference between values of D(t,w) and D(t, w - l), is compared with a tolerance level to test if the system has reached a steady state equilibrium. If so, the simulation is terminated. If not, the values of D(t,w) are used to updata t,(f,~). The computer program is described in detail in Cyna (1981).

APPENDIX

II

of key variables 43 travel time to cross the bottleneck 4 travel time from the origin to the bottleneck ‘2 travel time from the bottleneck to the destination f”(l) delay at the entrance to the bottleneck at time t. tttt) total travel time from origin to destination for an arrival at the bottleneck at time t N total flow of vehicles through the bottleneck T the longest feasible interval of time for the total flow to cross the bottleneck S the capacity of the bottleneck the rate of arrivals at the bottleneck at time t 41) the queue length at the entrance to the bottleneck at time t D(t) the average utility for a trip arriving at the bottleneck at time t v(t) the time at which a congestion period begins t, the time at which a congestion period ends t;

Definitions

t ; t*

A P&QJ R

the earliest arrival time at the bottleneck for an on-time arrival at the destination the latest arrival time at the bottleneck for an on-time arrival at the destination the time at the center of the on-time arrival period at the destination half the length of the on-time arrival period estimated parameters of the departure time choice mode1 the fraction of drivers who review their departure time choice every day.