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Dynamic network equilibrium research
Trrurrpn. Rex.-A Vol.19A.No.516,pi. 429431.1985 Pnnti mrhcU.S.A.
0191-26071115 13.00+.oo 0 1985Pergamon Press Ltd.
DYNAMIC NETWORK EQUILIBRIUM RESEARCH MOSHEBEN-AKIVA Dept. of Civil Engineering, M.I.T.,
INTRODUCTION In discussing Friesz’s comprehensive resource paper on network modeling, I will focus exclusively on the topic of dynamic network equilibrium. This topic deals with the dynamic nature of network performance under timevarying demands. I will not consider a related and equally important topic in demand modeling concerned with the dynamic day to day adjustment processes that result in transient disequilibria. Thus, the purpose of my discussion is to expand on Friesz’s coverage of dynamic network models and to put forward this topic as an important problem area with promising research opportunities. STATIC
EQUILIBRIUM
AND TRAFFIC
CONGESTION
At the outset I would like to emphasize that the standard static network equilibrium formulation fails to capture essential features of traffic congestion. A standard network equilibrium model represents the flow pattern during a fixed time interval-a peak period, or a peak hour. The temporal, or time of day, distribution of the traffic is assumed to be fixed and during the modeled time interval the traffic is assumed to be uniformfy distributed. However, observed traffic flows are not uniform over the rush hour. It is common knowledge, at least among trafftc engineers, that delays occur at different locations during varying time intervals. For example, some level of traffic congestion may be experienced from 7 to 8 a.m. on a suburban road and from 7 to 9:30 a.m. on a centrally located highway facility. Tittemore et al. (1972) performed a detailed analysis of the temporal distribution of vehicular travel in eight U.S. urban areas. They found that the percentage of daily travel occurring during the peak hour ranged from 712%; where the higher values are for rural and suburban highways which, in general, are less congested than urban roads. They also deduced, for example, that the significant differences between the St. Louis and Boston temporal traffic patterns are, to a large extent, due to a higher level of traffic congestion in Boston. Thus, we observe that higher levels of traffic congestion are associated with smaller peak hour fractions of daily traffic. We conclude that the level of traffic congestion appears to be an important determinant of the temporal distribution of personal travel in metropolitan areas. Recent evidence from on-going highway reconstruction projects in several large U.S. cities also demonstrates this point. Significant changes in the temporal distributions of traffic were observed during construction, when highway segments operate at reduced capacities. A typical change would be a decrease 429
Cambridge, MA 02139 U.S.A.
in the 8-9 a.m. traffic and an increase in traffic before 7 a.m. (See, for example, the discussion in Hendrickson and Plank, 1984.) The importance of the temporal variability of traffic congestion is also indicated by the few models of departure time choice that were estimated using data from travel surveys. For example, Hendrickson and Plank (1984) used data on work trips from Pittsburgh to estimate a model of the joint choice of travel mode and trip departure time. They concluded that “the departure time decision seems to be more elastic than the choice of mode.” (Hendrickson and Plank, 1984; p. 35). Clearly, a uniform peak period travel pattern is a fittitious concept. In some special circumstances with light traffic congestion it may provide a reasonable approximation. However, it may also result in significant errors when applied to a network with interdependent flows and more than one congested link. For example, in a recent paper Lisco (1983) argues that the standard traffic assignment techniques are unsuitable for an analysis of congestion problems on an urban expressway network. He conducted measurements of traffic delays on express highway facilities in and near downtown Boston and observed that in typical downtown area queuing situations, where one bottleneck highway segment can create queues stretching into many other segments, traffic assignments cannot indicate the locations and extents of queues or the delays associated with them. Because queuing can be of major importance in peak period expressway operations in downtown areas, the assignments can be grossly inaccurate in predicting peak period operating speeds.
(Lisco. 1983;p. 148) Lisco employed an empirical method to predict the maximum delays and queue lengths at bottlenecks. He assumed that there is a consistent relationship between the maximum delay and the ratio of average daily traffic (ADT) and. hourly capacity at a bottleneck. He applied this relationship to predict the pattern of traffic congestion for different configurations of the downtown expressway network. (A major disadvantage of this method is the need to conduct a large number of new delay measurements every time it is applied. Moreover, because of its empirical nature, the estimated relationship may be invalid for predictions when significant changes in travel patterns are anticipated.) More generally, highway travel times are convex functions of the traffic flows; therefore, by Jensen’s inequality (Rae, 1973), a static traffic assignment systematically underestimates travel times. (This theoretical observation was suggested by Bovy, 1984, who has shown that ran-
430
Transportation networks
dom errors in link travel times also result in underestimation of the shortest paths travel times.) Thus, it is evident temporal characteristics of traffic congestion that play a critical role in the determination of operating speeds are not captured by static network models. Yet, as Friesz noted in his paper; the existing theories and methods of dynamic network equilibrium that treat the duration and intensity of the peak period endogenously are rudimentary. Therefore, the main point of my discussion is to stress the need for research in the area of dynamic network models with the objectives of improving our understanding of congestion phenomena and developing analytical capabilities to address current and future congestion problems. A FRAMEWORK
FOR DYNAMIC
EQUILIBRIUM
with travel time and cost, to the user arriving at link ij at time t can be expressed by c ‘I(r) = CJ[X‘I(r)].
(4)
A deterministic queueing model provides a useful example of these expressions. Consider a traffic bottleneck with capacity s,, at the end of link ij . Let r,,(O) be the constant time to traverse link ij without delay. Eqn (2) becomes q,,(r) = min {X,,(WW).
s,,).
(2a)
1.
(3a)
Substitute (2a) in (3) to obtain
NETWORK
W)
= max {r,(O), X$)/s,,
MODELS
In order to substantiate the above position I will now proceed to sketch a framework for dynamic network equilibrium models as an extension of the static equilibrium formulation. A network equilibrium model consists of the following four elements:
Alternatively, suppose that the vehicles nor uniformly distributed over the length D,,(f) be the number of vehicles delayed to the bottleneck at time r. Eqns (l)-(3) dD,(Wdr
functions; 2. path choice criterion; 3. origin to destination demand model; 4. nodal flow conservation relationships.
1
Dynamic link performance Consider a link from node i to node j, denoted as ii. Let r,(t) be the arrival rate at link ij at time t and q,,(r) be the departure rate from link ij at time r. Denote by X,,(r) the number of users on link ij at time f. Assume that vehicles are uniformly distributed over the length of a link. Thus, the rate of change in the number of users on link u at time t is
(1)
The exit flow q,,(r) is assumed to be a function of the number of users X,,(r), or their density on the link, 4&) = WX,,Wl.
S I’
if D,,(r) = 0 and r,,[r - r,,(O)1
(2)
The time to traverse link 0 for a user arriving at i at time r is dependent on X,,(r), as follows
and r,(r) = r,,(O) + D&r +
t,,(O)Vs,.
Thus, the generalized
cost, or the disutility
(3) associated
(3b)
The conrinuousfluid approach can also be employed to model dynamic traffic flow on a link. In this approach the velocity of the vehicles over the link is not uniform. Point velocities are determined from a solution of a continuous flow conservation equation and an assumed speed/ density relationship. This approach is theoretically sound but a lot more complicated to solve than the simple deterministic queuing model. Parh choice A user is assumed to select the path with the minimum cost. The following recurrence formula defines the minimum cost for a trip at time r from node i to destination d as c, &) = min {c,,(r) + cj.d[f + r,,Wl,i in LA
=
c,.(r)
+
c,.,[t
+
(5)
t,.O)l,
where 15,~is the set of nodes that can be reached from i by one link and j* in L, is the next node on a minimum generalized cost path from i to d at time r. Note that the minimum path is not necessarily unique. The travel time on this path is defined by --r,.&r)“= t,.(r) + q.Ar + r,.(t)].
r,(r) = XjWq,W.
5 s,,
otherwise, (2b)
A dynamic formulation of these elements follows. The critical extensions of the static equilibrium problem are the time-dependent link performance functions and the temporal O/D demand relationships. A dynamic link performance function represents the queueing effects of timevarying arrival rate. The demand model is extended to include the distribution of departure times of the O/D trips as a function of the temporal distribution of travel times.
a;j(r)ldt = r,(t) - q&O.
r,,(O)l.
s,,(r) =
(lb)
h,(O)1- q,,(r),
= r,,lr -
r,,Lr -
I. link performance
on the link are of the link. Let at the entrance become
(54
Let p,,&) be the fraction of the trips to destination d arriving at i at time r that continue via link 0. The de-
431
Transportation networks
model and by Ben-Akiva ef al. (1983a and b) with a more general demand model. de Palma et al. (1984) have also analyzed a network of bottlenecks in series but with if c,(t) + c,,Jt + t,,(t)1 = c,Ad (6) a single destination. These studies have demonstrated the if c,,(r) + c,.& + #)I > ci.AO, interdependencies among congested links in terms of both the level and duration of traffic congestion. Simulation experiments with these simplified models have also exhibited the phenomenon of shifting peaks with the introjE, e,dt)= 1. duction of a peak-period toll and have consistently converged to a unique dynamic equilibrium.
terministic path choice model is given by >O,
p,,At)
=0,
and
Temporal distribution
of OID demand
Denote by qO,&) the departure rate at time t of trips from origin i to destination d. It is dependent on the temporal distributions of travel times and travel costs over the network. Thus, the temporal demand model for trips at time t from i to d can be expressed by
qo,Ad= Q“"[hAO, LAO, alli&H.
(7)
Flow conservation The arrival rate at link 0 at time t, r,(t), is obtained from the O/D demands and path shares, as follows
r,,(r) = 2
d
r;jAG
(8)
DIRECTIONS FOR
RESEARCH
Clearly, the state of the art of dynamic network equilibrium models is primitive and the models that exist can be generalized in many directions utilizing a variety of approaches. Therefore, a research program in this area should maintain a balance between theoretical and empirical investigations of the properties of alternative dynamic equilibrium models. Clearly, this requires that a suitable theoretical framework be developed. A key challenge of further research is to establish the feasibility of applying dynamic equilibrium models to more realistic networks. This would imply a qualitative increase in the degree of complexity of the problem relative to the simplified networks that have been investigated so far.
with r,,(t)
= P&)
[q‘&)
+
Acknowledgements-Andre de Palma and I have closely collaborated on this topic and the ideas in this paper originate from our joint work. I am indebted to him and to Piet Bovy for many
4t&l, tinL,B x
useful suggestions.
where L, is the set of nodes from which i can be reached by one link, p,+_,(t) is defined by (6). q,(t) is obtained from (7) and qbd(f) is the departure rate at time t from link ki to destination d. The departure rate q&t) is derived from
qk,Ar) = q~iWpd4
(9)
where q,,(t) is obtained from equation either (2). (2a) or (2b) and 7, the arrival time at k for a departure at time I from i, is the solution of the following implicit equation 7 = t - t&T) where t,,(7) is given by either (3). (3a) or (3b).
SIMPLIFIED NETWORKS The existing dynamic network equilibrium models are special cases of the above framework. Among these simplified dynamic models, existence and uniqueness conditions have only been established for the most simple network consisting of a single congested link. (See the review in Friesz’s paper.) A network with bottlenecks in parallel was investigated by Mahmassani and Hennan (1984) using a deterministic departure time and path choice
REFERENCES
Ben-Akiva M., de Palma A. and Kanaroglou P. (1983a) Dynamic Model of Peak Period Traffic Congestion with Elastic Arrival Rates. Unpublished paper, M.I.T., Cambridge, MA. Ben-Akiva M., de Palma A. and Kanaroglou P. (1983b) Capacity Constraints in Traffic Models with Elastic Demand. kape; prepared for presentation at the 10th Transportation Planning Research Colloquium, The Netherlands, December 14-16. Bovy P. H. L. (1984) Travel Time Errors in Shortest Route Predictions and All-or-nothing Assignments: Theoretical Analysis and Simulation Findings.” Memorandum nr. 28, Institute for Town Planning Research, Delft University of Technology, The Netherlands, April 1984. de Palma A., Ingenbleek J.-F.. Lefevre C. and Ben-Akiva M. (1984) Dynamic Model of Traffic Congestion in a Corridor. Unpublished paper, Universite Libre de Bruxelles. 1984. Hendrickson C. and Plank E. (1984) The flexibility of departure times for work trips. Transp. Res. HA, (I), 25-36. Lisco T. E. (1983) Procedure for predicting queues and delays on expressways in urban core areas. Transp. Res. Rec. 944,
148-154. Mahmassani H. and Herman R. (1984) Dynamic user equilibrium, departure times and route choice on idealized traffic arterials. Transp. Sri. 18, 362-384. Rao C. R. (1973) Linear Statistical Inference and its Applkarions, 2nd Ed. Wiley, New York. Tittemore L. H., Birdsall M. R., Hill D. M. and Hammond R. H. (1972) An Analysis of Urban Area Travel by Time of Day, FHWA, US Dept. of Transportation, Washington, DC.
0191-26071115 13.00+.oo 0 1985Pergamon Press Ltd.
DYNAMIC NETWORK EQUILIBRIUM RESEARCH MOSHEBEN-AKIVA Dept. of Civil Engineering, M.I.T.,
INTRODUCTION In discussing Friesz’s comprehensive resource paper on network modeling, I will focus exclusively on the topic of dynamic network equilibrium. This topic deals with the dynamic nature of network performance under timevarying demands. I will not consider a related and equally important topic in demand modeling concerned with the dynamic day to day adjustment processes that result in transient disequilibria. Thus, the purpose of my discussion is to expand on Friesz’s coverage of dynamic network models and to put forward this topic as an important problem area with promising research opportunities. STATIC
EQUILIBRIUM
AND TRAFFIC
CONGESTION
At the outset I would like to emphasize that the standard static network equilibrium formulation fails to capture essential features of traffic congestion. A standard network equilibrium model represents the flow pattern during a fixed time interval-a peak period, or a peak hour. The temporal, or time of day, distribution of the traffic is assumed to be fixed and during the modeled time interval the traffic is assumed to be uniformfy distributed. However, observed traffic flows are not uniform over the rush hour. It is common knowledge, at least among trafftc engineers, that delays occur at different locations during varying time intervals. For example, some level of traffic congestion may be experienced from 7 to 8 a.m. on a suburban road and from 7 to 9:30 a.m. on a centrally located highway facility. Tittemore et al. (1972) performed a detailed analysis of the temporal distribution of vehicular travel in eight U.S. urban areas. They found that the percentage of daily travel occurring during the peak hour ranged from 712%; where the higher values are for rural and suburban highways which, in general, are less congested than urban roads. They also deduced, for example, that the significant differences between the St. Louis and Boston temporal traffic patterns are, to a large extent, due to a higher level of traffic congestion in Boston. Thus, we observe that higher levels of traffic congestion are associated with smaller peak hour fractions of daily traffic. We conclude that the level of traffic congestion appears to be an important determinant of the temporal distribution of personal travel in metropolitan areas. Recent evidence from on-going highway reconstruction projects in several large U.S. cities also demonstrates this point. Significant changes in the temporal distributions of traffic were observed during construction, when highway segments operate at reduced capacities. A typical change would be a decrease 429
Cambridge, MA 02139 U.S.A.
in the 8-9 a.m. traffic and an increase in traffic before 7 a.m. (See, for example, the discussion in Hendrickson and Plank, 1984.) The importance of the temporal variability of traffic congestion is also indicated by the few models of departure time choice that were estimated using data from travel surveys. For example, Hendrickson and Plank (1984) used data on work trips from Pittsburgh to estimate a model of the joint choice of travel mode and trip departure time. They concluded that “the departure time decision seems to be more elastic than the choice of mode.” (Hendrickson and Plank, 1984; p. 35). Clearly, a uniform peak period travel pattern is a fittitious concept. In some special circumstances with light traffic congestion it may provide a reasonable approximation. However, it may also result in significant errors when applied to a network with interdependent flows and more than one congested link. For example, in a recent paper Lisco (1983) argues that the standard traffic assignment techniques are unsuitable for an analysis of congestion problems on an urban expressway network. He conducted measurements of traffic delays on express highway facilities in and near downtown Boston and observed that in typical downtown area queuing situations, where one bottleneck highway segment can create queues stretching into many other segments, traffic assignments cannot indicate the locations and extents of queues or the delays associated with them. Because queuing can be of major importance in peak period expressway operations in downtown areas, the assignments can be grossly inaccurate in predicting peak period operating speeds.
(Lisco. 1983;p. 148) Lisco employed an empirical method to predict the maximum delays and queue lengths at bottlenecks. He assumed that there is a consistent relationship between the maximum delay and the ratio of average daily traffic (ADT) and. hourly capacity at a bottleneck. He applied this relationship to predict the pattern of traffic congestion for different configurations of the downtown expressway network. (A major disadvantage of this method is the need to conduct a large number of new delay measurements every time it is applied. Moreover, because of its empirical nature, the estimated relationship may be invalid for predictions when significant changes in travel patterns are anticipated.) More generally, highway travel times are convex functions of the traffic flows; therefore, by Jensen’s inequality (Rae, 1973), a static traffic assignment systematically underestimates travel times. (This theoretical observation was suggested by Bovy, 1984, who has shown that ran-
430
Transportation networks
dom errors in link travel times also result in underestimation of the shortest paths travel times.) Thus, it is evident temporal characteristics of traffic congestion that play a critical role in the determination of operating speeds are not captured by static network models. Yet, as Friesz noted in his paper; the existing theories and methods of dynamic network equilibrium that treat the duration and intensity of the peak period endogenously are rudimentary. Therefore, the main point of my discussion is to stress the need for research in the area of dynamic network models with the objectives of improving our understanding of congestion phenomena and developing analytical capabilities to address current and future congestion problems. A FRAMEWORK
FOR DYNAMIC
EQUILIBRIUM
with travel time and cost, to the user arriving at link ij at time t can be expressed by c ‘I(r) = CJ[X‘I(r)].
(4)
A deterministic queueing model provides a useful example of these expressions. Consider a traffic bottleneck with capacity s,, at the end of link ij . Let r,,(O) be the constant time to traverse link ij without delay. Eqn (2) becomes q,,(r) = min {X,,(WW).
s,,).
(2a)
1.
(3a)
Substitute (2a) in (3) to obtain
NETWORK
W)
= max {r,(O), X$)/s,,
MODELS
In order to substantiate the above position I will now proceed to sketch a framework for dynamic network equilibrium models as an extension of the static equilibrium formulation. A network equilibrium model consists of the following four elements:
Alternatively, suppose that the vehicles nor uniformly distributed over the length D,,(f) be the number of vehicles delayed to the bottleneck at time r. Eqns (l)-(3) dD,(Wdr
functions; 2. path choice criterion; 3. origin to destination demand model; 4. nodal flow conservation relationships.
1
Dynamic link performance Consider a link from node i to node j, denoted as ii. Let r,(t) be the arrival rate at link ij at time t and q,,(r) be the departure rate from link ij at time r. Denote by X,,(r) the number of users on link ij at time f. Assume that vehicles are uniformly distributed over the length of a link. Thus, the rate of change in the number of users on link u at time t is
(1)
The exit flow q,,(r) is assumed to be a function of the number of users X,,(r), or their density on the link, 4&) = WX,,Wl.
S I’
if D,,(r) = 0 and r,,[r - r,,(O)1
(2)
The time to traverse link 0 for a user arriving at i at time r is dependent on X,,(r), as follows
and r,(r) = r,,(O) + D&r +
t,,(O)Vs,.
Thus, the generalized
cost, or the disutility
(3) associated
(3b)
The conrinuousfluid approach can also be employed to model dynamic traffic flow on a link. In this approach the velocity of the vehicles over the link is not uniform. Point velocities are determined from a solution of a continuous flow conservation equation and an assumed speed/ density relationship. This approach is theoretically sound but a lot more complicated to solve than the simple deterministic queuing model. Parh choice A user is assumed to select the path with the minimum cost. The following recurrence formula defines the minimum cost for a trip at time r from node i to destination d as c, &) = min {c,,(r) + cj.d[f + r,,Wl,i in LA
=
c,.(r)
+
c,.,[t
+
(5)
t,.O)l,
where 15,~is the set of nodes that can be reached from i by one link and j* in L, is the next node on a minimum generalized cost path from i to d at time r. Note that the minimum path is not necessarily unique. The travel time on this path is defined by --r,.&r)“= t,.(r) + q.Ar + r,.(t)].
r,(r) = XjWq,W.
5 s,,
otherwise, (2b)
A dynamic formulation of these elements follows. The critical extensions of the static equilibrium problem are the time-dependent link performance functions and the temporal O/D demand relationships. A dynamic link performance function represents the queueing effects of timevarying arrival rate. The demand model is extended to include the distribution of departure times of the O/D trips as a function of the temporal distribution of travel times.
a;j(r)ldt = r,(t) - q&O.
r,,(O)l.
s,,(r) =
(lb)
h,(O)1- q,,(r),
= r,,lr -
r,,Lr -
I. link performance
on the link are of the link. Let at the entrance become
(54
Let p,,&) be the fraction of the trips to destination d arriving at i at time r that continue via link 0. The de-
431
Transportation networks
model and by Ben-Akiva ef al. (1983a and b) with a more general demand model. de Palma et al. (1984) have also analyzed a network of bottlenecks in series but with if c,(t) + c,,Jt + t,,(t)1 = c,Ad (6) a single destination. These studies have demonstrated the if c,,(r) + c,.& + #)I > ci.AO, interdependencies among congested links in terms of both the level and duration of traffic congestion. Simulation experiments with these simplified models have also exhibited the phenomenon of shifting peaks with the introjE, e,dt)= 1. duction of a peak-period toll and have consistently converged to a unique dynamic equilibrium.
terministic path choice model is given by >O,
p,,At)
=0,
and
Temporal distribution
of OID demand
Denote by qO,&) the departure rate at time t of trips from origin i to destination d. It is dependent on the temporal distributions of travel times and travel costs over the network. Thus, the temporal demand model for trips at time t from i to d can be expressed by
qo,Ad= Q“"[hAO, LAO, alli&H.
(7)
Flow conservation The arrival rate at link 0 at time t, r,(t), is obtained from the O/D demands and path shares, as follows
r,,(r) = 2
d
r;jAG
(8)
DIRECTIONS FOR
RESEARCH
Clearly, the state of the art of dynamic network equilibrium models is primitive and the models that exist can be generalized in many directions utilizing a variety of approaches. Therefore, a research program in this area should maintain a balance between theoretical and empirical investigations of the properties of alternative dynamic equilibrium models. Clearly, this requires that a suitable theoretical framework be developed. A key challenge of further research is to establish the feasibility of applying dynamic equilibrium models to more realistic networks. This would imply a qualitative increase in the degree of complexity of the problem relative to the simplified networks that have been investigated so far.
with r,,(t)
= P&)
[q‘&)
+
Acknowledgements-Andre de Palma and I have closely collaborated on this topic and the ideas in this paper originate from our joint work. I am indebted to him and to Piet Bovy for many
4t&l, tinL,B x
useful suggestions.
where L, is the set of nodes from which i can be reached by one link, p,+_,(t) is defined by (6). q,(t) is obtained from (7) and qbd(f) is the departure rate at time t from link ki to destination d. The departure rate q&t) is derived from
qk,Ar) = q~iWpd4
(9)
where q,,(t) is obtained from equation either (2). (2a) or (2b) and 7, the arrival time at k for a departure at time I from i, is the solution of the following implicit equation 7 = t - t&T) where t,,(7) is given by either (3). (3a) or (3b).
SIMPLIFIED NETWORKS The existing dynamic network equilibrium models are special cases of the above framework. Among these simplified dynamic models, existence and uniqueness conditions have only been established for the most simple network consisting of a single congested link. (See the review in Friesz’s paper.) A network with bottlenecks in parallel was investigated by Mahmassani and Hennan (1984) using a deterministic departure time and path choice
REFERENCES
Ben-Akiva M., de Palma A. and Kanaroglou P. (1983a) Dynamic Model of Peak Period Traffic Congestion with Elastic Arrival Rates. Unpublished paper, M.I.T., Cambridge, MA. Ben-Akiva M., de Palma A. and Kanaroglou P. (1983b) Capacity Constraints in Traffic Models with Elastic Demand. kape; prepared for presentation at the 10th Transportation Planning Research Colloquium, The Netherlands, December 14-16. Bovy P. H. L. (1984) Travel Time Errors in Shortest Route Predictions and All-or-nothing Assignments: Theoretical Analysis and Simulation Findings.” Memorandum nr. 28, Institute for Town Planning Research, Delft University of Technology, The Netherlands, April 1984. de Palma A., Ingenbleek J.-F.. Lefevre C. and Ben-Akiva M. (1984) Dynamic Model of Traffic Congestion in a Corridor. Unpublished paper, Universite Libre de Bruxelles. 1984. Hendrickson C. and Plank E. (1984) The flexibility of departure times for work trips. Transp. Res. HA, (I), 25-36. Lisco T. E. (1983) Procedure for predicting queues and delays on expressways in urban core areas. Transp. Res. Rec. 944,
148-154. Mahmassani H. and Herman R. (1984) Dynamic user equilibrium, departure times and route choice on idealized traffic arterials. Transp. Sri. 18, 362-384. Rao C. R. (1973) Linear Statistical Inference and its Applkarions, 2nd Ed. Wiley, New York. Tittemore L. H., Birdsall M. R., Hill D. M. and Hammond R. H. (1972) An Analysis of Urban Area Travel by Time of Day, FHWA, US Dept. of Transportation, Washington, DC.