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Departure rate and route assignment of commuter traffic during peak period
Tm~pn. Ref.-B. Vol. 238. Printed in Great Britain.
No.
5. pp.
337-344.
1989
0191-1615189 $3.00~ .OO ‘0 19R9 Pergamon Press plc
DEPARTURE RATE AND ROUTE COMMUTER TRAFFIC DURING
ASSIGNMENT OF PEAK PERIOD?
A~AHIRU SULE ALFA Transport Institute and Department of Civil Engineering, University of Manitoba, Winnipeg, Manitoba R3T 2N2. Canada (Received 16 November
1987; in revised form 29 November 1988)
Abstract-An incremental heuristic is presented for estimating the departure rate of commuters in a network. Other results that can be obtained from the model include time-dependent flows of traffic and travel times for each link in the network. The model is able to handle multiple origin-destination pairs in a complex network. The size of the network it can handle is only limited by the size of the memory of the computer and available computer time.
INTRODUCTION
There is a need to understand the relation between the commuters’ departure time from home and their desired arrival times at their destination. Not only will this understanding assist the traffic planner in assessment of the effect of varying work start times on traffic congestion, but it will also improve the understanding of how commuters dynamically respond to changes in the transport system such as changes in roadway capacity. Several models have been developed with the hope of using them for estimating how commuters select their departure times. Most of the models are limited to one origin-destination pair examples. A review of these models can be found in Alfa (1986). There have been recent advancements in how this problem has been approached (Mahmassani and Chang, 1986, 1987). Their 1987 paper is a significant improvement over the pre-1986 literature in that it considered several origins, but it was limited to only one destination. Ben-Akiva, de Palma, and Kanaroglan (1986) did consider only one origin-destination pair with several distinct parallel routes. Newell (1987) considered the situation involving several classes of travellers who attach different cost parameters. This is a significant step forward in looking at this problem. His model was, however, still limited to the one origin-destination case. Most real-life situations involve several origin-destination pairs with multiple interconnecting and overlapping routes. None of the existing models can be used to study such realistic systems. This paper presents a heuristic approach that can be used to analyze a practical network. It is based on the incremental approach. Practical solution of the problem receives more attention in this paper than theoretical elegance. THE
MODEL
Definitions Consider a network G = {N. L}, where N is the set of nodes in the network and L is the set of links formed by pairs of nodes. Define a set of origin/destination nodes NP such that NP C N. Observe the system at equally spaced discrete time epochs sequentially numbered 0, 1, 2, . . . , k. Let d,,(n) be the demand for travel from origin r to destination s at time epoch n(r, s E Np). Further define a three-dimensional matrix D, with [D],$,, = d,,(n) and a two-dimensional matrix Q with [Q],, = d,, = X,, d,,(n). Because travel demand is time-dependent, traffic flow and travel time along a link will also be time-dependent. Let xii(n) be the traffic flow along link ij(i, j E L) at time epoch n. Define a three-dimensional matrix X such that (Xlic = xii(n). Also, let tij(n) IThis research was supported by Grant No. A6584 from the National Science and Engineering Research Council of Canada. Part of this paper was included in the International Conference on Roads and Road Transport Problems, Delhi, December 1988. 337
338
A. SULE ALFA
be the travel time along link ij at time epoch n and define a three-dimensional matrix T such that [T];,, = t,,(n). For each link ij, let S,j and b;j be the associated capacity and “zero-flow” travel time, respectively. Let S and B be the set of capacities and “zero-flow” travel times associated with L. Each traveiler in the system selects his or her departure time from home and route for travel according to some set strategies. Travel time in the network is a function of flows along the links in the network, associated capacities, and zero-flow travel times. Hence, T =
f&X, S, B).
(1)
Flows along the links in the network, on the other hand, are a function of temporal and spatial distribution of demand, travel time along the links, and route selection strategies. Hence.
X = fx(D, ‘0
(2)
Temporal distribution of demand is a function of total demand between origin-destination pairs, travel times along links, and departure time and route selection strategies. Hence,
D = fdQ3 T). Equations (l), (2), and (3) are the system equations describing the system. If we assume that modal choice by commuter is a long-term decision-making process, while departure time and route selection process are short-term, then we can assume that mode split is fixed. Hence, the demand D is the auto trips in which case Q, the total trip interchange matrix, is fixed. An unanswered question is whether there is an equilibrium situation in which there exists equilibrium temporal distribution of demand D”, flow X*, and travel times T*, such that one may substitute eqn (3) into eqn (2) and the resulting equation into eqn (1) to obtain
T* = fdfx[f,,(Q,
T*), T*], S, Bl,
which can be simplified to
‘I’* = fa(Q, T*,
S,
B).
(44
The existence of such an equilibrium in a complex network has not been proven theoretically, and is very difficult to prove. Taking a practical point of view, one may then ask whether it is conceptually reasonable to assume existence of such an equilibrium. It has been assumed by some researchers (Leonard, Tough, and Baguley, 1978; Yagar, 1971; Alfa, 1987) that if D is fixed, then there exist equilibrium flow X* and travel time T* that satisfy the relationship T* = fb(D, T*, S, B,). Examples in their reported hypothetical examples supported the assumption. However, if D is not fixed, i.e. if commuters keep changing their departure times, then the problem has a new dimension. It is not known if there exists an equilibrium, and, even if it exists, its uniqueness can not be assumed. The results of a recent empirical study by Alfa and Eden (1988) showed that the day-to-day departure times of individual commuter studied varies over the 40 days monitored. Fourteen individual commuters were monitored and only for two of them were the variations insignificant. In this paper, we assume that the departure time for a commuter does not necessarily remain the same for each day. It varies from day to day. We therefore study the departure rates on this basis.
Commuter traffic during peak period
339
In solving this problem, we need to know the functional relationships that describe eqns (l), (2), and (3). The exact relationships describing eqns (1) and (2) are documented in Alfa (1987). The functional form for eqn (3) will now be described. Departure
time selection
Consider a traveller going from origin r to destination s. Let 7l be his or her desired earliest time of arrival at the destination, and let 12 be the desired latest arrival time at the destination. We assume that part of the traveller’s objective is to arrive at any time between T, and T?. If this traveller departs from r at time n and r:(n) is the travel time at time n along the minimum time path, then he or she will arrive at destination s at time n + t:(n). If n + t:(n) < T,, this traveller is early by T, - n - t:(n), and would attach a cost C,(T, - n - r:(n)). On the other hand, if n + t:(n) > 72 then the traveller is late by n + t:(n) - T2 and would attach a cost C,(n + t:(n) - T?), There is no cost associated with arriving between TV and TV. The traveller would, however, also attach a cost C,(lz(n)) to the time spent in the system. The total cost associated with this travelier’s decision is C(n, rs, TV, T*) and iS given as C(n,
?-s’s,
71,
72) = c&z(n))
+ cc(T~- n - t:(n))+
+ C,(n + tl:(n)
-
Tz)+ y
(9
where (e)+ = max(O, e). By assuming that a traveller chooses to depart from origin r at time n and travel along the shortest path if, and only if, departing at any time other than time n would result in larger total cost, we state that if there exists a deterministic user equilibrium then
ccn,
rs,
71,
72)
=
c;
if &(n,
TV, T?) > 0 and
C(n, rs,
71,
72)
>
c;
if &(n,
TV,
T?)
=
0; 0 C C C =:,
(6)
where &(n, T,, TV) is the number of travellers who wish to arrive at s at a time between and T? and who choose to depart from r at time n, where
TV
d,,(n)
= c 2 d&, TI ‘2
71, 7~) and d,, = 2 ddn). n
However, the existence of a deterministic user equilibrium has not been proven. Existence and uniqueness of a deterministic user equilibrium have been established only for the simple network case with only one origin-destination pair (see Smith, 1984; Daganzo, 1985), and not for the multiple origin-destination pairs with a complex network. What makes it difficult to study the problem of a complex network is how to estimate the travel times the travellers would experience during their travel. We get around this problem by analyzing the system on a day-to-day basis and assuming that the travellers base their decisions for each day on previous-day information. We therefore do not need to know the travel time information for the day of analysis, but only for the previous day. In this case, we can modify the decision-making process. Let Q(n) be the travel time at epoch n from origin r to destination s on the yth day. We assume that for the (y + 1)th day, the traveller assumes that the travel times are about the same as they were on the yth day, and therefore bases his or her decision on that. After the (y + 1)th day, the traveller may discover that the travel times were not the same as the yth day, and therefore updates his or her knowledge. Let C?+,(n, rs, TV, T?) be the traveller’s anticipated total cost for day y + 1 if he or she departs from origin r at time n. Then C,+,(n,
rs, 71, 72) = C&,:Y(n))
+ Cd71 -
n -
tl:y(n))’
+ C,(n + tzy(n)
-
R)+. G-4
340
A. SULE ALFA
Hence,
where d;,?‘(n) = 2 c d&+l(n, 71, T?) and d,, = 2 d;;‘(n). n
11 ‘2
d;;*(n) is the travel demand between origin r and destination s at time n on the (y + 1)th day, and d;,+‘(n, 71, TJ is the number of this demand that departed at time n. Our interest in this paper is thus to investigate how d;,(n) changes from day to day. An algorithm based on the incremental principle is presented for analyzing this problem. Day-to-day decision process
The total demand between origin r and destination s is d,,. Let drr(T1, TJ be the travellers among d,, who wish to arrive at destination s at a time between times 7l and TV. Hence,
2
c drs(T,r72) =
TI
4,
‘I
and
2
dX(n, 71, 72) = d,(Tl,
n
72).
v Y.
c&?l,
I-S,
Let ‘%&(T,,
Tz,
n)
=
1 { 0
if C&Z,
TS,
71,
T?)
<
71,
71);
v
f2 +
m
otherwise.
Then
If we further define t:;(w) as the time at which a vehicle starts from node r on day y if it is to arrive at node j at time w, and further define the following parameters, 1
0
if link ij is along the minimum time path of the origin-destination pair rs, starting from r at time n, on day y otherwise,
and Atw =
1 0
if t:/(w) = n otherwise,
where Ai, = 1. Then the traffic flow along link ij at time n on day y, x;;(m) is evaluated as
The travel times for each link for each day and minimum time paths can be obtained as in Alfa (1987), which are then used to obtain the anticipated total costs CY+,(. . .)
Commuter traffic during peak period
341
for the (y + 1)th day. The departure rates of travellers for the (y + 1)th day are then evaluated using eqn (7). There are complexities involved with evaluating eqns (7) and (8). For eqn (7). it is difficult to estimate dX(n, T], T?) from eqn (6a). The difficulties with eqn (5) have been discussed in Alfa (1987). For these reasons, only a heuristic procedure is used to evaluate them. The heuristic is based on an incremental approach, and the algorithm is as follows. The algorithm
The algorithm divides the demand &(T,, TJ into small equal groups and assume that these groups travel together. It then determines the optimal departure time for the first group for all origin-destination pairs and for all destination target time pairs. It then uses the result in Alfa (1987) to assign this intermediate time varying demands to appropriate routes. The travel times in the network are updated and then used to decide the departure times of the next groups. These are added to the intermediate time-varying demands and another assignment comes out with the updated time-varying demands. This process is continued until all the groups have been considered. At this point the result becomes a representation of one day process (or one iteration). Using the latest information another day’s process is repeated. A detailed outline of the algorithm is given below. Divide the demand &(T,, TJ into small equal groups of size
Set N = maximum number of days to be studied. Set y := 1. Step I
(i) Assume no flows in the network, i.e. x;;(m) = 0, V i, j, m. Compute the travel times t;(n), V n for all links. Set all demands equal to zero, i.e. dX(n) = 0, V r, s, n. (ii) Find the minimum time paths for all origin-destination pairs, rs, and for all starting time, n (using the latest travel times calculated). (iii) For each origin-destination pair, rs, and each pair of desired intervals of time of arrival at s, T, and r?, find the time n with the minimum total cost, C?(n, rs, T,, T?), such that cx&(~,, T?) = 1. If there is more than one time epoch with equally minimum total cost, choose the first one (as an alternative, any of the time epochs may be selected at random, but it was not done that way in this paper). (iv) If d,,(T,, T? ) > 0, then d,,(T,, 72) : = d,,(T,, 71) -
Ad,, (719 Tz).
Otherwise.
d,(q, 72) : = d,s(T,,Tz). (v) Calculate
Step 2 Let d,(n)
:= d&(n). Divide the demand d,s(n) into small equal groups of size Ad,, (n); 0 < Ad,, (n) 4&r), V r, s, n.
(i) Assume no flows in the network, i.e. x$(m) = 0, V i, j. m. Compute the travel times t;;(n). V n for all links. (ii) Find the minimum time paths for all origin-destination pairs and for all starting times (using the latest travel times calculated).
A. SULE ALFA
342
bij
hj - ‘Zeroflow’ travel time for link ij
(sij 1
sij - Capacity
of link ij
Fig. 1 Example network. (iii) For each origin-destination pair r, s and each starting time n, if d_(n) > 0, assign Ad,, (n) to its shortest path. Otherwise, do not assign any traffic to the route. (iv) If d,,(n) > 0, d,(n) := d,,(n) - Ad, (n), calculate
x$(m) := x;(m)
+ 2 c 2 Ad,3 (n)G~.,(n)A;,,, n=o Vr VI
for all i, j, m, using latest travel time calculated. Otherwise, GO TO Step 3. (v) Using xt(m) as current flows in the network, calculate new travel times t;(n), V i, j, n, using the equation in Alfa (1987). (vi) If d,,(n) > 0, V r, s, n, GO TO (ii) in Step 2. Step 3 If &(rr, ~1) > 0, V r, s, T~, -r2, GO TO Step 1 (ii). Otherwise, dX(n), x$(m) and t$(n) V i, j, n, m are the desired results for day y. If y = N, STOP. Otherwise, set y : = y + 1, set d$(n) := 0, V r, s, n, and return to Step 1 (ii). This algorithm is now used with a hypothetical example to study how the departure process changes from day to day. Hypothetical example Consider the five-node network shown in Fig. 1, which has four origin-destination pairs ((1, 3), (1, 4), (2, 3) and (2, 4)). The origin-destination demand based on desired Table 1. Day-to-day departure rates for O-D pair (1, 4) for 10 days Number of departures d;.,(n)
‘(n)
1
2
3
4
5
6
7
8
9
10
Average number of departure
1 :
0 0
0 0
0 0
0 0
0 0
0 0
0 0
8
0 0
0 0
0.0 0.0
4
3
2
Y
;
2
20
;
:
:
2
!!
; 7 8
0 0 0
01 0 0
0 0 0
01 0 0
1 : 0
t!l 0 0
0
t!l 0 0
01 0 0
01 0 0
0:7 0.0 0.0 0.0
Day (iteration) no. - (y)
Time eooch
;
Commuter traffic during peak period
343
Table 2. Day-to-day departure rates for O-D pair (2, 3) for 10 days Number of departures di.,(rt) Time epoch (n)
Day (iteration) no. - (y) 1
2
0 0
0
1 1 1 8 0
3
4
5
8
0 0
0 0
0 0
1
2
: 1
;
8 0
; 0
: 0 1 0
: 0 0 0
arrival times at destination
6
7
8
0
0
0 2 0 1 0 0 0
0 0 1 0 2 0 0
8 1 :, 0 1 0
9
10
ii
0 0
2 1 0 0 0 0
2 0 1 0 0 0
Average number of departure 0.0 0.0 1.1 0.9 0.4 0.4 0.2 0.0
are given as follows: d,.,(6. c&,(6. &(6. d,,(6.
8) 8) 8) 8)
= = = =
&(7. 9) = 2 c&(7. 9) = 0 &(7. 9) = 1 dJ7. 9) = 4
3. 3. 2, 0,
For simplicity, let the costs attached to earliness, lateness, and delays be simple rates, i.e. C,(i) = C, x (i), C,(i) = C, x (i) and C,(i) = C, x (i) and choose C, = 1, C, = 2, and C, = 5 with units of cost, such as dollars. This example was analyzed for the first 10 days (or 10 iterations). The resulting departure process from day-to-day for some selected origin-destination pairs and travel times for selected links for each time epoch are shown in Tables 1, 2, 3, and 4. There does not seem to be an indication of a unique value for departure rate for each time epoch for the 10 days, except where the departure rate is zero. The same is noticed for travel times on the links. For this example, there is no evidence of steady state occurring after 10 days. The question then is whether there is a steady state in real life, or not. If there is steady state, one may then ask whether the lack of convergence after 10 days’ analysis is because 10 days is too short for stability or maybe the model presented is not a representation of the happenings in real life. If, on the other hand, there is no steady state, which the author has an inclination to believe, then we should be studying the average or expected departure rate at each time epoch. In this case, we should be taking an average of the 10 days observations. The assumption of nonexistence of steady state is supported by the results of the empirical study by Alfa and Eden (1988). However, one may assume that the departure rate has a probability distribution, and this distribution may be stable. Hence, there may exist a stochastic equilibrium of departure process. The expected departure rates for the selected origin-destination pairs and expected travel times for the selected links are shown in Tables 1, 2, 3, and 4. Table 3. Day-to-day travel times for link (1, 2) for 10 days Travel times I;;(II) Time epoch (n)
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
1 2 2 2 2 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 2 1 1 1 1 1
1 2 2 2 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1 1 1
1 2 2 2 1 1 1 1 1 1
1 I 1 1 1 1 1 1 1 1
1 2
Day (iteration) no. - (y)
2’ 3 2 1 1 1 1 1
: 2 1 1 1 1 1
Average travel time 1.0 1.5 1.5 1.7 1.4 1.0 1.0 1.0 1.0 1.0
A. SULE ALFA
344
Table 4. Day-to-day travel times for link (2. 5) for 10 days Travel times r:;(n)
1
2
3
4
5
6
7
8
9
10
Average travel time
: 3 4
1 2 i
1 1 :
1 1 2
1 1 2
1 1 2
1 3 2
1 1 2
1 1 2
1 1 2
1 3 2
1.0 1.5 1.9
:
1 1 1 1 1
1 2 I 1 I
3 2 1 1 1 1
3 2 1 I 1
3 2 2 2 1 1
21 1 1 1 1
23 1 1 1 1
3 2 1 1 1
32 2 I 1
21 1 1
2.5 1.8 1.5 1.2 1.0 1.0
Time epoch (n)
7
8 9 10
Day (iteration) no. - (y)
I I
CONCLUSION
The model presented in this paper can be used to simulate commuters’ departure rate from home and route selection during the peak period as function of network characteristics, desired arrival times at the destinations, perceived costs associated with late/early arrivals at destinations, and travel time plus delays. The algorithm is based on heuristic approach which uses incremental approach. It does not dwell on existence or nonexistence of steady state but simulates the system for many days and then averages the results. The advantage of this mode1 is that, given the true cost parameters, it can be used in its present form for analyzing a realistic network which none of the other models can do at the moment. The next aspect of this research is to determine the cost parameters associated with late/early arrivals at the destinations and cost of travel time and delays. One of the weaknesses of this model is related to how to decide on the number of days for which the system is to be simulated. This weakness will be investigated in due course by the author. Acknowledgemenf-The author wishes to acknowledge Ruth Eden and Derek Durant, who assisted with running of the computer programs, and Tim Gopaul, who assisted with writing of the computer program. REFERENCES Alfa A. S. (1986) A review of models for the temporal distribution of peak traffic demand. Transpn. Res., 20B. 491-499. Alfa A. S. (1987) Deterministic model for dynamic assignment of time-varying demand. Civil Engng. Sysreems, 4, 191-198. Alfa A. S. and Eden R. J. (1988) Variability in commuter departure times: An empirical study. In Proceedings of the 1988 Annual Canadian Society for Civil Engineering Conference, 561-582. Calgary, May 23-27. 1988. Canadian Society for Civil Engineering. Ben-Akiva M., de Palma A. and Kanaroglan P. (1986) Dynamic model of peak period traffic congestion with elastic arrival rates. Trunspn. Sci., 20. 164-181. Daganzo C. F. (1985) The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transpn. Sci.. 19, 29-37. Leonard D. R., Tough J. B. and Baguley P. C. (1978) CONTRAM: A traffic assignment model for predicting flows and queues during peak period. Transport and Road Research Laboratory Report. LR 841. TRRL, Crowthorne. Mahmassani H. S. and Chang G.-L. (1986) Dynamic aspects of departure-time choice behaviour in a commuting system: Theoretical framework and experimental analysis. Transpn. Res. Rec., 1037. 88-101. Mahmassani H. S. and Chang G.-L. (1987) On boundedly rational user equilibrium in transportation systems. Transpn. Sci., 21. 89-99.
Newell G. F. (1987) The morning commute for nonidentical travellers. Transpn. Sci., 21, 74-88. Smith M. J. (1984) The existence of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transpn. Sci., 18. 385-394.
Yagar S. (1971) Dynamic traffic assignment by individual path minimization and queueing. Transpn. Res.. 5. 179-196.
No.
5. pp.
337-344.
1989
0191-1615189 $3.00~ .OO ‘0 19R9 Pergamon Press plc
DEPARTURE RATE AND ROUTE COMMUTER TRAFFIC DURING
ASSIGNMENT OF PEAK PERIOD?
A~AHIRU SULE ALFA Transport Institute and Department of Civil Engineering, University of Manitoba, Winnipeg, Manitoba R3T 2N2. Canada (Received 16 November
1987; in revised form 29 November 1988)
Abstract-An incremental heuristic is presented for estimating the departure rate of commuters in a network. Other results that can be obtained from the model include time-dependent flows of traffic and travel times for each link in the network. The model is able to handle multiple origin-destination pairs in a complex network. The size of the network it can handle is only limited by the size of the memory of the computer and available computer time.
INTRODUCTION
There is a need to understand the relation between the commuters’ departure time from home and their desired arrival times at their destination. Not only will this understanding assist the traffic planner in assessment of the effect of varying work start times on traffic congestion, but it will also improve the understanding of how commuters dynamically respond to changes in the transport system such as changes in roadway capacity. Several models have been developed with the hope of using them for estimating how commuters select their departure times. Most of the models are limited to one origin-destination pair examples. A review of these models can be found in Alfa (1986). There have been recent advancements in how this problem has been approached (Mahmassani and Chang, 1986, 1987). Their 1987 paper is a significant improvement over the pre-1986 literature in that it considered several origins, but it was limited to only one destination. Ben-Akiva, de Palma, and Kanaroglan (1986) did consider only one origin-destination pair with several distinct parallel routes. Newell (1987) considered the situation involving several classes of travellers who attach different cost parameters. This is a significant step forward in looking at this problem. His model was, however, still limited to the one origin-destination case. Most real-life situations involve several origin-destination pairs with multiple interconnecting and overlapping routes. None of the existing models can be used to study such realistic systems. This paper presents a heuristic approach that can be used to analyze a practical network. It is based on the incremental approach. Practical solution of the problem receives more attention in this paper than theoretical elegance. THE
MODEL
Definitions Consider a network G = {N. L}, where N is the set of nodes in the network and L is the set of links formed by pairs of nodes. Define a set of origin/destination nodes NP such that NP C N. Observe the system at equally spaced discrete time epochs sequentially numbered 0, 1, 2, . . . , k. Let d,,(n) be the demand for travel from origin r to destination s at time epoch n(r, s E Np). Further define a three-dimensional matrix D, with [D],$,, = d,,(n) and a two-dimensional matrix Q with [Q],, = d,, = X,, d,,(n). Because travel demand is time-dependent, traffic flow and travel time along a link will also be time-dependent. Let xii(n) be the traffic flow along link ij(i, j E L) at time epoch n. Define a three-dimensional matrix X such that (Xlic = xii(n). Also, let tij(n) IThis research was supported by Grant No. A6584 from the National Science and Engineering Research Council of Canada. Part of this paper was included in the International Conference on Roads and Road Transport Problems, Delhi, December 1988. 337
338
A. SULE ALFA
be the travel time along link ij at time epoch n and define a three-dimensional matrix T such that [T];,, = t,,(n). For each link ij, let S,j and b;j be the associated capacity and “zero-flow” travel time, respectively. Let S and B be the set of capacities and “zero-flow” travel times associated with L. Each traveiler in the system selects his or her departure time from home and route for travel according to some set strategies. Travel time in the network is a function of flows along the links in the network, associated capacities, and zero-flow travel times. Hence, T =
f&X, S, B).
(1)
Flows along the links in the network, on the other hand, are a function of temporal and spatial distribution of demand, travel time along the links, and route selection strategies. Hence.
X = fx(D, ‘0
(2)
Temporal distribution of demand is a function of total demand between origin-destination pairs, travel times along links, and departure time and route selection strategies. Hence,
D = fdQ3 T). Equations (l), (2), and (3) are the system equations describing the system. If we assume that modal choice by commuter is a long-term decision-making process, while departure time and route selection process are short-term, then we can assume that mode split is fixed. Hence, the demand D is the auto trips in which case Q, the total trip interchange matrix, is fixed. An unanswered question is whether there is an equilibrium situation in which there exists equilibrium temporal distribution of demand D”, flow X*, and travel times T*, such that one may substitute eqn (3) into eqn (2) and the resulting equation into eqn (1) to obtain
T* = fdfx[f,,(Q,
T*), T*], S, Bl,
which can be simplified to
‘I’* = fa(Q, T*,
S,
B).
(44
The existence of such an equilibrium in a complex network has not been proven theoretically, and is very difficult to prove. Taking a practical point of view, one may then ask whether it is conceptually reasonable to assume existence of such an equilibrium. It has been assumed by some researchers (Leonard, Tough, and Baguley, 1978; Yagar, 1971; Alfa, 1987) that if D is fixed, then there exist equilibrium flow X* and travel time T* that satisfy the relationship T* = fb(D, T*, S, B,). Examples in their reported hypothetical examples supported the assumption. However, if D is not fixed, i.e. if commuters keep changing their departure times, then the problem has a new dimension. It is not known if there exists an equilibrium, and, even if it exists, its uniqueness can not be assumed. The results of a recent empirical study by Alfa and Eden (1988) showed that the day-to-day departure times of individual commuter studied varies over the 40 days monitored. Fourteen individual commuters were monitored and only for two of them were the variations insignificant. In this paper, we assume that the departure time for a commuter does not necessarily remain the same for each day. It varies from day to day. We therefore study the departure rates on this basis.
Commuter traffic during peak period
339
In solving this problem, we need to know the functional relationships that describe eqns (l), (2), and (3). The exact relationships describing eqns (1) and (2) are documented in Alfa (1987). The functional form for eqn (3) will now be described. Departure
time selection
Consider a traveller going from origin r to destination s. Let 7l be his or her desired earliest time of arrival at the destination, and let 12 be the desired latest arrival time at the destination. We assume that part of the traveller’s objective is to arrive at any time between T, and T?. If this traveller departs from r at time n and r:(n) is the travel time at time n along the minimum time path, then he or she will arrive at destination s at time n + t:(n). If n + t:(n) < T,, this traveller is early by T, - n - t:(n), and would attach a cost C,(T, - n - r:(n)). On the other hand, if n + t:(n) > 72 then the traveller is late by n + t:(n) - T2 and would attach a cost C,(n + t:(n) - T?), There is no cost associated with arriving between TV and TV. The traveller would, however, also attach a cost C,(lz(n)) to the time spent in the system. The total cost associated with this travelier’s decision is C(n, rs, TV, T*) and iS given as C(n,
?-s’s,
71,
72) = c&z(n))
+ cc(T~- n - t:(n))+
+ C,(n + tl:(n)
-
Tz)+ y
(9
where (e)+ = max(O, e). By assuming that a traveller chooses to depart from origin r at time n and travel along the shortest path if, and only if, departing at any time other than time n would result in larger total cost, we state that if there exists a deterministic user equilibrium then
ccn,
rs,
71,
72)
=
c;
if &(n,
TV, T?) > 0 and
C(n, rs,
71,
72)
>
c;
if &(n,
TV,
T?)
=
0; 0 C C C =:,
(6)
where &(n, T,, TV) is the number of travellers who wish to arrive at s at a time between and T? and who choose to depart from r at time n, where
TV
d,,(n)
= c 2 d&, TI ‘2
71, 7~) and d,, = 2 ddn). n
However, the existence of a deterministic user equilibrium has not been proven. Existence and uniqueness of a deterministic user equilibrium have been established only for the simple network case with only one origin-destination pair (see Smith, 1984; Daganzo, 1985), and not for the multiple origin-destination pairs with a complex network. What makes it difficult to study the problem of a complex network is how to estimate the travel times the travellers would experience during their travel. We get around this problem by analyzing the system on a day-to-day basis and assuming that the travellers base their decisions for each day on previous-day information. We therefore do not need to know the travel time information for the day of analysis, but only for the previous day. In this case, we can modify the decision-making process. Let Q(n) be the travel time at epoch n from origin r to destination s on the yth day. We assume that for the (y + 1)th day, the traveller assumes that the travel times are about the same as they were on the yth day, and therefore bases his or her decision on that. After the (y + 1)th day, the traveller may discover that the travel times were not the same as the yth day, and therefore updates his or her knowledge. Let C?+,(n, rs, TV, T?) be the traveller’s anticipated total cost for day y + 1 if he or she departs from origin r at time n. Then C,+,(n,
rs, 71, 72) = C&,:Y(n))
+ Cd71 -
n -
tl:y(n))’
+ C,(n + tzy(n)
-
R)+. G-4
340
A. SULE ALFA
Hence,
where d;,?‘(n) = 2 c d&+l(n, 71, T?) and d,, = 2 d;;‘(n). n
11 ‘2
d;;*(n) is the travel demand between origin r and destination s at time n on the (y + 1)th day, and d;,+‘(n, 71, TJ is the number of this demand that departed at time n. Our interest in this paper is thus to investigate how d;,(n) changes from day to day. An algorithm based on the incremental principle is presented for analyzing this problem. Day-to-day decision process
The total demand between origin r and destination s is d,,. Let drr(T1, TJ be the travellers among d,, who wish to arrive at destination s at a time between times 7l and TV. Hence,
2
c drs(T,r72) =
TI
4,
‘I
and
2
dX(n, 71, 72) = d,(Tl,
n
72).
v Y.
c&?l,
I-S,
Let ‘%&(T,,
Tz,
n)
=
1 { 0
if C&Z,
TS,
71,
T?)
<
71,
71);
v
f2 +
m
otherwise.
Then
If we further define t:;(w) as the time at which a vehicle starts from node r on day y if it is to arrive at node j at time w, and further define the following parameters, 1
0
if link ij is along the minimum time path of the origin-destination pair rs, starting from r at time n, on day y otherwise,
and Atw =
1 0
if t:/(w) = n otherwise,
where Ai, = 1. Then the traffic flow along link ij at time n on day y, x;;(m) is evaluated as
The travel times for each link for each day and minimum time paths can be obtained as in Alfa (1987), which are then used to obtain the anticipated total costs CY+,(. . .)
Commuter traffic during peak period
341
for the (y + 1)th day. The departure rates of travellers for the (y + 1)th day are then evaluated using eqn (7). There are complexities involved with evaluating eqns (7) and (8). For eqn (7). it is difficult to estimate dX(n, T], T?) from eqn (6a). The difficulties with eqn (5) have been discussed in Alfa (1987). For these reasons, only a heuristic procedure is used to evaluate them. The heuristic is based on an incremental approach, and the algorithm is as follows. The algorithm
The algorithm divides the demand &(T,, TJ into small equal groups and assume that these groups travel together. It then determines the optimal departure time for the first group for all origin-destination pairs and for all destination target time pairs. It then uses the result in Alfa (1987) to assign this intermediate time varying demands to appropriate routes. The travel times in the network are updated and then used to decide the departure times of the next groups. These are added to the intermediate time-varying demands and another assignment comes out with the updated time-varying demands. This process is continued until all the groups have been considered. At this point the result becomes a representation of one day process (or one iteration). Using the latest information another day’s process is repeated. A detailed outline of the algorithm is given below. Divide the demand &(T,, TJ into small equal groups of size
Set N = maximum number of days to be studied. Set y := 1. Step I
(i) Assume no flows in the network, i.e. x;;(m) = 0, V i, j, m. Compute the travel times t;(n), V n for all links. Set all demands equal to zero, i.e. dX(n) = 0, V r, s, n. (ii) Find the minimum time paths for all origin-destination pairs, rs, and for all starting time, n (using the latest travel times calculated). (iii) For each origin-destination pair, rs, and each pair of desired intervals of time of arrival at s, T, and r?, find the time n with the minimum total cost, C?(n, rs, T,, T?), such that cx&(~,, T?) = 1. If there is more than one time epoch with equally minimum total cost, choose the first one (as an alternative, any of the time epochs may be selected at random, but it was not done that way in this paper). (iv) If d,,(T,, T? ) > 0, then d,,(T,, 72) : = d,,(T,, 71) -
Ad,, (719 Tz).
Otherwise.
d,(q, 72) : = d,s(T,,Tz). (v) Calculate
Step 2 Let d,(n)
:= d&(n). Divide the demand d,s(n) into small equal groups of size Ad,, (n); 0 < Ad,, (n) 4&r), V r, s, n.
(i) Assume no flows in the network, i.e. x$(m) = 0, V i, j. m. Compute the travel times t;;(n). V n for all links. (ii) Find the minimum time paths for all origin-destination pairs and for all starting times (using the latest travel times calculated).
A. SULE ALFA
342
bij
hj - ‘Zeroflow’ travel time for link ij
(sij 1
sij - Capacity
of link ij
Fig. 1 Example network. (iii) For each origin-destination pair r, s and each starting time n, if d_(n) > 0, assign Ad,, (n) to its shortest path. Otherwise, do not assign any traffic to the route. (iv) If d,,(n) > 0, d,(n) := d,,(n) - Ad, (n), calculate
x$(m) := x;(m)
+ 2 c 2 Ad,3 (n)G~.,(n)A;,,, n=o Vr VI
for all i, j, m, using latest travel time calculated. Otherwise, GO TO Step 3. (v) Using xt(m) as current flows in the network, calculate new travel times t;(n), V i, j, n, using the equation in Alfa (1987). (vi) If d,,(n) > 0, V r, s, n, GO TO (ii) in Step 2. Step 3 If &(rr, ~1) > 0, V r, s, T~, -r2, GO TO Step 1 (ii). Otherwise, dX(n), x$(m) and t$(n) V i, j, n, m are the desired results for day y. If y = N, STOP. Otherwise, set y : = y + 1, set d$(n) := 0, V r, s, n, and return to Step 1 (ii). This algorithm is now used with a hypothetical example to study how the departure process changes from day to day. Hypothetical example Consider the five-node network shown in Fig. 1, which has four origin-destination pairs ((1, 3), (1, 4), (2, 3) and (2, 4)). The origin-destination demand based on desired Table 1. Day-to-day departure rates for O-D pair (1, 4) for 10 days Number of departures d;.,(n)
‘(n)
1
2
3
4
5
6
7
8
9
10
Average number of departure
1 :
0 0
0 0
0 0
0 0
0 0
0 0
0 0
8
0 0
0 0
0.0 0.0
4
3
2
Y
;
2
20
;
:
:
2
!!
; 7 8
0 0 0
01 0 0
0 0 0
01 0 0
1 : 0
t!l 0 0
0
t!l 0 0
01 0 0
01 0 0
0:7 0.0 0.0 0.0
Day (iteration) no. - (y)
Time eooch
;
Commuter traffic during peak period
343
Table 2. Day-to-day departure rates for O-D pair (2, 3) for 10 days Number of departures di.,(rt) Time epoch (n)
Day (iteration) no. - (y) 1
2
0 0
0
1 1 1 8 0
3
4
5
8
0 0
0 0
0 0
1
2
: 1
;
8 0
; 0
: 0 1 0
: 0 0 0
arrival times at destination
6
7
8
0
0
0 2 0 1 0 0 0
0 0 1 0 2 0 0
8 1 :, 0 1 0
9
10
ii
0 0
2 1 0 0 0 0
2 0 1 0 0 0
Average number of departure 0.0 0.0 1.1 0.9 0.4 0.4 0.2 0.0
are given as follows: d,.,(6. c&,(6. &(6. d,,(6.
8) 8) 8) 8)
= = = =
&(7. 9) = 2 c&(7. 9) = 0 &(7. 9) = 1 dJ7. 9) = 4
3. 3. 2, 0,
For simplicity, let the costs attached to earliness, lateness, and delays be simple rates, i.e. C,(i) = C, x (i), C,(i) = C, x (i) and C,(i) = C, x (i) and choose C, = 1, C, = 2, and C, = 5 with units of cost, such as dollars. This example was analyzed for the first 10 days (or 10 iterations). The resulting departure process from day-to-day for some selected origin-destination pairs and travel times for selected links for each time epoch are shown in Tables 1, 2, 3, and 4. There does not seem to be an indication of a unique value for departure rate for each time epoch for the 10 days, except where the departure rate is zero. The same is noticed for travel times on the links. For this example, there is no evidence of steady state occurring after 10 days. The question then is whether there is a steady state in real life, or not. If there is steady state, one may then ask whether the lack of convergence after 10 days’ analysis is because 10 days is too short for stability or maybe the model presented is not a representation of the happenings in real life. If, on the other hand, there is no steady state, which the author has an inclination to believe, then we should be studying the average or expected departure rate at each time epoch. In this case, we should be taking an average of the 10 days observations. The assumption of nonexistence of steady state is supported by the results of the empirical study by Alfa and Eden (1988). However, one may assume that the departure rate has a probability distribution, and this distribution may be stable. Hence, there may exist a stochastic equilibrium of departure process. The expected departure rates for the selected origin-destination pairs and expected travel times for the selected links are shown in Tables 1, 2, 3, and 4. Table 3. Day-to-day travel times for link (1, 2) for 10 days Travel times I;;(II) Time epoch (n)
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
1 2 2 2 2 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 2 1 1 1 1 1
1 2 2 2 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1
1 1 1 1 1 1 1 1 1 1
1 2 2 2 1 1 1 1 1 1
1 I 1 1 1 1 1 1 1 1
1 2
Day (iteration) no. - (y)
2’ 3 2 1 1 1 1 1
: 2 1 1 1 1 1
Average travel time 1.0 1.5 1.5 1.7 1.4 1.0 1.0 1.0 1.0 1.0
A. SULE ALFA
344
Table 4. Day-to-day travel times for link (2. 5) for 10 days Travel times r:;(n)
1
2
3
4
5
6
7
8
9
10
Average travel time
: 3 4
1 2 i
1 1 :
1 1 2
1 1 2
1 1 2
1 3 2
1 1 2
1 1 2
1 1 2
1 3 2
1.0 1.5 1.9
:
1 1 1 1 1
1 2 I 1 I
3 2 1 1 1 1
3 2 1 I 1
3 2 2 2 1 1
21 1 1 1 1
23 1 1 1 1
3 2 1 1 1
32 2 I 1
21 1 1
2.5 1.8 1.5 1.2 1.0 1.0
Time epoch (n)
7
8 9 10
Day (iteration) no. - (y)
I I
CONCLUSION
The model presented in this paper can be used to simulate commuters’ departure rate from home and route selection during the peak period as function of network characteristics, desired arrival times at the destinations, perceived costs associated with late/early arrivals at destinations, and travel time plus delays. The algorithm is based on heuristic approach which uses incremental approach. It does not dwell on existence or nonexistence of steady state but simulates the system for many days and then averages the results. The advantage of this mode1 is that, given the true cost parameters, it can be used in its present form for analyzing a realistic network which none of the other models can do at the moment. The next aspect of this research is to determine the cost parameters associated with late/early arrivals at the destinations and cost of travel time and delays. One of the weaknesses of this model is related to how to decide on the number of days for which the system is to be simulated. This weakness will be investigated in due course by the author. Acknowledgemenf-The author wishes to acknowledge Ruth Eden and Derek Durant, who assisted with running of the computer programs, and Tim Gopaul, who assisted with writing of the computer program. REFERENCES Alfa A. S. (1986) A review of models for the temporal distribution of peak traffic demand. Transpn. Res., 20B. 491-499. Alfa A. S. (1987) Deterministic model for dynamic assignment of time-varying demand. Civil Engng. Sysreems, 4, 191-198. Alfa A. S. and Eden R. J. (1988) Variability in commuter departure times: An empirical study. In Proceedings of the 1988 Annual Canadian Society for Civil Engineering Conference, 561-582. Calgary, May 23-27. 1988. Canadian Society for Civil Engineering. Ben-Akiva M., de Palma A. and Kanaroglan P. (1986) Dynamic model of peak period traffic congestion with elastic arrival rates. Trunspn. Sci., 20. 164-181. Daganzo C. F. (1985) The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transpn. Sci.. 19, 29-37. Leonard D. R., Tough J. B. and Baguley P. C. (1978) CONTRAM: A traffic assignment model for predicting flows and queues during peak period. Transport and Road Research Laboratory Report. LR 841. TRRL, Crowthorne. Mahmassani H. S. and Chang G.-L. (1986) Dynamic aspects of departure-time choice behaviour in a commuting system: Theoretical framework and experimental analysis. Transpn. Res. Rec., 1037. 88-101. Mahmassani H. S. and Chang G.-L. (1987) On boundedly rational user equilibrium in transportation systems. Transpn. Sci., 21. 89-99.
Newell G. F. (1987) The morning commute for nonidentical travellers. Transpn. Sci., 21, 74-88. Smith M. J. (1984) The existence of a time-dependent equilibrium distribution of arrivals at a single bottleneck. Transpn. Sci., 18. 385-394.
Yagar S. (1971) Dynamic traffic assignment by individual path minimization and queueing. Transpn. Res.. 5. 179-196.