A stochastic process approach to the analysis of temporal dynamics in transportation networks

Tranrpn. Res:6. Vol. 23B. No. I. pp. I-17. Printed in Great Britain.

0191.2615:89 13 00 c .@I L 1989 Pergamon Press plc

1989

A STOCHASTIC PROCESS APPROACH TO THE ANALYSIS OF TEMPORAL DYNAMICS IN TRANSPORTATION NETWORKS Department

of Transportation

ENNIO CASCETTA Engineering, Universita di Napoli, Via Claudio, 21, Napoli, Italy.

(Received 5 December 1986; in revised form 10 January 1988) Abstract-Equilibrium analyses of transportation networks are by their nature “static,” with equilibrium configuration defined as “fixed” or “autoreflexive” points, i.e. flow patterns reproducing themselves on the basis of the assumptions made on users’ behavior once reached by the system. In this paper it is argued that no transportation system remains in the same state over successive periods because of the action of several causes (e.g. temporal fluctuation of level and composition of demand, users’ choices, and travel costs). This implies that the sequence of states occupied by the system over successive epochs or times of similar characteristics (e.g. AM peak hour of working days) is the realization of a stochastic process, the type of which depends on, among other things, the choice mechanism followed by travelers. Stationarity of the stochastic process within fixed potential demand and network structures is considered to be a desirable property because it allows a flow pattern distribution to be associated to each demand-network system independently of its starting configuration and elapsed time. Furthermore, this stationarity makes it possible to define expected path and link flows and compare them with those of stochastic user equilibrium (SUE). In this paper rather general sufficient conditions for the process stationarity are given, essentially calling for a “stable” choice mechanism of potential users. In the following a particular model of temporal dynamics (STODYN), based upon a number of simplifying assumptions on users’ behavior common to most assignment models, is described. Exact and approximate relationships between STODYN steady-state expected flows and SUE average flows are also analyzed both in the case of unique and multiple equilibria. The possible use of STODYN as an assignment model giving unique average flows along with their variances and covariances is then discussed. The model takes into account stochastic fluctuations of demand and can be easily extended to other “dimensions” such as distribution and modal choice. Some results of an empirical analysis comparing STODYN average flows with SUE and observed flows on two urban car networks are also reported.

1. INTRODUCTION

Most research on transportation networks deals with the definition of equilibrium points and the study of their properties (existence, uniqueness, computational algorithms, etc.). Users’ equilibrium models can be classified as deterministic or stochastic depending on the assumptions made on users’ preception of costs within a utility maximizing or rational type of behavior. Deterministic models assume users endowed with an exact knowledge of alternative path costs while stochastic user equilibrium (SUE) models assume that users choose on the basis of “perceived” costs that are r.v. distributed across the user population, and possibly time, with “true” costs as mean values. Equilibrium analysis is by its nature static. Equilibrium configurations are defined in terms of fixed point conditions, i.e. flow patterns reproducing themselves once reached by the system on the basis of the assumed users’ behavior. This is clearly expressed in the definition of the (stochastic) user equilibrium configuration as a state in which no user can reduce his/her (perceived) travel cost by unilaterally changing route. Equilibrium analyses do not make any assumption on how users gain information about alternative costs or whether an equilibrium state is reached by the system independently of its starting configuration and the users’ information acquisition mechanism. Stochastic equilibrium models virtually ignore random fluctuations in the number of users actually choosing each path under the assumption that origin-destination (O/D) demand flows are large enough (Daganzo and Sheffi, 1977). In many applications, however, hourly demand flows are of few units for most O/D pairs. TR(B)

23:1-A

1

2

E.

CASCElTA

SUE path and link flows can still be interpreted as (time) average values if the equilibrium is unique and average travel costs can be approximated by costs computed with average flows due to the reduced variability of link flows with respect to path flows (see, for instance, Cascetta, 1987). The latter assumption, though acceptable in many cases, can be questioned when temporal fluctuations of link flow are sensible. This can be the case for elastic demand equilibrium models with highly variable demand types (e.g. non-work trips) or when the system oscillates among multiple equilibria. This point will be dealt with in greater detail in Section 5. Some references exist in the literature dealing with the interperiodical (e.g. among different days) dynamics of the demand-network system, both in the context of deterministic (Beckman et al., 1956; Smith, 1979, 1984) and stochastic (Daganzo, 1977; Horowitz, 1984) equilibria. Recently, the problem of dynamic behavior in transportation networks was suggested as an open research opportunity in the NSF Congress on Transportation Research (Friesz, 1985; Horowitz, 1985). The standard approach to system dynamics has been to find conditions that ensure the convergence (or not) to an equilibrium, or, to put it differently, that ensure system stability. In particular, the paper of Horowitz (1984) shows that for a two-link network SUE may not be “stable” under a range of reasonable assumptions on learning mechanisms. A noticeable exception is represented by the paper of Daganzo (1977), which models the system evolution as a Markov chain assuming that path choices in each period depend on costs incurred in the previous period. Daganzo also establishes the coincidence of time-average flows and SUE flows for uncongested networks and, approximately, for large O/D demand flows. In this paper the system dynamics is studied taking into account explicitly stochastic fluctuations to explore the consequences of removing some of the assumptions underlying SUE models as well as the relationships between time average and SUE flows. The premise is that no transportation system remains in the same state? over successive time periods because of many factors, such as temporal fluctuations of costs, of level and composition of demand, and of users’ choices. It is also assumed that the state occupied by the system at any given time is not predictable a priori due to the intrinsic randomness of the above phenomena and that it “depends” on previous states through the dependence of travelers’ choices at time t on costs incurred in the past. The above assumptions imply that the system evolves over successive time periods as a stochastic process, the type of which depends on the particular choice mechanism followed by travelers. This is a distinctive feature of the proposed approach with respect to existing models. Stability is not defined as “the convergence of link flows over time to equilibrium values regardless of the initial condtions” (Horowitz, 1984), because flows are considered variable over time by the “nature” of the underlying processes determining them. In the proposed framework even equilibrium flows that take place at one point in time will not necessarily reproduce themselves in successive epochs. In this context the possible role of SUE flows is that of average values over time. In the first part of this paper sufficient conditions on users’ choice mechanism are given, ensuring that the stochastic process is “dynamically” stable, i.e. it admits a unique steady-state (or stationary) probability of occupying any state independently of the starting configuration. These conditions hold whatever network structure and cost functions are assumed. Existence and uniqueness of a stationary distribution is considered of some importance as this gives the possibility of associating a probability distribution of states and flows to a given demand-network system independently of its past history and starting configuration. tSystem states can be defined differently according to their particular use. In this paper different levels of representation will be used: route choices, path flows, and link flows as described in Section 2.

Temporal dynamics in transportation

3

networks

Furthermore, it Bllows expected path (link) flows to be defined and compared with equililbrium flows, the latter being of no significance in a nonstationary type of system evolution. In the second part of the paper a particular model of system temporal dynamics is proposed based on some assumptions on travelers’ path choice and learning models, comparable with those existing in the literature. Exact and approximate relationships between steady-state expected flows and SUE average flows are also investigated. Finally, the model of system dynamics is considered as an assignment model in its own right giving typical outputs as means, variances, and covariances of link flows with a complexity comparable to that of existing equilibrium models. An algorithmic scheme based on the process ergodicity is proposed and some results relative to its application to the urban car networks of two medium-sized Italian towns are briefly reported. 2. NOTATION

AND

STATEMENT

OF THE

PROBLEM

Let us consider a transport network (N, L, C), consisting of a set N of nodes, a set L of directed links, and a set C of link cost functions. Let NC be the subset of centroid nodes, i.e. nodes in which trips can originate and/ or terminate. Let d be the demand vector whose component d, is the number of (potential) travelers? moving between centroid pair (i, j) in a given reference period. In order to study the evolution of the system over a sequence of times-$ . . . , t 1, t, f + 1, . . . the state occupied by the system in each period must be defined. In this paper states will be defined at different levels of aggregation of individual units. The most basic representation is that which associates to each user the chosen route. In this case the system state is defined by a vector R of the route chosen§ by each potential user moving in the reference period; the uth component being: R(u) = [integer corresponding

to the route chosen by the uth user]

R(u) E I(u),

(2.1)

where Z(u) is the set of “feasible” alternative routes considered by the uth traveler. If RT = (2, 3, 1, . . .), then user 1 chooses route R(1) = 2, user 2 chooses route R(2) = 3, etc. The number of feasible path choice vectors is finite (nR) because both the number of potential users and that of feasible acyclic routes are finite. Let us denote by SR the set of feasible route vectors: SR

=

(RI,

R2,

.

.

.

, RnJ

(2.2)

At a more aggregate level the state occupied by the system can be defined by the path flow vector F, whose ith component is given by: F(i) = [no. of users following path i in the reference period].

(2.3)

If FT = (100, 98, 0, 3, . . .), then the number of users on the first path is F(1) = 100, the number on the second path is F(2) = 98, etc. tThe main emphasis of this paper is on path-related choices; this has led to assume a demand vector of potential users with options relative to whether to make a trip between a prefixed O-D pair or not and to the path to be followed. If the no-trip option is eliminated, as it is usually done in the context of assignment models, then d, represents the average (constant) number of trips between O-D pairs (i, j). Other choice dimensions such as destination and mode could be introduced into the analysis through a hypernetwork representation of the system (Sheffi and Daganzo, 1979). *Times or epochs can have either a “chronological” interpretation as successive reference periods of similar characteristics (e.g. the AM peak period of successive working days) or they can be defined as “fictitious” moments in which users acquire awareness of path attributes and make their choices. OIn the following, simple paths will be assumed as choice alternatives. Extensions of results to the case of multipath (or hyperpath) alternatives, as required by some assignment models for transit networks (Nguyen and Pallottino, 1985; Cascetta and Nuzzolo, 1986b) should be straightforward.

4

E.

CASCETTA

The number of feasible path flow vectors is finite (nf) and let SF.be the set of such vectors: SF = (F,, Fz, . . . , FJ.

(2.4)

A path flow vector Fk can be associated to each path choice vector R,: F,(i) = C,G(R,(u),

i)

Vi

(2.5)

where 6(a) is the Kroneker delta function equal to one when the two arguments are equal, zero otherwise. In general, many choice vectors can be associated to each path flow vector F,; in the following we will denote by IF, the subset of feasible route choice vectors giving rise to the vector Fk. The most aggregate level of system representation considered in this paper is the link flow vector v with components given by: u(i) = [no. of travelers using link i in the reference

period].

(2.6)

Here again the number of feasible link flow vectors is finite (n,), S, denoting the set of such vectors. Path and link flow vectors are related through the link-path incidence matrix A: v = AF.

(2.7)

Let Ivi denote the subset of feasible path flow vectors giving rise to the link flow vector vi. In Fig. 1 a small network is represented with five users moving between a single O/D pair and with a possible configuration of vectors R, F, and v. In the literature the system evolution is usually studied in the path-flow space (Daganzo, 1977; Smith, 1979; Horowitz, 1984). This, however, requires that all users are considered “indistinguishable, ” i.e. endowed with the same information and following the same choice model. In order to obtain sufficient conditions with a wide generality, in this paper the stochastic process will first be studied in the path choice vector space. Stationarity in path and link flow spaces follows directly as will be shown later. Because of several causes such as random fluctuations of level of service attributes (e.g. travel times), random events in the network determining the actual path choice,

PatI7

1 -II

1 2

n.

nodes

Link

1,3,4 1,2,3,4

1 2 3 4

R=

2 2 1 1

Fig. 1. An example of vectors R, F, and v

V’

nodc

n.

1,: 1 a: 2,: 3,’

2 3 :

-[I

Temporal dynamics in transportation

5

networks

variations in the trip/no-trip choice of users, etc., it can be assumed that the route chosen by each traveler at time I (including the no-trip option) is a discrete random variable assuming values in the set of feasible routes 1(u). If we denote by p:(k) the probability that the uth user follows route k at time f and assume that travelers choose independently, the probability of finding the system in each feasible state R,, at that time can be expressed as: Prob[R’ =

&I = n p’,M41. Id

(2.8)

The state occupied by the system at each time is a discrete random variable: this implies that the evolution of the transportation system over different times is the realization of a stochastic process with discrete time and state spaces. In the next section, sufficient conditions will be stated ensuring that such a process is “well behaved,” i.e. admits a unique stationary or steady-state probability distribution and is ergodic. 3. SUFFICIENT

CONDITIONS FOR THE STATIONARITY STOCHASTIC PROCESS

OF THE

Stationarity of the process describing the system evolution will be studied under the assumption of fixed potential demand and network configqations. In other words, it will be assumed that both the network and the number of potential users remain unchanged for a number of epochs large enough to allow a steady-state evolution eventually to take place. It will also be assumed that the number of potential users making travel decisions in each epoch is uniformly distributed over subintervals of the reference period so that usual definitions of path and link flows maintain their significance. The type of stochastic process describing the demand-network system evolution over time depends on the p:(k), or on how travelers make their choices at each time t. It seems reasonable to assume that users moving at each time do not know in advance travel costs over different routes but they base their choices on available information relative to costs incurred in previous epochs. This can be expressed by assuming that path choice probabilities p:(k) depend on the states occupied by the system in previous times t - 1, t - 2, . . . p:(k)

= p;(k)

[R’-‘, R’-2, . . .].

(3.1)

Expressions (2.8) and (3.1) imply that the system probability of occupying any state at time I depends on the states previously occupied. Note that the above assumptions are rather general and allow a number of very different behavioral rules to be taken into account. For instance, the no-trip probability may either be modelled as depending on previously experienced costs or held constant. Most information acquisition and choice models proposed in the literature also satisfy eqn (3.1). For instance, it can either be assumed that each user has a “starting” knowledge of path attributes that is updated every time he/she actually uses a path, or a “moving average” type of mechanism (see Section 4) can be assumed. Under the above assumptions the following two propositions hold. Proposition A

The stochastic process describing the time evolution of the demand-network system in the route choice space has a unique steady-state probability vector 72 = (rl, 7r2, . . . , 7~“~) and it is ergodic under the following (sufficient) conditions: (i) Path choice probabilities are time homogeneous, a temporal translation given the set of previous states p;(k)[R’-’

= R,,, R’-2 = Rk, . . .] = p:‘(k)[R”-’

i.e. they are invariant under

= R,,, R”-2 = Rk, . . .].

(3.2)

6

E.

(ii) Path choice probabilities

are positive for all feasible paths

p:(k) (iii) Path choice probabilities previous states p$c)[R’-‘,

CASCElTA

> 0

Vk E I(u).

(3.3)

depend on not more than a finite number (m) of

R’-2, . . .] = p:(k)[R’-‘,

. . . , R’-m].

(3.4)

Conditions (i) and (iii) ensure that the stochastic process is a m-dependent (or of order m) Markov chain with a time-homogeneous transition probability matrix. This, as well as the other properties of the stochastic process stated in Proposition A, is proved in the Appendix. Assumption (i) requires that the choice process is invariant under time and that the average demand level (given by the no-trip option) is constant. Assumption (ii) requires that all paths belonging to each user’s “feasible” set have a positive probability of being chosen at each time. Assumption (iii) requires that users “remember” a finite number of previous travel experiences (possibly large and not necessarily the same for all). It is this author’s opinion that all of the above assumptions can approximate many real-life cases, at least over a short-term scenario. It should also be noted that these assumptions underlie virtually all assignment models proposed in the literature. Proposition B

If the process in the route choice space is stationary and ergodic, the same properties hold for the processes in path and link flow spaces. The above proposition can be shown by using the relationships among route choice, path flow, and link flow vectors defined in Section 2. Steady-state probabilities of feasible flow vectors exist, are unique, and can be computed as:

and

4~0 = 2 4Fi). F,EIU,

Note, however, that under the assumptions made so far processes in flow spaces are not Markovian. In general, the probability of observing a given vector of feasible path (link) flows does not depend on previously occurred path (link) flows only, but also on how these vectors were composed by users moving in those periods [see expression

(3.111. Both properties stated in the above propositions are relevant. Existence and uniqueness of steady-state probabilities of observing every feasible state make it possible to associate a “dynamically stable” evolution law to each transportation system, regardless of its starting configuration, elapsed time, and type of cost functions. Ergodicity allows the computation of steady-state probabilities and moments of flows from a single realization of the process and is particularly useful from the computational point of view. Various stochastic processes with their moments can be obtained by assuming different choice and information acquisition models with various degrees of realism. In the next section one model of this kind will be proposed based on a set of simplifying assumptions giving rise to a model whose results, and particularly average flows, are comparable with those of stochastic equilibrium models. 4. A

MODEL

OF

SYSTEM

DYNAMICS

A model of system dynamics (STODYN) will be described based on a set of simplifying assumptions usually made in the context of stochastic assignment models and in other studies on system dynamics.

Temporal dynamics in transportation

networks

These assumptions essentially amount to consider travelers as indistinguishable, following the same information acquisition and choice models. They are:

7

i.e.

(i) All travelers moving between the same O/D pair (i, i) have the same set of “feasible” paths Iii. Note that a “dummy” path between each O/D pair can be included in Ziirepresenting the no-trip option so as to maintain random fluctuations of demand. (ii) All travelers are rational decision makers. At each time t they associate a perceived (generalized) travel cost C;(t) to each alternative path and choose so as to minimize this cost. Because of random fluctuations in some fundamental attributes making up the generalized cost (such as travel time, no. of stops, etc.), variation of “tastes” across the population and over time, incomplete information, etc., it is assumed that perceived path costs are random variables with mean value C;(f) = Ck+(t) + Ek(Q.

c;(t)

Under this assumption path choice probabilities p:(k)

= pi, = Prob[C;(t)

(4.1)

can be computed as:

< CL(t)

Vh # k h E r,].

(4.2)

The functional form of path choice probabilities depends on the joint probability density function assumed for residuals Ek. Logit and probit are the most popular models in the context of stochastic assignment deriving from independent Weibull and Multivariate Normal distributions, respectively (see, for instance, Daganzo and Sheffi, 1977 and Sheffi, 198.5). (iii) All travelers have the same information acquisition (learning) mechanism. It is assumed that at each time t they base their choices on a weighted average of costs actually incurred in a finite number (m) of previous times C,(t):

C;(t)

=

$ WiCk(t - i).

i=l

(4.3)

Because of congestion path costs are functions of path flows C,(r) = Ck(F’) +

ai,

where crl,t is the determination of a r.v. taking into account fluctuations of actual cost and possibly perception errors around the average value C,(F’). In this case the average generalized perceived cost at time t is

C;(t)

= i

w~[C~(F’-~) + CL;]

i=l

The above equation combined with expression (4.1) and (4.3) yields: C;(f)

= i

i=l

wiCk(Ftmi)

+

ok(‘).

(4.4)

The random term e,(t) is obtained as the weighted sum of random variables Ek and ak; this by the central limit theorem should further support the use of probit path choice models

for

pkij.

tin assignment models random variation of path travel costs are assumed as negligible (ai = 0). If this is not the case, path choice probabilities (4.2) should be computed conditional to a vector of cost dispersions around the mean.

8

CASCEITA

E.

Note that eqns (4.4) and (4.2) give the dependence of path choice probabilities on the states occupied by the system in m previous times [see expression (3.4)]. Various learning processes can be devised to support the above model. One possibility is to assume that in each period travelers form perceptions of link costs through experience for links actually used and through guess on travel cost or conversations with other travelers as far as nonchosen links are concerned (see Horowitz, 1984). Under the above assumptions all travelers of the same categoryt moving between the same O/D pair are completely equivalent. In this case the temporal evolution of the system can be studied more conveniently in the path flow space. In fact, the probability of finding the system in a particular path flow state Fk at time t is the sum of the probabilities of finding it in any path choice state belonging to IF, Prob[F’ = Fk] = 2

Prob(R’ = R,),

iElF,

and it is constant for all sequences of previous path choice vectors giving rise to the same sequence of path flows. The above are necessary and sufficient conditions for the Markov chain to be “lumpable” (Kemeny and Snell, 1960; Bath, 1984), i.e. the states in the path choice space can be lumped into path flow states with the resulting process still being Markovian. Conditions ensuring existence and uniqueness of steady-state probabilities in the path choice space can be transferred to the Markovian process in the path flow space. If users are assumed to behave independently the probability of finding the system in state Fh at time t is given by-the multinomial probability function: Prob(F’ = Fh) = fl d,! fl ij

(Pii~)F~‘k’lFh(k)!.

(4.5)

kW,

The stochastic process describing the system evolution is an homogeneous Markov chain in the path flows space; in fact:

m-dependent

Prob[F’ = F,,/F’-l = Fk, . . . , F’-” = F,, . . .] = Prob[F’ = F,/F’-’

= Fk, . . . , F’-” = F,]

(4.6)

and Prob[F’ = F,,/F’-’ = Fk, . . . , F’-” = F,] = Prob[F”

= FJF”-*

= Fkr . . . , F”-” = F,]

(4.7)

Assumptions (i)-(iii) are such that the sufficient conditions given in Proposition A of Section 3 are respected so that a unique steady-state probability vector of feasible path flow exists rr = (ni) and moments of path and link flow vectors can be computed. For instance, means can be obtained as F’

=

E[F] = CiTiFi

=

AF’,

Fi E SF

(4.8)

and V+

where A is the link-path incidence matrix [see eqn (2.7)]. An alternative expected flows will be given in Section 5.

(4.9) expression of

tin this paper users are assumed as belonging to one category. Extension of results to the case of many users’ categories (e.g. differentiated by trip purpose) is straightforward.

Temporal dynamics in transportation

Note that existence and uniqueness of steady-state on cost functions.

9

networks

expected flows do not require

anyassumptions

5. RELATIONSHIPS

BETWEEN

STODYN

AND

SUE

EXPECTED

FLOWS

In this section relationships between STODYN and SUE expected flows will be examined. For.analytical convenience a one-step learning mechanism will be considered first. In this case the stochastic process is a homogeneous Markov chain: Prob[F’ = F,,/F’-l = Fk, F’-’ = F,, . . .] = Prob[F’ = Fh/Ft-* = Fk] = nk,,, where ?rk,, are the elements of the transition probability matrix and steady-state abilities IT; of feasible path flow Fi are the unique solution of the linear system n; = ~k~k~ki

vi

(5.1) prob-

(5.2)

On the other hand, SUE expected path and link flows are solutions of the following fixed point equations: F* = P(C(AF*))

(5.3)

and V*

=

AP((C(v*)),

(5.4)

where P(.) is a path choice probability matrix of dimensions (no. of paths) x (no. of O/D pairs) whose elements are the probabilities of choosing path k between O/D pair (i, j). These probabilities as obtained by a “random unility” model are functions of alternative path generalized travel costs and, because of congestion, of link volumes (see Daganzo and Sheffi, 1977). Existence of SUE flows is assured under mild conditions by Brauwer’s fixed point theorem (Daganzo, 1982; Sheffi, 1985). Pkij

Proposition C Steady-state expected flows of a one-step STODYN model do not coincide with SUE flows but for constant costs or for linear path choice probability and link cost functions. In general, however, they can be considered coincident within the limits of a first-order approximation of the vector function P(C(F)). In fact, the expected STODYN path flow vector is given by: F’ = ZiniF;.

(5.5)

Because of the “fixed point” property of steady-state probabilities nTTi, [eqn (5.2)], expression (5.5) becomes: (5.6) where I;ilTkiFi= xi Prob[F’ = F,/F’-’ = Fk]Fi = E[F’IF’-’ = Fk] = P(C(Fk))d.

(5.7)

Equation (5.7) expresses the expected value of path flows given the cost corresponding to Fkt as the product of the path choice probability matrix computed with C(F,) and the O-D demand vector d. tFor notational convenience,

path costs are expressed directly as functions of path flows.

E. CASCEl-t'A

10

By substituting expression (5.7) into (5.6) it results: F’

= TknkP(C(Fk))d

= E[P(C(F)]d.

(5.8)

Comparison of (5.8) with (5.3) shows that F’ is a SUE flow vector if the expected value of the function P(F) is equal to the function of the expected flow F’:

W(W)1

= P[C(EP))I.

(5.9)

Condition (5.9) is verified for linear (or affine) functions or for uncongested networks in which average costs are independent of link flows. In the usual case of nonlinear path choice and cost functions it can be shown that expression (5.9) holds as an approximation once that the vector function P(C(F)) is expanded in Taylor’s series? up to its linear term and that the approximation error is smaller the lower the variance of F components, i.e. the smaller the variability of the path flows over time. In fact it results:

Pk;j(C(F))= J’kij(C(F’)) + G[pk,(C(F’))]‘(F - F’) + l/2@ - F+)‘H[p;,(c(F+))](F -

F’)

+ 0(/F

-

F+lZ),

where G is the gradient and H is the Hessian of pk;, with respect to F. Taking expectation of both sides and ignoring higher-order terms it yields:

E[Pki,(C(F))I= Pki,(C(F+))+ 112traceP-=c,), where ZF is the variance-covariance matrix of path flows temporal fluctuations. The above results can be immediately extended to link flow v’ and v* via expressions (4.9) and (5.4). In this case the approximation argument receives support from the reduced variability or link flows with respect to path flows. It should, however, be noted that the above (approximate) coincidence of STODYN and SUE expected flows does not extend to second-order moments; higher variances and an autocorrelation structure should be expected for STODYN link flows. If m previous periods are remembered it can be shown by paralleling the demonstration given in the one step case that the STODYN expected path flow vector is:

F’ = &,k ... ~q.k....,jP(‘+‘,C(Fq) + . . . + wmC(Fj))d, ..I

wherenq.k.....j is the steady-state probability of the sequence of vectors

(5.10) (F,, Fk, . . . ,

Fj)*

Expression (5.10), as in the left-hand side term of (5.8), gives F’ as the product of the demand vector d by the expected path choice probability matrix computed over all possible sequence of m states. Conditions for exact and approximate coincidence with SUE flows are still valid if weights sum up to one. In the limiting case of a number of remembered costs tending to infinity with uniform weights, users tend to base their choices on average costs, which are still different from costs computed for average flows in the case of nonlinear cost functions. Also in this case STODYN and SUE expected flows are only approximately equal. The difference between expected link costs E[c(u)] and costs computed for the expected flows c(E[u]) can be quantified by expanding a generic separable link cost function in Taylor series up to its second-order term: c(u) = c(E[u])

+ c’(E[u])(u

- E[u]) + 1/2c”(E[u])(u

- E[u])~ -t- o(u - E[u])~,

Komponents of P(C(F)) are assumed continuous with their first- and second-order assumed as a continuous vectorial variable.

derivatives;

F is

Temporal dynamics in transportation

11

networks

where c’(.) and c”(.) are first- and second-order derivatives with respect to link volume LJ.Taking expectations of both sides and ignoring higher-order terms it yields: E[c(u)]

= c(E[u])

+ 1/2C”(E[U])U~.

For strictly convex cost functions, cl’(.) is greater than zero so that E[c(u)] is larger than c(E[u]); in any case the difference is proportional to the variance of link volume fluctuations. Numerical results for some of the most popular travel time functions, i.e. Bureau of Public Roads (BPR)-type and Webster’s delay formulas, show that E[c(u)] and c(E[u]) substantially coincide for volume/capacity ratios up to 0.8:0.90 for all values of uV. For volume/capacity ratios close to one and for link volumes standard deviation-to-mean ratios of 0.10:0.20 percentual differences between the two costs are comprised in the range 20:50 percent of c(E[u]). Experimental values of the between-day fluctuations of link volumes are usually lower than 0.10 so that the approximation of E[c(u)] with c(E[u]) can be accepted for most practical uses. However, higher values (0.10:0.15) can be observed for particularly variable demand types, so that elastic demand assignment models reproducing those temporal fluctuations should take into account the difference between E[c(u)] and c(E[u]). It should finally be noted that STODYN average flows, because of their uniqueness, can diverge substantially from SUE ones when multiple stochastic equilibria exist. In Figs. 2 and 3, SUE and STODYN average flows are reported for a two-link, single O/D pair network in the case of unique and multiple equilibria. Equilibrium flows were obtained analytically with a logit path choice model pkod

=

exp[-i3ck]/& exp[-o.3ck]

The STODYN model used the same path choice function and a learning mechanism based on m = 10 previous costs with uniform weights. Average flows were computed by a Monte Carlo simulation technique; figures in brackets are standard deviations of estimated time averages. For further details on the algorithm see Section 6. Results show that STODYN flows closely approximate equilibrium ones when they are unique but give “intermediate” values when many equilibria exist. This can be a definite advantage of the proposed model in those cases in which uniqueness of SUE cannot be proved a priori(e.g.for nonseparable, nonsymmetrical cost functions). A

cl

d

2

c* * Vl

SUE (doo=5)

STODYN

(don=51

cz

-

7C1+2.~(v~/(0.75.do~) 12C1+2.6(vz/(O.75

SUE (do,=Jo)

I

I

j-1 dm,b))4]

STODYN

(d,z,,,=50

1

Vi

2.93

2.90

(sd-0.00316)

29.27

29.30

(sd-0.01049)

v2

2.07

2.10

(sd=0.00316)

20.73

20.70

(sd=0.01049)

Fig. 2. SUE and STODYN average flows in the single equilibrium case.

E. CASCETI-A

L-4 c2

Cl

v,&-----A

E’JEl

(do,=

10)

2

CZ = 0.7v,+7 cz

5

c,

=

-8.464797~~+31.9296 (2/3)vz+(10/3)

SUEZ(d,,=lO)

v:.

<

3.132

vz

> 3.132

STODYN(don=lo)

Vl

3.60

9.50

8.81

(sd=0.01788)

v2

6.40

0.50

1.19

(sd=O.

01788)

Fig. 3. SUE and STODYN average flows in the multiple equilibria case.

6. STODYN

AS

A

DYNAMICAL

MODEL

OF

STOCHASTIC

ASSIGNMENT

The model of system evolution described in Section 4 can be seen as an assignment model insofar as it produces typical outputs of stochastic models: average flows and second-order moments. Results of Section 5 showed that STODYN expected flows do not necessarily coincide with SUE ones so that it is theoretically correct to consider them as different assignment models. It has also been noted that the existence and uniqueness of STODYN steady-state expected flows do not require any assumptions on cost functions, as is the case for equilibrium flows. Ergodicity of the process in the link flow space allows the construction of algorithmic schemes for computing mean and second-order moments of link flows based on the simulation of a realization of the process. These algorithms have the following general structure: Step 1: Compute (4.4)

average perceived

costs by using last m iteration flows via eqn

Cb = XiWjCk(V’-i). Step 2: Simulate path choices of travelers via a random utility model (4.2) and compute resulting link flows; Step 3: Average link volumes over all iterations. VI+ = 1ltC;u;. If the stop criterion is not met go to step 1 Different methods with different approximation levels can be devised for the most demanding Step 2. One possibility is to obtain link flows via a logit or probit-based Stochastic Network Loading algorithm (Dial or Monte Carlo). In this case the sequence of link flows is a pseudo-realization of a stochastic process; however, some numerical tests show that the influence of this approximation over final flows is practically negligible. A more “elastic,” but computationally more demanding, alternative is to simulate at each iteration path choice of each user (or “platoons” of them) by a Monte Carlo

Temporal dynamics in transportation

technique,

13

networks

e.g. simulating perceived link costs with average values given by i

bqc(v’-i).

i=l

Step 3 applies only to a stationary realization of the link flow process. A number of initial iterations are discarded until the stationarity hypothesis is not rejected by suitable statistical tests. A more detailed discussion of different algorithms can be found in Cascetta (1986). Stochastic fluctuations of the demand level can be easily introduced by simulating at each step the actual number of trips moving between each O/D pair as a r.v. with an average value given by the trip table. An empirical analysis of STODYN predicted vs. observed flows and a comparison with most popular deterministic and stochastic models was recently carried out for the car networks of two medium-sized Italian towns. In the following, a brief summary of results relevant to the problem dealt with in this paper will be given; further details relative to data bases, implemented algorithms, statistical analyses of results for equilibrium as well as “network loading” models can be found in Cascetta and Nuzzolo (1986a). Data relative to the urban areas of Parma and Foggia sample and network characteristics are reported in Table 1. The road network model of each area included arterials and collectors only. Travel time as obtained by separable functions was used as the only measure of generalized cost, thus giving rise to unique equilibrium flows. Assigned O/D trip matrices were relative to the AM peak hour (7:30-8:30) as obtained by household and cordon surveys. The demand was considered fixed over time. Counts over a number of selected links were carried out on a single day. Various equilibrium models were used for the two networks, including a probitbased SUE and a probit-based STODYN with uniform weights and m = 7. Parameters

Table 1. Main features of the two Italian towns used in this study

No. of sampled households

3.800

1.600

Household survey rate

0.06

0.04

Cordon rate

0.10

0.30

survey

Network nodes

No.

of observed links

264

245

E. CASCETTA

3000 2750 2500

estimated I[ /

2250 2000 1750 150&z! 1250 1000 750 500 +

250 &

observed I”’

1

fi

J

11

J

model.

Fig. 4. SUE probit

of stochastic models were calibrated by minimizing the sum of square errors and resulted very stable for the two networks. Average flows obtained by stochastic models were compared with observed ones for volume classes. Scatter diagrams of observed vs. modelled flows for STODYN and SUE models are reported in Figs. 4 and 5, respectively. As a performance indicator percentual root mean square error (RMSE%) was used: RMSE%

3000

= (Z,(Oi - uf)‘ln)“2

estimated

2750

[

2500

t

+ +

2250 -2000 1750 1500

750 500 250

4

I ;

/ / + ++

L

+ +++

1250 1000

(6.1)

9

urn

t

+ ++ 4

%* /

+

+

/

Temporal dynamics in transportation

networks

15

Table 2. RMSE% statistic values for observed flow classes: Town of Parma (v/h)

CLASSES

F

L

0

W

< 250

I

250 500

500 750

750

1000

1500

1000

1500

>

n= 7

n=lR

n=30

n=l7

n=19

n= 4

n=95

0.81

0.44

0.35

0.28

0.30

0.10

0.32

0.81

0.48

0.37

0.28

0.28

0.09

0.32

MODELS

SUE

Probit

STODYN

TOT

where uf and 6, are modelled and observed flows, respectively, n is the number of observed links, and u, is the average observed flow for class M. Values of RMSE% for different flow classes are reported in Tables 2 and 3 for Parma and Foggia, respectively. The analysis showed that SUE and STODYN expected flows were on the whole largely comparable (as expected by the results of Section 5) even if some differences appeared at the level of individual links. Although STODYN gives also the possibility of computing flow temporal variances within the assignment routine, single-day counts did not allow to compare observed and predicted second-order moments. CONCLUSIONS

In this paper the dynamics of the demand-network system were studied under the assumption that the states occupied by the system vary randomly over successive times. This assumption implies that the system evolves as a stochastic process, the type of which depends on the actual choice mechanism followed by travelers. In order to model the stochastic process in a “closed form” and to compare expected flows with those given by a static equilibrium model (SUE) stationary processes were considered. General sufficient conditions for stationarity and a particularly simple model of temporal dynamics (STODYN) based on “standard” path choice and information acquisition models were proposed. STODYN expected flows were compared with SUE flows and it was shown that the two coincide “theoretically” only under rather unrealistic linearity assumptions or for uncongested networks, even if they can be considered coincident within the limits of a first-order approximation the closer the larger the number of previous costs remembered by users in making their choices. However, it has been shown that STODYN and SUE average flows can substantially diverge when multiple equilibria exist. The possibility of using STODYN as an assignment model of a fixed demand giving unique first- and second-order moments of link flows was also explored. Some results of an empirical analysis carried out comparing observed flows with average flows predicted by STODYN and SUE in two Italian urban networks were reported. Table 3. RMSE% statistic values for observed flow classes: Town of Foggia FLOW

CLASSES

(v/h1

MODELS < 250

SUE

Probit

STODYN

250 500

500 750

750 >

TOT

n= 8

n=22

n=lb

n= 7

n=53

1.24

0.42

0.47

0.30

0.48

1.20

0.47

0.46

0.38

0.50

E. CASCE~TA

16

The main conclusions were that STODYN expected flows are reasonable approximations of real ones, largely comparable on the whole with SUE flows as theoretically expected in those cases. A number of points remain open to further research. Alternative, and possibility more realistic, choice and learning models can be specified and compared with the ones considered in this paper. In particular, the single-user formulation of the model given in Sections 2 and 3 allow to take into account phenomena as habit or un-uniform knowledge of the network. The model could possibly be extended to take into account intraperiodical fluctuations of demand, thus resulting in a “doubly dynamic” model. Last, but not least, a sounder experimental basis is needed. Comparisons of observed and predicted flows should be extended to second-order moments and travelers’ path choice and learning mechanisms should be investigated in a disaggregate way.

Acknowledgemenn-Comments of Paolo Ferrari and Agostino Nuzzolo on this paper are highly appreciated. The comments of an anonymous referee on an earlier version of this paper are also kindly acknowledged. This research was partly supported by the “Progetto Finalizzato Trasporti,” C.N.R. Subproject 2 under Grant No. 85.00128.93.

REFERENCES Bath U. N. (1984) Elements of Applied Stochastic Processes 2nd Ed. John Wiley, New York. Beckman M. S., McGuire C. B. and Winsten C. B. (1956) Studies in the Economics of Transportation. Yale University Press, New Haven, Connecticut. Cascetta E. (1986) A Dynamical Model of Stochastic Assignment to Transportation Networks. Internal report, Dept. of‘Transpn. l&g., University of Naples, Italy. Cascetta E. and Nuzzolo A. (1986a) An Emoirical Analvsis of Assionment Models for Urban Car Networks. IV National Congress of PFT/CNR, To&o, Italy. ’ Cascetta E. and Nuzzolo A. (1986b) A behavioral scheme for modelling route choice in transit networks. IV National Congress of PFT/CNR, Torino, Italy. Cascetta E. (1987) Static and dynamic models of stochastic assignment to transportation Networks. In Flow Conrrol of Congested Networks. Series F, vol. 38 (Edited by Szaego G.. Bianco L., Odoni A). SpringerVerlag, New York. Cox, D. R. and Miller H. D. (1972) Theory of Stochastic Processes. Chapmman and Hall, London. Daganzo C. and Sheffi Y. (1977) On stochastic models of traffic assignment. Transpn. Sci. 11, 253-274. Daganzo C. (1977) On achieving stochastic user equilibrium on a transportation network. WP7704, University of California, Berkeley, California. Daganzo C. (1982) Unconstrained extremal formulation of some transportation equilibrium problems. Transpn Sci. 16, 332-360.

Friesz T. L. (1985) Transportation network equilibrium, design and aggregation: Key developments and research opportunities. Transpn. Res. 19A, 413-426. Horowitz J. L. (1984) The stability of stochastic equilibrium in a two-link transportation network. Transpn. Res. 18B. 13-28. Horowitz J. L. (1985) Travel and location behavior: State of the art and research opportunities. Trunspn. Res. 19A, 441-453.

Kemenv J. G. and Snell J. L. (1960) Finite Markov Chains. Van Nostrand C., London. Nguyen S. and Pallottino S. (1985) A traffic equilibrium model for transit networks. Euro VII, Bologna, Italy. Sheffi Y. (1985) Urban Transoorturion Networks. Prentice-Hall. Enelewood Cliffs, New Jersev. Sheffi Y. and Daganzo C. (1979) Hypernetworks and supply demand equilibrium with disaggregate demand models Trunspn. Res. Rec. 673(l), 13-120. Smith M. J. (1979) The existence, uniqueness and stability of traffic equilibria. Transpn. Res. 13B. 295-304. Smith M. J. (1984) The stability of a dynamic model of traffic assignment. An application of a method of Lyapunov. Transpn Sci. 18, 245-252.

APPENDIX The stochastic process describing the system evolution in the route choice space is an m-dependent chain under the conditions stated in Proposition A of Section 3. In fact, by using expressions (2.8). (3.2). and (3.4) it results: Prob[R’ = R,/R’-i

Markov

= R,, , . . , Rfmm= R,, . . .]

= Prob[R’ = Rh/R’-’ = Rr, . .

, RI-“’ = R,]

(A.1)

Temporal dynamics in transportation

networks

17

and Prob[R’ = R,/R’-’

. , R’-” = R,]

= Rk,

= Prob(R”

= RJR”-’

, R”-” = R,]

= R,,

(A.2)

which are the defining equations for such a process. An homogeneous m-dependent Markov chain can be reduced to an homogeneous (l-dependent) Markov chain by an appropriate transformation of the state space. A transformed state R’ is defined as a collection of m feasible route choice states: R’ h3. i = (RI,, R,, .

9R,).

and the number of such states is n;. One-step transition probabilities between transformed states can thus be computed as: nh. ,,.,+ , = Prob[R”

= R& ,iR”-’

= R;

,,]

1R,. 41

= fLp,(Rq(n))[R,,

(A.3)

Condition (3.3) ensures that each pair of transformed states communicate, i.e. there is a positive probability of passing from each transformed state to every other in a finite number of steps. In fact from expressions (A.3) and (3.3) it follows that n* I,.,/*I > 0 vq E 3,

(A.4

and transition between each pair of transformed states can occur via at least one sequence of intermediate transformed states obtained by shifting the first m-l indexes and replacing the first as in eqn (A.4) (transitivity of communication relation). Because all states communicate the chain is irreducible. It can be shown that irreducible a-periodic finite Markov chains possess the following properties (Cox and Miller. 1972). pg. 124; Bath, 198-t. pg. 92): (i) It exists a unique “steady-state” or “stationary” probability distribution n - {Pi,,...} giving the longterm probability of the system occupying any transformed state independently of the initial one. The vector n is the solution of the linear system (fixed point of the linear function):

where II is the one-step transition probability matrix with elements given by expression (A.3) and nr is a row vector. (ii) The process is ergodic, i.e. the fraction of times in which each transformed state is occupied gives, in the limit. its steady-state probability lim

nqh ..,, II

=

71yh.,,,

(‘4.6)

+.’

where n+, , is the number of times state qh...j is occupied in I periods. Properties (i) and (ii) can be immediately transferred probabilities can be obtained and computed as:

to the original route vector states; steady-state

r,r = x;h.... n$l.. ,

(A.7)

7rs = lim n,/r CI

(AS)

Equations (A.7) and (A.8) can be used to compute means and second-order discrete random variable “route choices.”

TRIB) 23:1-B

moments of the vectorial