A dynamic space-time network flow model for city traffic congestion

A space-time network is used to model traffic flows over time for a capacitated road transportation system having one-way and two-way streets. Also, for the first time, traffic signal lights which change the network structure are explicitly incorporated into the model.


I. INTRODUCTION
A fundamental problem faced when planning for the construction of new roads and the improvement of existing roads is to predict in detail the resulting traffic patterns.A model which can estimate, with reasonable accuracy, future patterns will assist in avoiding the creation of new bottlenecks.The resulting problem is generally known as the traffic assignment problem.
In the literature the problem of estimating traffic flows in road networks is generally formulated as an equilibrium problem.Knight [18] has given the following intuitive description underlying this equilibrium approach: "Suppose that between two points there are two highways, one of which is broad enough to accommodate without crowding all the traffic which may care to use it, but is poorly graded and surfaced, while the other is a much better road, but narrow and quite limited in capacity.If a large number of trucks operate between the two termini and are free to choose either of the two routes they will tend to distribute themselves between the roads in such proportions that the cost per unit of transportation, or effective return per unit investment, will be the same for every truck on both routes.
As more trucks use the narrower and better road, congestion develops, until at a certain point it becomes equally profitable to use the broader but poorer road." Wardrop [25 ] formalized the notion of a network equilibrium into the following two principles: , and Florian [12].Nguyen [22] has developed efficient algorithms that are able to treat large scale problems encountered in practice.Florian and Nguyen [11] present the results of a validation study using the constant demand model.Gartner [13,14] provided a survey of work done on the elastic demand problem.Leblanc and Farhangian @192 and Dafermos [8] have made recent extensions.An introduction to the field is provided by Potts and Oliver [23].
In both the constant demand and elastic demand models a static set of origin destination demands is assumed to exist.This may be a valid long run assumption but it is not in the long run that road network congestion occurs.Let c(i,j) be the capacity of flow on any green arc connecting i to *j and let k(i,j) be the normal flow time from i to j.Then the process described above implies that the total time required for X flow units to travel from i to j given they start at time t is given by the following function: t),j) if 0 *X((i,t),j) < c(i,j) (a) (b) if c(i,j) *X((i,t),j) < 2c(i,j) t),j) -2c(i f j)] * (c) if 2c(i,j) *X((i,t),j) < 3c(i,j)

V = {[(i,t),i]/i e D, t « u (*,i),...,T}
The arcs in V all have length zero and infinite capacity.They allow the flow to arrive at a destination at various times.(j,t+k(i,j))] and of the red arcs c[(i,t), (i,t+l)] are the maximum rates in units/unit time at which units may enter an arc.These rates can be specified in various ways depending on the specific application.

FG(i,t) -{[(i,t),(j,t+k(i,j
))] e 5} for ieN and fixed t.  (  Figure 4 shows the graph G associated with the graph P of the example.

TG(i,t) = {[(j,t-k(j,i)),(i,t)] c G} for
The time horizon T has been selected to be 5. Table 3 contains a listing of the arcs (except for the dummy arcs) with their associated lengths and capacities.

"
The first criterion is quite a likely one in practice, since it might be assumed that traffic will tend to settle down into an equilibrium situation in which no driver can reduce his journey time by choosing a new route.On the other hand, the second criterion is the most efficient in the sense that it minimizes the vehicle hours spent on the journey."The first principle implies that each individual selects the shortest route available under given traffic conditions.The resulting pattern is generally called user optimal.The second implies a central authority imposing a set of routes which minimizes total travel time.The resulting pattern in this case is generally called system optimal.In Wardrop f s description of the problem the required number of trips between each origin destination pair is a specified constant.This problem is known as the fixed demand traffic assignment problem.The elastic demand traffic assignment problem was first enunciated by Beckman, McGuire and Winston [1].In this problem.thenumber of trips between each origin destination pair is a function of the travel time between each origin destination pair.Both of these formulations have been studied extensively.A survey of work done on problem formulation is given by Beckman [2].The basic paper on the constant demand model was written by Dafermos and Sparrow [3].This work has been extended by Dafermos [4,5,6,7] bottleneck situations which arise are short term dynamic phenomenon.Rush hour does not occur on the average or in the long run and yet these types of peak load conditions are the most important to analyze in judging traffic patterns on a street network.By assuming traffic is in equilibrium one assumes away the dynamics of demand which is the crux of the problem which planners need to resolve.Hence the appropriateness of the equilibrium assumption must be questioned in many applications.Recent papers by Heydecker [15] and Steinberg and Zangwill [24] also indicate that equilibrium solution concepts may not always be applicable to real traffic situations.Traffic assignment models are often intended for analysis of rush hour situations in which the rate of traffic flow into the network will be increasing for some period until a peak rate is achieved and possibly maintained for a brief period, followed by a decrease.There may be multiple peaks in the rate of traffic flow as various factories or office buildings release their workers.Furthermore the road network itself is dynamic.As traffic signals change available routes are being opened and closed.Hence it is not likely that there will ever be an equilibrium state achieved in the network.One model has been proposed by Merchant and Nemhauser [20] which considers dynamic flows.It allows multiple origins and a single destination and treats congestion explicitly.The model is nonlinear and nonconvex.Proof is given that it can be solved for a global optimum using a one pass simplex algorithm.The model minimizes the total cost of the flows.An efficient solution procedure for the model is presented by Ho [16].In the present paper we propose a space-time network flow model which explicitly incorporates all the characteristics of road networks generally considered in the literature together with dynamic aspects not usually considered.Congestion is modeled by a linear space time network and it can be shown that, from a given start time, the time to travel a street is a piecewise linear convex function of the flow on the street at the start time.Each route has a capacity at any given time which is determined by the physical space available at that time on that route.The model accepts a variable set of inputs over time.Finally the dynamics of a road network caused by the changing of traffic signals are for the first time explicitly incorporated in a road network flow model.Two types of solutions are proposed.The first for a multiple source single destination space time network is termed a network flow solution and requires a set of dynamic flows over the time horizon of the model which minimize total travel time.This solution is computed by use of the network code NETFLO, due to Kennington and He 1 gas on [17], which is very fast.The second, for a single source single destination space time network termed a shortest path solution, gives a set of dynamic paths from a single source to a single destination which are the shortest available at a given time relative to traffic leaving the source in all prior time periods.This solution is obtained by repeated use of a shortest 2 path algorithm which is known to have time complexity 0(n ) where n is the number of nodes of the network.In the second section of this paper the model is stated as a space time network model.The space time network model itself has more general application than simply traffic flow analysis, for instance, to the Job Shop Scheduling problem.In section three the model is a-plied specifically to the dynamic traffic flow problem.In sections four and five the space time network flow and shortest path solution methods are discussed.Finally in section six two hypothetical street networks are given and solutions are determined for each situation.II.A TWO ATTRIBUTE DYNAMIC NETWORK FLOW MODEL In this section a two attribute network flow model which is dynamic, i.e., changes over time, will be developed.The model is dynamic in two senses.First the input and output of the model can vary over time.Second, the network itself may change over time.In the present setting the other attribute beside time is space, but applications are under development where the second attribute may be other than space.For clarity in the description, the model attributes will be referred to as space and time, but it should be remembered that the space attribute could be interpreted as other attributes, also in other models there could be other attributes.In the space-time setting the network is constructed from nodes and directed arcs having the following properties.Each node represents both time and location.Each directed arc represents a directed path of length k time units connecting a pair of nodes.
sum of these terms.The first term is the normal travel time fcr the first n*c(i,j) flow units.The second term is the sum of the first n-1 integers multiplied by the capacity c(i,j) of the green arcs between i and j.This term is the additional time required for the (n-l) # c(i,j) units which exceed the normal capacity between i and j.The last term is the marginal travel time for the units exceeding n»c(i,j) multiplied by the number of units in excess.That is the flow units in the interval [n*c(i,j)+l,B] require the normal travel time plus n time units to travel from i to j.In expression (d) the upper bound Bon X((i,t),j), the flow starting at time t from location i to j, is the capacity of the red arc at location i (this capacity does not appear in the expression) plus the capacity c(i,j) of the green arc connecting i and j.The red arc capacity can be interpreted as the maximum length of a queue before the green arc.Note that k(i,j) and c(i,j) are nonnegative.Hence the function F is nonnegative.Also note that for each interval [m c(i,j),(m+l)c(i,j)} for m = 0,1...n the slope of the function is a positive constant k(i,j) + m and as m increases the slope increases.Clearly the function F is piecewise linear convex.Hence the next theorem follows immediately.THEOREM: The time required for X units of flow starting at time t to travel from i to j in the space time network is a piecewise linear convex function of X given by the function F above.A dynamic space time network is constructed for a graph P = (S,N,D,A), where S is a set of source locations, N is a set of intermediate locations, D is a set of destination locations and A is a set of ordered pairs (i,j) i 4 j ieSUN, jeNUD.For each ordered pair (i,j) e A, let k(i,j) be the normal travel time from i to j.This defines the length of arc (i,j) in P. Let c(i,j) be the flow capacity in flow units per unit time for arc (i,j) e A. Let T be the length of the time horizon over which the dynamic flow model will be constructed.Let u(i,j) be the length of the shortest path from i e S to j e SUNUD.Let u(*,j) -min u(i,j) j e S U N U D. ieS Based on this data the space time network G • (D,N,G,R,V) can be constructed where D is the set of destination locations of P, hi is the space time node set of G, 6 is the green arc set of G, R is the red arc set of G, and V is a dummy arc set of G hi = {(i,t)/i eSUNJD, t« 0,...,T} the results of solving a network flow problem associated with this network can only be meaningfully interpreted when quantities sent from a source can be identified as to their final destination.Hence if there are multiple destinations in the set D the associated network flow problem must be formulated as a multicommodity network flow model to insure units leaving a specific source arrive at a specific destination.The multiconmodity case will be treated in a later paper.In the rest of this paper the focus will be on situations where D is a singleton.Then a standard network flow problem can be solved while insuring all units arrive at the appropriate destination.Next the network flow problem associated with G will be defined.Let P -(S,N,D,A) be as the above graph where D • {d} and let G = (D,N,G,R,V) be the associated space time network.Let x[(i,t),(j,t+k(i,j))] be the flow in arc [ (i,t) , (j ,t4k(i,j))] e G. Let x[(i,t),(i,t+l)] be the flow in arc [(i,t), (i,t+l)] e R Let x[(d,t),d] be the flow in arc [(d,t),d] e V.This is the flow arriving at the destination at time t.It is an instantaneous flow from location d at time t to a timeless representation of location d.Let b(s,t) be the supply of flow units at source seS at time t.Let G(s,t) = {[(s,t)(j,t+k(s,j))] e G} for seS and fixed t.
t,d),d] ^0 for {(t,d) | t = 0,...,T} The objective function (1) calls for the minimization of the sum of total flow time in the network.The constraint set specifies the following.Equation (2) says that the sum of flows leaving a source location s at time t via all green and red arcs emanating from that source must equal the supply b(s,t) at the source node.This must be true for all source locations and times.Equation (3) says that the sum of flows into any node other than a source or destination node must be equal to the sum of flows out of the node.Equation (4) indicates that the sum of flows into the destination at any specific time t must be equal to the flow out of the destination at that time t and into the dummy destination.Equation (5) states that the sum of the flows into the dummy destination must equal the sum of all supplies from all source locations in all time periods.Finally (6) and (7) specify that the flows on all green and red arcs are nonnegative and capacitated.As stated at the outset of the section, the model is dynamic in two aspects.The first is that inputs at source locations can be varied in any fashion over time.The output is also free to arrive at the destination in any feasible manner where feasibility is determined by the arc capacities.The second dynamic aspect is that the network itself can be made to vary over time.This can be done by changing the capacities or. the arcs.An arc can be closed to flow at a specific time simply by setting it f s capacity at zero.Similarly if conditions change so that at a specific time capacity on a path in the network is reduced then the capacity of the corresponding arcs can be reduced accordingly.III.DYNAMIC TRAFFIC FLOW MODEL Traffic flow has three features that car readily be captured by the dynamic two attribute network model.First traffic flow is dynamic because the quantity of traffic entering a given street network varies over time.During rush hour the entering traffic may well exceed the normal capacity of the street causing long delays, where only a few minutes before the rush hour period traffic might be light.This characteristic of traffic flow is accounted for in the model by allowing varying source quantities to flow into the model over time.Second as the quantity of traffic in the streets increases, congestion results, causing the travel time between any pair of locations to increase as a function of the number of vehicles on the road.This is precisely what occurs in the model.As green arcs between i and j reach their capacity traffic will be forced to utilize red arcs at i causing the travel time between i and j to increase as a function of the units traveling between i and j at a given time.Different travel time functions can be fitted by changing the capacity settings on the green arcs.Third, when traffic signals change the street network changes its structure.When a signal is red in a given direction the intersection has been closed in that direction until the signal becomes green again.When the signal changes the intersection is reopened in the original direction and it is closed in the direction perpendicular to the original.As a traffic signal changes from one time period to the next the effect is to change the capacity of the routes controlled by the signal.For all routes using the direction in which the signal is red, the route capacity has been temporarily reduced to zero.Changes in route structure as a result of traffic signal operation can be incorporated into the model by varying the capacities of green arcs over time.This feature of the model will be demonstrated by an example.Consider a two way intersection which normally requires one time unit to cross in any direction.Such an intersection can be represented by 8 nodes and 12 green arcs as in Figure 1.

1 Table 1 1 of the previous example a location where 2 way traffic crosses has four points of entry to the intersection where traffic streams may separate and four points of exit from 8 in Figure 1 )Figure 2 .FIGURE 2 .From this street diagram the graph P given in Figure 3 FIGURE 3 . 2 . 3 TABLE 2 . 3 Utilizing
Figure 2.

FIGURE 4 .
FIGURE 4. Dynamic graph for the example.Arcs from q to r have varying capacities as shown depending on the stop light settings.

Figure 6 shows the graph of the number of departures from the source in each period from time 0 to time 10 .The restrictions caused by the traffic signals are the cause of the irregular output. This is evidenced by the fact that the time from trough to trough and from peak to peak in the arrival graphs is 3 .
Figure 6 shows the graph of the number of departures from the source in

Figure 7
Figure 7 shows the average travel times for each departing group under

Figure 9 FIGURE 5 .FIGURE 8 .
Figure 9 shows the graph of the total number of departures from all sources