FLOW OVER TIME PROBLEM WITH INFLOW-DEPENDENT TRANSIT TIMES

Network flow over time is an important area for the researcher relating to the traffic assignment problem. Transmission times of the vehicles directly depend on the number of vehicles entering the road. Flow over time with fixed transit times can be solved by using classical (static) flow algorithms in a corresponding time expanded network which is not exactly applicable for flow over time with inflow dependent transit times. In this paper we discuss the time expanded graph for inflow-dependent transit times and non-existence of earliest arrival flow on it. Flow over time with inflow-dependent transit times are turned to inflow-preserving flow by pushing the flow from slower arc to the fast flow carrying arc. We gave an example to show that time horizon of quickest flow in bow graph G B was strictly smaller than time horizon of any inflow-preserving flow over time in G B satisfying the same demand. The relaxation in the modified bow graph turns the problem into the linear programming problem.


INTRODUCTION
Due to increasing population and the economic activities ever day, numbers of vehicles are increasing rapidly.But limited capacity of roads causes major traffic problem in most of the cities.To solve this problem, a better traffic management system and route guidance is essential.Dynamic network flow theory, traffic simulation, models based on fluid dynamics, control theory and variational inequalities are common approaches to study the problem.Simulation optimization is the process of finding the best input variable values from among all possibilities without explicitly evaluating each possibility.The objective of simulation optimization is to minimize the resources spent while maximize the information obtained in a simulation experiment.Fluid dynamics is specially based on differential equations.It defines continuous dynamic behavior of fluids or traffic flow in small size network only.Similarly, optimal control theory is an approach to the dynamic optimization without being constrained to interior solutions.However, it calculates on differentiability.
Time is an essential component for the flow of vehicles that travel through a road network.Thus, transmission time plays an important role on the vehicle rout problems.In this paper, we focus on flow-dependent transit times for the purpose of our study.Actually, flow dependent transit times can be divided in two ways: inflow-dependent transit times and load-dependent transit times.We focused our study in inflow-dependent transit times.In inflowdependent transit times, transit time on arc only depends on the current rate of inflow in to that arc (Kӧhler et al. 2002).In the context of road traffic, this assumption means that the time needed to drive through a street depends on the number of cars entering the road at that moment in time.That is, transit time is a function of inflow rate.In load-dependent transit times, total amount of flow on the arc is used as input of the transit time function (Kӧhler & Skutella 2005).
An important application of flow over time problem is evacuation planning problem.In continuous time setting, different dynamic network flow problems have been solved for evacuation planning problems.The continuous dynamic flow was studied and continuous contraflow models were introduced (Pyakurel & Dhamala 2015, 2016, 2018, Dhamala & Pyakurel 2013).Efficient algorithms were presented to solve maximum dynamic, quickest and earliest arrival contraflow problems using natural transformation of Fleischer and Tardos (1998) by reversing the direction of arcs at time zero.This paper is structured as follows: The basic concepts and flow models were given in first section.This section was followed by flow models of flow over time with inflow-dependent transit times which is presented by fan graph and bow graph.Next section included approximation to the quickest flow problem and nonexistence of earliest arrival flow with inflow-dependent transit times.This was followed by modified bow graph and quickest inflow preserving flow.Finally the last section included concluding remarks.

FLOW MODELS
Network flows are related with directed graphs G = (N,A), where N stands for nodes and A for arcs.Each arc a ∈ A has positive capacity denoted by u a and a non-negative, left-continuous and non-decreasing transit time function τ a : [0, u a ] → R + .Note that the non-decreasing function τ is left continuous if and only if sup{τ (x ' )|x ' < x} = τ (x).If arc a = (v,w) ∈ A then we denote head(a) for head node w and tail(a) for tail node v of arc a. Actually, the transit time τ a of an arc a is the amount of time it takes to reach from the tail to the head of that arc.In general, a flow entering arc a at time θ arrives at head (a) at time θ + τ a .We say that any flow over time has time horizon T if no flow is entering an arc a after the time T − τ a .
Similarly, the transit time of a path P in G with static flow x is defined as .In case of constant transit time of a path P, it is simply denoted by .Source nodes has supply D ≥ 0 and sink node t has demand D ≤ 0. Nodes that are neither source nor sink are called intermediate nodes.For the node v, we denote δ + (v) and δ − (v) for the set of arcs leaving and entering the node v respectively.Sometimes we consider flows with costs.Then, each arc a ∈ A has associated cost coefficients c a ≥ 0, where c a is interpreted as the cost per flow unit for sending through the arc.Temporally repeated flow: Let x be a feasible static flow over time in G with path decomposition (x p ) p∈P , where P is a set of paths such that the transit time τ p of every path p ∈P is bounded above by T. The temporally repeated flow f with transit time (τ a ) a∈A and time horizon T, as in (Hall et al. 2003a(Hall et al. , 2003b)), is defined as follows: 1.For every path p ∈ P, flow f enters path p ∈ P at constant rate x p starting at time zero and ending at time T − τ p (x).    Example 1: Let G consists of a path P of length 2 denoted by a 1 and a 2 with respective capacities u a 1 =2 and u a 2 = 2.The first arc on P has transit time function

The transit time of every arc
The second arc has constant transit time τ a 2 = 0 (Fig. 4).
There exist a flow over time in G B which sends D = 9 units of flow from s to t within given time T = 5.For, using temporally repeated flow which sends flow at rate 1 into the s-t path containing lower bow arc b 1 during time interval (0,5) and flow rate 1 into the s-t path containing upper bow arc b 2 during time interval (0,4).We define an inflow-preserving flow over time f in G B which satisfies demand 8 within time T = 5.It sends flow at rate 2 into bow arc b 2 during (0,3), then after it sends flow at the rate 1 into bow arc b 1 until time 5.  Lemma1: The value is bounded from above by .
Proof: Since is an increasing function of T. So it is suffices to show that | .

Non-existence of earliest arrival flows
Earliest arrival flows are those maximum s-t flows over time that sends, for each time θ [0, T), the maximum amount of flow from s to t.Although maximum flow for a fixed time horizon T exists, it may not be true for each θ [0, T).Gale (1959) showed the existence of earliest arrival flows for general networks with constant transit times on the arcs and, more generally, for networks with timedependent (but not flow-dependent) transit times and capacities on the arcs.In case of flow-dependent transit times there exists an s-t flow over time that sends the maximum amount of flow from s to t for any time horizon T. But there may not be such a maximum s-t flow that is maximal for each θ [0, T ) (Baumann & Kӧhler 2007).We will show this by the following simple example.
Example 2: Consider the one-arc network together with the simple linear transit time function given by τ(x) = 2x for 0 ≤ x ≤ 2 (Fig. 6) and a capacity two.
We consider a flow model with inflow-dependent transit times.Let T = 6 be considered time horizon.When sending flows from s to t at a flow rate of 2 in time interval (0,2) and at flow rate linearly decreasing from 2 to 0 in the time interval (2,6), then the flow of 8 units has been reached to the sink t by time T = 6.In fact, this is the maximum amount of flow that can be sent from s to t in this time horizon.

Fig. 6. Dynamic network with inflow-dependent transit times as in example 2
To construct an earliest arrival flow, we have to make sure that the maximum possible amount of flow has reached the sink for any θ (0, T).To show that this is not possible for this example we examine just two values of θ.
Sending flow at the flow rate linearly decreasing from 2 to 0 in time interval (0,4) shows that an earliest arrival flow must send at least 4 units of flow to t up to time θ = 4.In fact, sending any flow in this time interval at a higher flow rate would result in a decrease of the flow value reaching t up to time θ = 4.It follows easily that any flow sending the maximum amount of flow up to θ = 4 into t cannot send more than 5 units of flow into t up to θ = 6.However, the maximum flow for time horizon 6 is 8. Thus we have the following conclusion: Theorem 1: For inflow-dependent transit times, earliest arrival flow does not exist in general.

The modified bow graph
The modified bow graph, denoted by G B = (N B ,A B ), is defined on the same node set as G, i.e., N B = N, and is obtained by creating several copies of an arc, one for possible transit times on the arc as described in previous section.Thus arc a is replaced by creating m parallel bow arcs b 1 , b 2 , ...., b m .The transit time of bow arc b j is τ j and capacity u j for j =1, 2, ..., m (Fig. 7).We denote the set of bow arcs corresponding to arc a by and refer to as the expansion of arc a.The cost coefficient to every arc e ∈ are identical to those of arc a, i.e., c e = c a .For every arc e , let a(e) denote the original arc a.The main difference between this modified bow graph and previously defined bow graph is as follows: In the modified model, we omit the regulating arcs which, in the previous model, limit the amount of flow entering the bow arcs.In particular, all bow arcs representing the same original arc share capacity.In the modified model, capacities are directly assigned to the bow arcs.They no longer share capacities.Moreover, we include arc costs in the modified model.
Static network flow: A static network flow x: A → R + in G assigns to every arc a, a non-negative flow value x a which satisfies the flow conservation The static flow x is called feasible, if it satisfies the capacity constraints 0 ≤ , for all arcs The value of an s-t flow x is defined as In case of the flow with cost, the cost of a static flow x is defined as Continuous flow over time: The continuous flow over time f in G is a Lebesgue measurable function f a : A×R + → R + , for every a ∈A.Here, f a (θ) is the rate of flow per unit time that enters arc a at time θ.Clearly f a (θ) = 0 for θ 0. If the flow is allowed to storage at intermediate nodes, we write it as for all ζ ∈ [0, T ) and v ∈ N \ {s, t}.An equality holds for v ∈ N \ {s, t} at time ζ = T.The flow f is said to be feasible if 0 ≤ f a (θ) ≤ u a for all θ ∈ R + and a ∈ A. The flow over time f satisfies supplies and demands if for every v ∈ {s, t}.The value of s-t flow over time f is given by Here, | f | is the total amount of flow leaving the source node s until time T. Due to flow conservation, this value is equals to the total amount of flow arriving in the sink node t until time T. The cost of s-t flow over time f is defined as a ∈ A is fixed to τ a , i.e., at every point in time θ ∈ [0, T), flow units entering arc a at time θ reaches head (a) at time θ + τ a .The value of a temporally repeated s-t flow f with underlying static flow (x p ) p∈P is given by Discrete flow over time: Assume that all transit times are integral values.A discrete flow over time f in G assigns to every arc a ∈ A, a function f a : A × Z + → R + .We say that the flow over time f has time horizon T, if no flow is entering an arc a after the time T-1-i.e., f a (θ) = 0 for all .Flow conservation constraints are as similar to the continuous flow over time except the integral over time is replaced by the sum.It is modeled as for all ζ ≤ T−1 and v ∈ N \ {s, t}.Also the equality holds for v ∈ N \ {s, t} at time T-1.The flow f is said to be feasible if 0 ≤ f a (θ) ≤ u a for all θ ∈ Z + and a ∈ A. The discrete flow f satisfies supplies and demands if for every v ∈{s, t}.The cost of a discrete flow over time f is defined as Time expanded graph: For a graph G = (N,A) with integral transit times on arcs and integral time horizon T, the T-time expanded graph of G, denoted by G(T), is obtained by creating T copies of N which are labeled as N(0), N(1), ...., N(T-1) together with θ th copy of node v denoted by v(θ), θ ∈{0,1,....,T−1}.For every arc a = (v,w) A and 0 ≤ θ < Tτ a , there is an arc a(θ) from v(θ) to w(θ + τ a ) with the same capacity as arc a.If the storage of flow at node v N is allowed, we include an infinite capacity holdover arc from v(θ) to v(θ +1) for all 0 ≤ θ < T-1, which models the possibility to hold flow at node v.
Figure 1(b) represents time expansion of Fig. 1(a) for time horizon T = 6.Upward holdover arcs are included in time expanded graph, if storage at intermediate nodes is allowed, otherwise they are omitted.

Fig. 1 .
Fig. 1.(b) represents the T-time expansion of network (a) for time horizon T = 6 Flow-dependent transit times Inflow-dependent transit times: The main objective of this section was to study about flow over time with inflow-dependent transit times, which is an extension of the flow over time with fixed transit times.Transit times are fixed in flow over time with fixed transit times so that the flow on arc a progresses at constant speed.Here, in inflow-dependent transit times (Kӧhler et al. 2002), transit times experienced by an infinitesimal unit of flow on an arc is determined when entering this arc and only depends on the inflow rate at that moment of time.In the flow over time with inflow-dependent transit times, flow entering arc a at time θ arrives head (a) at time θ + τ a (f a (θ)).In particular, the transit time of an arc only depends on the current flow rate.In time dependent flow, all arc must be empty from time T on, so for all arcs a A and θ R + we have θ + τ a (f a (θ)) < T whenever f a (θ) > 0.Flow conservation, in this case, is modeled as

Fig. 2 .
Fig. 2.An expansion of single arc a=(v,w) with given transit times by definition of fan graph

Fig. 4 .
Fig. 4. Bow graph according to the transit time function of example 1 Approximation to the quickest flow problem Quickest inflow-dependent flow problem is to determine an s-t flow over time with inflow-dependent transit times that satisfies D within minimum time .Ford and Fulkerson (1958) assumed be the minimum time horizon and x B be a static flow in G B such that the value of the resulting temporally repeated flow in G B is

Fig. 5 .
Fig. 5.(b) represents the inflow preserving temporally repeated flow of origin bow flow of (a) Since all flow-carrying arcs carry flow in x B as well so the transit time τ P of every path is bounded from above by the time horizon of a quickest flow in G B .Now the path decomposition of induces a temporally repeated flow in G B for any time horizon .We choose such that It is very essential to note that the flow over time is inflow-preserving.

Fig. 7 .
Fig. 7.An expansion of single arc a= (v,w) according to modified blow graph Relaxation: In this section, we discuss the relationship between flows over time in the bow graph G B and flows over time with inflow-dependent transit times in G. Any flow over time f with inflow-dependent transit times ( ) a∈A in G with time horizon T and cost C can be interpreted as a flow over time (with constant transit times) in G B with same time horizon T and cost C as