Dynamic Flows with Adaptive Route Choice

We study dynamic network flows and introduce a notion of instantaneous dynamic equilibrium (IDE) requiring that for any positive inflow into an edge, this edge must lie on a currently shortest path towards the respective sink. We measure current shortest path length by current waiting times in queues plus physical travel times. As our main results, we show: 1. existence and constructive computation of IDE flows for single-source single-sink networks assuming constant network inflow rates, 2. finite termination of IDE flows for multi-source single-sink networks assuming bounded and finitely lasting inflow rates, 3. the existence of IDE flows for multi-source multi-sink instances assuming general measurable network inflow rates, 4. the existence of a complex single-source multi-sink instance in which any IDE flow is caught in cycles and flow remains forever in the network.


Introduction
Dynamic network flows have been studied for decades in the optimization and transportation literature, see the classical book of Ford and Fulkerson [7] or the more recent surveys of Skutella [22] and Peeta [17]. A fundamental model describing the dynamic flow propagation process is the so-called deterministic queue model, see Vickrey [25]. Here, a directed graph G = (V, E) is given, where edges e ∈ E are associated with a queue with positive rate capacity ν e ∈ R >0 and a physical transit time τ e ∈ R >0 . If the total inflow into an edge e = vw ∈ E exceeds the rate capacity ν e , a queue builds up and agents need to wait in the queue before they are forwarded along the edge. The total travel time along e is thus composed of the waiting time spent in the queue plus the physical transit time τ e . A schematic illustration of the inflow and outflow mechanics of an edge e is given in Figure 1. The fluid queue model has been mostly studied from a game-theoretic perspective, where it is assumed that agents act selfishly and travel along shortest routes under prevailing conditions. This behavioral model is known as dynamic equilibrium and has been analyzed in the transportation science literature for decades, see Friesz et al. [8], Meunier and Wagner [16] and Zhu and Marcotte [26]. In the past years, however, several new exciting developments have emerged: Koch and Skutella [14] elegantly characterized dynamic equilibria by their derivatives, which gives a template for their computation. Subsequently, Cominetti, Correa and Larré [5] derived alternative characterizations and proved existence and uniqueness in terms of experienced travel times of equilibria even for multi-commodity networks. Very recently, Cominetti, Correa and Olver [6] shed light on the behavior of steady state queues assuming single commodity networks and constant inflow rates. Sering and Vargas-Koch [21] analyzed the impact of spillbacks in the fluid queuing model and Bhaskar et al. [1] devised Stackelberg strategies in order to improve the efficiency of dynamic equilibria.
The concept 'dynamic equilibrium' assumes complete knowledge and simultaneous route choice by all travelers. Complete knowledge requires that a traveler is able to exactly forecast future travel times along the chosen path effectively anticipating the whole evolution of the flow propagation process across the network. This assumption has been justified by letting travelers learn good routes over several trips and a dynamic equilibrium then corresponds to an attractor of the underlying learning dynamic. While certainly relevant, this concept may not accurately reflect the behavioral changes caused by the wide-spread use of navigation devices. As also discussed in Marcotte et al. [15], Hamdouch et al. [11] and Unnikrishnan and Waller [24], drivers may not always learn good routes over several trips but are now informed in real-time about the current traffic situations and, if beneficial, reroute instantaneously no matter how good or bad that route was in hindsight. Also, the information available to a navigation device is usually not complete, that is, congestion information is available only as an aggregate (estimated waiting times for road traversal) but the individual routes and/or source and destinations of travelers are unknown -for good reason. 1 In this paper, we consider an adaptive route choice model, where at every node (intersection), travelers may alter their route depending on the current network conditions, that is, based on current travel times and queuing delays. The needed information is anonymous and indeed available by navigation devices. We assume that, if a traveler arrives at the end of an edge, she may change the current route and opt for a currently shorter one. This type of reasoning does neither rely on personalized information nor on the capability of unraveling the future flow propagation process. We term a dynamic flow an instantaneous dynamic equilibrium (IDE), if for every point in time and every edge with positive inflow (of some commodity), this edge lies on a currently shortest path towards the respective sink. In the following, we illustrate IDE in comparison to classical dynamic equilibrium with an example.

An Example
Consider the network in Figure 2 (left). There are two source nodes s 1 and s 2 with constant inflow rates u 1 (θ) ≡ 3 for times θ ∈ [0, 1) and u 2 (θ) ≡ 4 for θ ∈ [1,2). Commodity 1 (red) has two simple paths connecting s 1 with the sink t. Since both have equal length ( e τ e = 3), in an IDE both can be used by commodity 1. In Figure 2, the flow takes the direct edge to t with a rate of one, while edge s 1 v is used at a rate of two. This is actually the only split possible in an IDE, since any other split (different in more than just a subset of measure zero of [0, 1)) would result in a queue forming on one of the two edges, which would make the respective path longer than the other one. At time θ = 1, the inflow at s 1 stops and a new inflow of commodity 2 (blue) at s 2 starts. This new flow again has two possible paths to t, however, here the direct path ( e τ e = 1) is shorter than the alternative ( e τ e = 4). So all flow enters edge s 2 t and starts to form a queue. At time θ = 2, the first flow particles of commodity 1 arrive at s 2 with a rate of 2. Since the flow of commodity 2 has built up a queue of length 3 on edge s 2 t by this time, the estimated travel times e (τ e + q e (θ)) are the same on both simple s 2 -t paths. Thus, the red flow is split evenly between both possible paths. This results in the queue-length on edge s 2 t remaining constant and therefore this split gives us an IDE flow for the interval [2,3). At time θ = 3, red particles will arrive at s 1 again, thus, completing a full cycle (namely s 1 , v, s 2 , s 1 ). This example shows that IDE flows may involve a flow decomposition along cycles. In contrast, the (classical) dynamic equilibrium flow will just send more of the red flow along the direct path s 1 , t since the future queue growth at edge s 2 t of the alternative path is already anticipated.
Note, that cycles can appear even in the case of only a single commodity (and therefore a single source and sink) and a constant inflow rate over a single interval -see Example 3.8 for such an instance. This shows that the differences between the two equilibrium concepts are quite fundamental and occur even in very simple examples.

Related Work
In the transportation science literature, the idea of an instantaneous user or dynamic equilibrium has already been proposed since the late 80's, see Ran and Boyce [18, § VII-IX], Boyce, Ran and LeBlanc [2,19], Friesz et al. [9]. These works develop an optimal control-theoretic formulation and characterize instantaneous user equilibria by Pontryagin's optimality conditions. However, not much is known regarding IDE existence and their structural properties. In fact, the underlying equilibrium concept of Boyce, Ran and LeBlanc [2,19] and Friesz et al. [9] is different from ours. While the verbally written concept of an IDE is similar to the one we use here, the mathematical definition of an IDE in [2,9,19] requires that instantaneous travel times are minimal only for used paths towards the sink. A path is used, if every arc of the path has positive flow. As, for instance, the authors in Boyce, Ran and LeBlanc [2, p.130] admit: "Specifically with our definition of a used route, it is possible that no route is ever 'used' because vehicles stop entering the route before vehicles arrive at the last link on the route. Thus, for some networks every flow can be in equilibrium." Ran and Boyce [18,§ VII,pp.148 ] present a link-based definition of IDE. They define node labels at nodes v ∈ V indicating the current shortest travel time from the source node to some intermediate node v and require that whenever edge vw has positive flow, edge vw must be contained in a shortest s-w path, where s is the flow's source. This is different from our definition of an IDE, because we require that whenever there is positive inflow into an edge vw, it must be contained in a currently shortest v-t path, where t is the sink of the considered inflow.
Another important difference to our model is the assumed time horizon. The previous works [2,9,18,19] all assume a finite time horizon on which the control problems are defined, thus, only describing the flow trajectories over the given time horizon. All numerical studies and simulation results appearing in these works further implicitly assume that for given finitely lasting bounded inflow rates, there exists a finite time horizon [0, T ] with T large enough so that eventually all travelers reach their destination. Our results reveal that this is in fact not true: there are multi-commodity instances with finitely lasting bounded inflows that admit IDE flows cycling forever. For the discrete version of this model using the natural discrete version of our equilibrium concept, such a behavior was already discovered in Ismaili [12,Theorem 8], though his instance makes critical use of edges with zero transit time (which we do not allow) and the instantaneous travel time always increases when there is positive inflow into an edge, even when the edge inflow rate is smaller than the rate capacity. In particular, players may observe increased instantaneous travel time, although no player is in fact delayed. This is not possible in our model. Based on this flexibility, the generated instance is considerably simpler than ours but one can show that IDE flows in our sense do terminate in that instance.

Our Results
In this paper we introduce the concept of an instantaneous dynamic equilibrium (IDE for short). We call a feasible flow over time an IDE, if at any point in time, every edge with positive inflow (of some commodity) lies on a currently shortest path towards the respective sink.
Our first main result (Theorem 3.4) shows that IDE exist for multi-source single-sink networks with piecewise constant inflow rates (generating the volume of agents originating at the sources). The existence proof relies on a constructive method extending any IDE flow up to time θ to an IDE flow on a strictly larger interval θ + for some > 0.
The key insight for the extension procedure relies on solving a sequence of nonlinear programs, each associated with finding the right outflow split for given node inflows. We also show that such solutions can be found by a simple water filling procedure. With the extension property we can apply a limit argument on the real numbers implying the existence of IDE on the whole R ≥0 .
Given that, unlike the classical dynamic equilibrium, IDE flows may involve cycling behavior (see the example in Figure 2), we turn to the issue of whether it is possible that positive flow volume remains forever in the network (assuming finitely lasting bounded inflows). Our second main result (Theorem 4.6) shows that for multi-source single-sink networks, this is impossible: Even for arbitrary bounded and finitely lasting inflow rate functions, there always exists a finite time T > 0 at which the network is cleared, that is, all flow particles have reached their destination.
We then turn to general multi-commodity networks. For given piece-wise constant network inflow rates, we also can extend an IDE flow to a strictly larger time interval by determining the edge inflow rates and the derivatives of the current shortest path distances. This extension relies on a solution of a system of equations, which is guaranteed to exist by Kakutani's fixed point theorem. The equations are inspired by those defining a "thin flows with resetting" for Nash flows over time that were introduced by Koch and Skutella [14] and refined by Cominetti et al. [4,5]. Again, a limit argument proves the existence of IDE flows over the whole R ≥0 (Theorem 5.5). Furthermore, we give a brief idea, how these extensions can be obtained by a mixed integer formulation. We then extend this result to arbitrary locally-integrable network inflow rate functions by using a theorem about the existence of a solution to a variational inequality in infinite-dimensional function spaces (Theorem 5.8).
Finally, we show that for bounded and finitely lasting inflow rates, termination in finite time is not guaranteed anymore as soon as the network contains more than one sink (Theorem 6.1). We construct a quite complex instance where all IDE flows are caught in cycles and travel forever. This instance reveals that the assumption of a finite time horizon [0, T ] made previously in the transportation literature cannot be made without loss of generality. We also show that the instance can be modified in such a way that only a single-source is needed.

The Flow Model
In the following, we describe a fluid queuing model as used before by Koch and Skutella [14] and Cominetti, Correa and Larré [5] and introduce the notation we will use throughout this paper.
We consider a finite directed graph 2 G = (V, E) with positive rate capacities ν e ∈ R >0 and positive transit times τ e ∈ R >0 for every e ∈ E. 3 There is a finite set of commodities I = {1, . . . , n}, each with a commodity-specific source node s i ∈ V and a commodity-specific sink node t i ∈ V . We will always assume that there is at least one s i -t i path for each i ∈ I. The (infinitesimally small) agents of every commodity i ∈ I enter the network according to a integrable network inflow rate function u i : The queue length of edge e at time θ is given by where denote the cumulative (edge) inflow and cumulative (edge) outflow. We implicitly assume f − i,e (θ) = 0 for all θ ∈ [0, τ e ), which will ensure together with Constraint (4) (see below) that the queue lengths are always non-negative. Furthermore, we define the cumulative network inflow rate by U i (θ) := θ 0 u i (z)dz and, for the sake of simplicity, we denote the aggregated flow over all commodities by f + e := i∈I f + i,e and f − e := i∈I f − i,e , as well as, A feasible flow over time satisfies the following conditions (2), (3), (4), and (6). The flow conservation constraints are modeled for a commodity i ∈ I and all nodes v = t i as where δ + v := { vu ∈ E } and δ − v := { uv ∈ E } are the sets of outgoing edges from v and incoming edges into v, respectively. For the sink node t i of commodity i we require We assume that the queue operates at capacity which can be modeled by Since Finally we want the flow to follow a strict FIFO principle on the queues, which can be formalized by the following condition (see [20]): where ϑ := min { ϑ ≤ θ | ϑ + τ e + qe(ϑ) νe = θ } is the earliest point in time a particle can enter edge e in order to leave it at time θ. The quotient qe (θ) νe is hereby the current waiting time to be spent in the queue of edge e. In other words, Constraint (6) ensures that the share of commodity i of the aggregated outflow rate at some point in time θ equals the share of commodity i of the aggregated inflow rate at the time the particles entered the edge.
Note however, that all results within this paper also hold if we relax the FIFO condition (6) to the following condition This condition allows for overtaking within a queue and only prevents flow from changing its commodity on an edge by requiring the total amount of flow of every commodity that has left an edge to not exceed the total amount of flow of this commodity that has reached the head of this edge up to that point in time. We assume that, whenever an agent arrives at an intermediate node v at time θ, she is given the information about the current queue lengths and transit times q e (θ), τ e , e ∈ E, and, based on this information, she computes a shortest v-t path and enters the first edge on this path (breaking potential ties arbitrarily). We define the instantaneous travel time of an edge e at time θ as We can now define commodity-specific node labels i,v (θ) corresponding to current shortest path distances from v to the sink t i . For i ∈ I, v ∈ V and θ ∈ R ≥0 , define We say that edge e = vw is active for i ∈ I at time θ, if i,v (θ) = i,w (θ) + c e (θ) and we denote the set of active edges for commodity i by E i θ ⊆ E. We call a v-t i path P an active v-t i path for commodity i at time θ, if all edges of P are active for i at θ or, equivalently, e∈P c e (θ) = i,v (θ). For differentiation we call paths that are minimal with respect to the transit times τ physical shortest paths. Now we are ready to formally define an instantaneous dynamic equilibrium for multicommodity flows over time: In other words, a feasible flow over time f is an IDE, if, whenever flow of commodity i enters an edge e = vw at some point θ, this edge is contained in the set of active edges E i θ , i.e., e lies on a currently shortest path from v to t i . Note that, while the set of active edges E i θ changes over time, the set of nodes, from which t i is reachable via does not. In particular, this means that, whenever flow of commodity i enters an active edge uv, v = t i , at least one of the edges in δ + v will be active by the time the flow reaches v -although possibly different edges than those active when the flow left u.

Existence and Computation of IDE Flows in Single-Sink Networks
In this and the following chapter we only consider single-sink networks, i.e., networks where all commodities have one common sink node t. In this case all commodities have the same label function, which we will denote by v . Since the origin of a particle in the network is not important for its route to the sink, we do not distinguish the commodities, and instead, only consider the aggregated flow functions f + e and f − e . Furthermore, we restrict the network inflow functions u i to be right-constant, where a function u : [a, b) → R is right-constant, if for every θ ∈ [a, b) there exists an ε > 0 such that u is constant on [θ, θ + ε). For this case, we will now describe an algorithm computing an IDE flow.
Let f = (f + , f − ) denote a feasible flow over time. We denote by the current inflow at node v at time θ. Moreover, let δ + v (θ) := δ + v ∩ E θ denote the set of outgoing edges of v that are active at time θ. The main idea of our algorithm works as follows. Starting from time θ = 0 we compute inductively a sequence of intervals [0, θ 1 ), [θ 1 , θ 2 ), . . . with 0 < θ i < θ i+1 and corresponding constant edge inflows (f + e (θ)) e for θ ∈ [θ i , θ i+1 ) that form together with the corresponding edge outflows (f − e (θ)) e an IDE. Suppose we are given an IDE flow up to time θ k , that is, a tuple (4) and (10). Note that this is enough information to compute F + e (θ k ) and F − e (θ k + τ e ), and thus also q e (θ k ), c e (θ k ) and v (θ k ) for all e ∈ E and v ∈ V . We now describe how to extend this feasible flow over time to the interval [θ k , θ k + ε) for some ε > 0. The idea is that whenever there is positive inflow b − v (θ k ) > 0 into some node v ∈ V , we assign this inflow to outgoing edges that are currently active. Since the node labels at the heads of these edges depend themselves on queue dynamics at other nodes along a currently shortest path towards t, we need to handle time-varying labels w (θ) when distributing the flow among the edges in δ + v (θ). In the following, we describe how to define the flow-split in order to maintain the invariant that flow is only assigned to edges that are active for at least some interval even if adjacent labels vary linearly over time.
Suppose that the labels of nodes w i change linearly after θ k , that is, there are constants Our goal is to find constant edge inflow rates f + vw i (θ) during [θ k , θ k + ε) satisfying the supply b − v (θ) and, for some ε > 0, fulfilling the following invariant for all i ∈ [p k ] and θ ∈ [θ k , θ k + ε ): Note that a constant inflow rate f + vw i implies by (5) that the queue length q vw i is piecewise linear. Hence, the instantaneous travel time c vw i is also piecewise linear on [θ k , θ k + ε ) for some ε > 0, with derivative and, thus, a flow with constant inflow rates satisfies (13) and (14) for This condition ensures that whenever an edge vw i has positive inflow, the remaining distance towards t grows from θ k onwards at the lowest speed. For a given value b − v (θ k ) and given vector (a w i ) i∈[p k ] we consider the following optimization problem in variables x vw i for i ∈ [p k ] in order to obtain inflow rates that satisfy the conditions (15) and (16).
Each function g vw i maps an edge inflow rate z to the change of the queue size if a constant flow with rate z would enter the edge vw i , i.e., (5)). (15) and (16) for all i ∈ [p k ]. Proof. The objective function is continuous and the feasible region is non-empty and compact. Hence, by the theorem of Weierstraß at least one optimal solution exists. More-over, the objective is differentiable over the feasible domain, thus, first order optimality conditions hold. Assigning a multiplier λ ∈ R to (17) and taking partial derivatives of the Lagrangian over the positive orthant, we obtain
Proof. First, it is possible to determine the queue lengths at time θ k using Constraint (1) and from those the labels v (θ k ) can be obtained. Applying Lemma 3.1 on the nodes in order of increasing v (θ k ) values, we obtain the outflow rates and, therefore, the slope a v of label v for some interval right after θ k . More precisely, we start with t (since where ε := ε. Now we take some node v such that there exists an ε > 0 and for all nodes w with strictly smaller label at time θ k and all edges e ∈ δ + w , we have already defined f + e on some interval [θ k , θ k + ε ) and f − e on some interval [θ k + τ e , θ k + τ e + ε ) in such a way that on the interval [θ k , θ k + ε ) 1. the labels w (θ) change linearly with slope a w , 2. no additional edges are added to the sets δ + w (θ) of active edges leaving w, 3. the functions f + e and f − e for e ∈ δ + w are constant and right-constant, respectively, and 4. the functions f + e and f − e for e ∈ δ + w satisfy Constraints (2), (4) and (10).
. . , vw p k } be the set of active edges at v at time θ k . Then, at time θ k , each w i must have a strictly smaller label than v. Hence, they satisfy Properties 1.-4. We can now apply Lemma 3.1 to determine the flows f + vw i (θ k ). Additionally, we set f + e (θ k ) = 0 for all non-active edges leaving v. Assuming that this flow remains constant on the whole interval [θ k , θ k + ε ), we can determine the first timeθ ≥ θ k , where an additional edge vw ∈ δ + v or wv ∈ δ + w with w ∈ V a node with already defined label w for the coming time interval becomes newly active. This can only happen after some positive amount of time has passed, i.e., for someθ > θ k , because • at time θ k the edge was non-active, and therefore • all labels change linearly (and thus continuously) and • c vw or c wv is changing piecewise linearly, since the length of its queue does so as well (as both f + vw and f − wv are piecewise constant).
If the differenceθ − θ k is smaller than the current ε , we take it as our new ε , otherwise we keep it as it is. In both cases, . This guarantees that the label of v changes linearly on this interval, no additional edges become active and the functions f + e are constant. Also, f + e satisfies Constraints (2) and (10) by definition. Finally, we define f − e as follows: Then, f − e is right-constant and together with f + e satisfies Constraint (4). In summary, using this procedure we can extend f node by node to an IDE flow up to θ k + ε for some ε > 0.
Remark 3.3. The above lemma is the key for the following existence result of IDE flows. For the extension property, we used that travel times are strictly positive, because otherwise, for a given IDE flow up to time θ k , the edge outflow rates are not well-defined for any proper interval after θ k assuming zero travel times. Existence of IDE flows is not ruled out for τ e ∈ R ≥0 , but this would require the use of a more intricate way of describing the extensions or a completely different approach.
This set is not empty so we setθ 1 Let θ * be the limit of these two sequences. By taking pointwise limits of the sequence (f i ) i∈N we can construct a flow f * such that (f * , θ * ) ∈ F 0 . By Lemma 3.2 we can extend f * by some ε but this is a contradiction to the definition ofθ i for all i withθ i ∈ [θ * , θ * + ε). Hence,θ 0 = ∞, which finishes the proof.
Remark 3.5. While there always exists an IDE flow, this flow does not have to be unique. In fact, neither the flow f itself nor the label functions v and the time of termination need to be unique. This is in contrast to dynamic equilibria, where at least the label functions are uniquely determined (see Cominetti et al. [5,Theorem 6]). An example for non-uniqueness is illustrated by the instance in Figure 3.
s v (1,2) w (1,2) t (1,2) (1, 1) Remark 3.6. The task to find suitable inflow rates x e at a given time θ k can also be formulated globally as follows: Find two vectors (x e ) e∈E θ k , (a v ) v∈V such that: where This formulation is very similar to the thin flow with resetting formulation of dynamic equilibria (see [14], [5]) and in the same way the existence of a solution can be shown via Kakutani's fixed point theorem. We will extend this formulation to a multi-commodity version in order to show the existence of IDE flows in a multiple sink setting in Section 5.
We have proven now that an IDE flow always exists, but in order to actually compute an IDE flow we need to make Lemma 3.1 constructive, i.e. provide an algorithm to find an optimal solution to ( (15) and (16).
) is a convex program there are various methods available -but thanks to the particular nice structure of (OPT-b − v (θ k )) even a simple water filling procedure suffices: First, note that we have and thus, setting . Without loss of generality we assume that the nodes w i are sorted such that (15) and (16). Algorithm 1: Water filling procedure for flow distribution as well as Constraints (15) and (16).
Proof. First, note that the functions h i are non-decreasing, continuous and unbounded on R ≥0 . Thus, all maxima in Algorithm 1 are well defined. Since the functions are even piecewise linear, all these maxima can actually be determined in constant time.
It remains to show that the returned which is a straight forward calculation.
Thus, we can indeed compute an IDE flow for single-sink networks with right constant network inflow rates, provided that 1. the IDE flow eventually terminates and 2. the ε in Lemma 3.2 does not become arbitrarily small (and therefore no limit process is needed in Theorem 3.4).
While the general validity of the second assumption currently remains an open question, we will show in the following Section that the first assumption holds for all IDE flows in single-sink networks. We now give an example for the computation of an IDE: Example 3.8. We use the same graph as in the example in Section 1.1, but with different rate capacities and only a single commodity with source node s, sink node t and a constant network inflow rate of 16 over the interval [0, 1] (see the top left picture in Figure 5). f + sv (0) = 7 + 7. This ensures that both edges remain active up to time θ 1 = 1, where the inflow into node s stops and the first flow particles arrive at node v. Then, the only node with positive b − v is v, but since v has only one edge leaving this node, no flow distribution is necessary. At time θ 2 = 2, the flow arriving at node w only has one active edge, and therefore enters the edge towards t at a rate of 7. At time θ 3 = 2.5, the edge ws becomes active and the outflow from node w (at rate 7) needs to be redistributed. Algorithm 1 gives us the inflow rates f + wt (2.5) = 1 and f + ws (2.5) = 6. From time θ 4 = 3.5 onward, flow arrives at node s at rate 6 and at node w at rate 7. Since s is currently closer to t than w ( s (3.5) = 3 < 4 = w (3.5)), we start by distributing the outflow of s. From s only the edge towards t is active, so all flow enters this edge. This means that a queue will start to build up on the edge st at a rate of 5, and therefore a s = 5. Thus, Algorithm 1 gives us the inflow rates f + wt (3.5) = 6 and f + ws (3.5) = 1. At time θ 5 = 4, the only active edge for the flow arriving at node s is st. At time θ 6 = 4.5, the edge sv becomes active, too.
Thus, we need to compute a new distribution for the flow arriving at node s (at rate 1). Since we have no outflow from node w, the queue on edge wt decreases at a rate of 1 leading to a w = −1, and therefore a v = −1. Algorithm 1 then gives us the inflow rates f + st (4.5) = 0 and f + sv (4.5) = 1. From time θ 7 = 5 on only the physically shortest paths are active, so all flow particles will stay on their currently chosen path towards t. The IDE flow computed by this procedure is displayed in Figure 5.

Termination of IDE Flows in Single-Sink Networks
In this section, we investigate the question, whether an IDE flow vanishes within finite time given finitely lasting network inflow rates. More precisely, given a time θ 0 , such that supp(u i ) ⊆ [0, θ 0 ] for every i ∈ I, we ask whether there exists a timeθ ≥ θ 0 , such that all injected flow actually reaches the sink within timeθ. To answer this question, we first need to introduce some additional notation. For every edge e ∈ E, we define a function F ∆ e denoting the total amount of flow currently on edge e (either waiting in its queue or traveling along the edge) for any time θ. As in [21], we call these the edge load functions: specifies the total amount of flow in the network at time θ. Furthermore, we define a function Z indicating the amount of flow that already reached the sink t by time θ: Note that for IDE flows the subtrahend is always 0 since edges leaving t are never active. For every subset W ⊆ V and any time θ a direct computation shows that we have Since is always non-negative by Constraint (3), it follows immediately that the total amount of flow in the network is non-increasing after time θ 0 . More generally, since all F + e are non-decreasing, for all In particular, forθ ≥ θ 0 with F ∆ (θ) = 0, we have F ∆ (θ) = 0 for all θ ≥θ.
We show next that for acyclic networks every feasible flow over time terminates. This intuitive result will serve as a building block for the more general result that in a singlesink network all IDE flows terminate. Proof. Since the graph is acyclic, we can consider a topological order < on V , i.e., uv ∈ E =⇒ u < v. Without loss of generality, let t be reachable from every node, since the part of the graph that cannot reach t will never be used by any feasible flow over time and can therefore be removed. Thus, t must be the last (smallest) element in this order. We will also use the notation e ≤ w to indicate that an edge e = uv lies before w, i.e., u, v ≤ w. Now, given a feasible flow over time f we can define a function F ∆ ≤w for every node w that gives us the total edge load before w at any time θ, i.e., F ∆ ≤w (θ) = e≤w F ∆ e (θ). As a first step, we show that after θ 0 , if there is no flow before some node w (i.e., on edges e ≤ w), there will be no flow on any edge leaving w some time later, or, more formally: Claim 1. Let e = wv ∈ E and θ w ≥ θ 0 such that we have F ∆ ≤w (θ w ) = 0. Then, there exists a time θ wv ≥ θ w such that for all θ ≥ θ wv , we have F ∆ e (θ) = 0.
Proof of Claim 1. We first show that F + e (θ) = F + e (θ w ) for all θ ≥ θ w . We define the set W := { u ∈ V | u ≤ w }, so that we have wv ∈ δ + W , as well as δ − W = ∅ and since F + e is non-decreasing, t / ∈ W and U i (θ) = U i (θ w ) for all i ∈ I. Hence, we have F + e (θ) = F + e (θ w ).

By Lemma 4.2 there exists a time θ wv
Proof of Claim 2. We show this by induction on the number of nodes greater than v in the given topological order on V . Our base case is, that there are no nodes w < v. Then there are also no edges e ≤ v, and therefore F ∆ ≤v (θ) = 0 holds for all θ ≥ θ 0 . So, we can assume that for all w < v there are already times θ w ≥ θ 0 with F ∆ ≤w (θ) = 0 for all θ ≥ θ w . Then for every edge wv ∈ E, Claim 1 gives us a time θ wv ≥ θ w with F ∆ e (θ) = 0 for all θ ≥ θ wv . Setting Finally, the lemma follows directly from Claim 2 by setting v = t, as then we have In the next step we show that if the sum of all edge loads between a node v and the sink t are small enough (and, thus, in particular all queues on edges between v and t are small), then an IDE flow can not be diverted away from the physically shortest paths towards t. Since those physically shortest paths form a time independent acyclic subgraph, we will be able to apply Lemma 4.3 to the flow inside this subgraph. For the next lemma, we need the minimal non-zero difference between two path lengths τ ∆ := min { τ (P ) − τ (P ) > 0 | u ∈ V, P, P two u-t paths } and the minimal rate capacity ν min := min { ν e | e ∈ E }.

Lemma 4.4.
If, for some node v ∈ V and some time θ ∈ R ≥0 , every physical shortest v-t path (i.e., w.r.t. τ ) has total flow volume of less than τ ∆ · ν min , then all active v-t paths at time θ are also physical shortest v-t paths, i.e., if e∈P F ∆ e (θ) < τ ∆ · ν min for all physical shortest v-t paths P , then the following holds: P is an active v-t path at time θ =⇒ P is a physical shortest v-t path.
Proof. Let P be a physically shortest v-t path and P an active v-t path. Then we have This implies 0 ≤ e∈P τ e − e∈P τ e < τ ∆ , and therefore e∈P τ e = e∈P τ e as τ ∆ is the smallest nonzero distance between two physical path lengths. Thus, P is also a shortest path w.r.t. the physical transit times τ .
Proof. By (24) we also have F ∆ (θ) ≤ F ∆ (θ) < τ ∆ ·ν min for all θ ≥θ. So from Lemma 4.4 we know that afterθ only shortest paths can be active, and therefore the flow only uses a time independent acyclic subgraph of G. Thus, by Lemma 4.3 the flow terminates. Proof. Let W ⊆ V be a subset of nodes with the following properties: 1. For every w ∈ W , all physical shortest w-t paths only use edges in E(W ).
We show that for every such W = V , there exists a node v ∈ V \ W such that W ∪ { v } also has the two properties. Since W = { t } satisfies the two properties this shows that W = V exhibits those as well and, in particular, there exists some time for all edges e ∈ E, and therefore e∈E F ∆ e (θ V ) < τ ∆ · ν min . Hence, by Corollary 4.5, f terminates.
Let W V be a set of nodes fulfilling both properties and v ∈ V \ W be the node with the shortest distance to t with respect to τ of all nodes in V \ W . Then, all physically shortest v-t paths only use edges from E(W ∪ { v }), so the first property holds for W ∪ { v }. Since the second property holds for W we know that for all θ ≥ θ W , we have Lemma 4.4 implies that for every node w ∈ W , all active edges leaving w have to be in E(W ), i.e., δ + w ∩E θ ⊆ E(W ). Since f is an IDE flow this implies f + e (θ) = 0 for all e ∈ δ + W , and thus F + e (θ) = F + e (θ W ) for all those edges. We now assume by contradiction that the second property does not hold for W ∪{ v }, so there is an edge e ∈ δ + v ∩δ − W and a sequence of times θ 1 < θ 2 < . . . with F ∆ e (θ k ) ≥ τ ∆ ·ν min |E| for all k ∈ N and θ k → ∞ for k → ∞.

From Lemma 4.2, we get times
By possibly taking a subsequence, we can assume θ k−1 ≤ θ k for all k, and thus, Hence F − e (θ k ) tends to infinity as k grows larger. On the other hand (22) states that which in turn is bounded from above as all positive summands are bounded as well: since θ k ≥θ and W has the second property.
• Z(θ k ) This is a contradiction. So the second property must also hold for W ∪ { v }, which concludes the proof.

Existence of IDE Flows in Multi-Sink Networks
We now turn to the general model with multiple sinks. In this section, we will give two different proofs for the existence of IDE flows in such networks. The first one is similar to the proof for single-sink networks in Section 3 using the concept of thin flows, and thus, constructive in the same sense. The second proof uses an infinite dimensional variational inequality. While not constructive, it allows for much more general network inflow rate functions u i (and interestingly, avoids the need for a limit argument to extend the flow for all times altogether).

Right-Constant Network Inflow Rates
First, we want to show that IDE flows exist in networks with multiple sinks, still under the assumption that the network inflow rates u i are right-constant. As before, we show that for a given IDE flow up to some point in time θ k it is possible to extend it by some ε > 0. In contrast to the case of a single sink, we need to determine all inflow rates f + i,e and the slopes a i,k of the current shortest path distances at the same time.
First of all, we denote the current inflow at a node v for commodity i by where 1 v=s i = 1 if v = s i and 1 v=s i = 0 otherwise. With this we can extend the idea of IDE thin flows (inspired by the thin flows with resetting for dynamic equilibria; see [14], [5]) introduced in Remark 3.6 to the multi-commodity setting.
x i,e = 0 for all i ∈ I and e ∈ E \ E i θ k , (TF2) As a first step, we show that such an IDE thin flow always exists for every reasonable network (i.e., every node v with b − i,v (θ k ) > 0 can reach t i within E i θ k ) and all current inflow vectors b − (θ k ). We do this by utilizing the following fixed point theorem by Kakutani [13]: (Kakutani's Fixed Point Theorem). Let K be a compact, convex and non-empty subset of R n , n ∈ N, and Γ : K → 2 K , such that for every x ∈ K the image Γ(x) is non-empty and convex and the set { (x, y) | x ∈ K, y ∈ Γ(x) } is closed. Then there exists a fixed point x * of Γ, i.e., x * ∈ Γ(x * ).
With this theorem we can prove the following lemma:

there exists an IDE thin flow (x, a).
Proof. For every vector x ∈ R I×E ≥0 satisfying (TF1) and (TF2) there exist uniquely defined node labels a ∈ R I×V ≥0 that fulfil (TF3) and (TF4). Existence follows since E i θ k is acyclic and the uniqueness follows from the fact that for every v there is a v-t i -path within E i θ k . This mapping x → a is continuous. So the only difficult part is to satisfy (TF5).
Let K be the set of x vectors satisfying (TF1) and (TF2), i.e., Clearly, K is compact, convex and non-empty. We define a set-valued function Γ : K → 2 K as follows: where a are the label corresponding to x. Then Γ(x) is non-empty and convex. For non-emptiness note that every node v with b − i,v > 0, except t i , has at least one outgoing edge with a i,v = ge j∈I x j,e /νe +a i,w , so y can send everything into this edge. Convexity is clear as well since x determines which edges can be used and which not and these are fixed within Γ(x).
Finally, we show that { (x, y) | x ∈ K, y ∈ Γ(x) } is a closed set. Therefore, let (x n , y n ) n∈N be a sequence in this set that converges in R I×E × R I×E . Since K is compact, both sequences separately converge to some points x and y in K. Let (a n ) n∈N be the sequence of associated node labels of x n and a the node label of x. Since x → a is continuous we have a = lim n→∞ a n . We need to show that y ∈ Γ(x). Suppose for contradiction that there is a commodity i ∈ I and an e = vw ∈ E i θ k with y i,e > 0 and a i,v < ge j∈I x j,e /νe + a i,w . But since g e is continuous, there has to be an n 0 ∈ N such that y n i,e > 0 and a n i,v < ge j∈I x n j,e /νe + a n i,w for all n ≥ n 0 . This is a contradiction to y n ∈ Γ(x n ).
Hence, by Kakutani's fixed point theorem (Theorem 5.2) there exists an x * ∈ K with x * ∈ Γ(x * ), which forms together with the associated node label a * an IDE thin flow.
Consider an IDE flow f up to time θ k where the inflow rates f + i,e are right-constant. Due to the continuity of q e and a i,v we can determine the active arcs E i θ k , as well as, the current node inflows b − i,v (θ k ) since the feasibility conditions (4) and (6) determine unique outflow rates f − i,e (θ k ) for given inflow rates f + i,e from the past. In order to extend f , we consider an IDE thin flow (x, a) and extend the inflow rates and current shortest path distances for all i ∈ I, e ∈ E and v ∈ V by We call this extended flow over time an α-extension.
To ensure that we end up with an IDE up to time θ + α, the following requirements on the size α of the extension phase [θ k , θ k + α) must be satisfied: First of all, the queues can never be negative, and therefore, the phase ends as soon as a queue depletes: for all e ∈ E with q e (θ k ) > 0. (25) Furthermore, the phase ends as soon as an inactive edge gets active. Since queues can build up on inactive edges as well (due to flow of other commodities), we need to take into account the changing rate of a queue, as well. Hence, for all i ∈ I and e = vw ∈ E \ E i θ k we have: Finally, the current node inflow should stay constant during a phase: We call α > 0 feasible if it satisfies (25), (26) and (27). It is easy to see that such a feasible α > 0 always exists since i,v (θ k ) < τ e + qe(θ k ) νe + i,w (θ k ) for all i ∈ I and e = vw ∈ E \ E i θ k . Furthermore, the functions b − i,v are rightconstant, since f + i,e as well as u i are right-constant. Since τ e > 0 for all e ∈ E we have that b − i,v (θ k ) is well-defined and constant on some small interval [θ k , θ k + ε). Proof. First note, that the feasibility conditions are satisfied, since the outflow rates f − i,e are exactly defined that way. Furthermore, flow conservation holds since Next we show that the labels satisfy Equation (9). Given a point in time θ k + ξ with ξ ∈ [0, α) we have by (5) applied on the total inflow rate f + e (θ k + ξ) that Note that we have q e (θ k + ξ) = q e (θ k ) + ξ · g e j∈I x j,e since q e (θ k + ξ) is constant for ξ ∈ [0, α).
For non-active arcs e = vw / ∈ E i θ k we have by (26) that For active arcs e = vw ∈ E i θ k we have by (TF4) that Since there has to be one active arc that satisfies (TF4) with equality, the same arc satisfies the inequality above with equality, which shows that (9) holds. In other words, the extended labels denote the current shortest path distances in the α-extension.
Finally, we show that the α-extension satisfies the IDE condition (10): For all ξ ∈ [0, α) and all arcs e = vw ∈ E we have that f + i,e (θ k +ξ) > 0 implies that x i,e > 0, and therefore, which shows e ∈ E i θ k +ξ . Hence, the α-extension is indeed an IDE flow up to time θ k+1 = θ k + α. The proof is exactly the same as the proof for Theorem 3.4 but this time we extend the IDE flow by Lemma 5.4.
Since Kakutani's fixed point theorem doesn't give much insight into how to construct such multi-commodity thin flows, we want to give a brief idea how to do this with a mixed integer program. For this we introduce two different types of boolean decision variables. For every i ∈ I and e ∈ E i θ k we have y i,e ∈ { 0, 1 } and for every e ∈ E with q e (θ k ) = 0 we have z e ∈ { 0, 1 }.
When these decision variables are guessed correctly, the task to find an IDE thin flow is a simple linear program. It remains open if the complete thin flow can be computed efficiently.

Arbitrary Network Inflow Rates
Next, we want to show that multi-sink IDE flows exist even with very general network inflow functions. Recall that the L 2 -space is defined by where two function are equal if they are equal almost everywhere. Together with the scalar product x, y = b a x(ξ)·y(ξ) dξ, it forms a Hilbert space. If we have vectors of functions f, g ∈ L 2 ([a, b)) d , for d ∈ N, the scalar product is ([a, b)) d . For a subset K ⊆ L 2 ([a, b)) d we call a mapping A : K → L 2 ([a, b)) d weak-strong-continuous at f ∈ K, if for every f k ∈ K that converges weakly to f , we have that A(f k ) converges to A(f ) with respect to the L 2 -norm. Lemma 5.6. Given a network with a finite set of commodities I with arbitrary sources and sinks and bounded network inflow functions u i such that u i [a,b) ∈ L 2 ([a, b)) for all a < b and i ∈ I. Let f be an IDE flow up to time φ ≥ 0. Then, for any 0 < ε < min { τ e | e ∈ E }, we can extend f to an IDE flow up to time φ + ε.
In order to prove this, we utilize the following variational inequality: ([a, b)) d and a mapping A : K → L 2 ([a, b)) d , then the variational inequality VI(K, A) is the following: Find g ∈ K such that A(g), g − g ≥ 0 for all g ∈ K. (VI) Conditions to guarantee the existence of such an element g are given by Brézis [3,Theorem 24] (see also [23]): Theorem 5.7. Let K be a nonempty, closed, convex and bounded subset of ([a, b)) d be a weak-strong-continuous mapping. Then, the variational inequality VI(K, A) has a solution g * ∈ K.
Proof of Lemma 5.6.
where f − i,e (θ) is uniquely defined according to Constraints (4) and (6). Note that by the choice of ε these functions f − i,e , and hence b − i,v , do not depend on any inflow function during [φ, φ + ε). We set d = |I| · |E| and define K to be the set of all feasible flows over time with FIFO, described by the edge inflow functions only, on the given network during the interval [φ, φ + ε) that satisfy the inflow functions, i.e., Note that K is indeed a nonempty, closed, convex and bounded subset of We define the following mapping A : Here e = uv, i,v (θ) and i,u (θ) are the current shortest paths distances from v (u, respectively) to t i at time θ and q e (θ) is the queue length at edge e at time θ all considering the feasible flow over time f extended by g. This mapping A is indeed weak-strong-continuous. As shown by Cominetti et al. [5,Lemma 4] the mapping (g i,e ) i∈I,e∈E → (q e ) e∈E is weak-strong-continuous and, since (q e ) e∈E → ( i,v ) i∈I,v∈V is (strong-strong)-continuous, it follows immediately that A is weak-strong-continuous. Applying Theorem 5.7 provides a solution g * for VI (K, A).
We have to show that f extended by g * is a multi-commodity IDE flow. By the definition of b − i,v the flow conservation is satisfied. Suppose that constraint (10) does not hold for almost all θ ∈ [φ, φ + ε). Then there is an edge e, a commodity i, and a set of times Θ ⊆ [φ, φ + ε) of positive measure, such that g * e,i (θ) > 0 and e ∈ E i θ for all θ ∈ Θ. It follows that h * e,i (θ) > 0 for all θ ∈ Θ. Since all functions of g * and h * are non-negative we have: We define a new flow g that fulfils A(g * ), g = 0. Note that for any flow, and especially for g * we have the following property: For every node v, every commodity i and every time θ there exists an outgoing edge e ∈ δ + v that is active, i.e., e ∈ E i θ . This follows immediately from the fact, that E i θ connects every node v with t i . Furthermore, the sets Θ i,e := { θ ∈ [φ, φ + ε) | e ∈ E i θ } are, by their definition and the continuity of the label functions i,v , a union of closed intervals, and therefore measurable.
We now define g ∈ K as follows. At every node v, for every commodity i and at every point in time θ, we send all arriving flow at v of commodity i into an edge e ∈ E i θ , where E i θ are the active edges according to g * . It is easy to check that we have A(g * ), g = 0. Combining these we get which is a contradiction to (VI). To show that constraint (10) is fulfilled for every θ, recall that set Θ 0 of the points in time where this is not satisfied has measure zero. It is possible to modify the edge inflow rates at every θ ∈ Θ 0 , such that flow conservation and (10) is fulfilled by sending all flow into edges in E i θ . This has no impact on the queues or the shortest path distances. Proof. Starting with the empty flow which is a feasible IDE flow for the empty set [0, 0), we can repeatedly extend it for ε := 1 2 min { τ e | e ∈ E } with Lemma 5.6 to obtain a sequence (f k ) k∈N , where f k is an IDE flow for [0, k · ε). Taking the pointwise limit for every inflow rate function gives us an IDE flow for all times. Note that u i (θ) and f − i,e (θ) ≤ ν e are bounded at every point in time. Hence, the flow conservation constraint implies that the inflow rate functions f + i,e are bounded as well.

Termination of IDE Flows in Multi-Sink Networks
We show that there are instances in which all IDE flows do not terminate. We first observe that while the proofs of Lemma 4.3 and Corollary 4.5 can easily be adapted to the multi-sink case (so it is still true that all flows in an acyclic network and all IDE flows with total volume less than τ ∆ ν min eventually terminate), this is not true for the proof of Theorem 4.6.
Theorem 6.1. There is a multi-source multi-sink network with two sinks and all edge transit times and rate capacities equal to 1, where any IDE flow does not terminate.
To construct such an instance we make use of several gadgets. The first one, gadget A, will serve as the main building block and is depicted in Figure 7. It consists of two cycles with one common edge v 1 v 2 and one player i with sink node t (outside the gadget and reachable from the nodes v 2 , v 5 and v 7 via some paths P 2 , P 5 and P 7 , respectively) and a constant network inflow rate of 2 on the interval [0, 1) at node v 1 . Our goal will be to embed this gadget into a larger instance in such a way, that for any IDE flow, the flow associated with player i will exhibit the following flow pattern for all h ∈ N (see Figure 6): 1. On the interval [5h, 5h + 1): All flow generated at v 1 (for h = 0) or arriving at v 1 (for h > 0) enters the edge to v 2 at a rate of 2, half of it directly starting to travel along the edge, half of it building up a queue of length 1 at time 5h + 1.

On the interval [5h + 1, 5h + 2):
The flow arriving at node v 2 enters the edge to v 3 because v 2 , v 3 , v 4 , v 5 , P 5 is currently the shortest path to t. The length of the queue of edge v 1 v 2 decreases until it reaches 0 at time 5h + 2.

On the interval [5h + 2, 5h + 3):
The flow arriving at node v 2 enters the edge to v 6 because v 2 , v 6 , v 7 , P 7 is currently the shortest path to t.

On the interval [5h + 4, 5h + 5):
The flows arriving at nodes v 5 and v 7 enter the respective edges towards node v 1 because v 5 , v 1 , v 2 , P 2 as well as v 7 , v 1 , v 2 , P 2 are currently the shortest paths to get to t.

5.
On the interval [5h + 5, 5h + 6): There is a total inflow of 2 at node v 1 , which enters the edge to v 2 . Thus, the pattern repeats.
The effect of this behavior is, that other particles outside the gadget, who want to travel through this gadget along the central vertical path, will estimate an additional waiting time as indicated by the diagram displayed inside gadget A in Figure 7 (next to the vertical red path). Now, in order to actually guarantee the described behavior, we need to embed gadget A into a larger instance in such a way, that for any IDE flow the following assumptions hold: 1. The only edges leaving A are the start edges of the four dashed paths indicated in Figure 7.

2.
The three (blue) paths P 2 , P 5 and P 7 are of the same length L (w.r.t. τ e ).

For all
• the unique shortest v 5 -t path for all θ ∈ [5h + 4, 5h + 5] is v 5 , v 1 , v 2 , P 2 and • the unique shortest v 7 -t path for all θ ∈ [5h + 4, 5h Note that at time θ = 5h + 2 we do not require that there is only one unique v 2 -t path. This is due to the fact that waiting times always change continuously and therefore when the shortest v 2 -t path changes from one path to another, there needs to be a time where both paths are equally long. Thus, there cannot always be a unique shortest path. However, this does not influence the overall flow pattern, since those discrete points in time form a set of measure zero and thus only allow for flow of volume zero to escape the overall flow pattern.
In order to satisfy the assumptions 1.-3., we will now construct three types of gadgets B 2 , B 5 and B 7 for the three paths P 2 , P 5 and P 7 , each of equal length and on which any IDE flow induces waiting times as shown by the respective diagrams on the right side in Figure 7.
To build these gadgets we need time shifted versions of gadget A, which we denote by A +k . Such a gadget is constructed the same way as gadget A above, with the only difference that the support of the network inflow rate function u i is shifted to the interval [k mod 5, k mod 5 + 1). Gadget B 2 now consists of the concatenation of four gadgets of type A +0 , four gadgets of type A +1 and four gadgets of type A +2 in series along their vertical paths through them with three edges between each two gadgets (see Figure 8).
Similarly, gadget B 5 consists of three copies of A +3 -type gadgets, three copies of A +4type gadgets and additional 6 · 4 edges to ensure that the vertical path has the same length as the one of gadget B 2 . Finally, gadget B 7 consists of three copies of A +3 -type gadgets, three copies of A +4 -type gadgets, two copies of A +5 -type gadgets, one copy of A +6 -type gadgets and additional 3 · 4 edges to ensure that the vertical path has the same length as the one of gadget B 2 . We again use the notation B +k j to refer to a time shifted version of gadget B j -i.e. with all used gadgets A shifted by additional k time steps. Next, we build a gadget C by just taking one copy of each B +k j for all j ∈ { 2, 5, 7 } and k = 0, 1, 2, 3, 4 (see Figure 9). Figure 9: Gadget C Finally, taking two copies of this gadget, C and C , and two additional nodes, t and t , where t will be the sink node for all players in C and t the sink node for all players in C , we can build our entire graph as indicated by Figure 10. We connect the top edges of the gadgets B +k j in gadget C with the sink t and use those gadgets' respective vertical paths as the P +k j paths for gadget C and vice versa.

Figure 10: The whole graph
In order to prove the correctness of our construction (i.e. that any IDE flow on this instance does not terminate) we need the following important observation: Observation 6.2. If a flow in some A +k -type gadget (with k ∈ { 0, 1, 2, 3, 4 }) follows the desired flow pattern for all unit time intervals between k and some θ ∈ N 0 , θ ≥ k, the induced waiting time on edge v 1 v 2 of this gadget (and therefore on the vertical path through this gadget) will follow the waiting time function indicated by the diagram in Figure 7 (shifted by k) for the next unit time interval [θ, θ + 1), independent of the evolution of the flow in this interval.
Proof of Observation 6.2. If (θ −k) ≡ 0 mod 5, then over the following interval [θ, θ +1) we will have an inflow of 2 at node v 1 . Either because it originates here (if θ = k) or because (by assumption) the flow pattern already holds for the previous unit time interval and, thus, flow has entered edges v 5 v 1 and v 7 v 1 , both at rate 1, in this interval. Since the edge v 1 v 2 is the only one leaving v 1 , all flow will enter this edge and thereby starting to build up a queue of length 1 at time θ + 1.
If (θ − k) ≡ 1 mod 5, we start with a queue of length 1 at edge v 1 v 2 , which linearly decreases to 0 over the course of the interval as no new flow arrives at v 1 and flow leaves the edge at v 2 at rate 1. As all edges leaving v 2 are currently empty, no new queues can form, regardless of which edge the flow actually uses.
In all other cases, we start with empty queues at all edges. As no flow arrives at node v 1 and at all other nodes flow arrives with at most rate 1 (since all other nodes only have one incoming edge with rate capacity 1), no new queues can form. We now want to prove that any IDE flow in the constructed instance does not terminate. To do that we will take a generic A +k -type gadget from this instance and show by induction that the flow originating in v 1 of this gadget will follow the flow pattern described at the beginning of the construction and indicated in Figure 6.
Proof of Theorem 6.1. LetÃ be a copy of gadget A, time shifted by some k ∈ { 0, 1, 2, 3, 4 } and w.l.o.g. in C. We then need to show that all flow in this gadget will obey the pattern described above (time shifted by k) for all unit time intervals between k and θ for all θ ∈ N 0 , θ ≥ k. As our induction basis we take θ = k, for which the claim trivially holds.
For our induction step we assume that the flow in this gadget (and therefore all other gadgets A in the instance) follows the desired pattern for all unit time intervals up to some θ ∈ N 0 , θ ≥ k and want to show that it also does so for the next unit time interval [θ, θ + 1).
Case 1: (θ − k) ≡ 0 mod 5 See the respective case in the proof of Observation 6.2.
Case 2: (θ − k) ≡ 1 mod 5 Over the following unit time interval, flow will arrive at node v 2 at a rate of 1 while the queue on edge v 1 v 2 decreases. We need to show that all arriving flow will enter edge v 2 v 3 , as v 2 , v 3 , v 4 , v 5 , P 5 is currently the shortest v 2 -t path (w.r.t. instantaneous travel time). By induction and Corollary 6.3, we already know that all paths P +k j will exhibit the waiting time pattern indicated by the respective diagram on the right in Figure 7 (shifted by k). As, by construction, all those paths have a common length L (w.r.t. τ e ), we can calculate the length of all possible v 2 -t paths: • The path v 2 , v 3 , v 4 , v 5 , P 5 has length 3 + L and an additional waiting time of 0.
• The path v 2 , P 2 has length L and an additional waiting time of 4.
• The path v 2 , v 6 , v 7 , P 7 has length 2+L and an additional waiting time between 2 and 1.
• All paths leaving gadgetÃ through the vertical path have a length of at least 3 + 1 + L (three edges between the current gadget and the next one, one edge through this gadget and L edges for whatever gadget B +k j is finally used to get to t) and possibly additional waiting times.
So the path beginning with edge v 2 v 3 has the shortest total instantaneous travel time and even uniquely so for all times except θ + 1. Therefore all flow arriving at v 2 must enter edge v 2 v 3 for the whole interval [θ, θ + 1).
Case 3: (θ − k) ≡ 2 mod 5 Over the following unit time interval, flow will arrive at rate 1 at node v 3 , which has to enter edge v 3 v 4 as this is the only one leaving v 3 , and at node v 2 . We now need to show that all this flow enters edge v 2 v 6 (except possibly at time θ). We will do this in the same way as in case 2, i.e. by calculating the instantaneous travel times for all relevant paths with the help of Corollary 6.3: • The path v 2 , v 3 , v 4 , v 5 , P 5 has length 3 + L and no additional waiting time.
• The path v 2 , P 2 has length L and an additional waiting time of 4.
• The path v 2 , v 6 , v 7 , P 7 has length 2+L and an additional waiting time between 1 and 0.
• All paths leaving gadgetÃ immediately again have a length of at least 4 + L.
So all flow must enter edge v 2 v 6 as the path beginning with this edge is the unique shortest v 2 − t path.
Case 4: (θ − k) ≡ 3 mod 5 Over the following unit time interval, flow arrives at rate 1 at the nodes v 4 and v 6 . Since those only have one edge leaving them, the flow will just follow the only possible path.
Case 5: (θ − k) ≡ 4 mod 5 Over the following unit time interval, flow arrives at rate 1 at the nodes v 5 and v 7 . We need to show that all this flow enters the edges v 5 v 1 and v 7 v 1 , respectively. So as in case 2 and 3 we need to calculate the instantaneous travel times on all relevant v 5 -t and v 7 -t paths -again using Corollary 6.3: • The path v 5 , v 2 , P 2 has length 2 + L and an additional waiting time of 0.
• The path v 5 , P 5 has length L and an additional waiting time of 3.
• The path v 7 , v 2 , P 2 has length 2 + L and an additional waiting time of 0.
• The path v 7 , P 7 has length L and an additional waiting time of 3.
With the induction completed we have shown that for any IDE flow and any copy of gadget A within the given instance all flow generated at v 1 will arrive back at its start node after five unit time steps at which point the whole network is in exactly the same state as before. Thus, every IDE flow cycles and does not terminate.
Although we made use of several distinct source nodes, this is in fact not necessary in order to get a network with non-terminating IDEs. I.e. Theorem 6.1 can be strengthened as follows.
Theorem 6.4. There exists a single-source multi-sink network with two sinks, where any IDE flow does not terminate.
Proof. To prove this theorem we extend the network from Theorem 6.1 in such a way that after some initial warm-up time any IDE flow behaves exactly as in the initial network. First, we make sure that all network inflow rate functions u i have the interval [0, 1) as their support. This can be accomplished by introducing a new source nodes i for every commodity i and connecting that new source node with an edge of length r i and rate capacity 2 to the original source node s i (see Figure 11). Also note that the two sink nodes t and t have no outgoing edges. Next, we add one nodeŝ, which will be our super source. Then for every commodity i we add two distinct nodes v i , w i and edges as indicated in Figure 12 (where w i is always connected to t i , the sink node of commodity i). Finally, we replace the commodities I by two new commodities 0 and 0 with common source nodeŝ. Commodity 0 has t as its sink node and the following network inflow rate function where for every commodity i ∈ I with sink node t, P i is a shortests i -t path in G. Commodity 0 has t as its sink node and an analogous inflow rate functionû 0 . Now at time θ = 0, the shortestŝ-t paths areŝ, w i , t andŝ, v i , w i , t (for i with t i = t) with length 3 since every other path has to go through G and therefore has a length of at least 4. As the inflow rate of commodity 0 exactly matches the total rate capacity of the first edges of all those paths, the flow of commodity splits between these edgesŝ, w i andŝ, v i using all of them with full capacity.
At time θ = 1, the flow arriving at node v i with rate 6 enters the edge v i w i and starts forming a queue on that edge. At time θ = 2, flow arrives with rate τ (P i ) + 2 at node (1 , 1) Figure 12: The modified network with only one single source and two sinks. After an initial warmup phase (see Figure 13) the flow inside G behaves just as in the original network from the proof of Theorem 6.1.
w i and starts entering the edge w i ,t i , building up a queue there. At time θ = 3, this queue has reached a length of τ (P i ) + 1, at which point the paths w i , t and w i ,s i , P i , t have the same instantaneous travel time (note that by this time no flow has yet entered G and, thus, no waiting times occur within G). Thus, from now on the flow arriving at node w i at rate 3 splits between the edges w i , t and w i ,s i proportional to the respective rate capacities. Thus, between time θ = 5 and θ = 6, flow arrives at nodes i at a rate of 2.
As the same flow evolution happens for commodity 0 , at time θ = 5, the new network is in the same state as the original network at time θ = 0. In particular, all the flow entering G at one of the nodess i will stay inside the network forever and so the flow will never terminate.

Summary and Open Problems
We introduced in this paper the concept of IDE flows and investigated two key questions: existence and termination of IDE flows. Regarding the former, we gave in Section 3 an extension-algorithm leading to the existence of IDE flows for single-sink instances. While the extension-algorithm is constructive, it is not clear if finitely many calls of the algorithm suffice to compute an IDE -at the moment existence relies on a limit argument. Especially for restricted graph classes (series-parallel graphs or acyclic graphs) we expect finiteness. For multi-source multi-sink instances, we gave in Section 5 a general existence theorem -also based on an extension property. For piece-wise constant network inflow rate functions this extension can be achieved by solving a set of equations (called IDE thin flows), which we can obtain algorithmically by solving a linear mixed-integer program. The extension property for general network inflow rates, however, relies on a solution to a variational inequality rendering it non-constructive. Regarding termination of IDE flows, we showed in Theorem 4.6 that for single-sink networks, all IDE flows terminate. In a forthcoming paper [10], we give quantitative upper bounds of O(U τ (G)) for the time such an IDE flow needs to terminate, where U := i∈I ∞ 0 u i (θ)dθ is the total network inflow and τ (G) := e∈E τ e the sum of all edge transit times. On the other hand, we can give lower bounds of order Θ(U log τ (G)), see [10] for details. In Section 6, we gave an example for a multi-sink network, where no IDE flow terminates. By Theorem 6.4, we know that only a single-source and two sinks are needed for this effect to appear (and by Theorem 4.6 we also know that this is the minimal number of sinks necessary). However, the underlying graph is quite complex and it would be interesting to see, whether there are certain graph classes beside acyclic