Long term behavior of dynamic equilibria in fluid queuing networks

by   Roberto Cominetti, et al.

A fluid queuing network constitutes one of the simplest models in which to study flow dynamics over a network. In this model we have a single source-sink pair and each link has a per-time-unit capacity and a transit time. A dynamic equilibrium (or equilibrium flow over time) is a flow pattern over time such that no flow particle has incentives to unilaterally change its path. Although the model has been around for almost fifty years, only recently results regarding existence and characterization of equilibria have been obtained. In particular the long term behavior remains poorly understood. Our main result in this paper is to show that, under a natural (and obviously necessary) condition on the queuing capacity, a dynamic equilibrium reaches a steady state (after which queue lengths remain constant) in finite time. Previously, it was not even known that queue lengths would remain bounded. The proof is based on the analysis of a rather non-obvious potential function that turns out to be monotone along the evolution of the equilibrium. Furthermore, we show that the steady state is characterized as an optimal solution of a certain linear program. When this program has a unique solution, which occurs generically, the long term behavior is completely predictable. On the contrary, if the linear program has multiple solutions the steady state is more difficult to identify as it depends on the whole temporal evolution of the equilibrium.



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1 Introduction

The theory of flows over time provide a natural and convenient model to describe the dynamics of a continuous stream of particles traveling from a source to a sink in a network, such as urban or Internet traffic. Probably the most basic model for the propagation of flow is the so-called

fluid-queue model in which each arc in the network consists of a fluid queue with an arc-specific capacity followed by a link with constant delay. Thus, the time to traverse an edge is composed of a flow-dependent waiting time in the queue plus a constant travel time after leaving the queue. This model was initially studied in the framework of optimization. Ford and Fulkerson (1958, 1962) considered a fluid queue model in a discrete time setting and designed an algorithm to compute a flow over time carrying the maximum possible flow from the source to the sink in a given timespan. Gale (1959) then showed the existence of a flow pattern that achieves this optimum simultaneously for all time horizons. These results were extended to continuous time by Anderson and Philpott (1994) and Fleischer and Tardos (1998). We refer to Skutella (2009) for an excellent survey. However, when network flows suffer from a lack of coordination among the participating agents, it is natural to take a game theoretic approach. As first described by Vickrey (1969) for a simple bottleneck model, in a dynamic network routing game each infinitesimal particle is interpreted as a player that seeks to complete its journey in the least possible time. Players are forward-looking and anticipate the congestion and queuing delays induced by others upon arrival to any edge in the network. Equilibrium occurs when each particle travels along a shortest path.

More formally, a fluid queuing network is a directed graph where each arc consists of a fluid queue with capacity followed by a link with constant delay (see Figure 1). A constant inflow rate enters the network at a fixed source and travels towards a terminal node . A dynamic equilibrium models the temporal evolution of the flows in the network. Loosely speaking, it consists of a flow pattern in which every particle travels along a shortest path, accounting for the fact that travel times depend on the instant at which a particle enters the network as well as the state of the queues that will be encountered along its path by the time at which they are reached. Intuitively, if the queues are initially empty, the equilibrium should start by sending all the flow along shortest paths considering only the free-flow delays . These paths are likely to become overloaded so that queues will grow on some of its edges and at some point in time new paths will become competitive and will be incorporated into the equilibrium. These new paths may in turn build queues so that even longer paths may come into play. Hence one might expect that the equilibrium proceeds in phases in which the paths used by the equilibrium remain stable. However, it is unclear if the number of such phases is finite and whether the equilibrium will eventually reach a steady state in which the queues and travel times stabilize.




Figure 1: An arc in the fluid queuing network.

Although dynamic equilibria have been around for almost fifty years (see, e.g., (Ford and Fulkerson 1958, Friesz et al. 1993, Gale 1959, Merchant and Nemhauser 1978a, b, Peeta and Ziliaskopoulos 2001, Ran and Boyce 1996, Vickrey 1969, Xu et al. 1999)), their existence has only been proved recently by Zhu and Marcotte (2000) though in a somewhat different setting, and by Meunier and Wagner (2010) who gave the first existence result for a model that covers the case of fluid queuing networks. These proofs, however, rely heavily on functional analysis techniques and provide little intuition on the combinatorial structure of dynamic equilibria, their characterization, or feasible approaches to compute them. Substantial progress was recently achieved by Koch and Skutella (2011) by introducing the concept of thin flows with resetting that characterize the time derivatives of a dynamic equilibrium, and which provide in turn a method to compute an equilibrium by integration. A slightly refined notion of normalized thin flows with resetting was considered by Cominetti et al. (2015), who proved existence and uniqueness, and provided a constructive proof for the existence of a dynamic equilibrium.

In recent work, further extensions and variants of the model have been studied. In particular, Sering and Skutella (2018) extend some of the known results about dynamic equilibria to the case in which there are multiple sources and terminal nodes. However, the multi-commodity case is largely open. Furthermore, Sering and Vargas Koch (2019) consider spill back effects to model the fact that in practice the queues cannot grow arbitrarily large and that their effect propagates back in the network. Graf and Harks (2019) consider a related model in which particles behave myopically and make routing decisions based on the current state of the network, without anticipating its evolution. Finally, we mention the work of Cao et al. (2017) who considered an atomic model and established that in series-parallel networks queues remain bounded in a dynamic equilibrium.

In this paper we focus on the long term behavior of dynamic equilibria in fluid queuing networks. Clearly if the inflow is very large compared to the queuing capacities, the queues will grow without bound, and no steady state can be expected. More precisely, let be an -cut with minimum queuing capacity ; if there are multiple options, choose (containing ) to be setwise minimal. If all the arcs in will grow unbounded queues, whereas for , it is natural to expect that the equilibrium should eventually reach a steady state, where queue lengths remain constant. This was not known—in fact, it was not even known that queue lengths remain bounded!

Our main goal in this paper is to show that both these properties do indeed hold: more precisely, when , the dynamic equilibrium reaches a steady state in finite time. At first glance, these convergence properties might seem “obvious”, and it might seem surprising that they are at all difficult to prove. We will present some examples that illustrate why this is not the case. For instance, it may occur that the flow across the cut may temporarily exceed its capacity by an arbitrarily large factor, forcing the queues to grow very large. This phenomenon may occur since the inflow entering the network at different points in time may experience different delays and eventually superpose at which gets an inflow larger than . In other cases some queues may grow during a period of time after which they reduce to zero and then grow again later on. In fact, we give a construction that shows that this can happen an exponential (in the input size) number of times during the evolution! Along the way to our main result, we provide a characterization of the steady state as an optimal solution of a certain linear programming problem and we discuss when this problem has a unique solution. Despite the fact that convergence to a steady state occurs in finite time, it remains as an open question whether this state is attained after finitely many phases.

The paper is structured as follows. Section 2 reviews the model of fluid queuing networks, including the precise definition of dynamic equilibrium and the main results known so far. Then, in Section 3 we discuss the notion of steady state and provide a characterization in terms of a linear program. Inspired by the objective function of this linear program, in Section 4 we introduce a potential function and we prove that it is a Lyapunov function for the dynamics. This potential turns out to be piecewise linear in time with finitely many possible slopes. We then prove that the potential remains bounded so that there is a finite time at which its slope is zero, and we show that in that case the system has reached a steady state. Further, we provide an explicit pseudopolynomial bound on the convergence time. Finally, in Section 5 we discuss the interesting (and perhaps surprising) examples alluded to earlier, as well as some remaining open questions.

2 Dynamic equilibria in fluid queuing networks

In this section we recall the definition of dynamic equilibria in fluid queuing networks, and we briefly review the known results on their existence, characterization, and computation. The results are stated without proofs for which we refer to Koch and Skutella (2011) and Cominetti et al. (2015).

2.1 The model

Consider a fluid queuing network with arc capacities and delays . The network dynamics are described in terms of the inflow rates that enter each arc at time , where is measurable.

Arc dynamics. If the inflow exceeds a queue will grow at the entrance of the arc.




Figure 2: Dynamics of an arc in the queuing network.

The queues are assumed to operate at capacity, that is to say, when the flow is released at rate , whereas when the queue is empty the outflow is the minimum between and the capacity . Hence the queue evolves from its initial state according to


These dynamics uniquely determine the queue lengths as well as the arc outflows


Flow conservation. A flow over time is a family of arc inflows such that flow is conserved at every node , namely for a.e.


Dynamic shortest paths. A particle entering an arc at time experiences a queuing delay plus a free-flow delay to traverse the arc after leaving the queue, so that it will exit the arc at time


Consider a particle entering the source node at time . If this particle follows a path , it will reach the end of the path at time


Denoting the set of all -paths, the minimal time at which node can be reached is


The paths attaining these minima are called dynamic shortest paths. The arcs in these paths are said to be active at time and we denote them by . Observe that can also be defined through the dynamic Bellman’s equations


so that is active precisely if .

Dynamic equilibrium. A dynamic equilibrium is a flow pattern that uses only dynamic shortest paths. More precisely, let be the set of entrance times at which the arc is active, and the set of local times at which will be active. A flow over time is called a dynamic equilibrium iff for almost every we have .

2.2 Characterization of dynamic equilibria

Since the inflows are measurable the same holds for and we may define the cumulative inflows and cumulative outflows as

These cumulative flows allow to express the queues as . It turns out that a dynamic equilibrium can be equivalently characterized by the fact that for each arc we have


In this case, the functions are static flows with


2.3 Derivatives of a dynamic equilibrium

The labels and the static flows are nondecreasing functions which are also absolutely continuous so that they can be reconstructed from their derivatives by integration.111These derivatives exist almost everywhere and are locally integrable. Moreover, from these functions one can recover the equilibrium inflows using the relation . Hence, finding a dynamic equilibrium reduces essentially to computing the derivatives .

Let be a point of differentiability and set and . From (9) we see that is a static -flow of size , namely,


while using (7), (4), (1) and the differentiation rule for a minimum we get




with the set of arcs with positive queue . In addition to this, the conditions for dynamic equilibria imply as well as


These equations fully characterize the derivatives of a dynamic equilibrium. In fact, for all subsets the system (10)-(13) admits at least one solution and moreover the component is unique. These solutions are called normalized thin flows with resetting (ntfr) and can be used to reconstruct a dynamic equilibrium by integration, proving the existence of equilibria. We refer to Cominetti et al. (2015) for the existence and uniqueness of ntfr’s and to Koch and Skutella (2011) for a description of the integration algorithm and how to find the equilibrium inflows .

Observe that there are only finitely many options for and . Since the corresponding is unique, it follows that the functions will be uniquely defined and piecewise linear with finitely many options for the derivatives. Although the static flows are not unique in general, one can still find an equilibrium in which these functions are also piecewise linear by fixing a specific in the ntfr for each pair .

2.4 A detailed example

We now work out the details of a small example that already provides some intuition on how the dynamic equilibria behaves. In particular this example exhibits an unexpected property, namely that the flow coming into the sink can be larger than the network inflow.

Example 1.

Consider the network consisting of the vertices with edges and inflow . Capacities are , , , and , and delays are , and .

In a dynamic equilibrium for this instance, flow is initially routed through the shortest path . Then queues grow in both edges until at time the path consisting of enters the shortest path network. From that point in time the flow splits in equal proportions between paths and implying that a queue starts growing on edge , the queue of starts decreasing, while the queue on continues to increase. Then, at time , the queue on has grown enough to make enter the shortest path network. At this point all edges except have a queue and the flow starts splitting as follows: take path , take path , and take path . Therefore all queues start decreasing until at time the queues on edges and deplete simultaneously. When this happens the shortest path network stays the same but the edges with queue change since only still has a queue. The new thin flow is thus computed, and the flow starts splitting evenly between the paths , , and (each gets flow ). This last phase constitute a steady state so it lasts forever. More precisely in this instance one can compute the derivative of the distance labels at node as

Interestingly, the amount of flow arriving at at time can readily be computed as . So that if we consider the local time at node this flow is then

This brings us to the surprising fact that for some time interval, the flow arriving at the sink is larger than the inflow.

3 Steady states

We say that a dynamic equilibrium attains a steady state if for sufficiently large times all the queues are frozen to a constant . This is clearly equivalent to the fact that the arc travel times become constant equal to with the corresponding queuing times.

Lemma 1.

A dynamic equilibrium attains a steady state iff there exists some such that for every node and all .


In a steady state we clearly have where is the minimum travel time from to with arc times , so that . Conversely, if all these derivatives are equal to 1 then for some constant and . Moreover, an arc with nonempty queue must be active so that which yields

which shows that all queues eventually become constant. ∎

Theorem 1.

Consider a steady state with queues and let be the minimum travel time from to under arc travel times , where . Let with for all be a corresponding ntfr and denote by the set of -flows of value . Then and are optimal solutions to the following pair of dual linear programs:


Clearly is feasible for (D). Also (10) gives , while (13) implies that if then . This implies that , so is feasible for (P). If then certainly the arc is active (formally, by (13)) and hence . And if , also implying that is active, then (11) implies that , which yields . This proves that and are complementary solutions, and hence are optimal for (P) and (D) respectively. ∎

According to this result, if a dynamic equilibrium eventually settles to a steady state then the corresponding queue lengths must be optimal for (D). Generically (after perturbing capacities) this linear program has a unique solution in which case the steady state is fully characterized. Otherwise, if (D) has multiple solutions it is not evident which queue lengths will be obtained in steady state. Note that even if the min cost flow for (P) is unique, this does not mean that only one steady state situation is possible because there may be flexibility in the queue lengths. For instance, if and the network has a single link from to of unit capacity, if we create a queue of some length at time 0 this queue will remain in the steady state solution. This point will be further discussed in Section 5.3.

Remark. It is not difficult to show that when we start with initial conditions where with optimal for (D), then the dynamic equilibrium is already at a steady state and the queues remain constant.

4 Convergence to a steady state

In this section we prove that a steady state exists and that it is actually reached in finite time. To this end we introduce a Lyapunov potential function that increases along the evolution of the dynamic equilibrium. The potential function is inspired from the previous dual program and is given by

Remark. The potential is the difference between the total travel time experienced by users leaving at time , and the total queue volumes, as seen by users leaving at time . We are not aware of a more insightful interpretation of it. In fact, none of the more “natural” quantities we tried as candidate potential functions (total delay, time spent queueing, total delay excluding queueing delays, …) are monotone.

Theorem 2.

For every that is a point of differentiability of , is nonnegative, and strictly positive unless the dynamic equilibrium has reached a steady state.


The queues can be expressed as , and therefore . To take the derivative we recall that is the set of active edges, i.e., those for which , while in the inequality is strict. Using the derivative of a max function and taking a ntfr at time , we thus obtain

Notice that the dependency of the ’s in the previous derivative is somewhat hidden in the set of active edges . Now, for we have if and if , so that letting we may write

Let us introduce a return arc with capacity and flow so that is a circulation. Let and for each define the function

Then the derivative can be expressed as

For the remainder of the proof, let (respectively denote the arcs in leaving (respectively entering) . Let and consider an arc . If then and therefore . Similarly, if then which implies and again . Hence for all . This equality also holds for the return arc , while in the remaining arcs . Hence


with strict inequality if is nonempty. It follows that for all we have

and therefore with strict inequality unless is empty for almost all . Since for to be empty we need that it either contains all vertices in or none of them, we have that if and only if all are equal, and hence (since ) all equal to . By Lemma 1, this exactly characterizes a steady state. ∎

Theorem 3.

Let be the minimal queuing capacity among all -cuts . If then the dynamic equilibrium attains a steady state in finite time.


From Theorem 2 it follows that there is some such that for every phase other than the steady state. This is simply because the thin flow depends only on the current shortest path network and the set of queuing edges , and so there are only finitely many possible derivatives.

Thus, in order to prove that a steady state is reached in finite time it suffices to show that remains bounded. To this end we note that the condition implies that (P) is feasible and hence it has a finite optimal value . The conclusion then follows by noting that the point with and is feasible for the dual (D) so that . ∎

Given that convergence to a steady state does happen in finite time, it is natural to ask for explicit bounds. It is easy to see that a polynomial bound (in the input size encoding) is impossible; simply consider a network consisting of two parallel links, one with capacity and length zero, the other with capacity and length . The first phase, where all traffic takes the shorter edge, lasts until time . However, we can give a pseudopolynomial bound on the convergence time (and hence, queue lengths). The following results shows this bound. We present it in a slightly more general setting that allows for rational inflow and capacities and arbitrary initial queues since we will need it in this form in Section 5.2.

Theorem 4.

Consider an instance for which all arc capacities as well as the inflow are multiples of , . We allow for an arbitrary initial state at time with possibly nonempty queues. Let and . Then assuming the dynamic equilibrium attains a steady state, it is reached by time , and moreover, the waiting time in any queue never exceeds .


We first remark that it suffices to prove the result for , i.e., integer capacities and inflow. For consider the instance where the inflow as well as all arc capacities have been scaled up by a factor

. The equilibrium flow on this new instance is obtained by scaling the equilibrium flow on the original one, and so the time to reach steady state, as well as the queue waiting times at any moment in time, are the same for both instances. Considering the impact of the scaling on the claimed bounds, the claim on this new instance thus implies the claim on the original instance, and so we assume

for the remainder.

We use the same notions defined in the proof of Theorem 2. Assume that a steady state is attained; thus (P) has a finite objective value, and this is at most . Thus for all . Initially,

So for all .

Consider some time which is a point of differentiability not in the steady state phase; so by Theorem 2. Our first goal will be to show that ; this clearly implies the bound on the time to reach steady state.

Let be any time before steady state is reached and for which is defined. Let and . From the proof of Theorem 2, we have for any (it is strictly negative and integral). And certainly for all and . Thus

To bound this, choose an arbitrary for which (recall that since we have not reached steady state). We have, following the lines of (14),


Since , , and , .

Our next goal is to bound the queue lengths. The only extra ingredient we need is a bound on the speed at which a queue can grow. There is always a ntfr which does not route flow along cycles (see Koch 2012, Theorem 6.64), so that for all . Therefore, . Hence, for all times before steady state is reached,

This implies the same bound on all queue waiting times. ∎

5 Some constructions and conjectures

While we have settled the finite-time convergence to a steady state, there are a number of questions about dynamic equilibria that remain open. In this section we provide some constructions exhibiting somehow surprising behavior. First we show that in a dynamic equilibrium the flow across a cut can be arbitrarily larger than the inflow. Then we build an instance for which the dynamic equilibria has exponentially many phases. We wrap up the section by discussing the possibility of characterizing the steady state queues, and some conjectures regarding more general steady state results.

5.1 Flow across a cut

As mentioned in the introduction a first conjecture would be that, similarly to what happens for static flows, the flow across any cut is always bounded by the inflow. This would provide a way to estimate the queues and to prove their boundedness. Unfortunately the property fails in a dynamic equilibrium. The reason for this is that flow entering the network at different times may experience different delays in such a way that they later superpose across an intermediate cut. In Example 

1 we constructed one such instance, where the peak outflow rate was a factor larger than the inflow rate. We will now show how to construct instances in which the outflow is arbitrarily larger than the inflow.

Figure 3: The modified instance.

The base of the construction, given in Figure 3, consists on appending to the instance in Example 1 (slightly rescaled for convenience) an extra sink and two arcs, and , connecting with . Since the outflow in Example 1 (with ) is

the outflow in the new instance will be

Indeed, while the flow leaving is , it will directly take arc and therefore immediately reach . Then the flow leaving increases to and it will continue to choose arc , though a queue will start to build up on that arc. At time (considering the local time at ) the queue in will be of size , implying a queuing time of . Since this is exactly the delay of arc , at this point enters the shortest path network. From this point the flow leaving increases to and thus the flow splits: flow units take arc while the remaining take arc . The latter pattern stays until time when the flow leaving changes to and thus it splits as taking arc and taking arc , at which point steady state has been reached.

Some observations are in order:

  • The length of the “pulse” in this construction is exactly , and so this can be made as large as required.

  • The full pulse is produced as long as the inflow is for a period of ; if it were to decrease or otherwise vary after this, it would only interfere with the final steady state phase.

  • All arcs in the gadget have capacity at least , meaning that no queues will form if the inflow is bounded by . This means that the instance would still produces a pulse if the inflow was at most for some initial period, and then equal to for a period (of length at least ).

Building on these observations, we now construct an instance where the “pulse” is arbitrarily larger than the inflow. More precisely, we will construct a gadget , for any and , with the following properties (we define for convenience).

  1. There is some value such that, assuming a constant inflow rate of in the interval , the inflow rate into the sink is at most for and exactly for (where is the local time at the sink).

  2. All arcs have capacity at least .

  3. The gadget has arcs, with all free-flow delays being multiples of and bounded by , and all arc capacities being multiples of and bounded by .

We have already seen in Figure 3 the construction of a gadget; all the required properties clearly hold. We construct a gadget for by chaining together in series a gadget (call it ) followed by a gadget (call it ). Properties (i)–(iii) are then mostly easy to verify inductively. An important observation is that since the outflow of before its pulse is at most , which is smaller than the smallest capacity arc in , this initial flow does not cause any disturbance. The bound on follows easily by making the stronger inductive claim that . Now , so this is correct for . For , the flow into increases to at time , meaning that the outflow increases to at time , and this the inductive claim holds.

5.2 Instances with an exponential number of phases

A natural hope would be that the number of phases is always polynomial in the input size (ideally as measured by the number of arcs in the instance, but failing that, as measured by the total encoding length of the instance). Unfortunately, as we show next, this is not the case and indeed the number of phases of a dynamic equilibrium may be exp