User equilibria for a parallel queueing system with state dependent routing (original) (raw)

The Downs-Thomson Paradox: Existence, Uniqueness and Stability of User Equilibria

Queueing Systems, 2005

Consider a network where two routes are available for users wishing to travel from a source to a destination. On one route (which could be viewed as private transport) service slows as traffic increases. On the other (which could be viewed as public transport) the service frequency increases with demand. The Downs-Thomson paradox occurs when improvements in service produce an overall decline in performance as user equilibria adjust. Using the model proposed by Calvert [10], with a •|M|1 queue corresponding to the private transport route, and a bulk-service infinite server queue modelling the public transport route, we give a complete analysis of this system in the setting of probabilistic routing. We obtain the user equilibria (which are not always unique), and determine their stability.

Queue Spillovers in Transportation Networks with a Route Choice

Transportation Science, 1998

This paper explores some of the traffic phenomena that arise when drivers have to navigate a network in which queues back up past diverge intersections. If a diverge provides two alternative routes to the same destination and the shorter route has a bottleneck that generates a queue, one would expect that queue to stabilize at an equilibrium level where the travel time on both routes is roughly equal. If the capacity of the longer route is unlimited then this network can accommodate any demand level.

To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems

International Series in Operations Research & Management Science, 2003

Preface xi 1. INTRODUCTION A non-cooperative game is defined as follows. Let N = {1,. .. , n} be a finite set of players and let A i denote a set of actions available to player i ∈ N. A pure strategy for player i is an action from A i. A mixed strategy corresponds to a probability function which prescribes a randomized rule for selecting an action from A i. Denote by S i the set of strategies available to player i. A strategy profile s = (s 1 ,. .. , s n) assigns a strategy s i ∈ S i to each player i ∈ N. Each player is associated with a real payoff function F i (s). This function specifies the payoff received by player i given that the strategy profile s is adopted by the players. Denote by s −i a profile for the set of players N \ {i}. The function F i (s) = F i (s i , s −i) is assumed to be linear in s i. This means that if s i is a mixture with 1 In case of periodicity, with period d, replace the limit by averaging the limits along d consecutive periods. Note that ∞ s=0 πs(δ) does not necessarily sum up to 1. On one hand, it can be greater than 1 (in fact, can even be unbounded) when more than one recurrent chain exists, and on the other hand it may sum up to 0. An example for the latter case is when λ > µ and δ(s) = join for all s ≥ 0. x F (x, y). We are interested in cases where x(y) is continuous and strictly monotone. Figure 1.1 illustrates a situation where a strategy corresponds to a nonnegative number. It depicts one instance where x(y) is monotone decreasing and another where it is monotone increasing. We call these situations avoid the crowd (ATC) and follow the crowd (FTC), respectively. The rationale behind this terminology is that in an FTC (respectively, ATC) case, the higher the values selected by the others, the higher (respectively, lower) is one's best response. 3 An interesting generalization to this rule is proposed by Balachandran and Radhakrishnan [19]. Suppose that waiting t time units costs Ce at for given parameters C > 0 and a ≥ 0. Then, the expected waiting cost of a customer is ∞ 0 Ce at w(t) dt where w(t) is the density function of the waiting time. In an M/M/1 system w(t) = (µ − λ)e −(µ−λ)t where λ is the arrival rate and µ is the service rate. In this case the expected cost equals C µ−a−λ. Note that the case of linear waiting costs is obtained when a = 0. 4 See Deacon and Sonstelie [43] and Png and Reitman [140] for empirical studies concerning this parameter. Examples for disciplines that are strong and work-conserving are FCFS, LCFS, random order, order which is based on customers payments, and EPS. Service requirements are assumed to be independent and identically distributed. Denote by µ −1 the (common) expected service requirement (i.e., µ is the rate of service). For stability, assume that the system's utilization factor ρ = λ µ is strictly less than 1 (sometimes, when individual optimization leads to stability, this assumption is removed). The following five results hold when the arrival process is Poisson with rate λ, the service distribution is exponential (an M/M/1 model) with rate µ, and the service discipline is strong and work-conserving. They also hold for M/G/1 models when the service discipline is either EPS or LCFS-PR. The probability that n (n ≥ 0) customers are in the system (at arbitrary times as well as at arrival times) is (1 − ρ)ρ n. (1.2) 11 When 3 5λ > 1, commuters appear at a rate so low that even when all of them use the shuttle service, the individual's best response is still to use the bus service. In other words, when λ < 3 5 , using the bus service is a dominant strategy. Chapter 2 OBSERVABLE QUEUES This chapter deals with queueing systems, where an arriving customer observes the length of the queue before making his decisions.

State-Dependent Signalling in Queueing Networks

Advances in Applied Probability, 1994

It has recently been shown that networks of queues with state-dependent movement of negative customers, and with state-independent triggering of customer movement have product-form equilibrium distributions. Triggers and negative customers are entities which, when arriving to a queue. force a single customer to be routed through the network or leave the network respectively. They are 'signals' which affect/control network behaviour. The provision of state-dependent intensities introduces queues other than single-server queues into the network.

Optimal routing in parallel, non-observable queues and the price of anarchy revisited

2010

We consider a network of parallel, non-observable queues and analyze the "price of anarchy", an index measuring the worst-case performance loss of a decentralized system with respect to its centralized counterpart in presence of non-cooperative users. Our analysis is undertaken from the new point of view where the router has the memory of previous dispatching choices, which significantly complicates the nature of the problem. In the regime where the demands proportionally grow with the network capacity, we provide a tight lower bound on the socially-optimal response time and a tight upper bound on the price of anarchy by means of convex programming. Then, we exploit this result to show, by simulation, that the billiard routing scheme yields a response time which is remarkably close to our lower bound, implying that billiards minimize response time. To study the added-value of non-Bernoulli routers, we introduce the "price of forgetting" and prove that it is bounded from above by two, which is tight in heavy-traffic. Finally, other structural properties are derived numerically for the price of forgetting. These claim that the benefit of having memory in the router is independent of the network size and heterogeneity, while monotonically depending on the network load only. These properties yield simple productforms well-approximating the socially-optimal response time.

A Tandem Queueing Network with Feedback Admission Control

Lecture Notes in Computer Science, 2007

Admission control in queueing networks is often based on partial information on the network state. This paper studies how the lack of state information affects performance by considering a simple model for admission control. The model is analyzed by studying a related censored process that has a matrix-geometric steady-state distribution.

Optimal Control of Service Rates in Networks of Queues

Advances in Applied Probability, 1987

We prove a monotonicity result for the problem of optimal service rate control in certain queueing networks. Consider, as an illustrative example, a number of ·/M/1 queues which are arranged in a cycle with some number of customers moving around the cycle. A holding cost hi (xi ) is charged for each unit of time that queue i contains xi customers, with hi being convex. As a function of the queue lengths the service rate at each queue i is to be chosen in the interval , where cost ci (μ) is charged for each unit of time that the service rate μis in effect at queue i. It is shown that the policy which minimizes the expected total discounted cost has a monotone structure: namely, that by moving one customer from queue i to the following queue, the optimal service rate in queue i is not increased and the optimal service rates elsewhere are not decreased. We prove a similar result for problems of optimal arrival rate and service rate control in general queueing networks. The results are ...

Geometric equilibrium distributions for queues with interactive batch departures

Annals of Operations Research, 1994

Gelenbe et al. consider single server Jackson networks of queues which contain both positive and negative customers. A negative customer arriving to a nonempty queue causes the number of customers in that queue to decrease by one, and has no effect on an empty queue, whereas a positive customer arriving at a queue will always increase the queue length by one. Gelenbe et al. show that a geometric product form equilibrium distribution prevails for this network. Applications for these types of networks can be found in systems incorporating resource allocations and in the modelling of decision making algorithms, neural networks and communications protocols.

Optimal Routing in Two Parallel Queues with Exponential Service Times

Discrete Event Dynamic Systems, 2006

In this paper we investigate the problem of the effective computation of the optimal routing sequence in a queuing system made of two queues with exponential service time in parallel. We first show that the optimal policy (minimizing the expected waiting time) is a Sturmian sequence and we establish several qualitative properties of this policy (monotonicity, continuity, convexity) Then, we propose an algorithm to compute the optimal routing sequence. We address the issues of time complexity as well as numerical stability of this algorithm. We then run an extensive set of experiments which show several interesting features of the optimal policy with apparent discontinuities and a fractal behavior.

Optima and equilibria for traffic flow on networks with backward propagating queues

Networks and Heterogeneous Media, 2015

The paper studies an optimal decision problem for several groups of drivers on a network of roads. Drivers have different origins and destinations, and different costs, related to their departure and arrival time. On each road the flow is governed by a conservation law, while intersections are modeled using buffers of limited capacity, so that queues can spill backward along roads leading to a crowded intersection. Two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination.