The price of anarchy in an exponential multi-server

Queueing Systems, 2015

We provide Game-theoretic analysis of the arrival process to a multi-server system with a limited queue buffer, which admits customers only during a finite time interval. A customer who arrives at a full system is blocked and does not receive service. Customers can choose their arrival times with the goal of minimizing their probability of being blocked. We characterize the unique symmetric Nash equilibrium arrival distribution and present a method for computing it. This distribution is comprised of an atom at time zero, an interval with no arrivals (a gap), and a continuous distribution until the closing time. We further present a fluid approximation for the equilibrium behaviour when the population is large, where the fluid solution also admits an atom at zero, no gap, and a uniform distribution throughout the arrival interval. In doing so, we provide an approximation model for the equilibrium behaviour that does not require a numerical solution for a set of differential equations, as is required in the discrete case. For the corresponding problem of social optimization we provide explicit analysis of some special cases and numerical analysis of the general model. An upper bound is established for the price of anarchy (PoA). The PoA is shown to be not monotone with respect to population size.

Probability in the Engineering and Informational Sciences, 2009

In most decision models dealing with unobservable stochastic congested environments, one looks for a (Nash) equilibrium behavior among customers. This is a strategy that, if adopted by all, then under the resulting steady-state conditions; the best response for an individual is to adopt this strategy too. The purpose of this article is to look for a simple decision problem but where the assumption of steady-state conditions is removed. Specifically, we consider an M/M/N/N loss model in which one pays for trying to get service but is rewarded only if one finds an available server. The initial conditions at time 0 are common knowledge and each customer possesses his arrival time as his private information. The equilibrium profile tells each arrival whether to try (randomization allowed) given his time of arrival. We show that all join up to some point of time. At this point, there is a quantum drop in the joining probability from one to some fraction. From then on, their joining proba...

IEEE INFOCOM 2014 - IEEE Conference on Computer Communications, 2014

In this paper, we consider the inefficiency of distributed routing in a network of parallel links with class-based traffic. Network link behavior is modeled by the M/M/1-GPS queue (i.e. when links use General Processor Sharing(GPS) scheduling scheme to serve packets). Each traffic type is guaranteed a minimum capacity rate on each link using GPS scheduling. We show under specific demand conditions that, among multiple equilibria the worst-case Nash equilibrium occurs when each class dispatcher utilizes all the links to fulfill its demand. Using this fact, we give an upper bound on the Price of Anarchy (PoA). Our results also indicate that, while the price of selfish behavior can be unbounded in a specific demand setting, there exist demand regimes where the bound on PoA is reasonable. These results also apply to the resource allocation or load balancing applications in the processor sharing 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.

Naval Research Logistics, 2003

Consider a distributed system where many gatekeepers share a single server. Customers arrive at each gatekeeper according to independent Poisson processes with different rates. Upon arrival of a new customer, the gatekeeper has to decide whether to admit the customer by sending it to the server, or to block it. Blocking costs nothing. The gatekeeper receives a reward after a customer completes the service, and incurs a cost if an admitted customer finds a busy server and therefore has to leave the system. Assuming an exponential service distribution, we formulate the problem as an n-person non-zero-sum game in which each gatekeeper is interested in maximizing its own long-run average reward. The key result is that each gatekeeper's optimal policy is that of a threshold type regardless what other gatekeepers do. We then derive Nash equilibria and discuss interesting insights.

Journal of Economic …, 2004

We study a class of queueing problems with endogenous arrival times that we formulate as non-cooperative n-person games in normal form with discrete strategy spaces, fixed starting and closing times, and complete information. With multiple equilibria in pure strategies, these queueing games give rise to problems of tacit coordination. We first describe and illustrate a Markov chain algorithm used to compute the symmetric mixed-strategy equilibrium solution. Then, we report the results of an experimental study of a large-scale (n=20) queueing game with fixed service time, FIFO queue discipline, and no balking, reneging, and early arrivals. Our results show consistent and replicable patterns of arrival that provide strong support for mixed-strategy equilibrium play on the aggregate but not individual level.

2013 5th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO), 2013

In the present paper we investigate the equilibrium customer behavior in a single server Markovian M 2 / M /1 queue with batch arrivals of two customers. We examine the various cases with respect to the level of information available to customers before they make this decision. More specifically, at their arrival epoch, the customers may or may not know the number of customers present in the system. In each of the two cases (observable case and unobservable case), we define the corresponding game, characterize customer equilibrium strategies, analyze the stationary behavior of the corresponding system. We also explore the effect of the information level on the equilibrium behavior and the social benefit via numerical comparisons. In other words, we analyze if the server is motivated to reveal information about the system state to the customers.

2005

We study computational and coordination efficiency issues of Nash equilibria in symmetric network congestion games. We first propose a simple and natural greedy method that computes a pure Nash equilibrium with respect to traffic congestion in a network. In this algorithm each user plays only once and allocates her traffic to a path selected via a shortest path computation. We then show that this algorithm works for series-parallel networks when users are identical or when users are of varying demands but have the same best response strategy for any initial network traffic. We also give constructions where the algorithm fails if either the above condition is violated (even for series-parallel networks) or the network is not series-parallel (even for identical users). Thus, we essentially indicate the limits of the applicability of this greedy approach. We also study the price of anarchy for the objective of maximum latency. We prove that for any network of m uniformly related links and for identical users, the price of anarchy is \({\it \Theta}({\frac{{\rm log} m}{{\rm log log} m}}\) ).

Queueing Systems, 2010

Consider a system of two queues in parallel, one of which is a •|M|1 single-server infinite capacity queue, and the other a •|G (N) |∞ batch service queue. A stream of general arrivals choose which queue to join, after observing the current state of the system, and so as to minimize their own expected delay. We show that a unique user equilibrium (user optimal policy) exists and that it possesses various monotonicity properties, using sample path and coupling arguments. This is a very simplified model of a transportation network with a choice of private and public modes of transport. Under probabilistic routing (which is equivalent to the assumption that users have knowledge only of the mean delays on routes), the network may exhibit the Downs-Thomson paradox observed in transportation networks with expected delay increasing as the capacity of the •|M|1 queue (private transport) is increased. We give examples where state-dependent routing mitigates the Downs-Thomson effect observed under probabilistic routing, and providing additional information on the state of the system to users reduces delay considerably. Keywords User equilibria • User optimal policies • Parallel queues • State dependent routing • Downs-Thomson paradox • Generalized threshold policies

Static & Dynamic Game Theory: Foundations & Applications, 2017