The Concert Queueing Game with a Random Volume of Arrivals

Queueing Systems, 2012

We consider the non-cooperative choice of arrival times by individual users, who seek service at a first-come first-served queueing system that opens up at a given time. Each user wishes to obtain service as early as possible, while minimizing the expected wait in the queue. This problem was recently studied within a simplified fluid-scale model. Here we address the unscaled stochastic system, assuming a finite (possibly random) number of homogeneous users, exponential service times, and linear cost functions. In this setting we establish that there exists a unique Nash equilibrium, which is symmetric across users, and characterize the equilibrium arrival-time distribution of each user in terms of a corresponding set of differential equations. We further establish convergence of the Nash equilibrium solution to that of the associated fluid model as the number of users is increased. We finally consider the price of anarchy in our system and show that it exceeds 2, but converges to this value for a large population size.

Proceedings of the 4th International ICST Conference on Performance Evaluation Methodologies and Tools, 2009

We introduce the concert or the cafeteria queueing problem: Fixed but a large number of users arrive into a queue which provides service starting at time 0. Users may arrive before 0. They incur a queued waiting cost α • W , where W is the time to wait in the queue until service, and service time cost β • (t + W), where t is the arrival time and t + W is the total time until service. Each user picks a mixed strategy for arrival to minimize E[αW + β(t + W)]. We analyze the system in an asymptotic regime and develop fluid limit for the resultant queueing system. The limiting system may be modeled as a non-atomic game for which we determine an equilibrium arrival strategy. In particular, we note that the equilibrium arrival strategy is to arrive uniformly between some τ0 < 0 and τ1 selected so that the queue is never empty. We note that larger the β/α, larger the queue. Furthermore, we note that the 'price of symmetric anarchy' of this system equals 2. In addition to modeling queue formation at large concerts or cafeterias in certain settings, the model may be relevant more generally, for instance, in explaining queue formation in DMV offices at opening time, and at retail stores at opening time during peak shopping season.

International Series in Operations Research &amp; 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.

Queueing Systems, 2021

Consider a population of customers each of which needs to decide independently when to arrive to a facility that provides a service during a fixed period of time, say a day. This is a common scenario in many service systems such as a bank, lunch at a cafeteria, music concert, flight check-in and many others. High demand for service at a specific time leads to congestion that comes at a cost, e.g., for waiting, earliness or tardiness. Queueing Theory provides tools for the analysis of the waiting times and associated costs. If customers have the option of deciding when to join the queue, they will face a decision dilemma of when to arrive. The level of congestion one suffers from depends on others behavior and not only that of the individual under consideration. This fact leads customers to make strategic decisions regarding their time of arrival. In addition, multiple decision makers that affect each other's expected congestion, call for non-cooperative game theoretic analysis of this strategic interaction. This common daily scenario has prompted a research stream pioneered by the ?/M/1 model of Glazer and Hassin [17] that first characterized an arrival process to a queue as a Nash equilibrium solution of a game. This survey provides an overview of the main results and developments in the literature on queueing systems with strategic timing of arrivals. Another issue is that of social optimality, namely the strategy profile used by customers that optimizes their aggregate utility. In particular, we review results concerning the price of anarchy (PoA), which is the ratio between the socially optimal and the equilibrium utilities.

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...