A survey of queueing systems with strategic timing of arrivals

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.

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.

International Journal of Mathematics in Operational Research, 2018

For an M/M/1 system, we analyse the strategic interactions of the social optimiser, the service provider and customers and their consequences on the system. The social optimiser chooses the type of information to make available to customers (make the system observable or unobservable), the service provider chooses the service rate with which he performs the service, and customers decide, according to the strategic choices of the first two agents, to use or not the system. As these agents are interacting in a common environment with respect to their objectives, we model the problem as a three-stage game between them. A resolution of the different stages will be made, which will give the overall solution to the considered problem, corresponding to the subgame perfect Nash equilibrium in behavioural strategies. A numerical analysis will be made where one can see the graphical solution of the game, comparisons and interpretations will be well established.

European Journal of Operational Research, 2013

This paper studies the effect of information suppression on Naor's model as well as on Edelson and Hildebrand's model under geometric distribution. We set the suitable non-cooperative games and search for their Nash equilibria under the observable and unobservable system. In each case, we analyze the effects of information level on the customers' equilibrium and socially optimal balking strategies as well as on the profit maximization of the system manager. The socially optimal behavior and the inefficiency of the equilibrium strategies are quantified via the price of anarchy measure. We discuss a comparison study of the profit maximization and social welfare under an imposed admission fee. Also, the impact of information on the selfish and social optimal joining rates is examined. Numerical results are presented to exemplify the impact of system parameters on the optimal behavior of customers under different information levels.

Queueing Systems

We study a Markovian single-server ticket queue where, upon arrival, each customer can draw a number from a take-a-number machine, while the number of the customer currently being served is displayed on a panel. The difference between the above two numbers is called the 'virtual queue length'. We consider a nonhomogeneous population of customers comprised of two types: 'regular', and 'strategic'. Upon arrival a regular customer, regardless of the value of the virtual queue length, draws a number from the machine, joins the queue and waits in the system until being served. A strategic customer, depending on the virtual queue length, may either join, leave, or go to 'orbit' for a random duration. If, upon return from orbit, a strategic customer realizes that s/he missed her/his turn, s/he balks. Otherwise, s/he joins the queue and waits to be served. We analyze this intricate stochastic system, calculate its steady state probabilities, derive the sojourn time's Laplace-Stieltjes transform of a regular and of a strategic customer, and calculate the system's performance measures. Finally, an economic analysis is performed to determine the optimal mean orbiting time of strategic customers for two types of objective functions. Numerical examples are presented.

Proceedings of the 6th International Conference on Performance Evaluation Methodologies and Tools, 2012

We consider the concert queueing game in the fluid framework, where the service facility opens at a specified time, the customers are particles in a fluid with homogeneous costs that are linear and additive in the waiting time and in the time to service completion, and wish to choose their own arrival times so as to minimize their cost. This problem has recently been analyzed under the assumption that the total volume of arriving customers is deterministic and known beforehand. We consider here the more plausible setting where this volume may be random, and only its probability distribution is known beforehand. In this setting, we identify the unique symmetric Nash equilibrium and show that under it the customer behavior significantly differs from the case where such uncertainties do not exist. While, in the latter case, the equilibrium profile is uniform, in the former case it is uniform up to a point and then it tapers off. We also solve the associated optimization problem to determine the socially optimal solution when the central planner is unaware of the actual amount of arrivals. Interestingly, the Price of Anarchy (ratio of the social cost of the equilibrium solution to that of the optimal one) for this model turns out to be two exactly, as in the deterministic case, despite the different form of the social and equilibrium arrival profiles.

2016

This thesis includes three essays exploring some economic implications of queueing. A preliminary chapter introducing useful results from the literature which help contextualize the original research in the thesis is presented first. This introductory chapter starts by surveying queueing results from probability theory and operations research. Then it covers a few seminal papers on strategic queueing, mostly but not exclusively from the economics literature. These cover issues of individual and social welfare in the context of First Come First Served (FCFS) and Equitable Processor Sharing (EPS) queues, with one or multiple servers, as well as a discussion of strategic interactions surrounding queue cutting. Then an overview of some important papers on the impact of queueing on competitive behaviour, mostly Industrial Organization economists, is presented. The first original chapter presents a model for the endogenous determination of the number of queues in an M/M/2 system. Customer...

European Journal of Operational Research, 2017

This work presents a variation of Naor's strategic observable model (1969), by adding a component of customer heterogeneity induced by the location of customers in relation to the server. Accordingly, customers incur a "travel cost" which depends linearly on the distance of the customer from the server. The arrival of customers with distances less than x is assumed to be a Poisson process with rate λ(x) = x 0 h(y)dy < ∞, where h(y) is a nonnegative "intensity" function of the distance y. In a loss system M/G/1/1 we define the threshold Nash equilibrium

Operations Research, 2014

In Naor's seminal queue-joining model, queue-joining probabilities decrease monotonously in the queue length; the longer the queue, the fewer consumers join. In practice, empirical evidence indicates that queue-joining probabilities may not always be decreasing in the queue length. For example, for restaurants, long queues may sometimes be more attractive than short queues. We rationalize non-monotonic strategies by relaxing the information assumptions in Naor's model. Instead of assuming that the expected service time and service value are common knowledge, we assume that they are unknown to consumers, but positively correlated. Under such informational assumptions, we show that equilibria may emerge for which the joining probability increases in the queue length. We refer to these as "sputtering equilibria." We discuss when and why such sputtering equilibria exist for discrete as well as continuously distributed priors on the expected service time (with positively correlated service value).

Mathematics

This paper considers a callback single-server system with an orbit and a First-Come First-Served (FCFS) service discipline. Customers (users, clients) that encounter a busy server are sent into orbit and then have the option to retry service after an exponential period of time. In addition, each customer entering the system uses a strategy and must independently decide when to arrive in the system within a fixed admission period of time so that the expected sojourn time is minimal. We interpret the arrival process as a Nash equilibrium solution of a noncooperative game when the arrival intensity is completely described by an unknown distribution function, and then we propose a way to find an equilibrium for the case when the client’s waiting time for service is obviously limited. The analytical solution for the equilibrium is illustrated numerically for two-person and three-person games.