Joining Longer Queues: Information Externalities in Queue Choice (original) (raw)
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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.
Equilibrium in Queues Under Unknown Service Rates and Service Value
SSRN Electronic Journal, 2000
We study a single queue joining equilibrium when there is uncertainty in the consumers' minds about the service rate and value. Without such uncertainty, the joining equilibria are characterized by means of a single threshold queue length above which consumers do not join . We show that in the presence of such uncertainty, the equilibrium joining strategy is not fully characterized by a single threshold. A "sputtering equilibrium" might exist. In the sputtering equilibrium, the queue length generally remains within a threshold, but reaches another, strictly higher, threshold, depending on the outcome of the randomized decision of the consumer arriving at the lower threshold. We discuss when and why sputtering equilibria exist.
Experimental Economics, 2005
We study a class of single-server queueing systems with a finite population size, FIFO queue discipline, and no balking or reneging. In contrast to the predominant assumptions of queueing theory of exogenously determined arrivals and steady state behavior, we investigate queueing systems with endogenously determined arrival times and focus on transient rather than steady state behavior. When arrival times are endogenous, the resulting interactive decision process is modeled as a non-cooperative n-person game with complete information. Assuming discrete strategy spaces, the mixed-strategy equilibrium solution for groups of n=20 agents is computed using a Markov chain method. Using a 2x2 between-subject design (private vs. public information by short vs. long service time), arrival and staying out decisions are presented and compared to the equilibrium predictions. The results indicate that players generate replicable patterns of behavior that are accounted for remarkably well on the aggregate, but not individual, level by the mixed-strategy equilibrium solution unless congestion is unavoidable and information about group behavior is not provided.
Equilibrium Play In Single-Server Queues With Endogenously Determined Arrival Times
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.
Equilibrium behavioural strategies in an M/M/1 queue
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.
Equilibrium in Queues Under Unknown Service Times and Service Value
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).
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...
Experimental Business Research, 2005
This chapter considers arrival time and staying out decisions in several variants of a queueing game characterized by endogenously determined arrival times, simultaneous play, finite populations of symmetric players, discrete strategy spaces, and fixed starting and closing times of the service facility. Experimental results show 1) consistent patterns of behavior on the aggregate level in all the conditions that are accounted for quite well by the symmetric mixed-strategy equilibrium of the stage game, 2) considerable individual differences in arrival time distributions that defy classification, and 3) learning trends across iterations of the stage queueing game in some of the experimental conditions. We propose and subsequently test a simple reinforcement-based learning model that, with a few exceptions, accounts for these major findings.
Strategic Dynamic Jockeying Between Two Parallel Queues
Probability in the Engineering and Informational Sciences, 2015
Consider a two-station, heterogeneous parallel queueing system in which each station operates as an independentM/M/1 queue with its own infinite-capacity buffer. The input to the system is a Poisson process that splits among the two stations according to a Bernoulli splitting mechanism. However, upon arrival, a strategic customer initially joins one of the queues selectively and decides at subsequent arrival and departure epochs whether to jockey (or switch queues) with the aim of reducing her own sojourn time. There is a holding cost per unit time, and jockeying incurs a fixed non-negative cost while placing the customer at the end of the other queue. We examine individually optimal joining and jockeying policies that minimize the strategic customer's total expected discounted (or undiscounted) costs over finite and infinite time horizons. The main results reveal that, if the strategic customer is in station 1 with ℓ customers in front of her, andq1andq2customers in stations 1 ...
Comments on: recent developments in the queueing problem
TOP, 2019
The problem of queues and waiting times is part of our daily life and so it is a situation that deserves a thorough study. Queueing theory mathematically studies the waiting lines and is part of the operations research field. This problem involves more complexity since it considers: the arrival process of the agents (customers) according to some probability distribution; the service time distribution and the number of available servers (in line or in parallel); and, finally, the queue discipline that determines the method used to serve the agents: first come, first served; last come, first served; etc. Chun presents a nice survey about the recent results on queueing problems where: there is only one server; the service time is the same for all agents (normalized to one); agents arrive according to some stochastic process; congestion may occur, and so the agents incur in waiting costs. The objective in this model, introduced by Dolan (1978), is to find an allocation rule that fixes the order in which the agents should be served and the monetary transfers. This problem can be addressed from several different approaches. We can assume an administrator is in charge of determining the order of the agents and the monetary transfers. But to do so, the administrator needs to know the waiting cost of each of the agents. We can assume that this is public information or we can assume it is private information and so an agent might reveal a different waiting cost if that is profitable for her. In the latter, it is important to provide incentives for the agents to reveal their true waiting costs. A queueing problem for a group of agents N is a vector θ = (θ i) i∈N , where θ i stands for the waiting cost of agent i. A solution, or a rule, for this problem is a pair (σ, t), where for each i, σ i denotes the position in the queue and t i denotes her monetary transfer. It is clear that to minimize the aggregated waiting cost, agents should be served according to their waiting cost in non-increasing order (queue efficiency). The property of queue efficiency means that the queue method applied in this problem is a priority queueing discipline that assigns a priority level, the waiting cost θ i in this case, to each customer and they are served following such priority on the first come first served basis.