Moshe Haviv

To queue or not to queue : equilibrium behavior in queueing systems

Preface. 1. Introduction. 2. Observable Queues. 3. Unobservable Queues. 4. Priorities. 5. Reneging and Jockeying. 6. Schedules and Retrials. 7. Competition Among Servers. 8. Service Rate Decisions. Index.

Price and delay competition between two service providers

Abstract In this paper we study situations in which two firms offer identical service for possibly different prices and response times. Customers’ choice between firms is based on their full price, which includes the service fee plus (expected) waiting costs. We consider a two level game. The first game is a non-cooperative game among customers who observe the prices (but not the queue sizes) and then decide if to give up the service (balk) or to join a queue. In the latter case, they need to decide which service provider to seek service from. The second game is played between the firms, who choose what prices to charge. In making their price selections, firms take into consideration the game played among customers. The new assumption here in contrast with existing literature is that customers belong to one of two classes, each of which is characterized by a waiting cost parameter. For this model, we propose a procedure for solving the former game analytically and the latter numerically. Various special cases are encountered, such as asymmetric price equilibria, continuum price equilibria, and cases in which demand for service increases with the service fee.

Perturbation bounds for the stationary probabilities of a finite Markov chain

This paper discusses perturbation bounds for the stationary distribution of a finite indecomposable Markov chain. Existing bounds are reviewed. New bounds are presented which more completely exploit the stochastic features of the perturbation and which also are easily computable. Examples illustrate the tightness of the bounds and their application to bounding the error in the Simon-Ando aggregation technique for approximating the stationary distribution of a nearly completely decomposable Markov chain. NEARLY COMPLETELY DECOMPOSABLE SYSTEMS; STOCHASTIC MATRICES

The price of anarchy in an exponential multi-server

We consider two criteria for routing selection in a multi-server service station: the equilibrium and social optimization. The ratio between the average mean waiting times in these two routings is called the price of anarchy (PoA). We show that the worst-case PoA is precisely the number of servers.

The Cost Allocation Problem for the First Order Interaction Joint Replenishment Model

We consider an infinite-horizon deterministic joint replenishment problem with first order interaction. Under this model, the setup transportation/reorder cost associated with a group of retailers placing an order at the same time equals some group-independent major setup cost plus retailer-dependent minor setup costs. In addition, each retailer is associated with a retailer-dependent holding-cost rate. The structure of optimal replenishment policies is not known, thus research has focused on optimal power-of-two (POT) policies. Following this convention, we consider the cost allocation problem of an optimal POT policy among the various retailers. For this sake, we define a characteristic function that assigns to any subset of retailers the average-time total cost of an optimal POT policy for replenishing the retailers in the subset, under the assumption that these are the only existing retailers. We show that the resulting transferable utility cooperative game with this characteristic function is concave. In particular, it is a totally balanced game, namely, this game and any of its subgames have nonempty core sets. Finally, we give an example for a core allocation and prove that there are infinitely many core allocations.

Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain

We implement and analyse aggregation/disaggregation procedures constructed to accelerate the convergence of successive approximation methods suitable for computing the stationary distribution of a finite Markov chain. We define six of these methods and analyse them in detail. In particular, we show that some existing procedures lie in the aggregation/disaggregation framework we set, and hence can be considered as special cases. Also, for all described methods, we identify cases where they are promising. Numerical examples for the applications of some of the methods for nearly completely decomposable stochastic matrices are given as well.

Singular Perturbations of Markov Chains and Decision Processes

In this survey we present a unified treatment of both singular and regular perturbations in finite Markov chains and decision processes. The treatment is based on the analysis of series expansions of various important entities such as the perturbed stationary distribution matrix, the deviation matrix, the mean-passage times matrix and others.

Equilibrium Threshold Strategies: The Case of Queues with Priorities

Multiplicity of solutions is typical for systems where the individuals tendency to act in a certain way increases when more of the other individuals in the population act in this way. We provide a detailed analysis of a queueing model in which two priority levels can be purchased. In particular, we compute all of the Nash equilibrium strategies (pure and mixed) of the threshold type.

Equilibrium strategies for queues with impatient customers

We consider a memoryless queue in which the reward of service completion for an individual reduces to zero after some time. Customers, while comparing expected holding costs and the rewards have to decide if to join the system at all and if they do when to renege. We show that a unique Nash equilibrium exists in which each of the customers joins with some probability and reneges as soon as the reward is zero.

Cooperation in Service Systems

We consider a number of servers that may improve the efficiency of the system by pooling their service capacities to serve the union of the individual streams of customers. This economies-of-scope phenomenon is due to the reduction in the steady-state mean total number of customers in the system. The question we pose is how the servers should split among themselves the cost of the pooled system. When the individual incoming streams of customers form Poisson processes and individual service times are exponential, we define a transferable utility cooperative game in which the cost of a coalition is the mean number of customers (or jobs) in the pooled system. We show that, despite the characteristic function is neither monotone nor concave, the game and its subgames possess nonempty cores. In other words, for any subset of servers there exist cost-sharing allocations under which no partial subset can take advantage by breaking away and forming a separate coalition. We give an explicit expression for all (infinitely many) nonnegative core cost allocations of this game. Finally, we show that, except for the case where all individual servers have the same cost, there exist infinitely many core allocations with negative entries, and we show how to construct a convex subset of the core where at least one server is being paid to join the grand coalition.