Optimal control of admission in service in a queue with impatience and setup costs

Abstract

We consider a single server queue in continuous time, in which customers must be served before some limit sojourn time of exponential distribution. A customer who is not served before this limit leaves the system: it is impatient. The fact of serving customers and the fact of losing them due to impatience induce costs. The fact of holding them in the queue also induces a constant cost per customer and per unit time. The purpose is to decide when to serve the customers so as to minimize costs. We use a Markov Decision Process with infinite horizon and discounted cost. Since the standard uniformization approach is not applicable here, we introduce a family of approximated uniformizable models, for which we establish the structural properties of the stochastic dynamic programming operator, and we deduce that the optimal policy is of threshold type. The threshold is computed explicitly. We then pass to the limit to show that this threshold policy is also optimal in the original model. A particular care is given to the completeness of the proof. We also illustrate the difficulties involved in the proof with numerical examples.

A Fuzzy-Based Control System for Customers’ Admission in a Two-Station Queue Network

“…A Poisson arrival queueing system is a fairly reasonable appropriation where the arrival and service rates are really more realistic than probabilistic. However, in many practical situations the parameters of arrival rate (λ) and service rate (µ) are frequently fuzzy and cannot be expressed in exact terms [28].…”