The Unreliable M/M/1 Retrial Queue in a Random Environment

• Published in: Stochastic models Vol. 28; no. 1; pp. 29 - 48
• Main Authors: Cordeiro, James D., Kharoufeh, Jeffrey P.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 01.01.2012; Taylor & Francis; Taylor & Francis Ltd
• Subjects: Distribution theory; Exact sciences and technology; LDQBD process; Mathematics; Operations research; Probability and statistics; Probability theory and stochastic processes; Retrial queue; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stochastic analysis; Stochastic processes; Studies; Unreliable server; Stochastic model; Probability theory; Retrial queue; LDQBD process; Failure rate; Mean estimation; Steady state; Statistical method; 90B22; Unreliable server; Analytical method; Distribution function; Environment; Matrix method; 60K25; Queue; Numerical stability
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1080/15326349.2011.614478

We examine an M/M/1 retrial queue with an unreliable server whose arrival, service, failure, repair, and retrial rates are all modulated by an exogenous random environment. Provided are conditions for stability, the (approximate) orbit size distribution, and mean queueing performance measures which are obtained via matrix-analytic methods. Additionally, we consider the problem of choosing arrival and service rates for each environment state with the objective of minimizing the steady state mean time spent in orbit by an arbitrary customer, subject to cost and revenue constraints. Two numerical examples illustrate the main results.