Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers

Abstract

This study investigates an infinite capacity Markovian queue with a single unreliable service station, in which the customers may balk (do not enter) and renege (leave the queue after entering). The unreliable service station can be working breakdowns even if no customers are in the system. The matrix-analytic method is used to compute the steady-state probabilities for the number of customers, rate matrix and stability condition in the system. The single-objective model for cost and bi-objective model for cost and expected waiting time are derived in the system to fit in with practical applications. The particle swarm optimisation algorithm is implemented to find the optimal combinations of parameters in the pursuit of minimum cost. Two different approaches are used to identify the Pareto optimal set and compared: the epsilon-constraint method and non-dominate sorting genetic algorithm. Compared results allow using the traditional optimisation approach epsilon-constraint method, which is computationally faster and permits a direct sensitivity analysis of the solution under constraint or parameter perturbation. The Pareto front and non-dominated solutions set are obtained and illustrated. The decision makers can use these to improve their decision-making quality.

Keywords:

• working breakdown
• impatient customers
• particle swarm optimisation
• epsilon-constraint method
• Pareto front

Acknowledgements

The author would like to thank the reviewers for their helpful comments and suggestions, which considerably improved this paper.

Additional information

Notes on contributors

Cheng-Dar Liou

Cheng-Dar Liou is an associate professor of the Department of Business Administration at National Formosa University, Taiwan. He received his PhD in Civil Engineering from National Chung Hsing University, Taiwan, in 2001. He also held an MSc of Management from Swansea University, UK, in 2008. His current research interests include applications of artificial intelligence, queueing theory, scheduling, and operations research.