Analysis of a queueing system in random environment with an unreliable server and geometric abandonments

¹ College of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, P.R. China.
² School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing 211800, P.R. China.
³ School of Science, Nanjing University of Science and Technology, Nanjing 210094, P.R. China.

^* Corresponding author: jtao0728@163.com

Received: 7 April 2017
Accepted: 1 March 2018

Abstract

This paper studies a single server queueing model in a multi-phase random environment with server breakdowns and geometric abandonments, where server breakdowns only occur while the server is in operation. At a server breakdown instant (i.e., an abandonment opportunity epoch), all present customers adopt the so-called geometric abandonments, that is, the customers decide sequentially whether they will leave the system or not. In the meantime, the server abandons the service and a repair process starts immediately. After the server is repaired, the server resumes its service, and the system enters into the operative phase i with probability q_i, i = 1, 2, …, d. Using probability generating functions and matrix geometric approach, we obtain the steady state distribution and various performance measures. In addition, some numerical examples are presented to show the impact of parameters on the performance measures.