Performance Analysis and Cost Optimization of the M/M/1/N Queueing System with Working Vacation and Working Breakdown

Xijuan Yang (), Yaqing Zhang, Bo Wang and Xue Jun Li
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Xijuan Yang: School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China
Yaqing Zhang: School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China
Bo Wang: School of Electronic and Information Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China
Xue Jun Li: Department of Electrical and Electronic Engineering, Auckland University of Technology, Auckland 1010, New Zealand

Mathematics, 2025, vol. 13, issue 18, 1-24

Abstract: This research advances steady state analysis and cost optimization of the M/M/1/N single vacation queueing system with setup time, working vacation, and working breakdown. The server works at a lower service rate instead of stopping work completely during both the vacation period and breakdown period—a key distinction from traditional vacation and breakdown strategies, where the server typically halts operations entirely. The setup time exists between the idle period and the regular busy period. The finite quasi birth-and-death process of this queueing system model is established. The stationary probability vector of the system is calculated using the matrix geometric method. Performance measures, such as output variance, availability, throughput rate, and stationary probabilities, are obtained using the theory of the fundamental matrix and covariance matrix. A cost optimization model based on system performance measures is established. The sparrow search algorithm is adopted to solve the cost optimization model. Through numerical experiments, the influences of system parameters on system performance measures and cost optimization function are analyzed, and the efficiency of the sparrow search algorithm for solving the cost optimization model is verified. The experimental results affirm the effectiveness and practicability of the proposed method, which provides a better theoretical basis for the practical application of the queueing system in communication engineering and production systems.

Keywords: working vacation; working breakdown; quasi birth-and-death; performance analysis; cost optimization; sparrow search algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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