Journal of Korean Society of Industrial and Systems Engineering (산업경영시스템학회지)

Society of Korea Industrial and System Engineering (한국산업경영시스템학회)
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Busy Period Analysis of the Geo/Geo/1/K Queue with a Single Vacation

단일 휴가형 Geo/Geo/1/K 대기행렬의 바쁜 기간 분석

  • Kim, Kilhwan (Department of Management Engineering, Sangmyung University)
  • Received : 2019.10.17
  • Accepted : 2019.11.21
  • Published : 2019.12.31
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Abstract

Discrete-time Queueing models are frequently utilized to analyze the performance of computing and communication systems. The length of busy period is one of important performance measures for such systems. In this paper, we consider the busy period of the Geo/Geo/1/K queue with a single vacation. We derive the moments of the length of the busy (idle) period, the number of customers who arrive and enter the system during the busy (idle) period and the number of customers who arrive but are lost due to no vacancies in the system for both early arrival system (EAS) and late arrival system (LAS). In order to do this, recursive equations for the joint probability generating function of the busy period of the Geo/Geo/1/K queue starting with n, 1 ≤ n ≤ K, customers, the number of customers who arrive and enter the system, and arrive but are lost during that busy period are constructed. Using the result of the busy period analysis, we also numerically study differences of various performance measures between EAS and LAS. This numerical study shows that the performance gap between EAS and LAS increases as the system capacity K decrease, and the arrival rate (probability) approaches the service rate (probability). This performance gap also decreases as the vacation rate (probability) decrease, but it does not shrink to zero.

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References

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