M/G/1 Feedback Queue with Two Stage Heterogeneous Service with Time Homogeneous Breakdowns and Deterministic Repair Times

M/G/1反饋排隊俱兩階段不同服務且具相同故障率及固定維修時間

S. Vanitha

《International Journal of Information and Management Sciences》 23:3 (2012/09) Pp. 305-322

https://doi.org/10.6186/IJIMS.2012.23.3.5

摘要

吾人探討反饋單一排隊，波松抵達，兩階段不同隨機服務時間，隨機故障。故障時，其維修時間為定常。第一階段服務後，服務必須繼續第二階段服務。若第二階段服務不滿意，顧客可能逕至隊尾，其機率為p，否則，永遠離開系統（機率為q = 1－p）。本文應用輔助變數法導得系統中人數之機率生成函數之公式以及系統中之平均人數。此外也討論了一些特具意義的特殊情況。

關鍵字

波松抵達 ； 閒置狀態 ； 穩定狀態 ； 固定維修 ； 輔助變數法

並列摘要

We analyze a single server feedback queue with Poisson arrivals, two stages of heterogeneous service with different (arbitrary) service time distributions subject to random breakdowns and just after a breakdown the server undergoes repairs of a fixed (constant) duration. After first-stage service, the server must provide the second stage service. However after the completion of second stage of service, if the customer is dissatisfied with his service, he can immediately join the tail of the queue as a feedback customer with probability p: Otherwise the customer may depart forever from the system with probability q = 1-p. The supplementary variable technique is employed to find explicitly the probability generating function of the number in the system and the mean number in the system. Some particular cases of interest are discussed as special cases.

並列關鍵字

Poisson arrivals ； idle state ； steady state ； deterministic repairs ； supplementary variable technique

參考文獻

Choudhury, G.(2005).An M/G/1 queueing system with two phase service under D policy.International Journal of Information and Management Sciences.16,1-17.

Choudhury, G.,Paul, M.(2006).On a single server queue with two stage heterogeneous service and deterministic server vacations.International Journal of System Science.32,837-844.

Cox, D. R.(1955).The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables.Proceedings of Cambridge Philosophical Society.(Proceedings of Cambridge Philosophical Society).

Cramer, M.(1989).Stationary distributions in a queueing system with vacation time and limited service.Queueing systems.4,57-68.

Doshi, B. T.(1985).A note on stochastic decomposition in a GI/G/1 queue with vacation or setup times.Journal of Applied Probability.22,419-428.