Optimal Control of the N Policy M/G/1 Queue with Server Breakdowns,Startup and Second Optional Service

此篇論文分析研究了含有故障服務者, 啟動時間及第二選擇服務之N-方策M/G/1排隊系統. 其中第一主要服務之服務時間假設為一般分配, 而第二選擇服務則假設為其它一般分配. 在完成第一主要服務後, 顧客有(1-θ)的機率離開或者有θ(01)時, 系統需要一個啟動時間後才開始服務顧客直到系統再次變空. 我們分析各種系統性能且正確的導出穩態結果. 發展每單位時間每位顧客的總期望花費在最小花費時決定N之最佳值. 敏感度分析的分析結果也被導出.This paper studies the N policy M/G/1 queue with server breakdowns, startup and second optional service. The service times of the first essential service are assumed to follow a general distribution and that of second optional service with another general distribution. A customer may leave the system either after the first essential service with probability (1-θ) or at the completion of the first essential service may immediately go for a second optional service with probability θ(01), the server requires a startup time before providing service until the system is again empty. We analyze various system performance and steady-state results are derived explicitly. The total expected cost per customer per unit time is developed to determine the optimal value N at a minimum cost. Analytic results for sensitivity analysis are also derived.1 Introduction 1 1.1 Problem Statement..........1 1.2 Literature Review...........3 1.3 Scope of Study ...........4 2 System Performance Measures 5 2.1 The ordinary M/G/1 queue.........5 2.1.1 Probability generating function of the number of customers in the system........6 2.1.2 Expected length of the complete, busy, and breakdown periods........7 2.2 The N policy M/G/1 queue.......8 2.2.1 Expected number of customers in the system.......8 2.2.2 Expected length of the idle period.........9 2.2.3 Expected length of the complete startup period......10 2.2.4 Expected length of the busy and breakdown periods........11 2.2.5 Expected length of the busy cycle........12 2.2.6 Long-run faction of time measures..........12 3 The Optimal N Policy 14 3.1 The cost function.........14 3.2 The optimal value of N..........15 4 Sensitivity Analysis and Numerical Results.........17 4.1 Analytical results for sensitivity analysis........17 4.1.1 Sensitivity analysis of system parameter λ........17 4.1.2 Sensitivity analysis of system parameter μ........18 4.1.3 Sensitivity analysis of system parameter α........19 4.1.4 Sensitivity analysis of system parameter β........19 4.1.5 Sensitivity analysis of system parameter γ........20 4.2 Numerical results..........21 5 Conclusions and Future Research 31 5.1 Conclusions........31 5.2 Future Research........31 References 3