具有第二次可選擇服務、服務者故障和啟動時間特徵之M/G/1排隊系統

Abstract

本論文主要研究 <N, p>-方策 和 <T, p>-方策 M/G/1 排隊系統具有第二次可選擇服務、服務者會故障以及啟動時間。所有抵達的顧客必須接受第一必要服務，而服務者在提供第一必要服務前，需要一啟動時間方可提供服務，當顧客接受完第一必要服務時，部分的顧客會繼續選擇接受第二種服務。所謂 <N, p>-方策是指，當系統顧客數到達某一數目 N 時，則服務者開始提供服務，直到系統中沒有顧客時，服務者擁有 p 的機率停止服務，亦有1- p 的機率服務者無法停止服務，即服務者保持著隨時可以提供服務狀態。而 <T, p>-方策則是指當系統中沒有人時，服務者會進行休假，假期時間長度為T，假期結束後服務者會返回系統察看是否有顧客，當系統至少有一個顧客時，服務者擁有 p 的機率對顧客提供服務，亦有 的機率服務者繼續離開系統，並進行另外一個時間長度 T 的休假，直到假期結束後才對顧客提供服務。對於 <N, p>-方策以及 <T, p>-方策的差異性，<N, p>-方策是服務者在服務結束階段時，對服務者隨機控制。而<T, p>-方策則是服務者在假期結束後返回系統，並發現系統內至少有一個顧客，服務者要準備提供服務階段時，對服務者隨機控制。 對於這兩個排隊系統，我們利用凸組合性質 (convex combination property) 以及 更新報酬定理 (renewal reward theorem) 來推論不同的系統績效，並建構成本函數來找尋聯合最佳門檻值 (N, p) 以及 (T, p)，經由推論後我們可以得到明確的封閉解 (closed-form) 以使得成本函數最小。由於排隊系統進行敏感度研究，可以提供系統分析者了解輸入參數對系統影響，因此，我們也將對最佳的門檻值進行敏感度分析，藉此分析來了解系統參數 (或是成本元素) 的變動後，對於最佳門檻值之影響，最後，我們有提供數值結果並討論之。 除此之外，要推論出這兩個排隊系統確切的穩態機率分配是相當困難的，然而在排隊理論中，最大熵值原理 (Maximum Entropy Principle) 已經被廣泛地利用來分析不同種類的排隊模型。在本篇論文裡，我們將利用最大熵值原理來近似這兩個排隊系統隊伍長度的穩態機率分配，然後利用推導出來的結果求取近似的隊伍中平均等候時間。給定特定的系統參數下，在不同的服務時間、修理時間以及啟動時間分配下，我們將對近似以及精確的隊伍中平均等候時間進行比較分析，藉由數值結果可以得知相對誤差是非常小的，因此，我們可以進一歩說明最大熵值法是相當準確地，而且處理較複雜的排隊模型時，確實是一個有用的方法。

In this dissertation, we analyze the <N, p>-policy and the <T, p>-policy M/G/1 queues with second optional service, server breakdowns and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service which follows a general distribution. By <N, p>-policy we mean that the server is turned on when N customers have accumulated, but the server is turned off with probability p as the system becomes empty. The so-called <T, p>-policy queue is characterized by the fact that if at least one customer is present in the queue after T time units have elapsed since the end of the busy period, the server can be switched on with probability p or leaves for a vacation of the same length T with probability (1-p). The main difference between the two randomized policies is that for an <N, p>-policy a decision maker selects actions randomly at completion epoch when there are no customers in the system, whereas for a <T, p>-policy a decision maker selects actions randomly at the beginning epoch of the service when at least one customer appears. For those two queueing systems, we develop various system performances by the convex combination property and the renewal reward theorem. The expected cost functions are established to determine the joint optimal threshold values of (N, p) and (T, p), respectively. Then we obtain the explicit closed-form of the joint optimal solutions for those two policies. Because of sensitivity investigation on the queueing system with critical input parameters may provide some answers for the system analyst. A sensitivity analysis is provided to discuss how the system performances can be affected by the input parameters (or cost parameters) in the investigated queueing service model. For illustration purpose, numerical results are also presented. Furthermore, it is rather difficult to derive the steady-sate probability explicitly for those two queueing systems. The maximum entropy approach has been widely applied in queueing theory to analyze different queueing models. Here, we use this approach to approximate the steady-state probability distributions of the queue length for those two queues. Then the approximate expected waiting time in the queue can be obtained by the maximum entropy solutions. Subsequently, we perform comparative analysis between the approximate results with established exact results for various service time, repair time and startup time distributions with specific parameter values. Our numerical investigations showed that the relative error percentages of the approximate method are quite small. Based on the numerical results, one can demonstrate that the maximum entropy approach is accurate enough and provides a helpful method for analyzing more complicated queueing models.