在N方策和T方策下具故障及啟動服務者M/G/1排隊之研究

Abstract

可控制排隊系統已被廣泛的應用於各領域，諸如製造/存貨系統、通訊及電腦網路系統等等，配合成本分析可幫助決策者在舜息萬變的資訊時代做出最佳決策，追求最大的利潤與效益。本論文主要是探討在N方策和T方策下服務者會故障及需一般啟動時間的M/G/1排隊系統，所謂N方策是指服務者一停止服務，要再重新開始提供服務，完全取決於等候線的顧客數是否達到N人，當顧客數達到N人時，服務者馬上開始提供服務直到系統中的所有顧客服務完成才停止服務；而所謂T方策是指服務者一停止服務，在一定時間T內至少有一位顧客到達，服務者才會開始提供服務直到系統中的所有顧客服務完成才停止服務，若在間T內無顧客到達，則必需等待直到下一個時間T內至少有一位有顧客到達。我們分別針對N方策和T方策，利用M/G/1排隊系統的隨機分解性質求出系統中的期望顧客數及系統期望的閒置、啟動、忙碌、故障期間長度，並推導出系統閒置、啟動、忙碌、故障狀態的機率，利用上述的期望值及機率，我們建構成本函數，分別求得最佳的N方策和T方策，並針對最佳的方策做敏感度分析，提供決策者在各種不同的參數下選擇最佳的方策。 此外，鑑於在N方策和T方策下服務者會故障及需一般啟動時間的M/G/1排隊系統中顧客數的機率分配無法確切求出，我們利用最大熵值法，在有限有用的資訊下諸如排隊系統中的期望顧客數、閒置、啟動、忙碌、故障狀態的機率，在最少偏誤的資訊下，求出估計的系統中顧客數的機率及近似的平均等候時間，並在各種不同分配下，比較最大熵值法解得近似的平均等候時間與真正的平均等候時間兩者之間的誤差。我們驗證得到最大熵值法是一個有用且夠精確的方法，可用來解決複雜的排隊問題。

The controllable queueing systems have been done by a considerable amount of work in the past and successfully used in various applied problems such as production/inventory systems, communication systems, computer networks and etc. To cooperate with the cost analysis, it can help the decision maker to make the optimal decision to obtain the maximal profit and efficiency for use. In this dissertation, we investigate an M/G/1 queue under the policy and the policy with sever breakdowns and general startup times. The N policy means the server returns to provide service only when the number of customers in the system reaches N (N>=1) until there are no customers present. The T policy means the server is turned on after a fixed length of time T repeatedly until at least one customer is present in the waiting line. Using the stochastic decomposition property of the M/G/1 queue, we derive various system performance measures, such as the expected number of customers, the expected length of the turned-off, startup, busy, and breakdown periods under the N policy and T policy, respectively. Then we deduce the probabilities of turned-off, startup, busy, and breakdown periods. We also construct the total expected cost function per unit time to determine the optimal threshold N and T , respectively, in order to minimize the cost function for the both policies. Sensitivity investigations on the optimal value of N and T for the both policies, respectively, are studied. Some numerical investigations are presented to demonstrate the analytical results obtained, and show how to make the decision based on minimizing the cost function. In addition, it is extremely difficult, not impossible, to obtain the explicit formulas such as the steady-state probability mass function of the number of customers and the expected waiting time for the N policy and the T policy M/G/1 queues with repair times and startup times are generally distributed. Under the given available information such as the queue length and the probabilities of idle, startup, busy and breakdown period, we use the maximum entropy principle to derive the approximate formulas for the steady-state probability distributions. We perform a comparative analysis between the approximate waiting time with established waiting time for various distributions, such as exponential (M), k-stage Erlang (Ek), and deterministic (D). We demonstrate that the maximum entropy approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems