具有第二次可選擇服務、服務者選擇休假之多個服務者排隊分析

Abstract

本論文主要為多個服務者排隊系統含有第二次可選擇服務、服務者選擇休假以及考量顧客重試行為等條件之分析研究。多服務者排隊模式在實務上較單一服務者模式更有彈性及適用性，以往多服務者排隊系統之數學分析技巧相對較為複雜且困難，而相關文獻也較少。所有抵達系統的顧客都必須接受服務者所提供的第一必要服務，當顧客接受完第一必要服務後，部分的顧客會選擇繼續接受第二種附加服務。所謂服務者選擇休假是指每位服務者在每服務完一位顧客後都有一定的機率會進行(僅)一次的休假，並於休假結束之後回到系統之中繼續提供服務或等待新顧客的到來，即單一次休假策略。當系統中的服務者都處於忙碌時，新到達的顧客將進入循環區(orbit)等待，於一段時間後再嘗試著進入系統之中接受服務，此循環將持續進行直到該顧客接受完服務並離開系統為止，此稱為顧客之重試行為。因循環區之中大多數顧客的嘗試都是失敗的重試行為，並不會造成系統狀態的變化，於是我們假設循環區中允許重試的顧客人數有一最大上限值N，同時可以簡化數學模式分析上的困難度。我們一共研究了M/M/c排隊系統含有第二次可選擇服務(及顧客重試行為)以及M/M/c 排隊系統含有服務者選擇休假(及顧客重試行為)等四個排隊模式。 對於這四個排隊系統，我們利用矩陣幾何法 (matrix-geometric method) 以及遞迴技巧 (recursive technique) 來推論其系統達穩態之條件及穩態機率解。除此之外，要推論出這四個排隊系統確切的比率矩陣 (closed-form of rate matrix) 是相當困難的，然而在使用矩陣幾何法時，比率矩陣為最重要之元件。在本篇論文裡，我們將利用一單調收斂之數列去求得比率矩陣之近似解，然後利用推導出來的結果去求取穩態機率的近似解。之後建構成本函數來找尋在不同條件設定下的最佳的服務者個數、平均服務率、平均休假率等系統參數，經由直接搜尋法 (direct search method) 及彷牛頓法 (Quasi-Newton method) 我們可以得到近似最佳解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，我們也將對近似解與最低成本進行敏感度分析，藉此分析來了解系統參數的變動後，對於近似解與最低成本之影響，最後，我們有提供數值結果並討論之。

In this dissertation, the optimization investigated multi-server queueing systems with the second optional service (SOS) channel, Bernoulli vacation policy, and customer retrial behaviors are investigated. Multi-server vacation models are more flexible and applicable in practice than single server models. For the multiple server queueing models, the mathematical analyses are complicated and difficult; hence there are only a limited number of studies. All arriving customers need the first essential service (FES) provided by the servers. As soon as the FES of a customer is completed, a customer may leave the system or opt for the SOS. Bernoulli vacation policy means that the server may take one and only one vacation of random length with certain probability at each service completion. As the completion of vacation, the server stays idly for the next new arriving customer or serves the customers waiting in the queue, if any. That is, the single vacation policy. If the customer finding all servers busy always joins the orbit and tries to enter the system for service later. This manner continues until the customer is eventually served then leave the system. This is so-called the customer retrial behaviors. Because most of retrial behaviors of the customers in the orbit are failed without the change of states, we assume that the number of customers who can generate retrial requests is restricted (truncated) to an upper bound value N. This setting makes the mathematical model easier to be analyzed. We investigate four queueing models include the M/M/c (retrial) queue with SOS channel, the M/M/c (retrial) queue with modified Bernoulli single vacation policy, and the M/M/c retrial queue with Bernoulli single vacation policy. For those four queueing systems, we develop the stability conditions and steady-state probability solutions by the matrix-geometric method and recursive technique. Furthermore, it is rather difficult to derive the closed-form solution of the rate matrix for those four queueing systems. The rate matrix is the most important component for implementing the matrix-geometric method to analyze the infinite capacity queueing system. Here, we employ a monotone and convergent sequence to approximate the rate matrix, and obtain the approximation solution of the steady-state probability. The expected cost functions are established to determine the optimal value of the number of servers, mean service rate, mean vacation rate and other system parameters. By implementing the direct search method and Quasi-Newton method, we can find the optimal solution heuristically so that the cost function is minimized. Because of sensitivity investigation on the queueing system with critical input parameters may provide some information for the system analyst. A sensitivity analysis is performed to discuss how the system performances and the optimal solutions are affected by the input parameters in the investigated queueing models. For illustration purpose, numerical results are also presented.