不耐煩顧客的M/M/S再嘗試模型中邊際機率與條件變異數的關係

Abstract

在現實生活中，等候模型被廣泛地使用於許多領域之中。在網路的問題中，我們可以將服務員視為伺服器，顧客則視為連接至伺服器的使用者。在實際的情形中，當使用者連不上伺服器時，通常會選擇重新連接伺服器，當使用者連續失敗幾次則可能選擇離開。這時候顧客便具有再嘗試且不耐煩的特性，所以我們便可以將不耐煩顧客的 M/M/S 再嘗試模型套用到網路的問題當中。在本文中，我們基於不耐煩顧客的 M/M/S 再嘗試模型的矩陣解來描述模型，並且導入條件機率及條件期望值的概念進行運算。最後我們給出幾條有關排隊人數或服務員忙碌人數的期望值及條件變異數的關係式和系統閒置機率的上下界，以利更好地刻畫整個網路的使用情形。

Queueing models has been widely used in many applications. In the communication network , we consider the user who tries to connect to a website or a server as a customer. Users in real world may retry to connect to a server when their connection failed. Moreover, when they failed to connect to a server several times, they will give up retrying to connect. Therefore, we can apply the M/M/S retrial queues with impatient customers to this problem. In the thesis, we describe the M/ M/S retrial queues with impatient customers by its infinitesemial generator. Then we apply the geometric form for this model and consider the conditional probability and the conditional expectation to provide some relations between the expectation of the queue length, the expectation of number of busy servers, and the variance of the queue length when all servers are occupied. Also, we provide bounds for the probability that the system is idle. If we want to estimate some system performance, these relations can let us easily to transform some ariables to other variables we needed.