双水平控制策略和延迟不中断单重休假的M/G/1排队系统分析

Analysis of M/G/1 Queueing System with Bi-level Control Policy and Delayed Uninterrupted Single Vacation

本文提出建立一个具有双水平控制策略和延迟单重休假且休假不中断的M/G/1排队系统模型.在这个系统模型中,服务员启动系统的阈值水平是1,而服务员开始对顾客服务的阈值水平是N(N≥1),即当服务员从休假归来时,若系统中已有顾客等待服务,服务员就立即启动系统,若系统中没有顾客等待,则服务员待在系统中直到有一个顾客到达时才启动系统.在系统的启动完成后,若系统中等待服务的顾客数大于等于N(N≥1),则服务员立即为在场的顾客服务直到系统再次变空,若系统中等待服务的顾客数小于N,则服务员待在系统中直到累计有N个顾客时立即开始服务.运用全概率分解技术和Laplace变换工具,讨论系统队长的瞬态和稳态性质,导出系统瞬态队长分布的Laplace变换的递推表达式,并通过直接计算获得系统队长的稳态分布的递推表达式,以及稳态队长分布的概率母函数和平均队长的表达式.最后,给出了几种特殊情况的注解.

This paper proposes to establish an M/G/1 queueing model with a bi-level control policy and a delayed uninterrupted single vacation. In this system, the startup threshold of the system is 1, and the threshold for starting services is N(N ≥ 1). That is, when the server comes back to the system, he/she will immediately start the system if there is at least 1 customer in the system, otherwise, he/she doesn’t start the system until a customer arrives. After the start of the system is completed, if the number of customers waiting for service in the system is greater than or equal to N, the server immediately starts to serve until the system becomes empty. If the number of customers waiting for service in the system is less than N, he/she will stay in the system until there are N customers and start serving immediately.Using the total probability decomposition technique and the tool of the Laplace transform, we study the transient and equilibrium properties of the queue-length. The expressions of the Laplace transformation of the transient queue-length distribution with respect to time t are obtained. Furthermore, the recursive expressions of the steady queue-length distribution, the probability generating function of the steady queue-length distribution and the expected size of the queue-length are derived by a direct calculation.Finally, some notes for several special cases are given.