Min(N,D)-策略下M/G/1排队系统的离去过程

Departure Process of M/G/1 Queueing System under Min(N,D)-policy

考虑Min(N,D)-策略下M/G/1排队系统的离去过程.运用全概率分解技术和更新过程理论,从任意初始状态出发,讨论系统在有限时间区间内离去顾客的平均数,得到了离去顾客平均数的瞬态表达式和稳态表达式,并给出了离去过程、服务员状态过程和服务员忙期中的服务更新过程之间的重要关系,该关系揭示了离去过程的随机分解特性:离去顾客的平均数被分解为两部分,一部分是服务员忙的概率,另一部分是服务员忙期中的离去顾客平均数,从而简化了对离去过程的研究.最后,得到了便于有效计算离去顾客平均数的渐近展开式,以及一些特殊情形下的相应结果.在排队网络中,由于一个排队系统的输出即为下游排队系统的输入,希望本文所得结果为排队网络的研究提供有用的信息.

This paper considers the departure process of the M=G=1 queue with Min(N;D)-policy. Using the total probability decomposition technique and the renewal process theory, we discuss the expected number of departures occurring in finite time interval from an arbitrary initial state. Both the transient expression and the steady state expression of the expected departure number are obtained. The important relation among departure process, server state process and renewal process of service during server busy period is discovered. The relation displays the stochastic decomposition characteristic of the departure process, i.e., the expected departure number is decomposed into two parts: one is the server busy probability, and the other is the expected departure number during server busy period, which simplifies the discussion on the departure process. Furthermore, the approximate expansion for convenient calculation of the expected departure number is obtained. Finally, we derive the corresponding results for some special cases. Since the departure process also often corresponds to an arrival process in downstream queues in queueing network, it is expected that the results obtained in this paper would provide useful information for queueing network.