SMAP/INID/1系统的瞬时队长分布

Distribution of the transient queue length in SMAP/INID/1

研究了一个有如下特征的排队系统:该系统的到达间隔及服务时间均为相互独立的随机变量,但不一定同分布。特别地,到达间隔分布与系统的瞬时输入量有关。这个系统是GI/G/1系统的拓广。该系统的瞬时队长过程一般不是一个马尔可夫过程,难于直接求取它的分布。利用补充变量技术,可以得到一个多维马尔可夫过程,使得上述系统的瞬时队长过程构成多维过程的一个分量过程,这样,便可借助马尔可夫过程理论及马氏骨架过程理论,得到一组柯尔莫哥洛夫向后方程及向后方程组,导出排队系统的瞬时队长分布的积分表示。在各到达间隔与服务时间均具密度函数的条件下,该积分表示的被积项能够递归地求取。此结论类似于A.S.Alfa等处理GI/G/1系统时所得结论。

This paper studies a queueing system which the interarrival times are mutually independent but not have to be identically distributed random variables, and so are the service times, especially, the distributions of the interarrival times depend on arrival quantity of the above queueing system at corresponding times. This queueing system is the generalization of GI/G/1 queue. Generally speaking, the queueing process in the above queueing system is not Markovian, so it is hard that its distribution to be directly evaluated. Using supplementary variable technique in stochastic models, a multi-dimensional Markov process including the queueing process in the queue mentioned here as a component is constructed. The Kolmogorov backward equations are easily established by the Markov processes and Markov skeleton processes. By the above backward (equations,) the integral representation of the transient distribution of the queueing process is obtained. On certain conditions, the integratedterm in the above representation can be calculated which is similar to the conclusion given by A.S.Alfa, et al.