Stochastic Comparisons Of Queuing Service System

The question of stochastic order has been the difficult important topic in the probability theory, and the stochastic comparison of Key indicators in queuing theory has a great practical significance and application value.This paper we studies the stochastic comparison of queue length and queue waiting time in queuing system.First we assume that the service time brought in by a customer and the subsequent interarrival time are independent. We consider two important classes of single-server bulk queuing models:M M/Γ(α,β)/1 andΓ(α,β)/M/1.In each class we compare two systems and prove that one is more congested than the other if their basic random variables are stochastically ordered in an appropriate manner. And we verify these results by random simulation.In the third chapter a queuing system with partial correlation is considered. We will further discuss the queuing system with multi-window. Comparison for waiting time of G/G/k. We show that in this case stronger dependence between interarrival and service times leads to decreasing waiting times in the increasing convex ordering sense. Extend the conclusion of Muller (2000).Also we all know the better server of peak period is very important. In the last chapter the number of customers of peak period during the waiting time is discussed in the bulk queue G/G/1.Based on the dependency of random variables in the sense of supermodel ordering, the monotonic of the number of customers of peak period during the waiting time is obtained, in the sense of increasing convex ordering, of G/G/1 on the dependency between the waiting time and the number of customers who come during the period. At last, we extend the conclusion to multi-window.