Strong Approximation And A Law Of The Iterated Logarithm For Tandem Queue

In this paper, we study the strong approximation and a law of the iterated logarithm (LIL) for multi-station tandem queue. Two types of tandem queue model are considered, one is single-sever tandem queue, and the other one is multi-sever tandem queue. Tandem queuing system has been widely used in the field of port transportation and computer communication. Except fluid limit and diffusion approximation, strong approximation and the law of the iterated logarithm are two other important ways of approximation approach.For strong approximation of single-server tandem queuing model, the condition of traffic intensity is not qualified. Using stochastic process limit, we obtain the strong approximation of queue length, busy time, workload, idle time, and departure processes. For strong approximation of multi-server tandem queuing model, we obtain the strong approximation of queue length and weighed cumulative idle processes under traffic intensity larger than or equal1. As a result, we approximate the original discrete tandem queuing system to a continuous Brownian motion system, which offer a basis for further theoretical research.For the LIL of single-server tandem queuing model, we assume five different conditions of traffic intensity. According to the results of strong approximation and the LIL of Brownian motion, we transfer the LIL of queue length, busy time, workload, idle time, and departure processes into problems related to reflected Brownian motion, and we get the LIL of these five processes as a result. For the LIL of multi-server tandem queuing model, we concluded the LIL of queue length and weighed cumulative idle processes under traffic intensity larger than1. From the result of limit value, we can have the conclusion that the asymptotic behavior of the tandem queuing system is closely related to the traffic intensity of queuing system.