Two Types Of Approximation For The Single-Station Queue

In this paper, we study two types of approximation for a single-station queue. We focus on the approximation of a single-station multi-server queue model. Firstly, we obtain the strong approximation for queue length process and the weighted cumulative idle time process of queueing model by using strong limit theory. Secondly, by using the result of the strong approximation and properties of reflected Brownian motion, we obtain the functional law of iterated logarithm for the queue length process and the weighted cumulative idle time process.On basis on the network of large call center in life, we model it as a single-station multi-server queue. We transfer the performance analysis of the network of large call center into the corresponding performance analysis of queueing model. Different from others we analyze the model by stochastic limit method in almost surely convergence regime. Specifically, we use strong limit theorem to obtain the strong approximation of the queue length process and the weighted cumulative idle time process. The results asymptotically describe the two indexes. As a result, we transfer the original discrete queueing system into a continuous Brownian motion, which offers a basis for further research.As we know, the functional law of iterated logarithm is a classic problem in probability theory. It can also characterize the asymptotic behavior of the queueing network. By using the result of the strong approximation and properties of reflected Brownian motion, we transfer the problem of functional law of iterated logarithm for the queue length process and the weighted cumulative idle time process into problems related to reflected Brownian motion. In the last, we get the functional law of iterated logarithm of the queue length process and the weighted cumulative idle time process of a single-station multi-server queue and determine the parameters. From the final limit, we can have the conclusion that the asymptotic behavior of the queueing system is closely related to traffic intensity of the corresponding queueing system.