The Strong Approximation Of The Customer's Stay Time In The Two-Stage Tandem Service System

This paper considers two-stage tandem queue,which proves the strong approximation and the functional law of the logarithm rate of stay time under heavy traffic conditions.The stay time here refers to the time from the arrival of the system to the departure of the system.Two-stage tandem queue system means that the customer arrives at the system from outside,and then passes through two service stations in series.The customer receives the service according to the rule of first come first service and leaves the system after one service is respectively finished.The arrival process and service process of the two-stage tandem queue are an update process.The arrival interval and service interval are a series of independent and identically distributed random variables.Therefore,this paper first introduces the basics about the Poisson process,the update process,and the law of large numbers.Because of the strong approximation of staying time is a Brownian motion,this paper introduces the basic knowledge of Brownian motion,and the definitions of oblique reflection mapping and continuous mapping used in solving the strong approximation and the functional law of large numbers of the stay time.The following article introduces model of the single stage queue system.The single stage queue model is the basis of the two-stage tandem queue system.It also complies with the service rules of the first come first serverd.This part solves the queue length and workload processes,idle and busy time in the underload,critically loaded and overloaded regimes at first and then their strong approximation and functional law of the iterated logarithm.Finally,the paper introduces the model of the two-stage tandem queue system,and solves the the queue length and workload processes,idle and busy time in the underload,critically loaded and overloaded regimes.The innovation part of the article is to solve the strong approximation of the stay time under heavy traffic conditions.The approximation result is stronger than the fluid limit supplementing the results of the predecessors and inspiration to the strong approximation and functional law of the iterated logarithm of stay time in the other model.