The Heavy-traffic Limits And The Simulation For M/M/n Queueing Models

The main purpose of this thesis is to explain the proof for standard multi-queueheavy-traffic stochastic limit of the workload process with martingale representation,including model M/M/n, model Erlang-A and Erlang-B.For model,through portraying and writing the expression of martingaleof the process of workload, and then applying the functional limit theorem ofmartingale and other characteristics, such as using random stopping theorem and localrandom square expression of variational theorem to prove the expression ofmartingale. To design the simulation algorithm, take model as an example,using the matlab software to simulate and analyzing the system workload condition ofconvergence when time tends to infinity. For model Erlang-A (M/M/n/M),the method of using similar writing expression of workload process martingale underthis model, by using the known conclusion model for heavy-traffic limitproved that using the matlab software for simulation and analysis of Erlang-A modelunder the workload of the process of convergence. Finally, for model Erlang-B(M/M/n/0), using the conclusions of the model Erlang-A, write the expression ofmartingale proof workload process, in the same way, carries on using matlab softwareto simulate and analyzing of workload in the process of convergence.To work with the process of convergence in various service system, through theexisting first, then by functional central limit theorem to prove that there has been aprevious research, this paper is a kind of new method which is martingale method toprove the convergence of the workload process of heavy-traffic limit and provide anew method to prove the various indicators of convergence for the further study.