Queue Models With Customer Churn In Martingale Method And Simulations

In queuing processes,customers would leave the queue due to limited waiting space,so it is of great significance to study the queue with limited waiting space.Functional central limit theorem,Markov queue and convergence of probability measure are the research tools to prove the convergence of performance indexes of queue model with limited waiting space under heavy traffic conditions.In this thesis,martingale method is used to study queue models with customer churn under heavy traffic conditions.This thesis mainly investigate four queue length processes with customer loss,which includs M/M/n/m_n,M/H*₂/n/m_n,G/M/n/m_n and G/H*₂/n/m_n models.Firstly,the convergence of M/M/n/m_n queue length process of queue models with customer loss is proved by functional central limit theorem and continuous mapping theorem.Then according to the relationship between performance functions,the martingale representations of queue length process of queue model are given by combining the definition of martingale and the decomposition representation of martingale process,the convergence limits of queue length process represented by martingale in the M/M/n/m_n and M/H*₂/n/m_nqueue models are studied by the stochastic boundedness of martingale process and the continuous mapping theorem.What's more,the arrival time interval which obeys the exponential distribution is changed to the arrival time interval which obeys the general distribution,then the convergence limits of queue length processes in the G/M/n/m_n and G/H*₂/n/m_n queue models are studied.In the last part,taking model M/M/n/m_n for example,a silicon method is adopted to study the queue length process of model M/M/n/m_n with customer loss,the relationships between loss ratio and space capacity and number of service stations are analyzed.