Two-parameter Heavy-traffic Limits For Special Queues With Dependent Service Times

In order to consider the impactions of dependent service times, the paper gives the two-parameter heavy-traffic limits for two special queue models. One is / /B D tG G ￥ queue model, and the other is the network model / / /tpD DtG G ￥®G ￥.Firstly, we consider a queue model, it is / /B D tG G ￥ with batch arrive and dependent service times distribution, possibly with time-varying arrival rates, and give the sequential empirical processes for the queue-length process.Secondly, we establish two-parameter heavy-traffic limits of queue-length and departure process for / /B D tG G ￥ queue model under with fluid scaled and diffusion scaled. We apply martingale and continuous mapping method to obtain FWLLN and FCLN for two-parameter stochastic processes and the Gaussian characterization of the limit processes.Finally, based on the two-parameter heavy-traffic limits for / /B D tG G ￥ queue model with dependent service times, we consider another special queue models, it is network model/ / /tpD D tG G ￥®G ￥ with two IS systems in series. We consider time-varying stochastic splitting after service completion at the first station, allowing only some of the customers to continue on to the second station. with probability p,and leave the network queue system with probability 1- p.Given the two-parameter heavy-traffic limits(FWLLN and FCLT) for the special models.This article about two models of the proposed methods are proved by means of martingale and continuous mapping, and involves the tightness of stochastic variable and convergence of finite-dimensional distribution.