Heavy-traffic For Queueing Models With Service Interruptions And Finited Waiting Room

In order to consider the impactions of the service interruptions, the paper gives thestochastic-process limits for G/M/n/m+Mqueueing models in heavy traffic. Basis on theG/M/n/mmodel, give with waiting room finite, service interruptions and asymptoticallynegligible service interruptions heavy traffic for the queue-length process respectively.The first, we consider a many-server G/M/n/m+Mqueueing models, allowingcustomer abandonment subject to exogenous regenerative service interruptions. Withunscaled service interruption times, in heavy traffic, we obtain a FWLLN for thequeue-length process where the limit is an ordinary differential equation in a two-staterandom environment.And then establish heavy-traffic limits for G/M/n/m+Mqueueing models withdiffusion scaled. With asymptotically negligible service interruptions, we obtain a FCLT forthe queue-length process, where the limit is characterized as the unique solution to astochastic integral equation with jumps.When the arrivals are renewal and the interruptioncycle time is exponential, the limit is a Markov process, being a jump-diffusion process in theQED regime.Finally, we consider a special class of service time distributions,denoted byH₂^*, whichare mixtures of an exponential distribution with probability p and a unit point mass at0with probability1-p. And give the heavy-traffic limits for G/H₂^*/n/mqueueing modelswith abandonments.This article three models of the proposed method is proved by means of martingale andcontinuous mapping method, so makes a simple introduction.