Positive Harris Recurrence and Diffusion Scale Analysis of a Push Pull Queueing Network

We consider a push pull queueing system with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull system was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the Kumar-Seidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Thus each server can either process jobs of one of the types, which it pulls from the other server, or jobs of the other type which it pushes out of the infinite supply towards the other server. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform an asymptotic analysis of the push pull network under these policies to quantify its behavior: We show that under fluid scaling the fluid model of the network is stable. We adapt the proofs of Dai, to show that as a result the queues of jobs waiting for pull operation are positive Harris recurrent. Finally we obtain the diffusion scale behavior of the network, in which we show that the queues are zero under diffusion scaling, and calculate the Brownian approximation of the output processes of the two types of jobs. The approximation shows that the two output streams are highly negatively correlated.