2015/12/09 by Aghajani, Reza, Ramanan, Kavita
#60G10 #60K25. Secondary 90B22 #FOS: Mathematics #Primary: 60H15 #Probability (math.PR)
paper · doi:10.48550/arxiv.1512.02929
We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by N identical servers in a First-Come-First-Serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as 1 - βN-1/2+ o(N-1/2) for some β> 0. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued Itô equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [1] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in 1981. The methods introduced here are more generally applicable for the analysis of a broader class of networks.