Abstract
In the seminal paper of Gamarnik and Zeevi (2006), the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steady-state approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks.
Highlights
This paper considers open single-class queueing networks that have d service stations
To prove (1.2), we show in Proposition 4.1 that φ(nk)(θ) and its boundary counterparts satisfy basic adjoint relationship (BAR) (2.30) asymptotically
To introduce the notation of heavy traffic, we introduce a sequence of generalized Jackson networks
Summary
This paper considers open single-class queueing networks that have d service stations. It may be helpful for the reader to compare the arguments that follow to the proof of Lemma 2.3 in [29, Appendix A.2], where the author seeks to find test functions for which the jump terms vanish.
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