On the large deviations approximation for the stationary distribution of skip-free regulated queueing networks

Large deviations papers like that of Ignatyuk et al. [Russ. Math. Surv. 49 (1994) 41–99] have shown that asymptotically, the stationary distribution of homogeneous regulated networks is of the form P{X(t)≈n}≈K e−αnwith the coefficient α being different in various “boundary influence domains” and also depending on some of these domains on n. In this paper, we focus on the case of constant exponents α and on a subclass of networks we call “strongly skip-free” (which includes all Jackson and all two-dimensional skip-free networks). We conjecture that an asymptotic exponent α is constant iff it corresponds to a large deviations escape path which progresses gradually (from the origin to the interior) through boundary facets whose dimension always increases by one. Solving the corresponding large deviations problem for our subclass of networks leads to a family of “local large deviation systems” (LLDSs) (for the constant exponents), which are expressed entirely in terms of the cumulant generating function of the network. In this paper, we show that at least for “strongly skip-free” Markovian networks with independent transition processes, the LLDS is closely related to some “local boundary equilibrium systems” (LESs) obtained by retaining from the equilibrium equations only those valid in neighborhoods of the boundary. asymptotic results require typically only that the cumulant generating function is well-defined over an appropriate domain, it is natural to conjecture that these LLDSs will provide the asymptotic constant exponents regardless of any distributional assumptions made on the network. y, we outline a practical recipe for combining the local approximations to produce a global large deviations approximation P{Xt≈n}≈∑jKj e−α(j)n, with the coefficients Kj determined numerically.