Exact tail asymptotics for a three-dimensional Brownian-driven tandem queue with intermediate inputs

Abstract

In this paper, we consider a three-dimensional Brownian-driven tandem queue with intermediate inputs, which corresponds to a three-dimensional semimartingale reflecting Brownian motion whose reflection matrix is triangular. For this three-node tandem queue, no closed form formula is known, not only for its stationary distribution but also for the corresponding transform. We are interested in exact tail asymptotics for stationary distributions. By generalizing the kernel method, and using extreme value theory and copula, we obtain exact tail asymptotics for the marginal stationary distribution of the buffer content in the third buffer and for the joint stationary distribution.

Kernel Method -- An Analytic Approach for Tail Asymptotics in Stationary Probabilities of 2-Dimensional Queueing Systems

“…We can then establish the fundamental form for the model: −H(x, y, z)φ(x, y, z) = H 1 (x, y)φ 1 (x, y, z) + H 2 (y, z)φ 2 (x, y, z) + H 3 (z)φ 3 (x, y, z), where φ(x, y, z) = E e xL 1 +yL 2 +zL 3 , φ i (x, y, z) = For this specific model, the analytic continuation of the unknown functions φ i , i = 1, 2, 3, and their asymptotic property at the dominant singularity were obtained in [10] based on relationships (or interplay) of the unknown functions. The key idea for the continuation is equivalent to a dimension reduction procedure.…”

“…One of such moelds was considered in Dai, Dawson and Zhao [10], which is a three-dimensional Brownian-driven tandem queue with intermediate inputs. The buffer content L i (t) at node i at time t ≥ 0, for i = 1, 2, 3, is described by…”

“…where X i (t) is the exogenous input to node i, assumed to be a Brownian process of the form: X i (t) = λ i t + B i (t), with λ i > 0 and B i (t) being a Brownian motion with variance σ 2 i and no drift, and Y i (t) is a regulator at node i, which is a minimal nondecreasing process for L i (t) to be nonnegative. Under a stability condition (for example, see [10]), let (L 1 , L 2 , L 3 ) ′ be the stationary vector of (L 1 (t), L 2 (t), L 3 (t)) having the same distribution π as the stationary distribution of the model, we are interested in the exact tail asymptotic of P (L 3 > t). Notice that the tail asymptotic property for P (L 1 > t), or P (L 2 > t) is trivial, or can be obtained through the kernel method directly, respectively.…”

“…The asymptotic analysis is also not simple, but the main idea is also valid for higher-dimensional tandem queues. The application of the Tauberian-like theorem remains straightforward to give the following tail asymptotic result: where both z * and z max are explicitly given in [10], and Ki , i = 1, 2, 3, are non-zero constants.…”

“…The Tauberian-like theorems are developed from the work in analytic combinatorics, for which references can be found in Flajolet and Sedgewick [24]. The extended version of the kernel method has been successfully applied to various models such as: Li and Zhao [51] for the generalized two-demand queueing model; Li, Tavakoli and Zhao [49] for genus 0 random walks in the quarter plane; Ye [80] and Ye, Li and Zhao [81] for a longer-queue-serve-first system; Zafari [82] for the generalized join-the-shortest-queue model considered earlier by several other researchers including [25], [35], [46] and [64]; Dai and Zhao [12] for a revisit of the wireless 3-hop networks with stealing considered in [28]; Song, Liu and Dai [72] for a discrete-time preemptive priority queueing model; Song, Liu and Zhao [73] for a revisit of the retrial queue with two input streams and two orbits considered by Avrachenkov, Nain and Yechiali [3]; Dai, Dawson and Zhao [9] for extending the kernel method to continuous random walks; Dai, Kong and Song [11] for a two-stage queue; Dai, Dawson and Zhao [10] for extending the kernel method to a 3-dimensional tandem queueing system; Li and Zhao [52] for a systematic treatment of the kernel method for random walks in the quarter plane; Li, Liu and Zhao [53] for a 2-demand model modulated by a two-state Markov chain, which is a special case of the random walk in the quarter plane modulate by a finite-state Markov chain; Song and Lu [74] for the Israeli queue with retrials and non-persistent customers.…”