Asymptotic analysis of Lévy-driven tandem queues

Abstract

We analyze tail asymptotics of a two-node tandem queue with spectrallypositive Lévy input. A first focus lies in the tail probabilities of the type, for α ∈ (0, 1) and x large, and Q i denoting the steadystate workload in the ith queue. In case of light-tailed input, our analysis heavily uses the joint Laplace transform of the stationary buffer contents of the first and second queue; the logarithmic asymptotics can be expressed as the solution to a convex programming problem. In case of heavy-tailed input we rely on sample-path methods to derive the exact asymptotics. Then we specialize in the tail asymptotics of the downstream queue, again in case of both light-tailed and heavy-tailed Lévy inputs. It is also indicated how the results can be extended to tandem queues with more than two nodes.

Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

“…Two-dimensional semimartingale reflecting Brownian motion (SRBM) in the quarter plane received a lot of attention from the mathematical community. Problems such as SRBM existence [39,40], stationary distribution conditions [19,22], explicit forms of stationary distribution in special cases [7,8,19,23,30], large deviations [1,7,33,34] construction of Lyapunov functions [10], and queueing networks approximations [19,21,31,32,43] have been intensively studied in the literature. References cited above are non-exhaustive, see also [42] for a survey of some of these topics.…”

Exact tail asymptotics in a priority queue—characterizations of the preemptive model

“…Asymptotics of the coefficients of ψ 0 (y) are obtained by using a Tauberian-type theorem, such as Theorem 4 in Bender [4] or Corollary 2 in Flajolet and Odlyzko [13], which is restated in the following lemma for convenience. The key idea used in the following analysis is the same as that used in Lieshout and Mandjes [31].…”

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

“…Recently, Lieshout and Mandjes [16,17] studied this asymptotic decay problem for a Lévy-driven two node tandem queue when there is no intermediate input, which means that the second queue has no exogenous input. In [16], the joint stationary distribution function was obtained in a closed form for the Brownian input case, and using these closed form expressions, the exact tail asymptotics were obtained in all directions for the Brownian input.…”

“…In [16], the joint stationary distribution function was obtained in a closed form for the Brownian input case, and using these closed form expressions, the exact tail asymptotics were obtained in all directions for the Brownian input. In [17], they use the joint Laplace transform due to Dȩbicki, Dieker and Rolski [8], which has closed form and is obtained for the general Lévy-driven n-node tandem queue. With the help of some sample path large deviations techniques, the rough decay rates were obtained in all directions for the general Lévy input case in [17].…”

“…In [17], they use the joint Laplace transform due to Dȩbicki, Dieker and Rolski [8], which has closed form and is obtained for the general Lévy-driven n-node tandem queue. With the help of some sample path large deviations techniques, the rough decay rates were obtained in all directions for the general Lévy input case in [17]. Here, the tail probability is said to have rough decay rate α if its logarithm at level x divided by x converges to −α as x → ∞, and said to have exact asymptotics h for some function h if the ratio of the tail probability at level x and h(x) converges to one.…”