A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications

The main contribution of this paper is to present a new sufficient condition for the subexponential asymptotics of the stationary distribution of a GI/GI/1-type Markov chain without jumps from level "infinity" to level zero. For simplicity, we call such Markov chains {\it GI/GI/1-type Markov chains without disasters} because they are often used to analyze semi-Markovian queues without "disasters", which are negative customers who remove all the customers in the system (including themselves) on their arrivals. In this paper, we demonstrate the application of our main result to the stationary queue length distribution in the standard BMAP/GI/1 queue. Thus we obtain new asymptotic formulas and prove the existing formulas under weaker conditions than those in the literature. In addition, applying our main result to a single-server queue with Markovian arrivals and the $(a,b)$-bulk-service rule (i.e., MAP/${\rm GI}^{(a,b)}$/1 queue), we obatin a subexponential asymptotic formula for the stationary queue length distribution.Comment: Submitted for revie

A tandem queue with Lévy input: a new representation of the downstream queue length.

In this paper we present a new representation for the steady state distribution of the workload of the second queue in a two-node tandem network. It involves the difference of two suprema over two adjacent intervals. In case of spectrally-positive

On exceedance times for some processes with dependent increments

Let ${Z_n}_{n\ge 0}$ be a random walk with a negative drift and i.i.d. increments with heavy-tailed distribution and let $M=\sup_{n\ge 0}Z_n$ be its supremum. Asmussen & Kl{\"u}ppelberg (1996) considered the behavior of the random walk given that $M>x$, for $x$ large, and obtained a limit theorem, as $x\to\infty$, for the distribution of the quadruple that includes the time \rtreg=\rtreg(x) to exceed level $x$, position Z_{\rtreg} at this time, position Z_{\rtreg-1} at the prior time, and the trajectory up to it (similar results were obtained for the Cram\'er-Lundberg insurance risk process). We obtain here several extensions of this result to various regenerative-type models and, in particular, to the case of a random walk with dependent increments. Particular attention is given to describing the limiting conditional behavior of $\tau$. The class of models include Markov-modulated models as particular cases. We also study fluid models, the Bj{\"o}rk-Grandell risk process, give examples where the order of $\tau$ is genuinely different from the random walk case, and discuss which growth rates are possible. Our proofs are purely probabilistic and are based on results and ideas from Asmussen, Schmidli & Schmidt (1999), Foss & Zachary (2002), and Foss, Konstantopoulos & Zachary (2007).Comment: 17 page

On the infimum attained by a reflected L\'evy process

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided L\'evy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of \prob{M(T_u)> u} (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios

On the tail asymptotics of the area swept under the Brownian storage graph

In this paper, the area swept under the workload graph is analyzed: with $\{Q(t) : t\ge0\}$ denoting the stationary workload process, the asymptotic behavior of $$\pi_{T(u)}(u):={\mathbb{P}}\biggl(\int_0^ {T(u)}Q(r)\,\mathrm{d}r>u\biggr)$$ is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of $\pi_{T(u)}(u)$ are given for the case that $T(u)$ grows slower than $\sqrt{u}$, and then logarithmic asymptotics for (i) $T(u)=T\sqrt{u}$ (relying on sample-path large deviations), and (ii) $\sqrt{u}=\mathrm{o}(T(u))$ but $T(u)=\mathrm{o}(u)$. Finally, the Laplace transform of the residual busy period are given in terms of the Airy function.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ491 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm