Useful martingales for stochastic storage processes with Lévy-type input and decomposition results
This paper has been withdrawn by the author as it has been divided into two separate papers.
math.PR↗
arXiv subjects
Onno Boxma
Publications and source records attributed to Onno Boxma.
Useful martingales for stochastic storage processes with Lévy-type input
In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that the (local) martingales obtained are in fact square integrable martingales which upon dividing by the time index converge to zero a.s. and in $L^2$. The reflected Lévy-type process is considered as an example.
math.PR↗
Decomposition results for stochastic storage processes and queues with alternating Lévy inputs
In this paper we generalize known workload decomposition results for Lévy queues with secondary jump inputs and queues with server vacations or service interruptions. Special cases are polling systems with either compound Poisson or more general Lévy inputs. Our main tools are new martingale results, which have been derived in a companion paper.
math.PR↗
Waiting time asymptotics in the single server queue with service in random order
We consider the single server queue with service in random order. For a large class of heavy-tailed service time distributions, we determine the asymptotic behavior of the waiting time distribution. For the special case of Poisson arrivals and regularly varying service time distribution with index -ν, it is shown that the waiting time distribution is also regularly varying, with index 1 -ν, and the pre-factor is determined explicitly. Another contribution of the paper is the heavy-traffic analysis of the waiting time distribution in the M/G/1 case. We consider not only the case of finite service time variance, but also the case of regularly varying service time distribution with infinite variance.
math.PR↗
Queue lengths and workloads in polling systems
We consider a polling system: a queueing system of $N\ge 1$ queues with Poisson arrivals $Q_1,...,Q_N$ visited in a cyclic order (with or without switchover times) by a single server. For this system we derive the probability generating function $\mathscr Q(\cdot)$ of the joint queue length distribution at an arbitrary epoch in a stationary cycle, under no assumptions on service disciplines. We also derive the Laplace-Stieltjes transform $\mathscr W(\cdot)$ of the joint workload distribution at an arbitrary epoch. We express $\mathscr Q$ and $\mathscr W$ in the probability generating functions of the joint queue length distribution at visit beginnings, ${\mathscr V}_{b_i}(\cdot)$, and visit completions, ${\mathscr V}_{c_i}(\cdot)$, at $Q_i$, $i=1,...,N$. It is well known that ${\mathscr V}_{b_i}$ and ${\mathscr V}_{c_i}$ can be computed in a broad variety of cases. Furthermore, we establish a workload decomposition result.
math.PR↗
Convergence of the all-time supremum of a Lévy process in the heavy-traffic regime
In this paper we derive a technique of obtaining limit theorems for suprema of Lévy processes from their random walk counterparts. For each $a>0$, let $\{Y^{(a)}_n:n\ge 1\}$ be a sequence of independent and identically distributed random variables and $\{X^{(a)}_t:t\ge 0\}$ be a Lévy processes such that $X_1^{(a)}\stackrel{d}{=} Y_1^{(a)}$, $\mathbb E X_1^{(a)}<0$ and $\mathbb E X_1^{(a)}\uparrow0$ as $a\downarrow0$. Let $S^{(a)}_n=\sum_{k=1}^n Y^{(a)}_k$. Then, under some mild assumptions, $Δ(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iffΔ(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $Δ(\cdot)$. We utilize this result to present a number of limit theorems for suprema of Lévy processes in the heavy-traffic regime.
math.PR↗