Queues and risk processes with dependencies

We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in [12]. In this setting, we obtain the steady state waiting time distribution and we show that the classical relation between the steady state waiting time and the workload distributions re- mains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally we show that there exist stochastic order relations between the waiting times under various instances of correlation

Ruin excursions, the G/G/Infinity queue and tax payments in renewal risk models

In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramer-Lundberg model with and without tax payments. We also provide a relation of the Cramer-Lundberg risk model with the G/G/infinity queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramer-Lundberg risk model

Queues and risk models with simultaneous arrivals

We focus on a particular connection between queueing and risk models in a multi-dimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline how the two-dimensional risk model with a general two-dimensional claim size distribution (hence without ordering of claim sizes) is related to a known Riemann boundary value problem

Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime

In this paper we derive a technique of obtaining limit theorems for suprema of L\'evy 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 L\'evy 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, $\Delta(a)\max_{n\ge 0} S_n^{(a)}\stackrel{d}{\to} R\iff\Delta(a)\sup_{t\ge 0} X^{(a)}_t\stackrel{d}{\to} R$, for some random variable $R$ and some function $\Delta(\cdot)$. We utilize this result to present a number of limit theorems for suprema of L\'evy processes in the heavy-traffic regime

Computable bounds in fork-join queueing systems

In a Fork-Join (FJ) queueing system an upstream fork station splits incoming jobs into N tasks to be further processed by N parallel servers, each with its own queue; the response time of one job is determined, at a downstream join station, by the maximum of the corresponding tasks' response times. This queueing system is useful to the modelling of multi-service systems subject to synchronization constraints, such as MapReduce clusters or multipath routing. Despite their apparent simplicity, FJ systems are hard to analyze. This paper provides the first computable stochastic bounds on the waiting and response time distributions in FJ systems. We consider four practical scenarios by combining 1a) renewal and 1b) non-renewal arrivals, and 2a) non-blocking and 2b) blocking servers. In the case of non blocking servers we prove that delays scale as O(logN), a law which is known for first moments under renewal input only. In the case of blocking servers, we prove that the same factor of log N dictates the stability region of the system. Simulation results indicate that our bounds are tight, especially at high utilizations, in all four scenarios. A remarkable insight gained from our results is that, at moderate to high utilizations, multipath routing 'makes sense' from a queueing perspective for two paths only, i.e., response times drop the most when N = 2; the technical explanation is that the resequencing (delay) price starts to quickly dominate the tempting gain due to multipath transmissions

Shot-noise queueing models

We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process

Useful martingales for stochastic storage processes with Lévy-input and decomposition results

In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that under some quite minimal conditions the local martingales are actually L^2 martingales which upon dividing by the time index converge to zero a.s. and in L^2. We apply these results to generalize known 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. Keywords: Lévy-type processes, Lévy storage systems, Kella-Whitt martingale, decomposition results, queues with server vacation

Useful martingales for stochastic storage processes with Lévy-input and decomposition results

In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that under some quite minimal conditions the local martingales are actually L^2 martingales which upon dividing by the time index converge to zero a.s. and in L^2. We apply these results to generalize known 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. Keywords: Lévy-type processes, Lévy storage systems, Kella-Whitt martingale, decomposition results, queues with server vacation