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

Marginal queue length approximations for a two-layered network with correlated queues

We consider an extension of the classical machine-repair model. As opposed to the classical model, we assume that the machines, apart from receiving service from the repairman, also supply service themselves to queues of products. The extended model can be viewed as a layered queueing network (LQN), where the first layer consists of two separate queues of products. Each of these queues is served by its own machine. The second layer consists of a waiting buffer and a repairman, able to restore the machines into an operational state. When a machine breaks down, it waits in the repair buffer for the repairman to become available. Since the repair time of one machine may affect the period of time the other machine is not able to process products, the downtimes of the machines are correlated. We explicitly model the correlation between the downtimes, which leads to correlation between the queues of products in the first layer. Taking these correlations into account, we obtain approximations for the marginal distributions of the queue lengths in the first layer, by the study of a single server vacation queue. Extensive numerical results show that these approximations are highly accurate

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

We study a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traf¿c limits for the distributions of the joint workload, waiting time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traf¿c limits. The marginals of these limiting distributions are shown to be exponential with rates that can be computed by matrix-analytic methods. Moreover, we show how to numerically compute the joint distributions, by viewing the limit processes as multi-dimensional semi-martingale re¿ected Brownian motions in the non-negative orthant

Two queues with random time-limited polling

In this paper, we analyse a single server polling model with two queues. Customers arrive at the two queues according to two independent Poisson processes. There is a single server that serves both queues with generally distributed service times. The server spends an exponentially distributed amount of time in each queue. After the completion of this residing time, the server instantaneously switches to the other queue, i.e., there is no switch-over time. For this polling model we derive the steady-state marginal workload distribution, as well as heavy traffic and heavy tail asymptotic results. Furthermore, we also calculate the joint queue length distribution for the special case of exponentially distributed service times using singular perturbation analysis

Quasi-stationary analysis for queues with temporary overload

Motivated by the high variation in transmission rates for document transfer in the Internet and file down loads from web servers, we study the buffer content in a queue with a fluctuating service rate. The fluctuations are assumed to be driven by an independent stochastic process. We allow the queue to be overloaded in some of the server states. In all but a few special cases, either exact analysis is not tractable, or the dependence of system performance in terms of input parameters (such as the traffic load) is hidden in complex or implicit characterizations. Various asymptotic regimes have been considered to develop insightful approximations. In particular, the so-called quasistationary approximation has proven extremely useful under the assumption of uniform stability. We refine the quasi-stationary analysis to allow for temporary instability, by studying the “effective system load” which captures the effect of accumulated work during periods in which the queue is unstable

Queues with regular variation

X+173hlm.;24c