Factorization identities for reflected processes, with applications

We derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Levy processes, diffusion processes, and certain types of growth-collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Levy case, our identities simplify to both the well-known Wiener-Hopf factorization, and another interesting factorization of reflected Levy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth-death processes and diffusion processes

An infinite-server queue influenced by a semi-Markovian environment

We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that's independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various "parts" of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that's either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little's law, to generate rigorous approximations of the steady-state queue length in the case that the amount of work brought by a given arrival is of an arbitrary distribution

An infinite-server queue influenced by a semi-Markovian environment

Abstract We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that's independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various "parts" of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that's either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little's law, to generate rigorous approximations of the steady-state queue length in the case that the amount of work brought by a given arrival is of an arbitrary distribution

A new look at transient versions of Little's law, and M/G/1 preemptive Last-Come-First-Served queues

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we will show that previous laws (see [6]) can be generalized; furthermore, within this framework a new law can be derived as well, which gives higher-moment expressions for very general types of queueing systems; in particular, the laws hold for systems that allow customers to overtake one another. What's especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide closed-form expressions of all moments of the number of customers in the system in an M=G=1 preemptive-LCFS queue at a time t > 0, for any initial condition and for any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling). The phrase \closed-form" is used here to stress that the moments can be expressed in terms of probabilities that consist of convolutions of busy periods and residual busy periods, and so moment-matching methods can be used to generate very simple approximations of these quantities, as in [3]. It is also worth noting that these results appear to be new for the M=M=1 queue as well (see [3], [4]), and so we use them to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see [1], [2])

Waiting times in polling systems with various service disciplines. EURANDOM report 2008-019, EURANDOM,

Abstract We consider a polling system of N queues Q 1 , . . . , Q N , cyclically visited by a single server. Customers arrive at these queues according to independent Poisson processes, requiring generally distributed service times. When the server visits Qi, i = 1, . . . , N , it serves a number of customers according to a certain visit discipline. This discipline is assumed to belong to the class of branching-type disciplines, which includes gated and exhaustive service. The special feature of our study is that, within each queue, we do not restrict ourselves to service in order of arrival (FCFS); we are interested in the effect of different service disciplines, like Last-ComeFirst-Served, Processor Sharing, Random Order of Service, and Shortest Job First. After a discussion of the joint distribution of the numbers of customers at each queue at visit epochs of the server to a particular queue, we determine the Laplace-Stieltjes transform of the cycle-time distribution, viz., the time between two successive visits of the server to, say, Q 1 . This yields the transform of the joint distribution of past and residual cycle time, w.r.t. the arrival of a tagged customer at Q1. Subsequently concentrating on the case of gated service at Q1, we use that cycle-time result to determine the (Laplace-Stieltjes transform of the) waiting-time distribution at Q1. Next to locally gated visit disciplines, we also consider the globally gated discipline. Again, we consider various non-FCFS service disciplines at the queues, and we determine the (LaplaceStieltjes transform of the) waiting-time distribution at an arbitrary queue