Another look at the transient behavior of the M/G/1 workload process

We use Palm measures, along with a simple approximation technique to derive new explicit expressions for all of the transient moments of the workload process of an M=G=1 queue. These expressions can also be used to derive a closed-form expression for the nth moment of the stationary workload, which solves the well-known Takacs recursion that generates the waiting time moments of an M=G=1 queue that serves customers in a first-come-first-serve manner

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

Time-dependent analysis of an M / M / c preemptive priority system with two priority classes

We analyze the time-dependent behavior of an $M/M/c$ priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least $c$ high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most $c - 1$ high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami's formula from the theory of $M/G/1$-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution: these results seem to yield the most explicit expressions known to date.Comment: 34 pages, 4 figure

Waiting times in polling systems with various service disciplines

We consider a polling system of N queues Q1,..., QN, 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-Come-First-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, Q1. 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 (Laplace-Stieltjes transform of the) waiting-time distribution at an arbitrary queue.

First passage times to congested states of many-server systems in the Halfin-Whitt regime

We consider the heavy-traffic approximation to the $GI/M/s$ queueing system in the Halfin-Whitt regime, where both the number of servers $s$ and the arrival rate $\lambda$ grow large (taking the service rate as unity), with $\lambda=s-\beta\sqrt{s}$ and $\beta$ some constant. In this asymptotic regime, the queue length process can be approximated by a diffusion process that behaves like a Brownian motion with drift above zero and like an Ornstein-Uhlenbeck process below zero. We analyze the first passage times of this hybrid diffusion process to levels in the state space that represent congested states in the original queueing system

Non-Stationary Queues with Batch Arrivals

Motivated by applications that involve setting proper staffing levels for multi-server queueing systems with batch arrivals, we present a thorough study of the queue-length process $\{Q(t); t \geq 0\}$, departure process $\{D(t); t \geq 0\}$, and the workload process $\{W(t); t \geq 0\}$ associated with the M$_{t}^{B_{t}}$/G$_{t}$/$\infty$ queueing system, where arrivals occur in batches, with the batch size distribution varying with time. Notably, we first show that both $Q(t)$ and $D(t)$ are equal in distribution to an infinite sum of independent, scaled Poisson random variables. When the batch size distribution has finite support, this sum becomes finite as well. We then derive the finite-dimensional distributions of both the queue-length process and the departure process, and we use these results to show that these finite-dimensional distributions converge weakly under a certain scaling regime, where the finite-dimensional distributions of the queue-length process converge weakly to a shot-noise process driven by a non-homogeneous Poisson process. Next, we derive an expression for the joint Laplace-Stieltjes transform of $W(t)$, $Q(t)$, and $D(t)$, and we show that these three random variables, under the same scaling regime, converge weakly, where the limit associated with the workload process corresponds to another Poisson-driven shot-noise process