Service Time Density Research Articles

Nonparametric Inference About Service Time Distribution from Indirect Measurements

SummaryIn studies of properties of queues, for example in relation to Internet traffic, a subject that is of particular interest is the ‘shape’ of service time distribution. For example, we might wish to know whether the service time density is unimodal, suggesting that service time distribution is possibly homogeneous, or whether it is multimodal, indicating that there are two or more distinct customer populations. However, even in relatively controlled experiments we may not have access to explicit service time data. Our only information might be the durations of service time clusters, i.e. of busy periods. We wish to ‘deconvolve’ these concatenations, and to construct empirical approximations to the distribution and, particularly, the density function of service time. Explicit solutions of these problems will be suggested. In particular, a kernel-based ‘deconvolution’ estimator of service time density will be introduced, admitting conventional approaches to the choice of bandwidth.

Bayesian estimation for the M/ G/1 queue using a phase-type approximation

This article deals with Bayesian inference and prediction for M/ G/1 queueing systems. The general service time density is approximated with a class of Erlang mixtures which are phase-type distributions. Given this phase-type approximation, an explicit evaluation of measures such as the stationary queue size, waiting time and busy period distributions can be obtained. Given arrival and service data, a Bayesian procedure based on reversible jump Markov Chain Monte Carlo methods is proposed to estimate system parameters and predictive distributions.

The MAP, M/G1, G2/1 queue with preemptive priority

We consider the MAP, M/G1, G2/1 queue with preemptive resume priority, where low priority customers arrive to the system according to a Markovian arrival process (MAP) and high priority customers according to a Poisson process. The service time density function of low (respectively: high) priority customers is g1(x) (respectively: g2(x)). We use the supplementary variable method with Extended Laplace Transforms to obtain the joint transform of the number of customers in each priority queue, as well as the remaining service time for the customer in service in the steady state. We also derive the probability generating function for the number of customers of low (respectively, high) priority in the system just after the service completion epochs for customers of low (respectively, high) priority.

Two Parallel M/G/1 Queues where Arrivals Join the System with the Smaller Buffer Content

We consider two parallel, infinite capacity, M/G/1 queues characterized by ( U_{1}(t), U_{2}(t) ) with U_{j}(t) denoting the unfinished work (buffer content) in queue j . A new arrival is assigned to the queue with the smaller buffer content. We construct formal (as opposed to rigorous) asymptotic approximations to the Joint stationary distribution of the Markov process ( U_{1}(t), U_{2}(t) ), treating separately the asymptotic limits of heavy traffic, light traffic, and large buffer contents. In heavy traffic, the stochastic processes U_{1}(t) + U_{2}(t) and U_{2}(t) - U_{1}(t) become independent, with the distribution of U_{1}(t) + U_{2}(t) identical to the heavy traffic waiting time distribution in the standard M/G/2 queue, and the distribution of U_{2}(t) - U_{1}(t) closely related to the tail of the service time density. In light traffic, we obtain a formal expansion of the stationary distribution in powers of the arrival rate.

On the complete monotonicity of the waiting time density in some GI/G/k systems

In this note the complete monotonicity of the waiting time density in GI/G/k queues is proved under the assumption that the service time density is completely monotone. This is an extension of Keilson's [3] result for M/G/1 queues. We also provide another proof of the result that complete monotonicity is preserved by geometric compounding.

A Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.

Stochastic comparisons for bulk queues

We consider two important classes of single-server bulk queueing models: M(X)/G(Y)/1 with Poisson arrivals of customer groups, and G(X)/m(Y)1 with batch service times having exponential density. In each class we compare two systems and prove that one is more congested than the other if their basic random variables are stochastically ordered in an appropriate manner. However, it must be recognized that a system that appears congested to customers might be working efficiently from the system manager's point of view. We apply the results of this comparison to (i) the family {M/G(s)/1,s ⩾ 1} of systems with Poisson input of customers and batch service times with varying service capacity; (ii) the family {G(s)/1,s ⩾ 1} of systems with exponential customer service time density and group arrivals with varying group size; and (iii) the family {M/D/s,s⩾ 1} of systems with Poisson arrivals, constant service time and varying number of servers. Within each family, we find the system that is the best for customers, but this turns out to be the worst for the manager (or vice versa). We also establish upper (or lower) bounds for the expected queue length in steady state and the expected number of batches (or groups) served during a busy period. The approach of the paper is based on the stochastic comparison of random walks underlying the models.

Queuing with reneging and multiple heterogeneous servers

AbstractA Poisson stream of items arrive at a multiple parallel serving facility consisting of s heterogeneous servers. The service time density functions are all negative exponential and may all have different means. An arrival balks (refuses to enter) if the queue size is equal to N. If the queue size is less than N, the arrival enters the queue and is served on a first‐come, first‐served basis unless it reneges (leaves the queue). Each entering item has a definite length of time it will wait in queue before reneging. This time is a random variable with negative exponential density function. If on arrival more than one server is free, the server to be used is chosen at random. The results obtained are for steady‐state and include (1) state probabilities; (2) the probability of j or more in system, j=s, s+I,…, s+N; (3) mean number in queue and system and mean number of busy servers; (4) probability that an arrival acquires service; (5) mean rate of loss due to balking and reneging; and (6) density functions and mean values of waiting times in queue for those who are served, those who renege, and all who enter. Certain important special cases are derived from the preceding, namely, (1) unbounded queues, (2) homogeneous servers, (3) no reneging, (4) no reneging and unbounded queues, and (5) single server. Finally, a comparison is made of a heterogeneous system with an equivalent homogeneous system.