The M/G/1 queue with permanent customers

Journal of Probability and Statistics

This paper presents analytically explicit results for the distribution of the number of customers served during a busy period for special cases of the M/G/1 queues when initiated with m customers. The functional equation for the Laplace transform of the number of customers served during a busy period is widely known, but several researchers state that, in general, it is not easy to invert it except for some simple cases such as M/M/1 and M/D/1 queues. Using the Lagrange inversion theorem, we give an elegant solution to this equation. We obtain the distribution of the number of customers served during a busy period for various service-time distributions such as exponential, deterministic, Erlang-k, gamma, chi-square, inverse Gaussian, generalized Erlang, matrix exponential, hyperexponential, uniform, Coxian, phase-type, Markov-modulated Poisson process, and interrupted Poisson process. Further, we also provide computational results using our method. The derivations are very fast and ...

ECMS 2014 Proceedings edited by: Flaminio Squazzoni, Fabio Baronio, Claudia Archetti, Marco Castellani, 2014

In this paper, we study a discrete-time first-comefirst-served queueing system with a single server and two types (classes) of customers, where the (average) service time of a customer is longer if its type differs from the type of the preceding customer. As opposed to traditional literature, the different types of customers do not occur randomly and independently in the arrival stream: we include a Markovian type of correlation in the types of consecutive customers instead. We deduce the probability generating function of the system content, from which we extract various performance measures, such as the mean values of the system content and the customer delay. We demonstrate that the interclass correlation in the arrival stream has a tremendous impact on the system performance, which highlights the necessity to include it in the performance assessment of the system.

2007

We consider an extension of the standard G/G/1 queue, described by the equation W D = max{0, B − A + Y W }, where P[Y = 1] = p and P[Y = −1] = 1 − p. For p = 1 this model reduces to the classical Lindley equation for the waiting time in the G/G/1 queue, whereas for p = 0 it describes the waiting time of the server in an alternating service model.

The paper studies a queuing model with Poisson arrival process and bulk service. The server serves the customers in batches of fixed size b, and the service time is assumed to be exponentially distribution. The model is analyzed to find the steady-state distribution of the number of customers stranded following each service. The approach adopted is based on discrete-time Markov chains, instead of Laplace transforms that is usually used in literature. A simulation study is carried out to estimate the expected number of stranded customers at any point of time, its variance and the downside risk for given values of the system parameters.

Mathematics

In this paper, we discuss the waiting-time distribution for a finite-space, single-server queueing system, in which customers arrive singly following a Poisson process and the server operates under (a,b)-bulk service rule. The queueing system has a finite-buffer capacity ‘N’ excluding the batch in service. The service-time distribution of batches follows a general distribution, which is independent of the arrival process. We first develop an alternative approach of obtaining the probability distribution for the queue length at a post-departure epoch of a batch and, subsequently, the probability distribution for the queue length at a random epoch using an embedded Markov chain, Markov renewal theory and the semi-Markov process. The waiting-time distribution (in the queue) of a random customer is derived using the functional relation between the probability generating function (pgf) for the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distri...

In this paper, a discrete-time single server queueing system with infinite buffer size and geometrically distributed arrivals is considered. We derive the functional equations and analyze the distribution of the number of customers served during a busy period for geometrically distributed service time as well as for deterministic service time. We also show that in the limiting case the results obtained in this paper are consistent with the corresponding continuous-time counterparts by Medhi [1].

This article discusses the steady state analysis of the M=G=2 queuing system with two heterogeneous servers under new queue disciplines when the classical First Come First Served ‘(FCFS)’ queue discipline is to be violated. Customers are served either by server-I according to an exponential service time distribution with mean rate l or by server-II with a general service time distribution BðtÞ. Sequel to some objections raised in the literature on the use of the classical FCFS queue discipline in heterogeneous service systems, two alternative queue disciplines (Serial and Parallel) are considered in this work with the objective that if the FCFS is violated then the violation is a minimum in the long run. Using the embedded method under the serial queue discipline and the supplementary variable technique under the parallel queue discipline, we present an exact analysis of the steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. Our work shows that one can obtain all stationary probabilities and other vital measures for this queue under certain simple but realistic assumptions.

Operations Research, 2004

For a broad class of discrete-and continuous-time queueing systems, we show that the stationary number of customers in system (queue plus servers) is the sum of two independent random variables, one of which is the stationary number of customers in queue and the other is the number of customers that arrive during the time a customer spends in service. We call this relation an invariance relation in the sense that it does not change for a variety of single-sever queues (with batch arrivals and batch services) and some of multi-server queues (with batch arrivals and deterministic service times) that satisfy a certain set of assumptions. Making use of this relation, we also present a simple method of deriving the stationary distributions of the numbers in queue and in system as well as some of their properties. This is illustrated by several examples, which show that new simple derivations of old results as well as new results can be obtained in a unified manner. Furthermore, we show that the invariance relation and the method we are presenting are easily generalized to analyze queues with BMAP (Batch Markovian Arrival Process) arrivals. Most of the results are presented under the discrete-time setting. The corresponding continuous-time results, however, are covered as well because deriving the results for continuous-time queues runs exactly parallel to that for their discrete-time counterparts.

Queueing Systems, 1994

We consider the standard GI/G/1 queue with unlimited waiting room and the first-in first-out service discipline. We investigate the steady-state waiting-time tail probabilities P(W > x) when the service-time distribution has a long-tail distribution, i.e., when the service-time distribution fails to have a finite moment generating function. We have developed algorithms for computing the waiting-time distribution by Laplace transform inversion when the Laplace transforms of the interarrival-time and service-time distributions are known. One algorithm, exploiting Pollaczek's classical contour-integral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient twoparameter family of long-tail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Pareto-like tails, i.e., are of order x − r for r > 1. We use this family of long-tail distributions to investigate the quality of approximations based on asymptotics for P(W > x) as x → ∞. We show that the asymptotic approximations with these long-tail service-time distributions can be remarkably inaccurate for typical x values of interest.

This paper deals with the steady-state solution of the queueing system: M X /H k /1/N with reneging in which (i) units arrive in batches of random size with the interarrival times of batches following negative exponential distribution, (ii) the batches are served in order of their arrival; and (iii) the service time distribution is hyperexponential with k branches. Recurrence relations connecting the various probabilities introduced are found. Some measures of effectiveness are deduced and some special cases are also obtained. Keywords: queueing system: M X /H k /1/N, hyperexponential DESCRIPTION OF THE SYSTEM Morse [4] discussed the steady-state queueing system in which the service channel consists of two branches, the units arrive singly and the capacity of the waiting space is infinite. Gupta and Goyal [1] studied a similar system by using the generating functions with k branches in the service channel, the units arrive singly and the capacity of the waiting space is finite. Habib [3] and Gupta and Goyal [2] treated the system M X /Hk/1. White et al. [6] solved the system: M/H2 /2/2 numerically. All the previous studies are without balking and reneging. In the present system, it is assumed that the units arrive at the system in batches of random size X, i. e., at each moment of arrival, there is a probability Cj = Pr (X= j) that j units arrive simultaneously, and the interarrival times of batches follow a negative exponential distribution with time independent parameter . Let  Cj t, (j = 1, 2,..., N), be the first order probability that a batches of j units comes in time t. The service channel is busy if a unit is present in any one of the k branches and in this case the arrival units form a queue and the capacity of the system is N. The unit at the head of the queue requires service in the r th branch with probability *. The service time distribution in the r th branch is * The variation of the subscripts i,j,r,s is from 1 to k, unless otherwise explicitly mentioned.