From the matrix-geometric to the matrix-exponential

We consider the single server queuesN/G/1 andGI/N/1 respectively in which the arrival process or the service process is a Neuts Process, and derive the matrix-exponential forms of the solution of relevant nonlinear matrix equations for such queues. We thereby generalize the matrix-exponential results of Sengupta forGI/PH/1 and of Neuts forMMPP/G/1 to substantially more general models. Our derivation of the results also establishes the equivalence of the methods of Neuts and those of Sengupta. A detailed analysis of the queueGI/N/1 is given, and it is noted that not only the stationary distribution at arrivals but also at an arbitrary time is matrix-geometric. Matrix-exponential steady state distributions are established for the waiting times in the queueGI/N/1. From this, by appealing to the duality theorem of Ramaswami, it is deduced that the stationary virtual and actual waiting times in aGI/PH/1 queue are of phase type.     

Buffer overflow calculations using an infinite-capacity model

Sufficient conditions are established for approximation of the overflow probability in a stochastic service system with capacity C by the probability that the related infinite-capacity system has C customers. These conditions are that (a) the infinite-capacity system has negligible probability of C or more customers; (b) the probabilities of states with exactly C customers for the infinite-capacity system are nearly proportional to the same probabilities for the finite- capacity system. Condition (b) is controlling if the probabilities for the infinite-capacity system are rescaled so that the probability of at most C customers is unity. For systems with precisely one state with C customers, such as birth-and-death processes, the latter approximation is exact even when condition (a) does not hold.     

Analysis of a finite MAP/G/1 queue with group services

The finite capacity queues, GI/PH/1/N and PH/G/1/N, in which customers are served in groups of varying sizes were recently introduced and studied in detail by the author. In this paper we consider a finite capacity queue in which arrivals are governed by a particular Markov renewal process, called a Markovian arrival process (MAP). With general service times and with the same type of service rule, we study this finite capacity queueing model in detail by obtaining explicit expressions for (a) the steady-state queue length densities at arrivals, at departures and at arbitrary time points, (b) the probability distributions of the busy period and the idle period of the server and (c) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures when services are of phase type are discussed. An illustrative numerical example is presented.     

Analysis of the queue-length distribution for the discrete-time batch-service Geo/G/1/K queue

In this paper, we consider a discrete-time finite-capacity queue with Bernoulli arrivals and batch services. In this queue, the single server has a variable service capacity and serves the customers only when the number of customers in system is at least a certain threshold value. For this queue, we first obtain the queue-length distribution just after a service completion, using the embedded Markov chain technique. Then we establish a relationship between the queue-length distribution just after a service completion and that at a random epoch, using elementary ‘rate-in = rate-out’ arguments. Based on this relationship, we obtain the queue-length distribution at a random (as well as at an arrival) epoch, from which important performance measures of practical interest, such as the mean queue length, the mean waiting time, and the loss probability, are also obtained. Sample numerical examples are presented at the end.     

Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37–54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded \(G{/}GI{/}n+GI\) queue with stationary arrivals, nonexponential service times, n identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits.     

M/M/∞ Queues in series with non-homogeneous inputs

This paper studies the transient behaviour of tandem queueing system consisting of an arbitrary number r of queues in series with infinite server service facility at each queue. Poisson arrivals with time dependent parameter and exponential service times have been assumed. Infinite server queues realistically describe those queues in which sufficient service capacity exist to prevent virtually any waiting by the customer present. The model is suitable for both phase type service as well services in series. Very elegant solutions have been obtained and it has been shown that if the queue sizes are initially independent and Poisson then they remain independent and Poisson for all t.     

An optimality criterion for a limited capacity queueing system

A single server, limited capacity queueing system with Poisson arrivals and exponential service is studied. The joint probability distribution of the number of times the system reaches its capacity in time interval (0t] and the number of customers in the system at time i has been obtained. From, the joint probability, the probability that the system has reached its capacity m times in time interval (0t] has been determined and the expectation and variance have been found explicitly. A criterion for the system to be optimum is established and is illustrated numerically.     

Approximations for multi-server queues: System interpolations

This paper provides a unifying method of generating and/or evaluating approximations for the principal congestion measures in aGI/G/s queueing system. The main focus is on the mean waiting time, but approximations are also developed for the queue-length distribution, the waiting-time distribution and the delay probability for the Poisson arrival case. The approximations have closed forms that combine analytical solutions of simpler systems, and hence they are referred to as system-interpolation approximations or, simply, system interpolations. The method in this paper is consistent with and generalizes system interpolations previously presented for the mean waiting time in theGI/G/s queue.     

Discrete NT-policy single server queue with Markovian arrival process and phase type service

We consider a discrete time single server queueing system in which arrivals are governed by the Markovian arrival process. During a service period, all customers are served exhaustively. The server goes on vacation as soon as he/she completes service and the system is empty. Termination of the vacation period is controlled by two threshold parameters N and T, i.e. the server terminates his/her vacation as soon as the number waiting reaches N or the waiting time of the leading customer reaches T units. The steady state probability vector is shown to be of matrix-geometric type. The average queue length and the probability that the server is on vacation (or idle) are obtained. We also derive the steady state distribution of the waiting time at arrivals and show that the vacation period distribution is of phase type.     

Vacations in GI X/M Y/1 systems and Riemann boundary value problems

We consider systems of GI/M/1 type with bulk arrivals, bulk service and exponential server vacations. The generating functions of the steady-state probabilities of the embedded Markov chain are found in terms of Riemann boundary value problems, a necessary and sufficient condition of ergodicity is proved. Explicit formulas are obtained for the case where the generating function of the arrival group size is rational. Resonance between the vacation rate and the system is studied. Complete formulas are given for the cases of single and geometric arrivals. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analyzing the discrete-timeG (G)/Geo/1 queue using complex contour integration

In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as theG ^(G)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases we rediscover a number of well-known results, such as Hunter's result for theG/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more generalG ^(G)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.Both authors wish to thank the Belgian National Fund for Scientific Research (NFWO) for support of this work.     

The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue

This paper studies a single server queueing system with multiple types of customers. The first part of the paper discusses some modeling issues associated with the Markov arrival processes with marked arrivals (MMAP[K], where K is an integer representing the number of types of customers). The usefulness of MMAP[K] in modeling point processes is shown by a number of interesting examples. The second part of the paper studies a single server queueing system with an MMAP[K] as its input process. The busy period, virtual waiting time, and actual waiting times are studied. The focus is on the actual waiting times of individual types of customers. Explicit formulas are obtained for the Laplace–Stieltjes transforms of these actual waiting times.     

Two Queues with Weighted One-Way Overflow

We consider an open queueing network consisting of two queues with Poisson arrivals and exponential service times and having some overflow capability from the first to the second queue. Each queue is equipped with a finite number of servers and a waiting room with finite or infinite capacity. Arriving customers may be blocked at one of the queues depending on whether all servers and/or waiting positions are occupied. Blocked customers from the first queue can overflow to the second queue according to specific overflow routines. Using a separation method for the balance equations of the two-dimensional server and waiting room demand process, we reduce the dimension of the problem of solving these balance equations substantially. We extend the existing results in the literature in three directions. Firstly, we allow different service rates at the two queues. Secondly, the overflow stream is weighted with a parameter p ∈ [0,1], i.e., an arriving customer who is blocked and overflows, joins the overflow queue with probability p and leaves the system with probability 1 − p. Thirdly, we consider several new blocking and overflow routines. An erratum to this article can be found at     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

Performance analysis of a batch service queue arising out of manufacturing system modelling

In this paper, we present an exact analysis of a queueing system with Poisson arrivals and batch service. The system has a finite numberS of waiting places and a batch service capacityb. A service period is initialized when a service starting thresholda of waiting customers has been reached. The model is denoted accordingly byM/G ^[a,b] /1–S. The motivation for this model arises from manufacturing environments with batch service work stations, e.g. in machines for computer components and chip productions. The method of embedded Markov chain is used for the analysis, whereby a representation of the general service time is obtained via a moment matching approach. Numerical results are shown in order to illustrate the dependency of performance measures on special sets of system parameters. Furthermore, attention is devoted to the issues of starting rules, where performance objectives like short waiting time, small blocking probability and minimal amount of work in progress are taken into account.     

On the GI/M/∞ queue with batch arrivals of constant size

In this note, the GI/M/ queue with batch arrivals of constant sizek is investigated. It is shown that the stationary probabilities that an arriving batch findsi customers in the system can be computed in terms of the corresponding binomial moments (Jordan's formula), which are determined by a recursive relation. This generalizes well-known results by Takács [12] for GI/M/. Furthermore, relations between batch arrival- and time-stationary probabilities are given.     

Maximum likelihood estimation for single server queues from waiting time data

Maximum likelihood estimators for the parameters of a GI/G/1 queue are derived based on the information on waiting times {W _t },t=1,...,n, ofn successive customers. The consistency and asymptotic normality of the estimators are established. A simulation study of the M/M/1 and M/E _k /1 queues is presented.     

Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time

We consider anM/G/1 queue with FCFS queue discipline. We present asymptotic expansions for tail probabilities of the stationary waiting time when the service time distribution is longtailed and we discuss an extension of our methods to theM ^[x]/G/1 queue with batch arrivals.     

Stationary Distributions of GI/M/c Queue with PH Type Vacations

We study a GI/M/c type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find waiting customers in the system at a vacation completion instant.The vacation time is a phase-type (PH) distributed random variable. Using embedded Markov chain modeling and the matrix geometric solution methods, we obtain explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. To compare the vacation model with the classical GI/M/c queue without vacations, we prove conditional stochastic decomposition properties for the queue length and the waiting time when all servers are busy. Our model is a generalization of several previous studies.