A NEW APPROACH FOR THE STUDY OF THE M X /G/1 QUEUE USING RENEWAL ARGUMENTS

The M ^X /G/1 queueing system as well as several of its variants have long ago been studied by considering the embedded discrete-time Markov chain at service completion epochs. Alternatively other approaches have been proposed such as the theory of regenerative processes, the supplementary variable method, properties of the busy period, etc. In this note we study the M ^X /G/1 queue via a simple new method that uses renewal arguments. This approach seems quite powerful and may become fruitful in the investigation of other queueing systems as well.     

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.     

Exact Convergence Rate for the Distributions of GI/M/c/K Queue as K Tends to Infinity

We obtain the exact convergence rate of the stationary distribution ^(K) of the embedded Markov chain in GI/M/c/K queue to the stationary distribution of the embedded Markov chain in GI/M/c queue as K. Similar result for the time-stationary distributions of queue size is also included. These generalize Choi and Kim's results of the case c=1 by nontrivial ways. Our results also strengthen the Simonot's results [5].     

A batch arrival retrial queue with two phases of service and Bernoulli vacation schedule

We consider an M X /G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system , including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and calculation of the first moment.     

The N threshold policy for the GI/M/1 queue

We study a GI/M/1 queue with an N threshold policy. In this system, the server stops attending the queue when the system becomes empty and resumes serving the queue when the number of customers reaches a threshold value N. Using the embeded Markov chain method, we obtain the stationary distributions of queue length and waiting time and prove the stochastic decomposition properties.     

A Polynomial Factorization Approach for the Discrete Time GIX/>G/1/K Queue

This paper proposes a polynomial factorization approach for queue length distribution of discrete time GI ^X /G/1 and GI _X /G/1/K queues. They are analyzed by using a two-component state model at the arrival and departure instants of customers. The equilibrium state-transition equations of state probabilities are solved by a polynomial factorization method. Finally, the queue length distributions are then obtained as linear combinations of geometric series, whose parameters are evaluated from roots of a characteristic polynomial.     

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.     

Analysis of the GI/Geo/1 queue with N-policy

We consider a discrete-time single server N  -policy GI/Geo/1$\mathit{GI}/\mathit{Geo}/1$ queueing system. The server stops servicing whenever the system becomes empty, and resumes its service as soon as the number of waiting customers in the queue reaches N. Using an embedded Markov chain and a trial solution approach, the stationary queue length distribution at arrival epochs is obtained. Furthermore, we obtain the stationary queue length distribution at arbitrary epochs by using the preceding result and a semi-Markov process. The sojourn time distribution is also presented.     

A Markov Renewal Approach to M/G/1 Type Queues with Countably Many Background States

We consider the stationary distribution of the M/GI/1 type queue when background states are countable. We are interested in its tail behavior. To this end, we derive a Markov renewal equation for characterizing the stationary distribution using a Markov additive process that describes the number of customers in system when the system is not empty. Variants of this Markov renewal equation are also derived. It is shown that the transition kernels of these renewal equations can be expressed by the ladder height and the associated background state of a dual Markov additive process. Usually, matrix analysis is extensively used for studying the M/G/1 type queue. However, this may not be convenient when the background states are countable. We here rely on stochastic arguments, which not only make computations possible but also reveal new features. Those results are applied to study the tail decay rates of the stationary distributions. This includes refinements of the existence results with extensions.     

Strong Stability of the Batch Arrival Queueing Systems

In this work, we use the strong stability method to study the batch arrival queue after a perturbation of the batch size distribution. We show that, under some hypotheses, the characteristics of the batch arrival queueing system M ^X /M/1 may be approximated by the correspondent characteristics of the system M ^Geo /M/1.

After clarifying the conditions of approximation, we obtain stability inequalities with an exact computation of constants.     

Some results for the queues Mx/M/c and GIx/ G/c

We develop for the queue M^x/M/c an upper bound for the mean queue length and lower bounds for the delay probabilities (that of an arrival group and that of an arbitrary customer in the arrival group). An approximate formula is also developed for the general bulk-arrival queue GI^x/G/c. Preliminary numerical studies have indicated excellent performance of the results.     

On bounds for bulk arrival queues

This paper gives a simple and effective approach pf deriving bounds for bulk arrival queues by making use of the bounds for single arrival queues. With this approach, upper bounds of mean actual/virtual waiting times and mean queue length at random epochs can be derived for the bulk arrival queues GI^X/G/1 and GI^X/G/c (lower bounds can be derived in a similar way). The merit of this approach is shown by comparing the bounds obtained with some existing results in the literature.     

Continuity theorems for the M/M/1/n queueing system

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution. Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.     

Performance Analysis of a Finite-Buffer Bulk-Arrival and Bulk-Service Queue with Variable Server Capacity

Abstract

In this paper, we consider a finite-buffer bulk-arrival and bulk-service queue with variable server capacity: M ^X /G ^Y /1/K + B. The main purpose of this paper is to discuss the analytic and computational aspects of this system. We first derive steady-state departure-epoch probabilities based on the embedded Markov chain method. Next, we demonstrate two numerically stable relationships for the steady-state probabilities of the queue lengths at three different epochs: departure, random, and arrival. Finally, based on these relationships, we present various useful performance measures of interest such as moments of the number of customers in the queue at three different epochs, the loss probability, and the probability that server is busy. Numerical results are presented for a deterministic service-time distribution – a case that has gained importance in recent years.     

Error bounds for augmented truncation approximations of continuous-time Markov chains

This paper considers the augmented truncation approximation of the generator of an ergodic continuous-time Markov chain with a countably infinite state space. The main purpose of this paper is to present bounds for the absolute difference between the stationary distributions of the original generator and its augmented truncation. As examples, we apply the bounds to an $M∕M∕s$ retrial queue and an upper Hessenberg Markov chain.     

A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.     

An <Emphasis Type="Italic">M</Emphasis><Superscript>[<Emphasis Type="Italic">X</Emphasis>]</Superscript>/<Emphasis Type="Italic">G</Emphasis>/1 retrial queue with server breakdowns and constant rate of repeated attempts

We consider an M ^[X]/G/1 retrial queue subject to breakdowns where the retrial time is exponential and independent of the number of customers applying for service. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest joins a retrial group (called orbit) to repeat his request later; otherwise, if the server is busy or down, all customers of the coming batch enter the orbit. It is assumed that the server has a constant failure rate and arbitrary repair time distribution. We study the ergodicity of the embedded Markov chain, its stationary distribution and the joint distribution of the server state and the orbit size in steady-state. The orbit and system size distributions are obtained as well as some performance measures of the system. The stochastic decomposition property and the asymptotic behavior under high rate of retrials are discussed. We also analyse some reliability problems, the k-busy period and the ordinary busy period of our retrial queue. Besides, we give a recursive scheme to compute the distribution of the number of served customers during the k-busy period and the ordinary busy period. The effects of several parameters on the system are analysed numerically. I. Atencia’s and Moreno’s research is supported by the MEC through the project MTM2005-01248.     

A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots

A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GI^X/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GI^X/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GI^X/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms.

     

Transient and Stationary Distributions for the GI/G/k Queue with Lebesgue-Dominated Inter-Arrival Time Distribution

In this paper, the multi-server queue with general service time distribution and Lebesgue-dominated iid inter-arival times is analyzed. This is done by introducing auxiliary variables for the remaining service times and then examining the embedded Markov chain at arrival instants. The concept of piecewise-deterministic Markov processes is applied to model the inter-arrival behaviour. It turns out that the transition probability kernel of the embedded Markov chain at arrival instants has the form of a lower Hessenberg matrix and hence admits an operator–geometric stationary distribution. Thus it is shown that matrix–analytical methods can be extended to provide a modeling tool even for the general multi-server queue.     

Algorithmic analysis of the Geo/Geo/c retrial queue

In this paper, we consider a discrete-time queue of Geo/Geo/c type with geometric repeated attempts. It is known that its continuous counterpart, namely the M/M/c queue with exponential retrials, is analytically intractable due to the spatial heterogeneity of the underlying Markov chain, caused from the retrial feature. In discrete-time, the occurrence of multiple events at each slot increases the complexity of the model and raises further computational difficulties. We propose several algorithmic procedures for the efficient computation of the main performance measures of this system. More specifically, we investigate the stationary distribution of the system state, the busy period and the waiting time. Several numerical examples illustrate the analysis.