An M/G/1 unreliable retrial queue with two phases of service and Bernoulli admission mechanism

This paper deals with the steady state behaviour of an M^X/G/1 retrial queue with an additional second phase of optional service and unreliable server where breakdowns occur randomly at any instant while serving the customers. Further concept of Bernoulli admission mechanism is also introduced in the model. This model generalizes both the classical M^X/G/1 retrial queue subject to random breakdown and Bernoulli admission mechanism as well as M^X/G/1 queue with second optional service and unreliable server. We carry out an extensive analysis of this model.     

On the single server retrial queue with priority customers

We consider anM ₂/G ₂/1 type queueing system which serves two types of calls. In the case of blocking the first type customers can be queued whereas the second type customers must leave the service area but return after some random period of time to try their luck again. This model is a natural generalization of the classicM ₂/G ₂/1 priority queue with the head-of-theline priority discipline and the classicM/G/1 retrial queue. We carry out an extensive analysis of the system, including existence of the stationary regime, embedded Markov chain, stochastic decomposition, limit theorems under high and low rates of retrials and heavy traffic analysis.Visiting from: Department of Probability, Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia.     

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.This research was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

Analysis of the discrete-time bulk-service queue Geo/G/1/N+B

This paper considers a discrete-time bulk-service queueing system with variable capacity, finite waiting space and independent Bernoulli arrival process: Geo/G^Y/1/N+B. Both the analytic and computational aspects of the distributions of the number of customers in the queue at post-departure, random and pre-arrival epochs are discussed.     

Some results on a generalized M/G/1 feedback queue with negative customers

This paper deals with a generalized M/G/1 feedback queue in which customers are either “positive" or “negative". We assume that the service time distribution of a positive customer who initiates a busy period is G _e (x) and all subsequent positive customers in the same busy period have service time drawn independently from the distribution G _b (x). The server is idle until a random number N of positive customers accumulate in the queue. Following the arrival of the N-th positive customer, the server serves exhaustively the positive customers in the queue and then a new idle period commences. This queueing system is a generalization of the conventional N-policy queue with N a constant number. Explicit expressions for the probability generating function and mean of the system size of positive customers are obtained under steady-state condition. Various vacation models are discussed as special cases. The effects of various parameters on the mean system size and the probability that the system is empty are also analysed numerically. AMS Subject Classification: Primary: 60 K 25 · Secondary: 60 K 20, 90 B 22     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

A MAP/G/1 Queue with Negative Customers

In this paper, we consider a MAP/G/1 queue with MAP arrivals of negative customers, where there are two types of service times and two classes of removal rules: the RCA and RCH, as introduced in section 2. We provide an approach for analyzing the system. This approach is based on the classical supplementary variable method, combined with the matrix-analytic method and the censoring technique. By using this approach, we are able to relate the boundary conditions of the system of differential equations to a Markov chain of GI/G/1 type or a Markov renewal process of GI/G/1 type. This leads to a solution of the boundary equations, which is crucial for solving the system of differential equations. We also provide expressions for the distributions of stationary queue length and virtual sojourn time, and the Laplace transform of the busy period. Moreover, we provide an analysis for the asymptotics of the stationary queue length of the MAP/G/1 queues with and without negative customers.     

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.     

Bayesian analysis of M/Er/1 and M/H_k/1 queues

This paper describes Bayesian inference and prediction for some M/G/1 queueing models. Cases when the service distribution is Erlang, hyperexponential and hyperexponential with a random number of components are considered. Monte Carlo and Markov chain Monte Carlo methods are used for estimation of quantities of interest assuming the queue is in equilibrium. This revised version was published online in June 2006 with corrections to the Cover Date.     

GI\M\n系統中大量服务的排队过程

<正> §1.引言 我們知道,描述一个排队过程,需要三个因素:輸入过程,排队紀律,及服务机构.所謂GI|M|n,就是指这样的一个排队过程,它的 i)輸入过程,各顾客到来的时間区間的长度t相互独立、相同分布.其分布記     

Analysis of stationary discrete-time GI/D-MSP/1 queue with finite and infinite buffers

This paper considers a single-server queueing model with finite and infinite buffers in which customers arrive according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process (D-MSP). This service process is similar to the discrete-time Markovian arrival process (D-MAP), where arrivals are replaced with service completions. Using the imbedded Markov chain technique and the matrix-geometric method, we obtain the system-length distribution at a prearrival epoch. We also provide the steady-state system-length distribution at an arbitrary epoch by using the supplementary variable technique and the classical argument based on renewal-theory. The analysis of actual-waiting-time (in the queue) distribution (measured in slots) has also been investigated. Further, we derive the coefficient of correlation of the lagged interdeparture intervals. Moreover, computational experiences with a variety of numerical results in the form of tables and graphs are discussed.     

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.     

On the Diophantine Queueing System, <Emphasis Type="Italic">D/D</Emphasis><Superscript><Emphasis Type="Italic">a,b</Emphasis></Superscript><Emphasis Type="Italic">/</Emphasis>1

This paper considers the solution of a deterministic queueing system. In this system, the single server provides service in bulk with a threshold for the acceptance of customers into service. Analytic results are given for the steady-state probabilities of the number of customers in the system and in the queue for random and pre-arrival epochs. The solution of this system is a prerequisite to a four-point approximation to the model GI/G ^a,b /1. The paper demonstrates that the solution of such a system is not a trivial problem and can produce interesting results. The graphical solution discussed in the literature requires that the traffic intensity be a rational number. The results so generated may be misleading in practice when a control policy is imposed, even when the probability distributions for the interarrival and service times are both deterministic.     

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.     

An M/G/1 Queueing Model with Gated Random Order of Service

We analyse an M/G/1 queueing model with gated random order of service. In this service discipline there are a waiting room, in which arriving customers are collected, and a service queue. Each time the service queue becomes empty, all customers in the waiting room are put instantaneously and in random order into the service queue. The service times of customers are generally distributed with finite mean. We derive various bivariate steady-state probabilities and the bivariate Laplace–Stieltjes transform (LST) of the joint distribution of the sojourn times in the waiting room and the service queue. The derivation follows the line of reasoning of Avi-Itzhak and Halfin [4]. As a by-product, we obtain the joint sojourn times LST for several other gated service disciplines.     

Single server retrial queues with two way communication

The main aim of this paper is to study the steady state behavior of an M/G/1-type retrial queue in which there are two flows of arrivals namely ingoing calls made by regular customers and outgoing calls made by the server when it is idle. We carry out an extensive stationary analysis of the system, including stability condition, embedded Markov chain, steady state joint distribution of the server state and the number of customers in the orbit (i.e., the retrial group) and calculation of the first moments. We also obtain light-tailed asymptotic results for the number of customers in the orbit. We further formulate a more complicate but realistic model where the arrivals and the service time distributions are modeled in terms of the Markovian arrival process (MAP) and the phase (PH) type distribution.     

Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

Two variants of an M/G/1 queue with negative customers lead to the study of a random walkX _n+1=[X _n + _n ]⁺ where the integer-valued _n are not bounded from below or from above, and are distributed differently in the interior of the state-space and on the boundary. Their generating functions are assumed to be rational. We give a simple closed-form formula for , corresponding to a representation of the data which is suitable for the queueing model. Alternative representations and derivations are discussed. With this formula, we calculate the queue length generating function of an M/G/1 queue with negative customers, in which the negative customers can remove ordinary customers only at the end of a service. If the service is exponential, the arbitrarytime queue length distribution is a mixture of two geometrical distributions.Supported by the European grant BRA-QMIPS of CEC DG XIII.     

Delay Distribution of (Im)Patient Customers in a Discrete Time D-MAP/PH/1 Queue with Age-Dependent Service Times

This paper presents an algorithmic procedure to calculate the delay distribution of (im)patient customers in a discrete time D-MAP/PH/1 queue, where the service time distribution of a customer depends on his waiting time. We consider three different situations: impatient customers in the waiting room, impatient customers in the system, that is, if a customer has been in the waiting room, respectively, in the system for a time units it leaves the waiting room, respectively, the system. In the third situation, all customers are patient – that is, they only leave the system after completing service. In all three situations the service time of a customer depends upon the time he has spent in the waiting room. As opposed to the general approach in many queueing systems, we calculate the delay distribution, using matrix analytic methods, without obtaining the steady state probabilities of the queue length. The trick used in this paper, which was also applied by Van Houdt and Blondia [J. Appl. Probab., Vol. 39, No. 1 (2002) pp. 213–222], is to keep track of the age of the customer in service, while remembering the D-MAP state immediately after the customer in service arrived. Possible extentions of this method to more general queues and numerical examples that demonstrate the strength of the algorithm are also included.     

Impact of Bursty Traffic on Queues

The impact of bursty traffic on queues is investigated in this paper. We consider a discrete-time single server queue with an infinite storage room, that releases customers at the constant rate of c customers/slot. The queue is fed by an M/G/∞ process. The M/G/∞ process can be seen as a process resulting from the superposition of infinitely many ‘sessions’: sessions become active according to a Poisson process; a station stays active for a random time, with probability distribution G, after which it becomes inactive. The number of customers entering the queue in the time-interval [t, t + 1) is then defined as the number of active sessions at time t (t = 0,1, ...) or, equivalently, as the number of busy servers at time t in an M/G/∞ queue, thereby explaining the terminology. The M/G/∞ process enjoys several attractive features: First, it can display various forms of dependencies, the extent of which being governed by the service time distribution G. The heavier the tail of G, the more bursty the M/G/∞ process. Second, this process arises naturally in teletraffic as the limiting case for the aggregation of on/off sources [27]. Third, it has been shown to be a good model for various types of network traffic, including telnet/ftp connections [37] and variable-bit-rate (VBR) video traffic [24]. Last but not least, it is amenable to queueing analysis due to its very strong structural properties. In this paper, we compute an asymptotic lower bound for the tail distribution of the queue length. This bound suggests that the queueing delays will dramatically increase as the burstiness of the M/G/∞ input process increases. More specifically, if the tail of G is heavy, implying a bursty input process, then the tail of the queue length will also be heavy. This result is in sharp contrast with the exponential decay rate of the tail distribution of the queue length in presence of ‘non-bursty’ traffic (e.g. Poisson-like traffic). This revised version was published online in August 2006 with corrections to the Cover Date.     

Increasing convex ordering of queue length in bulk queues

This paper considers single-server bulk queues M^(X)/G^(Y)/1 and G^(X)/M^(Y)/1. In the former queue, service times and service capacity are dependent, while in the latter queue, inter-arriving times and arriving group size are dependent. We show that stronger dependence between those leads to shorter queue lengths in the increasing convex ordering sense.