Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs

Retrial queues have been widely used to model many problems arising in telephone switching systems, telecommunication networks, computer networks and computer systems, etc. It is of basic importance to study reliability of retrial queues with server breakdowns and repairs because of limited ability of repairs and heavy influence of the breakdowns on the performance measure of the system. However, so far the repairable retrial queues are analyzed only by queueing theory. In this paper we give a detailed analysis for reliability of retrial queues. By using the supplementary variables method, we obtain the explicit expressions of some main reliability indexes such as the availability, failure frequency and reliability function of the server. In addition, some special queues, for instance, the repairable M/G/1 queue and repairable retrial queue can be derived from our results. These results may be generalized to the repairable multi-server retrial models.     

Monotonicity properties in various retrial queues and their applications

We consider several multi-server retrial queueing models with exponential retrial times that arise in the literature of retrial queues. The effect of retrial rates on the behavior of the queue length process is investigated via sample path approach. We show that the number of customers in orbit and in the system as a whole are monotonically changed if the retrial rates in one system are bounded by the rates in second one. The monotonicity results are applied to show the convergence of generalized truncated systems that have been widely used for approximating the stationary queue length distribution in retrial queues. AMS subject classifications: Primary 60K25     

A retrial queue with redundancy and unreliable server

Retrial queues are useful in the stochastic modelling of computer and telecommunication systems amongst others. In this paper we study a version of the retrial queue with variable service. Such a point of view gives another look at the unreliable retrial queueing problem which includes the redundancy model.By using the theory of piecewise Markovian processes, we obtain the analogue of the Pollaczek-Khintchine formula for such retrial queues, which is useful for operations researchers to obtain performance measures of interest.     

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The discriminatory processor sharing queues with multiple classes of customers (abbreviated as DPS queues) are an important but difficult research direction in queueing theory, and it has many important practical applications in the fields of, such as, computer networks, manufacturing systems, transportation networks, and so forth. Recently, researchers have carried out some key work for the DPS queues. They gave the generating function of the steady-state joint queue lengths, which leads to the first two moments of the steady-state joint queue lengths. However, using the generating function to provide explicit expressions for the steady-state joint queue lengths has been a difficult and challenging problem for many years. Based on this, this paper applies the maximum entropy principle in the information theory to providing an approximate expression with high precision, and this approximate expression can have the same first three moments as those of its exact expression. On the other hand, this paper gives efficiently numerical computation by means of this approximate expression, and analyzes how the key variables of this approximate expression depend on the original parameters of this queueing system in terms of some numerical experiments. Therefore, this approximate expression has important theoretical significance to promote practical applications of the DPS queues. At the same time, not only do the methodology and results given in this paper provide a new line in the study of DPS queues, but they also provide the theoretical basis and technical support for how to apply the information theory to the study of queueing systems, queueing networks and more generally, stochastic models.     

Ergodicity of the BMAP/PH/s/s+K retrial queue with PH-retrial times

Define the traffic intensity as the ratio of the arrival rate to the service rate. This paper shows that the BMAP/PH/s/s+K retrial queue with PH-retrial times is ergodic if and only if its traffic intensity is less than one. The result implies that the BMAP/PH/s/s+K retrial queue with PH-retrial times and the corresponding BMAP/PH/s queue have the same condition for ergodicity, a fact which has been believed for a long time without rigorous proof. This paper also shows that the same condition is necessary and sufficient for two modified retrial queueing systems to be ergodic. In addition, conditions for ergodicity of two BMAP/PH/s/s+K retrial queues with PH-retrial times and impatient customers are obtained. This revised version was published online in June 2006 with corrections to the Cover Date.     

排队系统中的泰勒展开方法

综述了排队系统中的泰勒展开方法。它由Gong和Hu在1990s首次提出,并在最近几年里有了一些新的发展。首先,通过GI/GI/1队列的简单例子介绍其基本原理;其次,展示如何应用该方法分析相关性队列和离去过程;然后,阐述如何基于该方法发展排队网络近似的高阶矩方法;最后,讨论未来的几个可能研究方向。     

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper, we develop a new method to obtain exact expressions for the Laplace–Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now.     

Transient distributions of level dependent quasi-birth-death processes with linear transition rates

Many queueing systems such asM/M/s/K retrial queue with impatient customers, MAP/PH/1 retrial queue, retrial queue with two types of customers andMAP/M/∞ queue can be modeled by a level dependent quasi-birth-death (LDQBD) process with linear transition rates of the form λ_k = α+ βk at each levelk. The purpose of this paper is to propose an algorithm to find transient distributions for LDQBD processes with linear transition rates based on the adaptive uniformizaton technique introduced by van Moorsel and Sanders [11]. We apply the algorithm to some retrial queues and present numerical results.     

G-networks with multiple classes of signals and positive customers

G-networks are queueing models in which the types of customers one usually deals with in queues are enriched in several ways. In Gnetworks, positive customers are those that are ordinarily found in queueing systems; they queue up and wait for service, obtain service and then leave or go to some other queue. Negative customers have the specific function of destroying ordinary or positive customers. Finally triggers simply move an ordinary customer from one queue to the other. The term “signal” is used to cover negative customers and triggers. G-networks contain these three type of entities with certain restrictions; positive customers can move from one queue to another, and they can change into negative customers or into triggers when they leave a queue. On the other hand, signals (i.e. negative customers and triggers) do not queue up for service and simply disappear after having joined a queue and having destroyed or moved a negative customer. This paper considers this class of networks with multiple classes of positive customers and of signals. We show that with appropriate assumptions on service times, service disciplines, and triggering or destruction rules on the part of signals, these networks have a product form solution, extending earlier results.     

Some Applications of Queueing Theory in Operational Research (1953-89)

An account of the development of queueing theory from an operationalresearch perspective is given. The theory discussed ranges fromthe simple queue, finite queues, and queues in tandem, to non-Poissoninput and service-queueing systems. A methodology for analysingthe systems is outlined, and appropriate references given. Allthese developments relate to models used in investigating realproblems undertaken by different operational research departments.     

Analysis of Markov Multiserver Retrial Queues with Negative Arrivals

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. In the paper we analyze multiserver retrial queues; i.e., any customer finding all servers busy upon arrival must leave the service area and re-apply for service after some random time. The control mechanism is such that, whenever the service facility is full occupied, an exponential timer is activated. If the timer expires and the service facility remains full, then a random batch of customers, which are stored at the retrial pool, are automatically removed. This model extends the existing literature, which only deals with a single server case and individual removals. Two different approaches are considered. For the stable case, the matrix–analytic formalism is used to study the joint distribution of the service facility and the retrial pool. The approximation by more simple infinite retrial model is also proved. In the overloading case we study the transient behaviour of the trajectory of the suitably normalized retrial queue and the long-run behaviour of the number of busy servers. The method of investigation in this case is based on the averaging principle for switching processes.     

On the stationary distribution of queue lengths in a multi-class priority queueing system with customer transfers

This paper deals with a multi-class priority queueing system with customer transfers that occur only from lower priority queues to higher priority queues. Conditions for the queueing system to be stable/unstable are obtained. An auxiliary queueing system is introduced, for which an explicit product-form solution is found for the stationary distribution of queue lengths. Sample path relationships between the queue lengths in the original queueing system and the auxiliary queueing system are obtained, which lead to bounds on the stationary distribution of the queue lengths in the original queueing system. Using matrix-analytic methods, it is shown that the tail asymptotics of the stationary distribution is exact geometric, if the queue with the highest priority is overloaded.      

M/G/1 retrial queueing systems with two types of calls and finite capacity

We consider anM/G/1 priority retrial queueing system with two types of calls which models a telephone switching system and a cellular mobile communication system. In the case that arriving calls are blocked due to the server being busy, type I calls are queued in a priority queue of finite capacityK whereas type II calls enter the retrial group in order to try service again after a random amount of time. In this paper we find the joint generating function of the numbers of calls in the priority queue and the retrial group in closed form. When ₁=0, it is shown that our results are consistent with the known results for a classical retrial queueing system.     

Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.     

A tandem GI/PH/1→•/PH/1/0 queue with blocking

Tandem queues are widely used in mathematical modeling of random processes describing the operation of manufacturing systems, supply chains, computer and telecommunication networks. Although there exists a lot of publications on tandem queueing systems, analytical research on tandem queues with non-Markovian input is very limited. In this paper, the results of analytical investigation of two-node tandem queue with arbitrary distribution of inter-arrival times are presented. The first station of the tandem is represented by a single-server queue with infinite waiting room. After service at the first station, a customer proceeds to the second station that is described by a single-server queue without a buffer. Service times of a customer at the first and the second server have PH (Phase-type) distributions. A customer, who completes service at the first server and meets a busy second server, is forced to wait at the first server until the second server becomes available. During the waiting period, the first server becomes blocked, i.e., not available for service of customers. We calculate the joint stationary distribution of the system states at the embedded epochs and at arbitrary time. The Laplace–Stieltjes transform of the sojourn time distribution is derived. Key performance measures are calculated and numerical results presented.     

Order independent loss queues

This paper presents an analysis of generalized Order Independent (OI) loss queues serving customers belonging to different types (classes) where limits are placed on the number of customers of each type that may be present in the system. We prove that such queues satisfy partial balance and we present their stationary distribution. OI loss queues can be used to model blocking systems with simultaneous resource possession with the option of queueing blocked customers. The OI loss queue thus extends previous loss models where customers are rejected when processing resources are not available.This work was supported by grants from the Foundation for Research Development.     

Solution for a retrial queueing problem in cellular networks with the Fractional Guard Channel policy

In this paper, we introduce a retrial queueing problem for wireless cellular networks applying the Fractional Guard Channel (FGC) admission control policy and propose a new algorithm to solve the retrial queue problem involving the FGC policy.A comparison between analytical and simulation results shows that our algorithm is accurate and fast to evaluate the performance of the system.     

Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

We propose an analytically tractable approach for studying the transient behavior of multi-server queueing systems and feed-forward networks. We model the queueing primitives via polyhedral uncertainty sets inspired by the limit laws of probability. These uncertainty sets are characterized by variability parameters that control the degree of conservatism of the model. Assuming the inter-arrival and service times belong to such uncertainty sets, we obtain closed-form expressions for the worst case transient system time in multi-server queues and feed-forward networks with deterministic routing. These analytic formulas offer rich qualitative insights on the dependence of the system times as a function of the variability parameters and the fundamental quantities in the queueing system. To approximate the average behavior, we treat the variability parameters as random variables and infer their density by using ideas from queues in heavy traffic under reflected Brownian motion. We then average the worst case values obtained with respect to the variability parameters. Our averaging approach yields approximations that match the diffusion approximations for a single queue with light-tailed primitives and allows us to extend the framework to heavy-tailed feed-forward networks. Our methodology achieves significant computational tractability and provides accurate approximations for the expected system time relative to simulated values.     

Stability analysis of GI/GI/c/K retrial queue with constant retrial rate

We consider a finite buffer capacity GI/GI/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has \(c\) identical servers and can accommodate up to \(K\) jobs (including \(c\) jobs under service). If a newly arriving job finds the primary queue to be full, it joins the orbit queue. The original primary jobs arrive to the system according to a renewal process. The jobs have i.i.d. service times. The head of line job in the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the length of the orbit queue. Telephone exchange systems, medium access protocols, optical networks with near-zero buffering and TCP short-file transfers are some telecommunication applications of the proposed queueing system. The model is also applicable in logistics. We establish sufficient stability conditions for this system. In addition to the known cases, the proposed model covers a number of new particular cases with the closed-form stability conditions. The stability conditions that we obtained have clear probabilistic interpretation.     

Approximation of multiserver retrial queues by means of generalized truncated models

It is well-known that an analytical solution of multiserver retrial queues is difficult and does not lead to numerical implementation. Thus, many papers approximate the original intractable system by the so-called generalized truncated systems which are simpler and converge to the original model. Most papers assume heuristically the convergence but do not provide a rigorous mathematical proof. In this paper, we present a proof based on a synchronization procedure. To this end, we concentrate on theM/M/c retrial queue and the approximation developed by Neuts and Rao (1990). However, the methodology can be employed to establish the convergence of several generalized truncated systems and a variety of Markovian multiserver retrial queues. J.R. Artalejo thanks the support received from DGES 98-0837.