Analysis of the M1,M2/G/1 G-queueing system with retrial customers

We consider a single server retrial queue with waiting places in service area and three classes of customers subject to the server breakdowns and repairs. When the server is unavailable, the arriving class-1 customer is queued in the priority queue with infinite capacity whereas class-2 customer enters the retrial group. The class-3 customers which are also called negative customers do not receive service. If the server is found serving a customer, the arriving class-3 customer breaks the server down and simultaneously deletes the customer under service. The failed server is sent to repair immediately and after repair it is assumed as good as new. We study the ergodicity of the embedded Markov chains and their stationary distributions. We obtain the steady-state solutions for both queueing measures and reliability quantities. Moreover, we investigate the stochastic decomposition law, the busy period of the system and the virtual waiting times. Finally, an application to cellular mobile networks is provided and the effects of various parameters on the system performance are analyzed numerically.     

Convex comparison of service disciplines in real time queues

We present a comparison of the service disciplines in real time queueing systems (the customers have a deadline before which they should enter the service booth). We state that giving priority to customers having an early deadline minimizes the average stationary lateness. We show this result by comparing adequate random vectors with the Schur convex majorization ordering.     

An M/G/1 queue with two phases of service subject to the server breakdown and delayed repair

This paper deals with the steady-state behaviour of an M/G/1 queue with an additional second phase of optional service subject to breakdowns occurring randomly at any instant while serving the customers and delayed repair. This model generalizes both the classical M/G/1 queue subject to random breakdown and delayed repair as well as M/G/1 queue with second optional service and server breakdowns. For this model, we first derive the joint distributions of state of the server and queue size, which is one of chief objectives of the paper. Secondly, we derive the probability generating function of the stationary queue size distribution at a departure epoch as a classical generalization of Pollaczek–Khinchin formula. Next, we derive Laplace Stieltjes transform of busy period distribution and waiting time distribution. Finally, we obtain some important performance measures and reliability indices of this model.     

On the single server retrial queue subject to breakdowns

This paper deals with a single server retrial queueing system subject to active and independent breakdowns. The objective is to extend the results given independently by Aissani [1] and Kulkarni and Choi [15]. To this end, we introduce the concept of fundamental server period and an auxiliary queueing system with breakdowns and option for leaving the system. Then, we concentrate our attention on the limiting distribution of the system state. We obtain simplified expressions for the partial generating functions of the server state and the number of customers in the retrial group, a recursive scheme for computing the limiting probabilities and closed-form formulae for the second order partial moments. Some stochastic decomposition results are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

A simulation tool to evaluate the performance of finite-source queueing models

This paper deals with a software tool to evaluate the main characteristics of a nonhomogeneous finite-source queueing model to describe the performance of a multi-terminal system subject to random breakdowns under FIFO, priority processor sharing, and polling service disciplines. The model studied here is actually a closed queueing network network with three nonindependent service stations (CPU, terminals, and repairman), and a finite number of customers (jobs), which have different service rates at the service stations. The aim of this paper is to introduce the FQM (finite-source queueing model) program package, which was developed at the Institute of Mathematics and Informatics of Lajos Kossuth University in Debrecen, Hungary, and to investigate the performance of the above-mentioned finite-source queueing models. At the end we give a sample result to illustrate the problem in question. Supported by the Hungarian National Foundation for Scientific Research (grant Nos. OTKA T014974/95 and T016933/95). Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

Equilibrium analysis of the observable queues with balking and delayed repairs

The equilibrium threshold balking strategies are investigated for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Upon arriving, the customers decide whether to join or balk the queue based on observation of the queue length and status of the server, along with the consideration of waiting cost and the reward after finishing their service. By using Markov chain approach and system cost analysis, we obtain the stationary distribution of queue size of the queueing systems and provide algorithms in order to identify the equilibrium strategies for the fully and partially observable models. Finally, the equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively, both with server breakdowns and delayed repairs.     

Workload in queues having priorities assigned according to service time

System designers often implement priority queueing disciplines in order to improve overall system performance; however, improvement is often gained at the expense of lower priority cystomers. Shortest Processing Time is an example of a priority discipline wherein lower priority customers may suffer very long waiting times when compared to their waiting times under a democratic service discipline. In what follows, we shall investigate a queueing system where customers are divided into a finitie number of priority classes according to their service times.We develop the multivariate generating function characterizing the joint workload among the priority classes. First moments obtained from the generating function yield traffic intensities for each priority class. Second moments address expected workloads, in particular, we obtain simple Pollaczek-Khinchine type formulae for the classes. Higher moments address variance and covariance among the workloads of the priority classes.This work was supported in part by National Science Foundation Grant DDM-8913658.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

A queueing model for a nonreliable multiterminal system with polling scheduling

This paper deals with a nonhomogeneous finite-source queueing model to describe the performance of a multiterminal system subject to random breakdowns under the polling service discipline. The model studied here is a closed queueing network which has three service stations. a CPU (single server), terminals (infinite server), a repairman (single server), and a finite number of customers (jobs) that have distinct service rates at the service stations. The CPU's repair has preemptive priority over the terminal repairs, and failure of the CPU stops the service of the other stations, thus the nodes are not independent. It can be viewed as a continuation of papers by the authors (see references), which discussed a FIFO (first-in, first-out) and a PPS (priority processor sharing) serviced queueing model subject to random breakdowns. All random variables are assumed to be independent and exponentially distributed. The system behavior can be described by a Markov chain, but the number of states is very large. The purpose of this paper is to give a recursive computational approach to solve steady-state equations and to illustrate the problem in question using some numerical results. Supported by the Hungarian National Foundation for Scientific Research (grant Nos. OTKA T014974/95 and T016933/95) Proceedings of the Seminar on Stability Problems for Stochastic Models. Hajdúszoboszló, Hungary, 1997, Part, II.     

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.     

Expected waiting times in polling systems under priority disciplines

In this paper, we analyse a service system which consists of several queues (stations) polled by a single server in a cyclic order with arbitrary switchover times. Customers from several priority classes arrive into each of the queues according to independent Poisson processes and require arbitrarily distributed service times. We consider the system under various priority service disciplines: head-of-the-line priority limited to one and semi-exhaustive, head-of-the-line priority limited to one with background customers, and global priority limited to one. For the first two disciplines we derive a pseudo conservation law. For the third discipline, we show how to obtain the expected waiting time of a customer from any given priority class. For the last discipline we find the expected waiting time of a customer from the highest priority class. The principal tool for our analysis is the stochastic decomposition law for a single server system with vacations.     

A composite step conjugate gradients squared algorithm for solving nonsymmetric linear systems

We propose a new and more stable variant of the CGS method [27] for solving nonsymmetric linear systems. The method is based on squaring the Composite Step BCG method, introduced recently by Bank and Chan [1,2], which itself is a stabilized variant of BCG in that it skips over steps for which the BCG iterate is not defined and causes one kind of breakdown in BCG. By doing this, we obtain a method (Composite Step CGS or CSCGS) which not only handles the breakdowns described above, but does so with the advantages of CGS, namely, no multiplications by the transpose matrix and a faster convergence rate than BCG. Our strategy for deciding whether to skip a step does not involve any machine dependent parameters and is designed to skip near breakdowns as well as produce smoother iterates. Numerical experiments show that the new method does produce improved performance over CGS on practical problems.Partially supported by the Office of Naval Research grant N00014-92-J-1890, the National Science Foundation grant ASC92-01266, and the Army Research Office grant DAAL03-91-G-150.     

The effects of service disciplines on the performance of a nonreliable terminal system

In this paper, we consider a stochastic queueing model for the performance evaluaton of a real-life computer system consisting of n terminals connected with a CPU. A user at terminal i has thinking and processing time depending on the index i. Let us suppose that the operational system is subject to random breakdowns, which may be software or hardware ones, stopping the service both at the terminals and at the CPU. The failure-free operation times of the system and the restoration times are random variables. Busy terminals are also subject to random breakdowns not affecting the system operation. The failure-free operation times and the repair times of a busy terminal i are random variables with distribution function depending on index i. The breakdowns are serviced by a single repairman providing preemptive priority to the system's failure, while the restoration at the terminals are carried out according to the FIFO rule. We assume that each user generates only one job at a time, and he waits at the CPU before he starts thinking again, that is, the terminal is inactive while waiting at the CPU, and it cannot break down. All random variables involved in the model construction are assumed to be exponentially distributed and independent of each other. The aim of this paper is to investigate the effect of different service disciplines, such as FIFO, processor sharing, priority processor sharing, and polling, on the main performance measures, such as utilizations, response times, throughput, and mean queue length. Supported by the Hungarian National Foundation for Scientific Research (grant No. ORKA T014974/95). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló. Hungary, 1997, Part II.     

Double-sided matching queues: Priority and impatient customers

We analyze a double-sided queue with priority that serves patient customers and customers with zero patience (i.e., impatient customers). In a two-sided market, high and low priority customers arrive to one side and match with queued customers on the opposite side. Impatient customers match with queued patient customers; when there is no queue, they leave the system unmatched. All arrivals follow independent Poisson processes. We derive exact formulae for the stationary queue length distribution and several steady-state performance measures.     

具有工作故障及有限容量的MAP/M/1排队系统性能分析

为了拓展随机排队理论,在具有工作故障的MAP/M/1排队的基础上,引入有限容量策略建立起一个新的排队模型.通过Uniformization Technique将连续时间排队模型转化成对应的离散时间排队模型,运用矩阵几何组合解给出系统中的顾客数量和服务器状态的联合稳态概率表达式,并给出基于稳态概率的性能指标.最后通过一些数值例子展示参数对性能指标的影响.     

Multi-server queueing systems with cooperation of the servers

In this paper, we consider an N-server queueing model with homogeneous servers in which customers arrive according to a stationary Poisson arrival process. The service times are exponentially distributed. Two new customer’s service disciplines assuming simultaneous service of arriving customer by all currently idle servers are discussed. The steady state analysis of the queue length and sojourn time distribution is performed by means of the matrix analytic methods. Numerical examples, which illustrate advantage of introduced disciplines comparing to the classical one, are presented.     

一类随机完工时间二阶矩的表示方法

给出一种运用机器的工作时间、故障时间和工件的加工时间的分布特征表示在一台具有Birge所定义的序列随机故障的机器上加工一个中断-重复型工件的完工时间的二阶矩的方法, 并通过举例说明所建立的表示在风险分析与决策优化方面的应用.     

Analysis of nonpreemptive priority queues with multiple servers and two priority classes

In this paper, we model a priority multiserver queueing system with two priority classes. A high priority customer has nonpreemptive priority over low priority customers. The approaches for solving the problem are the state-reduction based variant of Kao, the modified boundary algorithm of Latouche, the logarithmic reduction algorithm of Latouche and Ramaswami, and the power-series method of Blanc. The objectives of this paper are to present a power-series implementation for the priority queue and to evaluate the relative efficiencies of alternative procedures to compute various performance characteristics. In the paper, we find that at times the logarithmic reduction algorithm may not perform as well as expected and the power-series approach can occasionally pose numerical difficulties.     

The discrete-time preemptive repeat identical priority queue

Priority queueing systems come natural when customers with diversified delay requirements have to wait to get service. The customers that cannot tolerate but small delays get service priority over customers which are less delay-sensitive. In this contribution, we analyze a discrete-time two-class preemptive repeat identical priority queue with infinite buffer space and generally distributed service times. Newly arriving high-priority customers interrupt the on-going service of a low-priority customer. After all high-priority customers have left the system, the interrupted service of the low-priority customer has to be repeated completely. By means of a probability generating functions approach, we analyze the system content and the delay of both types of customers. Performance measures (such as means and variances) are calculated and the impact of the priority scheduling is discussed by means of some numerical examples.