Stability of a two-class two-server retrial queueing system

We consider a two-class two-server retrial queueing system. Customers of each class arrive according to a Poisson process and the service times of each class of customers are assumed to be exponentially distributed with service rates depending on both the customer’s class and the servers. We provide stability and instability conditions for this retrial queueing system.     

A discrete-time retrial queueing system with starting failures,Bernoulli feedback and general retrial times

It is well known that discrete-time queues are more appropriate than their continuous-time counterparts for modelling computer and telecommunication systems. This paper is concerned with a discrete-time Geo/G/1 retrial queue with general retrial times, Bernoulli feedback and the server subjected to starting failures. In this paper, we generalize the previous works in discrete-time retrial queue with unreliable server due to starting failures in the sense that we consider general service with Bernoulli feedback and general retrial times. We analyse the Markov chain underlying the considered queueing system and present some performance measures of the system in steady-state. We provide two stochastic decomposition laws and as application we give bounds for the distance between the system size distribution of our model and the corresponding model without retrials. Besides, some numerical results are given to illustrate the impact of the unreliability and the feedback on the performance of the system. We also investigate the relation between our discrete-time system and its continuous counterpart.     

On a Multi-Channel Retrial Queueing System

The paper considers Markov multi-channel queueing systems with a single retrial attempt to begin service process. The authors establish the conditions of the existence of stationary mode and propose efficient algorithms for calculation of the stationary probabilities with the use of continued fractions and explicit vector-matrix formulas.     

A BMAP/SM/1 Queueing System with Hybrid Operation Mechanism

A single-server queueing system BMAP/SM/1 is studied. It operates as a retrial queueing system under decentralized and centralized retrial strategies with and without loss of primary customers, as well as a system with waiting. The operation mechanism changes under the action of a random environment. The stationary state probability distributions at imbedded and arbitrary instants and main performance characteristics of the system are determined.__________Translated from Avtomatika i Telemekhanika, No. 5, 2005, pp. 111–124.Original Russian Text Copyright © 2005 by Klimenok.     

Simulation of queueing network with time varying arrival rates

Queueing models are important in the analysis of many aspects of system or human behaviour. In such widely divergent fields as administrative processes, health services, traffic control, production control, computer communication networks, time sharing, and architecture, queueing analysis is indisposable for better understanding and ultimately synthesis of the system.However straight mathematical analysis is difficult, as the difference equations describing a queueing network are non-linear and of infinite dimensional. Due to this difficulty, time varying cases are seldomly included in the study of analytic solutions of queueing systems. In existing literature, the mean arrival rates are assumed independent of time. This assumption is of dubious validity as the arrival rate in actual operations varies with a twenty four hours period. While one can assume further that at every instant in time, the equilibrium condition for constant arrival rate is reached for the current arrival rate, it has never been verified that such is the case.An alternative approach is to reduce the queueing problem to finite dimensional and to simulate it on computer as is proposed in the present paper. The basic assumptions are (1) time-varying random arrival rate with Poisson statistics and (2) exponentially distributed service time. An algorithm and flow diagram for doing this is given. With the simulation techniques developed in this paper, we can study the dynamics of varying arrival rate. Simulation of a simple network shows that as the arrival rate approaches capacity, the divergence between simulated results and previous analytical results for the equilibrium condition become increasingly large. A conclusion is therefore that except for systems with very light loading, analytical results based on constant arrival rate are not accurate. Simulation is necessary if one is to obtain a true picture of the system.     

A memory interference model for regularly patterned multiple streamvector accesses

Most existing analytical models for memory interference generally assume random bank selection for each memory access. In vector computers, however, memory accesses are typically regularly patterned with a number of data items being accessed concurrently from different banks. Very little is known about the queueing behavior of memory interferences in multiple stream vector accesses. This paper presents an analytical model for memory interferences due to vector accesses in multiple vector processor systems. The model captures the effects of both bank conflicts among elements within one vector access stream and conflicts among multiple vector access streams on system performance. The model is based on a closed queueing network assuming an ideal interconnection network. An approximation technique is proposed to solve the memory queueing system that serves customers in a complicated way (non-FIFO). We also carry out extensive simulation experiments to study memory interference and validate our analytical model. Simulation results and analytical results are in a very good agreement, indicating that the model is very accurate. We further validate our analysis by comparing the numerical results obtained from our analytical model with those measurement results that were published by other researchers. Based on our analytical model and simulations, we carry out performance evaluation of the multiple vector processor systems. Our numerical results show that memory access conflicts pose a severe limitation on the number of useful processors in the system, implying that memory system design is essential to high-performance computing     

A new computational algorithm for retrial queues to cellular mobile systems with guard channels

Retrial queueing models have been applied to evaluate the impact of new and handover calls on the call control mechanisms of cellular mobile networks. The fact that the retrial rate depends on the number of retried calls waiting in the system leads to an analytically intractable model. Therefore, approximation procedures should be used to compute the performability of the system. However, there exists a numerical problem concerning a recursive algorithm to compute the stationary probabilities for retrial queues modeling guard channels. Namely, the consideration of guard channels leads to calculations with negative terms in the recursive algorithm. Negative terms and extremely small values involved in the computation are the main causes for a numerical instability in the recursive algorithm.     

Multi-server retrial model with variable number of active servers

This paper deals with a multi-server retrial queueing model in which the number of active servers depends on the number of customers in the system. To this end, the servers are switched on and off according to a multi-threshold strategy. For a fixed choice of the threshold levels, the stationary distribution and various performance measures of the system are calculated. In the case of equidistant connection levels, the optimum threshold level is numerically computed.     

Fuzzy service control of queueing systems

We consider queueing systems in which the service rate is the controlled variable. The cost depends on the queue length and selected rate. The objective is to choose the service rate dynamically, based on the state of the system so as to minimize the average cost over an infinite horizon. Six classes, either known in the literature or new, are studied in detail: queueing systems with vacations, single-server queueing systems with and without switching costs, and tandem queueing systems with and without service costs. A novel approach is presented here using fuzzy control to solve these problems. Simulation shows that the approach is efficient and promising, especially in cases where analytical solutions do not exist.     

The BMAP/PH/N retrial queueing system operating in Markovian random environment

We consider the BMAP/PH/N retrial queueing system operating in a finite state space Markovian random environment. The stationary distribution of the system states is computed. The main performance measures of the system are derived. Presented numerical examples illustrate a poor quality of the approximation of the main performance measures of the system by means of the simpler queueing models. An effect of smoothing the traffic and an impact of intensity of retrials are shown.     

Retrial Queueing System <Emphasis Type="Italic">M</Emphasis> / <Emphasis Type="Italic">M</Emphasis> / 1 / 0 with Combined Service Discipline

The paper considers retrial queueing system M /M /1/ 0 with combined service discipline, namely, a customer from the orbit is serviced in its turn, but in case of a free channel an arrival from the original flow is serviced immediately. The author obtains the expressions for state probabilities as well as ergodicity conditions. The system is compared with the Lakatos-type system.     

Unstable Random Access Networks under a Static Conflict Warning Protocol

A model for a random access network controlled by a static conflict warning protocol is designed as a single-server queueing system with a retrial buffer. The domain of stable operation of the system, whose state probability distribution remains constant in a large time interval, is determined under asymptotic conditions. The mean stable operation time of the system is determined and shown to be large, almost infinite, for certain traffic intensities.     

The multi-server retrial system with Bernoulli feedback and starting failures

In this paper, we present a detailed analysis of a multi-server retrial queue with Bernoulli feedback, where the servers are subject to starting failures. Upon completion of a service, a customer would decide either to leave the system with probability p or to join the retrial orbit again for another service with complementary probability 1?p. We analyse this queueing system as a quasi-birth–death process. Specifically, the equilibrium condition of the system is given for the existence of the steady-state analysis. Applying the matrix-geometric method, the formulae for computing the rate matrix and stationary probabilities are obtained. We further develop the matrix-form expressions for various system performance measures. A cost model is constructed to determine the optimal number of servers, the optimal mean service rate and the optimal mean repair rate subject to the stability condition. Finally, we give a practical example to illustrate the potential applicability of this model.     

Queueing Theory for Semiconductor Manufacturing Systems: A Survey and Open Problems

This paper surveys applications of queueing theory for semiconductor manufacturing systems (SMSs). Due to sophisticated tool specifications and process flows in semiconductor manufacturing, queueing models can be very complicated. Research efforts have been on the improvement of model assumptions and model input, mainly in the first moment (averages) and the second moment (variations). However, practices show that implementation of classical queueing theory in semiconductor industry has been unsatisfactory. In this paper, open problems on queueing modeling of SMS are discussed. A potential solution is also proposed by relaxing the independent assumptions in the classical queueing theory. Cycle time reduction has constantly been a key focus of semiconductor manufacturing. Compared with simulation, queueing theory-based analytical modeling is much faster in estimating manufacturing system performance and providing more insights for performance improvement. Therefore, queueing modeling attracts generous semiconductor research grants. Unfortunately, existing queueing models focus on simple extensions of the classical queueing theory and fail to question its applicability to the complicated SMS. Hence, related researches have not been employed widely in the semiconductor industry. In this paper, we conduct a survey on the important works and also present some open problems. We also propose a novel solution by relaxing a key assumption in the classical queueing theory. We are currently funded by Intel to explore this potential solution, and we hope it can foster an interesting research field for the years to come.     

Tool supported reliability analysis of finite-source retrial queues

This paper deals with the role of performance modeling tools. It introduces 4 major tool developer centers and shows how a given tool can be applied to reliability investigations of finite-source retrial queueing system in steady state. Some numerical examples are given demonstrating the effect of failure and repair rates of server on the mean response time of sources.     

A finite waiting room queueing model with multiple servers having Markovian interruptions and its application to computer communications

A queueing model with finite waiting room, Poisson arrivals, multiple synchronous outputs and the number of active servers varying through the birth and death process is studied. Variation of active servers is obtained by providing a switch in each server. The switches are controlled through the birth and death process having finite population. The relationships between overflow probabilities, buffer size and expected queueing delay due to buffering are obtained. An efficient algorithm for computation of steady state probabilities is developed. Two digital voice-data systems; Data multiplexing in speech with synchronous Time Division Multiplexing (STDM) and Time Assignment Speech Interpolation (TASI) systems are considered for the application of the queueing model studied. The results of this study are portrayed on graphs and may be used as guidelines for designing a buffer in digital systems. The two voice-data systems using the developed model are simulated on the EC-1030 computer to check the validity of the analytical results. Although this problem arose in the study of data interpolation in STDM and TASI systems, the queueing model developed is quite general and may be useful for other industrial applications.     

Exploring gate-limited analytical models for high-performance network storage servers

Gate-limited service is a type of service discipline found in queueing theory and can be used to describe a number of operational environments, for example, large transport systems such as buses, trains or taxis, etc. Recently, there has been the observation that such systems can also be used to describe interactive Internet Services which use a Client/Server interaction. In addition, new services of this genre are being developed for the local area. One such service is a Network Memory Server (NMS) being developed here at Middlesex University. Though there are several examples of real systems that can be modelled using gate-limited service, it is fair to say that the analytical models which have been developed for gate-limited systems have been difficult to use, requiring many iterations before practical results can be generated. In this paper, a detailed gate-limited bulk service queueing model based on Markov chains is explored and a numerical solution is demonstrated for simple scenarios. Quantitative results are presented and compared with a mathematical simulation. The analysis is used to develop an algorithm based on the concept of optimum operational points. The algorithm is then employed to build a high-performance server which is capable of balancing the need to prefetch for streaming applications while promptly satisfying demand misses. The algorithm is further tested using a systems simulation and then incorporated into an Experimental File System (EFS) which showed that the algorithm can be used in a real networking environment.     

M/M/1 retrial queue with working vacations

In this paper we introduce the new M/M/1 retrial queue with working vacations which is motivated by the performance analysis of a Media Access Control function in wireless systems. We give a condition for the stability of the model, which has an important impact on setting the retrial rate for such systems. We derive the closed form solution in equilibrium for the retrial M/M/1 queue with working vacations, and we also show that the conditional stochastic decomposition holds for this model as well.     

On fluid queueing systems with strict priority

We consider priority fluid queueing systems where high-priority class has strict priority access to service. Sample path analysis tools, such as Poisson counter driven stochastic differential equation, are employed to study system queueing behavior in steady state. We are able to obtain various analytical results for different fluid traffic models and system configurations. Those results can be used as a general rule of thumb in buffer dimensioning and other traffic engineering issues.     

On the number of customers served in the M/G/1 retrial queue: first moments and maximum entropy approach

In this paper we present general results on the number of customers, I, served during the busy period in an M/G/1 retrial system. Its analysis in terms of Laplace transforms has been previously discussed in the literature. However, this solution presents important limitations in practice; in particular, the moments of I cannot be obtained by direct differentiation. We propose a direct method of computation for the second moment of I and also for the probability of k,k⩽4, customers being served in a busy period. Then, the maximum entropy principle approach is used to estimate the true distribution of I according to the available information.Scope and purposeWe consider an M/G/1 queue with retrials. Retrial queueing systems are characterized by the fact that, an arriving customer who finds the server busy is obliged to leave the service area and return later to repeat his request after some random time. We deal with I, the number of customers served during the busy period of a retrial queue, and obtain closed expressions for its main characteristics, which will be employed in order to estimate the true distribution of this random variable.