An assignment problem for a parallel queueing system with two heterogeneous servers

In this paper, we consider an optimization problem for a parallel queueing system with two heterogeneous servers. Each server has its own queue and customers arrive at each queue according to independent Poisson processes. Each service time is independent and exponentially distributed. When a customer arrives at queue 1, the customers in queue 1 can be transferred to queue 2 by paying an assignment cost which is proportional to the number of moved customers. Holding cost is a function of the pair of queue lengths of the two servers. Our objective is to minimize the expected total discounted cost. We use the dynamic programming approach for this problem. Considering the pair of queue lengths as a state space, we show that the optimal policy has a switch over structure under some conditions on the holding cost.     

Analysis of optimal and nearly optimal sequencing policies for a closed queueing network

We analyze sequencing policies designed to most effectively utilize the resources of a closed queueing network representation of a manufacturing system. A continuous time Markov decision process formulation is used to compare the performance of optimal sequencing policies and a heuristic developed by analyzing a heavy traffic approximation of the system.     

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.     

Queueing systems with different types of server interruptions

We consider a queueing system with disruptive and non-disruptive server interruptions. Both disruptive and non-disruptive interruptions may start when there is a customer in service. The customer repeats its service after a disruptive interruption, and continues its service after a non-disruptive interruption. Using a transform approach, we obtain various performance measures such as the moments of the queue content and waiting times. We illustrate our approach by means of some numerical examples.     

Design of manufacturing systems using queueing models

Design issues in various types of manufacturing systems such as flow lines, automatic transfer lines, job shops, flexible machining systems, flexible assembly systems and multiple cell systems are addressed in this paper. Approaches to resolving these design issues of these systems using queueing models are reviewed. In particular, we show how the structural properties that are recently derived for single and multiple stage queueing systems can be used effectively in the solution of certain design optimization problems.Supported in part by the Natural Sciences and Engineering Research Council of Canada via Operating and Strategic Grants on Modeling and Analyses of Production Systems and Modeling and Implementation of Just-in-Time Cells.Supported in part by the NSF Grants ECS-8811234 and DDM-9113008 and by Sloan Foundation Grants for the Consortium for Competitiveness and Cooperation and for the study on Competitive Semiconductor Manufacturing.     

On the control of queueing systems with failures and repairs

Several classes of problems can be modelled as queueing systems with failures where preventative maintenance has to be optimized: relevant examples are computer systems, databases and production lines. This paper studies a simple case of such a situation based on the M/M/1 queue where the state of the server can take two values (on, off); the transition on–off happens either when there is a failure or when one decides to maintain. The characterization and an approximation of an optimal policy are given. Extensions to more general models, including general Markov processes, are examined.     

Modelling of three-dimensional mechanical systems using point coordinates with a recursive approach

In this paper, the dynamic simulation of constrained mechanical systems that are interconnected of rigid bodies is studied using projection recursive algorithm. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. Closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. For the resulting open-chain system, the equations of motion are generated recursively along the serial chains. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

A class of renewal Interrupted Poisson Processes and applications to queueing systems

Switched Poisson Processes and Interrupted Poisson Processes are often employed to characterize traffic streams in distributed computer and communications systems, especially in investigations of overflow processes in telecommunication networks. With these processes, input streams having inter-segment correlations and high variance as well as state-dependent traffic can properly be modelled. In this paper we first derive an approximation method to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption. As a special case of this class of processes, the class of Interrupted Poisson processes is also included in the investigation. As a result, a generalization of the well-known class of Interrupted Poisson processes is obtained. It is shown that the renewal property is also given for this general class of Interrupted Poisson processes having generally distributed off-phase. To illustrate the accuracy of the presented renewal approximation of Generalized Switched Poisson processes and to show the major properties of the General Interrupted Poisson processes, applications to some basic queueing systems are discussed by means of numerical results.This work was done while the author was with Institute of Communications Switching and Data Technics, University of Stuttgart, Seidenstrasse 36, D-7000 Stuttgart 1, FRG.     

Performance optimization of queueing systems with perturbation realization

After the intensive studies of queueing theory in the past decades, many excellent results in performance analysis have been obtained, and successful examples abound. However, exploring special features of queueing systems directly in performance optimization still seems to be a territory not very well cultivated. Recent progresses of perturbation analysis (PA) and sensitivity-based optimization provide a new perspective of performance optimization of queueing systems. PA utilizes the structural information of queueing systems to efficiently extract the performance sensitivity information from a sample path of system. This paper gives a brief review of PA and performance optimization of queueing systems, focusing on a fundamental concept called perturbation realization factors, which captures the special dynamic feature of a queueing system. With the perturbation realization factors as building blocks, the performance derivative formula and performance difference formula can be obtained. With performance derivatives, gradient-based optimization can be derived, while with performance difference, policy iteration and optimality equations can be derived. These two fundamental formulas provide a foundation for performance optimization of queueing systems from a sensitivity-based point of view. We hope this survey may provide some inspirations on this promising research topic.     

Heavy traffic limit theorems for a sequence of shortest queueing systems

A sequence of shortest queueing systems is considered in this paper. We give weak convergence theorems for the queue length and waiting time processes when the sequence of traffic intensities associated with the sequence of shortest queueing systems approaches the critical value (=1) at appropriate rates.Research supported by the National Natural Science Foundation of China.     

Robustness of perturbation analysis estimators for queueing systems with unknown distributions

Sample-path-based stochastic gradient estimators for performance measures of queueing systems rely on the assumption that a probability distribution of the random vector of interest (e.g., a service or interarrival time sequence) is given. In this paper, we address the issue of dealing with unknown probability distributions and investigate the robustness of such estimators with respect to possibly erroneous distribution choices. We show that infinitesimal perturbation analysis (IPA) can be robust in this sense and, in some cases, provides distribution-independent estimates. Comparisons with other gradient estimators are provided, including experimental results. We also show that finite perturbation analysis (FPA), though only providing gradient approximations, possesses some attractive robustness properties with respect to unknown distribution parameters. An application of FPA estimation is included for a queueing system performance optimization problem involving customers with real-time constraints.This work was supported in part by the National Science Foundation Grant ECS-88-01912 and by the Office of Naval Research Contract N00014-87-K-0304.The authors wish to thank Dr. Jack Holtzman for several useful comments and suggestions.     

Relative priority policies for minimizing the cost of queueing systems with service discrimination

This paper considers several single-server two-class queueing systems with different cost functions. Customers in the two classes are discriminated by service rates and relative priorities. Most attention is focused on the ones with general quadratic bivariable and exponential cost functions that are usually applied in the relatively complicated systems. To the best of the authors’ knowledge, there is no literature analyzing these two kinds of cost functions on the subject of relative priority. We explicitly present the conditions under which relative priority outperforms absolute priority for reducing system cost and further provide the method to find the optimal DPS policy. Moreover, we also discuss variations where service rates of the two classes are decision variables under service equalization and service discrimination disciplines, respectively.     

A recursive method for the F-policy G/M/1/K queueing system with an exponential startup time

This paper deals with the optimal control of a finite capacity G/M/1 queueing system combined the F-policy and an exponential startup time before start allowing customers in the system. The F-policy queueing problem investigates the most common issue of controlling arrival to a queueing system. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distribution of the number of customers in the system. We illustrate a recursive method by presenting three simple examples for exponential, 3-stage Erlang, and deterministic interarrival time distributions, respectively. A cost model is developed to determine the optimal management F-policy at minimum cost. We use an efficient Maple computer program to determine the optimal operating F-policy and some system performance measures. Sensitivity analysis is also studied.     

Asymptotical properties of erlang queueing systems with a dependent Poisson input flow

The purpose of this paper is to obtain stationary characteristics of the mathematical model, which consists of two Erlang queueing systems with a bivariate Poisson input flow of customers. Functional theorems of diffusion approximation type are proved. This paper is supported by INTAS 96-0828.     

Effects of service disciplines inG/GI/s queueing systems

Transient extremal properties of some service disciplines are established in theG/GI/s queueing system for the minimization and maximization of the expectations of the Schur convex functions, convex symmetric functions and the sums of convex functions of the waiting times, response times, lag times and latenesses. When resequencing is required in the system, the FCFS and LCFS disciplines are shown to minimize and maximize, respectively, the expectations of any increasing functions of the end-to-end delays. All of these results are presented in terms of stochastic orderings. The paper concludes with extensions of the results to the stationary regime and to tandem as well as general queueing networks.This work was supported in part by the National Science Foundation under grant ASC 88-8802764.The work of this author was also partially supported by CEC DG-XIII under the ESPRIT-BRA grant QMIPS.     

Optimal control of a removable and non-reliable server in an M/M/1 queueing system with exponential startup time

We study a single removable and non-reliable server in the N policy M/M/1 queueing system. The server begins service only when the number of customers in the system reaches N (N1). After each idle period, the startup times of the server follow the negative exponential distribution. While the server is working, it is subject to breakdowns according to a Poisson process. When the server breaks down, it requires repair at a repair facility, where the repair times follow the negative exponential distribution. The steady-state results are derived and it is shown that the probability that the server is busy is equal to the traffic intensity. Cost model is developed to determine the optimal operating N policy at minimum cost.     

An optimization problem of manufacturing systems with stochastic machine breakdown and rework process

This paper is concerned with optimization of production run time that takes stochastic breakdown and the reworking of defective items into consideration. In a real‐life manufacturing process, production of imperfect quality items as well as random breakdowns of production equipment is inevitable. All defective items produced are assumed to be repairable through a rework process right after the regular production stops in each cycle. This research starts with derivations of the cost functions for production systems with breakdown (no‐resumption policy is considered) and without breakdown taking place, respectively. Then cost functions of both cases are integrated. Theorems on conditional convexity of the overall cost function and bounds for optimal production run time are proposed and proved. This study concludes that although the optimal run time cannot be expressed in a closed form, it falls within the range of bounds. Hence, it can be pinpointed by the use of the bisection method based on the intermediate value theorem. A numerical example is provided to demonstrate its practical usages. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of cyclic queueing networks with parallelism and vacation

The aim of this paper is to improve the machine interference model with vacation to deal with more recent problems of the communication area. To this scope the model is extended to include parallelism in the vacation station. The underlying Markov process is analyzed and a state arrangement is found that yields an efficient matrix-analytic technique that substantially lowers down the time- and space-complexity of standard methods. A numerical example of the method effectiveness is presented, and an example of resource allocation is introduced that finds applications in the QoS management of wireless networks. The author is thankful to the anonymous referees for the improvements their comments have earned to the quality of the presentation and to the completeness of the paper. The author is thankful to Giuseppe Iazeolla, whose careful reading of the original draft of this paper led to significant improvements in its overall quality. This work was partially supported by funds from the FIRB project “Performance Evaluation of Complex Systems: Techniques Methodologies and Tools” and by the University of Roma TorVergata project on High Performance ICT Platforms.     

Non‐recursive solution of sparse block Hessenberg systems

A double‐phase algorithm, based on the block recursive LU decomposition, has been recently proposed to solve block Hessenberg systems with sparsity properties. In the first phase the information related to the factorization of A and required to solve the system, is computed and stored. The solution of the system is then computed in the second phase. In the present paper the algorithm is modified: the two phases are merged into a one‐phase version having the same computational cost and allowing a saving of storage. Moreover, the corresponding non‐recursive version of the new algorithm is presented, which is especially suitable to solve infinite systems where the coefficient matrix dimension is not a priori fixed and a subsequent size enlargement technique is used. A special version of the algorithm, well suited to deal with block Hessenberg matrices having also a block band structure, is presented. Theoretical asymptotic bounds on the computational costs are proved. They indicate that, under suitable sparsity conditions, the proposed algorithms outperform Gaussian elimination. Numerical experiments have been carried out, showing the effectiveness of the algorithms when the size of the system is of practical interest. Copyright © 2004 John Wiley & Sons, Ltd.