On the perturbation analysis of discrete-event dynamic systems

The paper describes a new approach to the analysis and optimization of discrete-event dynamic systems, such as queueing networks.Dedicated to G. LeitmannThe work reported in this paper was supported in part by ONR Contracts Nos. N00014-75-C-0648 and N00014-79-C-0776 and in part by NSF Grant No. NSF-ECS-82-13680.     

Smoothed perturbation analysis for queues with finite buffers

Applying the technique of smoothed perturbation analysis (SPA) to theGI/G/1/K queue, we derive gradient estimators for two performance measures: the mean steady-state system time of a served customer and the probability that an arriving customer is rejected. Unbiasedness of the estimators follows from results of a previous general framework on SPA estimators. However, in that framework, the estimators often require the simulation of numerous additional sample subpaths, possibly making the technique practically infeasible in applications. We exploit some of the special structure of theGI/G/1/K queue to come up with an estimator which requires at most the simulation of a single additional sample subpath. By establishing certain regenerative properties, we provide a strong consistency proof for the estimator.     

Maximal coupling and rare perturbation sensitivity analysis

In this article, we introduce a new method for obtaining the ersatz sample derivatives useful in sensitivity analysis: the maximal coupling RPA method.     

Occupation times for the finite buffer fluid queue with phase-type ON-times

In this short communication we study a fluid queue with a finite buffer. The performance measure we are interested in is the occupation time over a finite time period, i.e., the fraction of time the workload process is below some fixed target level. Using an alternating renewal sequence, we determine the double transform of the occupation time; the occupation time for the finite buffer M/G/1 queue with phase-type jumps follows as a limiting case.     

Optimal regulation of the service rate for a queue with finite waiting space

The optimal control for the service rate of a single-server queue with limited waiting space is derived. Costs are associated with providing the service and with waiting. A comparison with the traditional steady-state result is made.     

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.     

Discrete-time analysis of a classified multi-server queue with burst arrivals and a shared buffer

We study a discrete-time, classified multi-server queue with a shared buffer. There arem servers and each server belongs to one ofk classes (mk), so thatk kinds of jobs can be served in the system. We characterize a bursty arrival process using bursts which consist of the same kind of jobs. Once the first job of a burst arrives at the queue, the successive jobs will arrive on every time slot until the last job of the burst arrives. The numbers of jobs of a burst and the inter-arrival times of bursts are assumed to be i.i.d., respectively, and the service time is assumed to be equal to one slot. We propose an efficient numerical method to exactly obtain the job loss probability, the waiting time distribution and the mean queue length. We apply this model to the ATM switch with a shared buffer and obtain the performance measures. Numerical results show the advantage of the ATM switch with a shared buffer compared to the one with output buffers.     

Consistency of infinitesimal perturbation analysis estimators with rates

We study a class of infinitesimal perturbation analysis (IPA) algorithms for queueing systems with load-dependent service and/or arrival rates. Such IPA algorithms were originally motivated by applications to large queueing systems in conjunction with aggregation algorithms. We prove strong consistency of these estimators through a type of birth and death queue. This work was supported in part by the NSF under Grants Nos. ECS85-15449 and CDR-8803012, by ONR under Contracts Nos. N00014-89-J-0075 and N00014-90-K-1093, and by the US Army under Contract No. DAAL-03-83-K-0171. This paper was written while the author was with the Division of Applied Sciences at Harvard University.     

Matrix analytic methods for a multi-server retrial queue with buffer

We present numerical methods for obtaining the stationary distribution of states for multi-server retrial queues with Markovian arrival process, phase type service time distribution with two states and finite buffer; and moments of the waiting time. The methods are direct extensions of the ones for the single server retrial queues earlier developed by the authors. The queue is modelled as a level dependent Markov process and the generator for the process is approximated with one which is spacially homogeneous above some levelN. The levelN is chosen such that the probability associated with the homogeneous part of the approximated system is bounded by a small tolerance and the generator is eventually truncated above that level. Solutions are obtained by efficient application of block Gaussian elimination.     

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.     

Multi-state throughput analysis of a two-stage manufacturing system with parallel unreliable machines and a finite buffer

This paper models and analyzes the throughput of a two-stage manufacturing system with multiple independent unreliable machines at each stage and one finite-sized buffer between the stages. The machines follow exponential operation, failure, and repair processes. Most of the literature uses binary random variables to model unreliable machines in transfer lines and other production lines. This paper first illustrates the importance of using more than two states to model parallel unreliable machines because of their independent and asynchronous operations in the parallel system. The system balance equations are then formulated based on a set of new notations of vector manipulations, and are transformed into a matrix form fitting the properties of the Quasi-Birth–Death (QBD) process. The Matrix-Analytic (MA) method for solving the generic QBD processes is used to calculate the system state probability and throughput. Numerical cases demonstrate that solution method is fast and accurate in analyzing parallel manufacturing systems, and thus prove the applicability of the new model and the effectiveness of the MA-based method. Such multi-state models and their solution techniques can be used as a building block for analyzing larger, more complex manufacturing systems.     

A nonstandard finite difference method for solving a mathematical model of HIV-TB co-infection

We design and analyze an unconditionally convergent nonstandard finite-difference method to study transmission dynamics of a mathematical model of HIV-TB co-infection. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves positivity of the solution which is one of the essential requirements when modelling epidemic diseases. Furthermore, we show that the numerical method is unconditionally stable. Competitive numerical results confirming theoretical investigations are provided. Comparisons are also made with other conventional approaches that are routinely used to solve these types of problems.     

Optimal (d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs

This paper considers a finite buffer M/M/c queueing system in which servers are unreliable and follow a (d, c) vacation policy. With such a policy, at a service completion instant, if the number of customers is reduced to c − d (c > d), the d idle servers together take a vacation (or leave for a random amount of time doing other secondary job). When these d servers return from a vacation and if still no more than c − d customers are in the system, they will leave for another vacation and so on, until they find at least c − d + 1 customers are in the system at a vacation completion instant, and then they return to serve the queue. This study is motivated by the fact that some practical production and inventory systems or call centers can be modeled as this finite-buffer Markovian queue with unreliable servers and (d, c) vacation policy. Using the Markovian process model, we obtain the stationary distribution of the number of customers in the system numerically. Some cost relationships among several related systems are used to develop a finite search algorithm for the optimal policy (d, c) which maximizes the long-term average profit. Numerical results are presented to illustrate the usefulness of such a algorithm for examining the effects of system parameters on the optimal policy and its associated average profit.     

Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier

Based on the matrix-analytic approach to fluid flows initiated by Ramaswami, we develop an efficient time dependent analysis for a general Markov modulated fluid flow model with a finite buffer and an arbitrary initial fluid level at time 0. We also apply this to an insurance risk model with a dividend barrier and a general Markovian arrival process of claims with possible dependencies in successive inter-claim intervals and in claim sizes. We demonstrate the implementability and accuracy of our algorithms through a set of numerical examples that could also serve as test cases for comparing other solution approaches.      

Numerical analysis of a discrete-time finite capacity queue with a burst arrival

We study a discrete-time, multi-server, finite capacity queue with a burst arrival. Once the first job of a burst arrives at the queue, the successive jobs will arrive every time slot until the last job of the burst arrives. The number of jobs and the inter-arrival time of bursts are assumed to be generally distributed, and the service time is assumed to be equal to one slot. We propose an efficient numerical method to exactly obtain the job loss probability, the waiting time distribution and the mean queue length using an embedded Markov chain at the arrival instants of bursts.     

Convergence of a stochastic approximation algorithm for the GI/G/1 queue using infinitesimal perturbation analysis

Discrete-event systems to which the technique of infinitesimal perturbation analysis (IPA) is applicable are natural candidates for optimization via a Robbins-Monro type stochastic approximation algorithm. We establish a simple framework for single-run optimization of systems with regenerative structure. The main idea is to convert the original problem into one in which unbiased estimators can be derived from strongly consistent IPA gradient estimators. Standard stochastic approximation results can then be applied. In particular, we consider the GI/G/1 queue, for which IPA gives strongly consistent estimators for the derivative of the mean system time. Convergence (w.p.1) proofs for the problem of minimizing the mean system time with respect to a scalar service time parameter are presented.     

A Cl closing lemma for equivariant endomorphisms under actions of finite groups

A C^l closing lemma is proved for equivariant endomorphisms under actions of finite groups. Our result shows that, for such an endomorphism, a nonwandering orbit together with its symmetric conjugates can be closed up under a Cl-small equivariant perturbation Project partially supported by the National Natural Science Foundation of China (Grant No. 19531070).     

对一类等待空间有限的抢占优先权排队的分析

讨论M/M/1抢占优先权排队模型, 且假设低优先权顾客的等待空间有限. 该模型可以用有限位相拟生灭过程来描述. 由矩阵解析方法, 对该拟生灭过程进行了分析, 并得到排队模型平稳队长的计算公式, 最后还用数值 结果说明了方法的有效性.     

The qualitative analysis of a dynamical system of differential equations arising from the study of multilayer scales on pure metals II

We provide a qualitative analysis of the -dimensional dynamical system:

where is an arbitrary positive integer. Under mild algebraic conditions on the constant matrix , we show that every solution , , extends to a solution on , such that , for . Moreover, the difference between any two solutions approaches as . We then use this result to give a complete qualitative analysis of a 3-dimensional dynamical system introduced by F. Gesmundo and F. Viani in modeling parabolic growth of three-oxide scales on pure metals.