Sensitivity analysis and optimization of throughput in a production line with blocking
TL;DR: Using this perturbation analysis estimate of gradient, the Robbins-Monro stochastic procedure can be applied in optimizing the system throughput and the results also hold for systems with general routing but in which no server can block more than one other server simultaneously.
Abstract: Production lines with limited storage capacities can be modeled as cyclic queueing networks with finite buffers and general service times. A new technique, called perturbation analysis of discrete event dynamic systems, is applied to these queueing networks. An estimate of the gradient of the system throughput is obtained by perturbation analysis based on only one sample trajectory of the system. We show that the estimate is strongly consistent. Using this perturbation analysis estimate of gradient, we can apply the Robbins-Monro stochastic procedure in optimizing the system throughput. Compared to the conventional Kiefer-Wolfowitz optimization procedure, this approach saves a large amount of computation. For a real system, the gradient estimate can be obtained without changing any parameters in the system. The results also hold for systems with general routing but in which no server can block more than one other server simultaneously.
Citations
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01 Jan 1989
TL;DR: Perturbation analysis of discrete event dynamic systems (DEDS) enables parameter sensitivities of DEDS to be obtained by observing a single sample path of the system by demonstrating the basic theoretical issues involved.
Abstract: Perturbation analysis (PA) of discrete event dynamic systems (DEDS) enables parameter sensitivities of DEDS to be obtained by observing a single sample path of the system. A simple GI/G/1 system is used to give an introduction to PA and illustrate the basic theoretical issues involved in this technique. The application of PA to networks of queues is covered briefly, and then some of the recent extensions to PA are discussed. It is shown that many interesting open questions remain for PA, and areas for research are indicated. >
223 citations
TL;DR: This work considers the problem of optimizing the steady-state mean of a controlled regenerative process using a stochastic optimization algorithm driven by infinitesimal perturbation analysis (IPA) derivative estimates and derives IPA derivative estimates for the problem and proves almost sure convergence of the algorithm.
Abstract: We consider the problem of optimizing the steady-state mean of a controlled regenerative process using a stochastic optimization algorithm driven by infinitesimal perturbation analysis (IPA) derivative estimates. We derive IPA derivative estimates for our problem and prove almost sure convergence of the algorithm. The generality of our formulation should encompass a wide variety of practical systems. We illustrate our framework and results via several examples. >
63 citations
TL;DR: It is shown that although the CT model conceptually represents a line with continuous fluid, it can be represented as a generalized semi-Markov process (GSMP) which leads to a simple and concise simulation algorithm for a CT model.
Abstract: We extend previous research on the use of models ofcontinuous tandem (CT) lines for performance analysis ofdiscrete tandem (DT) production lines. We formalize the translation of input parameters from the DT line to the CT model, as well as the translation of performance measures (PMs) obtained from the CT model back to the DT line. We show that although the CT model conceptually represents a line with continuous fluid, it can be represented as a generalized semi-Markov process (GSMP). This representation leads to a simple and concise simulation algorithm for a CT model. We investigate the accuracy of the CT model for prediction of PMs in the DT line, and show that, with proper translation of parameters and PMs, the CT model provides reasonable estimates for the DT line PMs. We provide preliminary results on gradient estimation for CT models via infinitestimal perturbation analysis. The aim of the paper is to provide a firm foundation for the future exploration of CT models as a means to parameter optimization for DT lines.
57 citations
TL;DR: In this article, a strong consistency of infinitesimal perturbation analysis (IPA) estimates for a class of stochastic processes satisfying certain convex properties is established.
Abstract: A strong consistency of infinitesimal perturbation analysis (IPA) estimates for a class of stochastic processes satisfying certain convex properties is established. The approach is based on convex analysis of sample paths. Applications of the results to tandem and cyclic queueing systems are discussed. Possible extensions to more general systems are also illustrated. >
43 citations
01 Apr 1994
TL;DR: A continuous-flow model for a serial production system with unreliable machines maintained by a limited number of repairmen, and finite storage between machines is developed, utilizing simulation and analysis.
Abstract: We consider a serial production system with unreliable machines maintained by a limited number of repairmen, and finite storage between machines. Processing times may be random variables with exponential or gamma distributions, or deterministic. We develop a continuous-flow model for such a system utilizing simulation and analysis. Random processing times are approximated by sums of deterministic variables using a simple probabilistic technique. The model observes a limited number of events which are sufficient to determine system performance and mean buffer levels. By appropriately reducing the rates of starved and blocked machines and using analysis to compute the times of next event at each machine and buffer, discrete part computations are avoided. It is demonstrated that this approximate model is highly accurate and faster by a factor of 3 or more when compared to conventional simulators. The paper addresses also optimal repair allocation to maximize the expected throughput of the system. Two different approaches are proposed: perturbation analysis and experimental evaluation of various nonpreemptive rules for assigning a repairman to failed machines. >
39 citations
References
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TL;DR: Many of the network results of Jackson on arrival and service rate dependencies, of Posner and Bernholtz on different classes of customers, and of Chandy on different types of service centers are combined and extended in this paper.
Abstract: We derive the joint equilibrium distribution of queue sizes in a network of queues containing N service centers and R classes of customers. The equilibrium state probabilities have the general form: P(S) - Cd(S) $f_1$($x_1$)$f_2$($x_2$)...$f_N$($x_N$) where S is the state of the system, $x_i$ is the configuration of customers at the ith service center, d(S) is a function of the state of the model, $f_i$ is a function that depends on the type of the ith service center, and C is a normalizing constant. We consider four types of service centers to model central processors, data channels, terminals, and routing delays. The queueing disciplines associated with these service centers include first-come-first-served, processor sharing, no queueing, and last-come-first-served. Each customer belongs to a single class of customers while awaiting or receiving service at a service center but may change classes and service centers according to fixed probabilities at the completion of a service request. For open networks we consider state dependent arrival processes. Closed networks are those with no arrivals. A network may be closed with respect to some classes of customers and open with respect to other classes of customers. At three of the four types of service centers, the service times of customers are governed by probability distributions having rational Laplace transforms, different classes of customers having different distributions. At first-come-first-served type service centers the service time distribution must be identical and exponential for all classes of customers. Many of the network results of Jackson on arrival and service rate dependencies, of Posner and Bernholtz on different classes of customers, and of Chandy on different types of service centers are combined and extended in this paper. The results become special cases of the model presented here. An example shows how different classes of customers can affect models of computer systems. Finally, we show that an equivalent model encompassing all of the results involves only classes of customers with identical exponentially distributed service times. All of the other structure of the first model can be absorbed into the fixed probabilities governing the change of class and change of service center of each class of customers.
2,416 citations
01 Apr 1955
TL;DR: In this paper, the authors generalized Erlang's idea of extending to cover all distributions with rational Laplace transforms; this involves the formal use of complex transition probabilities and properties of the method are considered.
Abstract: The exponential distribution is very important in the theory of stochastic processes with discrete states in continuous time. A. K. Erlang suggested a method of extending to other distributions methods that apply in the first instance only to exponential distributions. His idea is generalized to cover all distributions with rational Laplace transforms; this involves the formal use of complex transition probabilities. Properties of the method are considered.
603 citations
TL;DR: It is found that the distribution of customers in the closed queuing system is regulated by the stage or stages with the slowest effective service rate, which means that closed systems are shown to be stochastically equivalent to open systems in which the number of customers cannot exceed N.
Abstract: The results contained herein pertain to the problem of determining the equilibrium distribution of customers in closed queuing systems composed of M interconnected stages of service. The number of customers, N, in a closed queuing system is fixed since customers pass repeatedly through the M stages with neither entrances nor exits permitted. At the ith stage there are ri parallel exponential servers all of which have the same mean service rate µi. When service is completed at stage i, a customer proceeds directly to stage j with probability pij. Such closed systems are shown to be stochastically equivalent to open systems in which the number of customers cannot exceed N. The equilibrium equations for the joint probability distribution of customers are solved by a separation of variables technique. In the limit of N â ∞ it is found that the distribution of customers in the system is regulated by the stage or stages with the slowest effective service rate. Asymptotic expressions are given for the marginal distributions of customers in such systems. Then, an asymptotic analysis is carried out for systems with a large number of stages i.e., M â« 1 all of which have comparable effective service rates. Approximate expressions are obtained for the marginal probability distributions. The details of the analysis are illustrated by an example.
580 citations
TL;DR: This paper presents an efficient method for evaluating performance measures for a class of tandem queueing systems with finite buffers in which blocking and starvation are important and based on system characteristics such as conservation of flow.
Abstract: This paper presents an efficient method for evaluating performance measures for a class of tandem queueing systems with finite buffers in which blocking and starvation are important. These systems are difficult to evaluate because they have large state spaces and because they cannot be decomposed exactly. The approximate decomposition approach we describe is based on system characteristics such as conservation of flow. Comparisons with exact and simulation results indicate that the approach is very accurate.
517 citations
TL;DR: In this paper, the authors present a complete and novel solution to the well known buffer storage design problem in a serial production line, where the key ingredient of the solution is the efficient calculation of the gradient vector of the throughput with respect to the various buffer sizes.
Abstract: In this paper, we present a complete and novel solution to the well known buffer storage design problem in a serial production line. The key ingredient of the solution is the efficient calculation of the gradient vector of the throughput with respect to the various buffer sizes. Analytical and experimental results are presented.
295 citations