Showing papers in "Discrete Event Dynamic Systems in 1992"

Ordinal Optimization of DEDS

Abstract: In this paper we argue thatordinal rather thancardinal optimization, i.e., concentrating on finding good, better, or best designs rather than on estimating accurately the performance value of these designs, offers a new, efficient, and complementary approach to the performance optimization of systems. Some experimental and analytical evidence is offered to substantiate this claim. The main purpose of the paper is to call attention to a novel and promising approach to system optimization.

The Brownian approximation for rate-control throttles and the G/G/1/C queue

Abstract: This paper studies approximations to describe the performance of a rate-control throttle based on a token bank, which is closely related to the standard G/G/1/C queue and the two-node cyclic network of ·/G/1/∞ queues. Several different approximations for the throttle are considered, but most attention is given to a Brownian or diffusion approximation. The Brownian approximation is supported by a heavy-traffic limit theorem (as the traffic intensity approaches the upper limit for stability) for which an upper bound on the rate of convergence is established. Means and squared coefficients of variation associated with renewal-process approximations for the overflow processes are also obtained from the Brownian approximation. The accuracy of the Brownian approximation is investigated by making numerical comparisons with exact values. The relatively simple Brownian approximation for the job overflow rate is not very accurate for small overflow rates, but it nevertheless provides important insights into the way the throttle design parameters should depend on the arrival-process characteristics in order to achieve a specified overflow rate. This simple approximation also provides estimates of the sensitivity of the overflow rates to the model parameters.

Augmented infinitesimal perturbation analysis: An alternate explanation

Abstract: Augmented infinitesimal perturbation analysis (APA) was introduced by Gaivoronski [1991] to increase the purview of the theory of Infinitesimal Perturbation Analysis (IPA). In reference [Gaivoronski 1991] it is shown that an unbiased estimate for the gradient of a class of performance measures of DEDS represented bygeneralized semi-Markov processes (GSMPs) (cf. [Glynn 1989] can be expressed as a sum of an IPA-estimate and a term that takes into account the event order changes. In this paper we present an alternate approach to establishing the result of Gaivoronski, and from this we derive a necessary and sufficient condition for the validity of the IPA algorithm for this class of performance measures. Finally we validate our results by simulation examples.

A graph-theoretic optimal control problem for terminating discrete event processes

Abstract: Most of the results to date in discrete event supervisory control assume a “zero-or-infinity” structure for the cost of controlling a discrete event system, in the sense that it costs nothing to disable controllable events while uncontrollable events cannot be disabled (i.e., their disablement entails infinite cost). In several applications however, a more refined structure of the control cost becomes necessary in order to quantify the tradeoffs between candidate supervisors. In this paper, we formulate and solve a new optimal control problem for a class of discrete event systems. We assume that the system can be modeled as a finite acylic directed graph, i.e., the system process has a finite set of event trajectories and thus is “terminating.” The optimal control problem explicitly considers the cost of control in the objective function. In general terms, this problem involves a tradeoff between the cost of system evolution, which is quantified in terms of a path cost on the event trajectories generated by the system, and the cost of impacting on the external environment, which is quantified as a dynamic cost on control. We also seek a least restrictive solution. An algorithm based on dynamic programming is developed for the solution of this problem. This algorithm is based on a graph-theoretic formulation of the problem. The use of dynamic programming allows for the efficient construction of an “optimal subgraph” (i.e., optimal supervisor) of the given graph (i.e., discrete event system) with respect to the cost structure imposed. We show that this algorithm is of polynomial complexity in the number of vertices of the graph of the system.

Modeling operator/workstation interference in asynchronous automatic assembly systems

Abstract: In several instances ofdiscrete event dynamic systems (DEDS), jobs sometimes require service from two or more resources at the same time. When queueing network models are used to study DEDS, this feature ofsimultaneous resource possession is often ignored because it is difficult for the models to handle. In some DEDS, this feature of a job demanding several resources simultaneously can have a significant effect on system performance, especially if there is a limited amount of one or more of these resources. For example, in an asynchronous automatic assembly system, an assembly at a workstation needs an operator when it experiences a jam (a random phenomenon) in order to clear the jam. Due to the presence of a limited (small) number of operators, an assembly may have to wait for an operator. This waiting orinterference time has a significant effect on the system production rate. This paper develops an analytical approximation method that can be used to determine the steady-state performance of automatic assembly systems for a given assignment of operators. The analytical method involves the simultaneous solution of two “coupled” queueing models; one of the models calculates the waiting time for an operator resource, while the other computes the waiting time for a workstation resource. The solution technique developed can be adapted to study instances of simultaneous resource possession in other DEDS, such as flexible manufacturing systems and computer/communication networks.

Event Rates and Aggregation in Hierarchical Discrete Event Systems

Abstract: A discrete event system (DES) is a dynamical system whose evolution in time develops as the result of the occurrence of physical events at possibly irregular time intervals. Although many DES's operation is asynchronous, others have dynamics which depend on a clock or some other complex timing schedule. Here we provide a formal representation of the advancement of time for logical DES via interpretations of time. We show that the interpretations of time along with a timing structure provide a framework to study principles of the advancement of time for hierarchical DES (HDES). In particular, it is shown that for a wide class of HDES the event rate is higher for DES at the lower levels of the hierarchy than at the higher levels of the hierarchy. Relationships between event rate and event aggregation are shown. We define a measure for event aggregation and show that there exists an inverse relationship between the amount of event aggregation and the event rate at any two successive levels in a class of HDES. Next, we study how to design the timing structure to ensure that there will be a decrease in the event rate (by some constant factor) between any two levels of a wide class of HDES. It is shown that if the communications between the various DES in the HDES satisfy a certain admissibility condition then there will be a decrease in the event rate. These results for HDES constitute the main results of this paper, since they provide the first mathematical characterization of the relationship between event aggregation and event rates of the HDES and show how to design the interconnections in a HDES to achieve event rate reduction. Several examples are provided to illustrate the results.

New performance sensitivity formulae for a class of product-form queueing networks

Abstract: Perturbation analysis (PA) applies a dynamic point of view to the sample paths of stochastic systems; the realization factor, one of the main concepts of PA, measures the final effect of a perturbation on system performance and provides a novel approach in obtaining performance sensitivities. In this paper, we solve analytically the set of equations for realization factors of a two-server cyclic network. We prove an invariance property of the performance sensitivity for Norton's aggregation. Using the results, we derive closed-form formulae for the derivatives of performance measures in a closed queueing network with load-dependent exponential servers. The performance measures have two general forms: customer average and time average. In contrast with the usual approach based on product-form solutions, our results provide additional insights into the performance sensitivity of closed queueing networks and have immediate applications to problems of optimal control. The general formulae are expressed in terms of Buzen's algorithm with a computational complexity comparable to that of the formulae obtained by directly taking the derivatives of the product-form solutions.

Full-state perturbation analysis of discrete event dynamic systems

Abstract: Discrete event dynamic systems are studied within the framework of perturbation analysis in this paper. Perturbation is extended from the event times only to both event times and queue lengths. An approximate technique, full-state perturbation analysis (PA), is developed as an extension of the PA approach. Full-state PA is able to deal with problems involving queue length perturbations which often defy existing PA methods, while it still retains all the advantages of existing PA. Full-state PA is used to calculate the throughput sensitivity to the number of customers in closed queueing networks and the throughput sensitivity to routing change. Numerical examples are given. Experimental results verify the validity and accuracy.

Monotonicity properties of cost functions in queueing networks

Abstract: We establish monotonicity properties of cost functions in queueing networks with blocking, using sample path analysis. For any configuration of queueing stations with general interarrival times and exponentially distributed service times, we show that increasing the initial population and/or the arrival rates increases the expected cost (including blocking penalties for jobs that are rejected) incurred over a finite or an infinite horizon. Routing can be either probabilistic or state dependent. Furthermore, we show that this result also applies to networks with communication-type blocking where no jobs are ever lost. The criticality of the constraint imposed on the service times in demonstrated by a simple counterexample.