Paul J. Schweitzer

Problem Decomposition and Data Reorganization by a Clustering Technique

A new cluster-analysis method, the bond energy algorithm, has been developed recently; it operates upon a raw input object-object or object-attribute data array by permuting its rows and columns in order to find informative variable groups and their interrelations. This paper describes the algorithm and illustrates by several examples its use for both problem decomposition and data reorganization.

The Laurent expansion of pencils that are singular at the origin

Abstract This paper gives two new proofs of a theorem of Langenhop on the Laurent expansion of a matrix pencil that is singular at the origin. The second, based on a decomposition of Van Dooren, leads to a computational algorithm.

Iterative Aggregation-Disaggregation Procedures for Discounted Semi-Markov Reward Processes

The equation v = q + Mv, where M is a matrix with nonnegative elements and spectral radius less than one, arises in Markovian decision processes and input-output models. In this paper, we solve the equation using an iterative aggregation-disaggregation procedure that alternates between solving an aggregated problem and disaggregating the variables, one block at a time, in terms of the aggregate variables of the other blocks. The disaggregated variables are then used to guide the choice of weights in the subsequent aggregation. Computational experiments on randomly generated and inventory problems indicate that this algorithm is significantly faster than successive approximations when the spectral radius of M is near one, and is slower in unstructured problems with spectral radii in the neighborhood of 0.8. The algorithm appears promising for large structured problems, where it can often reduce computational time and main memory storage requirements and offer greater robustness to initial values.

Perturbation Theory and Undiscounted Markov Renewal Programming

A recently-developed perturbation formalism for finite Markov chains is used here to analyze the policy iteration algorithm for undiscounted, single-chain Markov renewal programming. The relative values are shown to be essentially partial derivatives of the gain rate with respect to the transition probabilities, and they rank the states by indicating desirable changes in the probabilistic structure. This both implies the optimality of nonrandomized policies and suggests a gradient technique for optimizing the gain rate with respect to a parameter. The policy iteration algorithm is shown to be a steepest-ascent technique in policy space: the successor to a given policy is chosen in a direction that maximizes the directional derivative of the gain rate. The occurrence during policy improvement of the gain and relative values of the original policy is explained by their essentially determining the gradient of the gain rate.

A Survey of Bottleneck Analysis in Closed Networks of Queues

Several of the principal results in bottleneck analysis for closed queueing networks are surveyed. Both product-form closed queueing networks, where exact bottleneck analysis is possible, and non-product-form closed queueing networks, where approximations are given for asymptotic bottleneck behavior, are considered. Algorithms for the asymptotic bottleneck analysis are presented and the switching surfaces of bottlenecks are described.

A unifying framework for the approximate solution of closed multiclass queuing networks

Queuing network models of modern computing systems must consider a large number of components (e.g., Web servers, DB servers, application servers, firewall, routers, networks) and hundreds of customers with very different resource requirements. The complexity of such models makes the application of exact solution techniques prohibitively expensive, motivating research on approximate methods. This paper proposes an interpolation-matching framework that allows a unified view of approximate solution techniques for closed product-form queuing networks. Depending upon the interpolating functional form and the matching populations selected, a large versatile family of new approximations can be generated. It is shown that all the known approximation strategies, including Linearizer, are instances of the interpolation-matching framework. Furthermore, a new approximation technique, based on a third-order polynomial, is obtained using the interpolation-matching framework. The new technique is shown to be more accurate than other known methods.

Technical Note—Mathematical Pitfalls in the One Machine Multiproduct Economic Lot Scheduling Problem

We consider a group of products sharing the same piece of production equipment, and show that two mathematical problem formulations in the literature are ill-posed. In both cases, the infimum of the cost function occurs at a boundary point that is not feasible.

Multiple Policy Improvements in Undiscounted Markov Renewal Programming

This paper examines, for undiscounted unichain Markov renewal programming, both the Hastings policy-value iteration algorithm and the case of multiple policy improvements between each policy evaluation. The modified policy improvement procedure proposed by Hastings either increases the gain rate or maintains it, and has a larger value improvement in some transient state than in all recurrent states. This prevents cycling and ensures convergence of the policy-value iteration algorithm. Multiple policy improvements, using either the unmodified or modified policy-improvement procedure, are shown to settle ultimately upon higher-gain policies, if any exist. The iterated policy improvements, each time using the improved values, also lead to upper and lower bounds on the maximal gain rate.

INITIAL PROVISIONING WITH SPARE DETERIORATION

Initial provisioning is analyzed for the case where parts in use and spares have exponential failure distributions with different failure rates. Expressions are given for system reliability and its asymptotic expansions for small time, large time, large numbers of spares, and small spare failure rate. The incremental reliability associated with each additional spare part is analyzed. Graphs are presented that yield, for arbitrary failure rates of spares and parts in use, the minimum number of spare parts needed to achieve system reliabilities of 90, 95, and 99 per cent.

Iterative bounds on the equilibrium distribution of a finite Markov chain

This article presents a new iterative method for computing the equilibrium distribution of a finite Markov chain, which has the significant advantage of providing good upper and lower bounds for the equilibrium probabilities. The method approximates the expected number of visits to each state between two successive visits to a given reference state. Numerical examples indicate that the performance of this method is quite good.