Waiting times for M/G/1 queues with service-time or delay-dependent server vacations

This article shows how to determine the stationary distribution of the virtual wait in M/G/1 queues with either one-at-a-time or exhaustive server vacations, depending on either service times or accrued workload. For the first type of dependence, each vacation time is a function of the immediately preceding service time or of whether the server finds the system empty after returning from vacation. In this way, it is possible to model situations such as long service times followed by short vacations, and vice versa. For the second type of dependence, the vacation time assigned to an arrival to follow its service is a function of the level of virtual wait reached. By this device, we can model situations in which vacations may be shortened whenever virtual delays have gotten excessive. The method of analysis employs level-crossing theory, and examples are given for various cases of service and vacation-time distributions. A closing discussion relates the new model class to standard M/G/1 queues where the service time is a sum of variables having complex dependencies. © 1992 John Wiley & Sons, Inc.     

The reporting time problem: Integration of intelligence with verification

This article is a theoretic study of the following problem in verification: Mobile units under control of an agent, whom we call the HIDER, travel on a known transportation network and must at the conclusion of their itinerary report locations at fixed time intervals to a monitoring authority, whom we call the SEEKER. The purpose of this reporting requirement is to verify that illegal units do not infiltrate the network from sources under the control of the HIDER. We assume that the SEEKER has an independent intelligence-gathering capability which gives sightings of both legal and illegal units. The purpose of this article is to quantify the advantage of frequent over infrequent reporting. © 1992 John Wiley & Sons, Inc.     

Evaluation of an integrated supplier-buyer quality-control system

In this article, a quality-control design framework that employs information for the supplier-buyer system is modeled. Significant operational savings may be obtained by using the integrated plans developed under this framework. This is especially true when the cost of a defective is high, and the variable sampling and rework costs are low. Analysis of the interaction of defective, rework, and variable sampling costs reveals that the savings are the result of a shift of control effort from the process-control to the lot-acceptance stage, which is the consequence of tradeoffs involving both stages. The managerial impact of adopting integrated plans is discussed.     

Distribution of the workload in multiclass queueing systems with server vacations

The steady-state workload at an arbitrary time is considered for several single-server queueing systems with nonpreemptive services for multiple classes of customers (arriving according to Poisson processes) and server vacation (switchover) times. The distribution of the workload at an arbitrary point during the vacation period is obtained for systems with setup times, and for polling systems with exhaustive, gated, or globally gated service disciplines. From the stochastic decomposition property, this workload is added to the workload in the corresponding M/G/1 system without vacations to give the workload at an arbitrary time in vacation systems. Dependence of the workload distribution on the vacation parameters is studied.     

Bayesian inference of survival curve from record-breaking observations: Estimation and asymptotic results

In a variety of industrial situations experimental outcomes are only record-breaking observations. The data available may be represented as X₁, K_1., X₂, K₂,…, where X₁, X₂,… are the successive minima and K₁, K₂, … are the number of trials needed to obtain new records. Samaniego and Whitaker [11, 12] discussed the problem of estimating the survival function in both parametric and nonparametric setups when the data consisted of record-breaking observations. In this article we derive nonparametric Bayes and empirical Bayes estimators of the survival function for such data under a Dirichlet process prior and squared error loss. Furthermore, under the assumptions that the process of observing random records can be replicated, the weak convergence of the Bayes estimator is studied as the number of replications grows large. The calculations involved are illustrated by adopting Proschan's [9] data on successive failure times of air conditioning units on Boeing aircraft, for our purpose. The nonparametric maximum likelihood estimators of the survival function for different choices of the prior are displayed for comparison purposes.     

Optimal clustering: A model and method

Classifying items into distinct groupings is fundamental in scientific inquiry. The objective of cluster analysis is to assign n objects to up to K mutually exclusive groups while minimizing some measure of dissimilarity among the items. Few mathematical programming approaches have been applied to these problems. Most clustering methods to date only consider lowering the amount of interaction between each observation and the group mean or median. Clustering used in information systems development to determine groupings of modules requires a model that will account for the total group interaction. We formulate a mixed-integer programming model for optimal clustering based upon scaled distance measures to account for this total group interaction. We discuss an efficient, implicit enumeration algorithm along with some implementation issues, a method for computing tight bounds for each node in the solution tree, and a small example. A computational example problem, taken from the computer-assisted process organization (CAPO) literature, is presented. Detailed computational results indicate that the method is effective for solving this type of cluster analysis problem.     

Parallel machine replacement

We consider the parallel replacement problem in which there are both fixed and variable costs associated with replacing machines. Increasing maintenance costs motivate replacements, and the fixed replacement cost provides incentive for replacing machines of different ages together in “clusters.” We prove two intuitive results for this problem. First, it is never optimal to split a cluster of like-aged machines, and second, it is never optimal to replace newer clusters before older clusters. By incorporating these two results into an algorithmic approach, we vastly reduce the amount of computation required to identify an optimal replacement policy.     

Estimating travel distances by the weighted lp norm

Empirical distance functions are used to estimate actual travel distances in a transportation network, to verify the accuracy of road mileage data, and to formulate continuous location models. In this article we consider the problem of fitting the weighted l_p norm to a given network. Mathematical properties are derived for two fitting criteria found in the literature. These properties are used to develop an accurate and efficient methodology to solve for the best-fitting parameter values. The directional bias of the l_p norm is analyzed for its effect on the range of search for the optimal p value. Concepts and methodology are applied to a case study of the road system in Southern Ontario. In conclusion, a general framework for other types of distance functions is briefly discussed.     

Economic lot scheduling of fully loaded processes with external setups

Without restricting the class of permissible schedules, we derive optimal schedules for economic lot scheduling problems that are fully loaded, have external setups, and have only two products. The fully loaded condition accurately represents certain types of bottlenecks. We show that the optimal schedule must have the Wagner-Whitin property. We also develop a measure of aggregate inventory, derive an optimal steady-state aggregate inventory policy, and provide conditions under which the aggregate inventory level of an optimal schedule must approach a steady state. By restricting the class of permissible schedules to rotation cycle schedules, we extend these results to more than two products.     

Estimating a survival curve when new is better than used in expectation

Let X be a positive random variable. The distribution F of X is said to be “new better than used in expectation,” or “NBUE,” if E(X) ⩾ E(X – t|X > t) for all t ⩾ 0. Suppose X₁, …, X_n, is a random sample from an NBUE distribution F. The problem of estimating F by a distribution which is itself NBUE is considered. The estimator G_n, defined as the NBUE distribution supported on the sample which minimizes the (sup norm) distance between the NBUE class and the empirical distribution function, is studied. The strong uniform consistency of G_n, is proven, and a numerical algorithm for obtaining G_n, is given. Our approach is applied to provide an estimate of the distribution of lifetime following the diagnosis of chronic granulocytic leukemia based on data from a National Cancer Institute study.