Martin A. Wortman

A maintenance strategy for systems subjected to deterioration governed by random shocks

The authors examine the time-stationary availability of maintained systems that deteriorate according to a random-shock process. System failures are not self-announcing; hence, failures must be detected via inspection. The approach considers randomly occurring shocks that cumulatively damage the system; shock magnitudes are taken as random. The authors develop an expression for computing system availability when inspections follow a renewal process. This expression leads to a proved proposition showing that, for any specified mean inspection rate, system availability is maximized by choosing deterministic inter-inspection times. >

An approach for computing tight numerical bounds on renewal functions

This method computes tight lower and upper bounds for the renewal function. It is based on Riemann-Stieltjes integration, and provides bounds for solving certain renewal equations used in the study of availability. An error analysis is given for the numerical bounds when inter-renewal time distributions are sufficiently smooth. Three examples are explored that demonstrate the accuracy of these computed numerical bounds.

ON MAINTAINED SYSTEMS OPERATING IN A RANDOM ENVIRONMENT

This paper examines the availability of a maintained system where the rate of deterioration is governed by an exogenous random environment. We provide a qualitative result that exposes the relationship between remaining lifetime, environment, and repairs. This result leads to simple bounds that can be used to choose inspection rates that guarantee a specified level of availability. The principal result requires no specific distributional assumptions, is intuitively appealing and can be directly applied by practitioners. Our development employs techniques from stochastic calculus.

The M/GI/ 1 Bernoulli feedback queue with vacations

Feedback may be introduced as a mechanism for scheduling customer service (for example in systems in which customers bring work that is divided into a random number of stages). A model is developed that characterizes the queue length distribution as seen following vacations and service stage completions. We demonstrate the relationship that exists between these distributions. The ergodic waiting time distribution is formulated in such a way as to reveal the effects of server vacations when feedback is introduced.

Analysis of asymmetric patrolling repairman systems

Abstract A system of N non-identical machines in a non-symmetric layout that are maintained by a single repairman is referred to as asymmetric patrolling repairman system. A probability model for a large class of such systems is developed. The model yields useful computational formulae for evaluating system performance measures that include mean down time and availability of a machine, repairman utilization, etc. The so called class admits exponential machine inter-failure times, generally distributed repairman walk times and machine repair times, and a variety of repair scheduling disciplines including most of those that appear in the literature. Many special cases of patrolling repairman systems that belong to this class can thus be examined under one unified model. This is a new result and it offers a common ground for performance comparisons between various systems.

Job flow control in assembly operations

The authors develop a state variable feedback control for a class of assembly operations. The objective of this control is to increase the number of jobs completed within a time horizon T while keeping the job cycle times within an acceptable range with a prespecified probability. The assembly operations are modeled as stochastic timed-event graphs which in the (max, +) algebra are characterized by a set of linear stochastic difference equations.

An approximation for computing the throughput of closed assembly-type queueing networks

In this paper, we examine throughput (mean number of completed assemblies per unit time) of closed assembly type queueing networks where machine processing times are drawn from general distributions. The system dynamics are characterized via a set of stochastic difference equations; it is shown that the system state can be modeled by a discrete index Markov chain on a continuous state space. Standard Markovian analysis is employed to derive an approximate expression for system throughput, following discretisation of state space. Four examples of CONWIP (CONstant Work IN Process) systems are given that illustrate the results.

Performance ofN machine centers ofK-out-of-M: G type maintained by a single repairman

We consider a system of N (nonsymmetric) machine centers of the K-out-of-M : G type that are maintained by a single repairman. [A machine center functions if and only if at least K of the M machines belonging to the center are good (G).] Such systems are commonly found in various manufacturing and service industries. A stochastic model is developed that accommodates generally distributed repair times and repairman walk times, and most repair scheduling disciplines. K-out-of-M : G type systems also appear as a modeling paradigm in reliability analysis and polling systems performance analysis. Several performance measures are derived for machine-repair systems having K-out-of-M-type centers. A simple example system is developed in detail that exposes the computations involved in modeling applications.

On the time-dependent occupancy distribution of the g/g/1 queuing system

This article offers an approach for studying the time-dependent occupancy distribution for a modest generalization of the GI/G/1 queuing system in which interarrival times and service times, although mutually independent, are not necessarily identically distributed. We develop and explore an analytical model leading to a computational approach that gives tight bounds on the occupancy distribution. Although there is no general closed-form characterization of probability law dynamics for occupancy in the GI/G/1 queue, our results offer what might be termed “near-closed-form” in that accurate plots of the transient occupancy distribution can be constructed with an insignificant computational burden. We believe that our results are unique; we are unaware of any alternative analytical approach leading to a numerical characterization of the time-dependent occupancy distribution for the G/G/1 queuing systems considered here. Our analyses employ a marked point process that converges to the occupancy process at any fixed time t; it is shown that this process forms a Markov chain from which the transient occupancy law is available. We verify our analytical approach via comparison with the well-known closed-form expressions for time-dependent occupancy distribution of the M/M/1 queue. Additionally, we suggest the viability of our approach, as a computational means of obtaining the time-dependent occupancy distribution, through straightforward application to a Gamma[x]/Weibull/1 queuing system having batch arrivals and batch job services.

The backlog process in queues with random service speed

This paper examines the backlog (total amount of unfinished work in the system) in a single server queue that works in a ‘random environment’. Specifically, the service speed is described by an exogenous (non-negative) ‘environment’ process. We characterize the time-dependent backlog via a stochastic integral equation and use this equation to compute stationary performance measures. For the M/G/1 queue, our results lead to a generalization of the Pollaczek-Khintchine transform equation for backlog that ‘explains’ why congestion is greater in a queue with random service speed than in an equivalent queue with constant service speed. We provide a numerical example that helps to illustrate our results.