Menachem Dishon

On the asymptotic behavior of the stochastic and deterministic models of an epidemic

Abstract The deterministic and stochastic versions of a generalization of Baileys simple epidemic model are considered. It is shown that the asymptotic values of the expected number of infected as calculated by the two theories do not necessarily agree. Nevertheless, when the mean time to epidemic extinction is calculated from the stochastic theory, it is shown that when the deterministic mean tends to zero, the mean time to extinction remains bounded, whereas when the deterministic mean tends to a finite limit, the mean time to extinction tends to infinity as the total population number grows without bound.

Some Economic Problems Related to Burn-In Programs

We consider the problems raised by the requirement of N components, some or all of which are subjected to a burn-in program lasting a time T. A model is first analyzed in which N components are initially subjected to burn-in and failures are replaced at the end of the program by components that have not been burnt in. Then generalizations of this model are analyzed. These include one in which M > N components are subjected to bum-in initially and one in which intermediate replacements can be made at times before T.

Burn-In Programs for Repairable Systems

Burn-in programs are often used for automotive or airplane engines in order to eliminate early failures due to ineffective adjustments and similar repairable sources of failure. We assume that there are two operating states: Good and Poor. Each has its own reliability characteristic, and neither necessarily has any statistical property that improves with bum-in. The purpose of the bum-in program is to uncover the Poor engines and then to repair them to the Good state. We calculate the mean time to failure of such engines when a burn-in program is used and derive conditions under which a burn-in program is justified.

Application of a singular perturbation expansion to the solution of certain Fokker-Planck equations

We show that a singular perturbation expansion for the solution of a parabolic equation can be applied to some Fokker-Planck equations arising in the analysis of the effects of noise on laser operations. A generalization to the approximate solution of the Smoluchowski equation, when diffusion is a small effect, is given.

A Model for Burn-In Programs for Components with Eliminateable Defects

We consider a component with a random number of defects which can be discovered and corrected during the course of a burn-in program. The failure rate after burn-in is constant (in time) and depends on the number of remaining defects. A sufficient condition is given for a burn-in program to lead to increased mean time to failure; it is given in terms of the prior distribution of the number of defects and the failure rates.

Resolution in Nonuniform Chromatographic Systems

Abstract The resolving power of a chromatographic column is commonly expressed in terms of the peak positions and standard deviations of the two peaks to be separated. Calculation of the resolution for Gaussian peaks is straightforward. When the column properties vary with position, the peaks are no longer generally Gaussian. We use a recently developed theory to calculate approximations to moments of the peak when diffusion is a relatively small effect, and thereby calculate an expression for the resolving power of the system.

A Communications Satellite Replenishment Policy

This paper considers a model for the replenishment of communications satellites which may fail at random when in orbit. It is assumed that when the satellite level reaches a value N, launchings are made until the level reaches M, at which time launchings are terminated. We calculate the probability that n or fewer satellites are functional under this regime. Although the terminology here refers only to satellite systems, the application of the present theory has a wider range. It can be used to calculate the relevant parameters for any (s, S) inventory system when orders are placed singly and the time between the order and filling of the order has a negative exponential distribution.