Cyrus Derman

On the Consecutive-k-of-n:F System

The linear (circular) k-of-n:F system has n linearly (circularly) ordered components. Each component either functions or fails. The system fails i.f.f.k consecutive components fail. This paper provides, in the i.i.d. case, recursive formulas and bounds for computing system reliability. It considers properties of system life distributions and, in the non i.i.d. case, questions of optimal system design.

Statistical Aspects of Quality Control

Introduction. Elements of Probability. Statistical Inference. Off-Line Quality Control. Shewhart Control Charts. More General Control Charts. Sampling Inspection by Attributes. Sampling Inspection by Variables--A Loss FunctionApproach. Subject Index.

Some contributions to the theory of denumerable Markov chains

? 1 deals with the statistical regularity properties of a denumerable number of particles, all moving about the states of a Markov chain according to the same transition probabilities. ?2 deals with the problem of obtaining a sharper version of a strong limit theorem proved independently by Harris and Levy. Introduction. We shall be concerned throughout with a sequence of random variables { X, I}, n = 0, 1, * , which assume only integer values(2) and which have the property that

On the optimal assignment of servers and a repairman

Abstract : Consider an N server queuing system in which service times of server i are exponentially distributed random variables with rate lambda sub i. Customers arrive in accordance with some arbitrary arrival process. If a customer arrives when all servers are busy, then he is lost to the system; otherwise, he is assigned to one of the free servers according to some policy. Once a customer is assigned to a server he remains in that status until service is completed. We show that the policy that always assigns an arrival to that free server whose service rate is largest (smallest) stochastically minimizes (maximizes) the number in the system. The result is then used to show that in an N component system in which the i superscript th power components up-time is exponential with rate lambda sub i and in which the repair times are exponential with rate mu, the policy of always repairing the failed components whose failure rate lambda is smallest stochastically maximizes the number of working components.

A solution to a set of fundamental equations in Markov chains

If the chain is stationary, uk is the probability that at any given time the chain will be in state k. When the states are null no such set of probabilities exist since the uks would all be zero. However, while no probability can be ascribed to the chain being in a given state at a given time, the question arises as to whether some states are visited more frequently than others and whether relative frequencies considered in some sense have properties analogous to the set { U7} Doeblin [3 ] considered ratios of the type given in (2). He proved the existence of a finite positive limit for all types of chains. We shall prove the following

A Stochastic Sequential Allocation Model

This paper considers the following model, described in terms of an investment problem. We have D units available for investment. During each of N time periods an opportunity to invest will occur with probability p. As soon as an opportunity presents itself, we must decide how much of our available resources to invest. If we invest y, then we obtain an expected profit P(y), where P is a nondecreasing continuous function. The amount y then becomes unavailable for future investment. The problem is to decide how much to invest at each opportunity so as to maximize total expected profit. When P(y) is a concave function, the structure of the optimal policy is obtained (§1). Bounds on the optimal value function and asymptotic results are presented in §2. A closed-form expression for the optimal value to invest is found in §3 for the special cases of P(y) = log y and P(y) = yα, for 0 < α < 1. §4 presents a continuous-time version of the model, i.e., we assume that opportunities occur in accordance with a Poisson ...

Optimal Repair Allocation in a Series System

In this paper we consider a problem on the optimal assignment over time of a single repairman to failed components in a series system. The following assumptions are made. The component failure and repair times are random variables with exponential distributions, independent of the state of other components. Component failures can occur even while the system is not functioning and it is permissible to reassign the repairman among failed components without penalty. It is shown that the policy which always assigns the repairman to the failed component with the smallest failure rate, among the failed ones, maximizes the availability of the system, irrespective of the values of the repair rates.

On the Use of Replacements to Extend System Life

We consider an extreme version of the replacement problem. A vital component of a system must be replaced before it fails, otherwise the system fails with no possibility of repairing the system. We assume that n spares are available and that the distribution of a component life is known; the objective is to schedule the replacements so that the expected life of the system is maximized. Our results range from an iterative formula for constructing the optimal schedule to more general mathematical properties of optimal schedules and expected times.

Renewal Decision Problem-Random Horizon

A system must operate for T units of time. T is a random variable with known distribution function F. A certain component is essential for the systems operation and when it fails must be replaced with a new component. There are n possible types of replacements. An unlimited supply of each type is assumed. A type i replacement costs cici > 0 and functions independently of T for an exponentially distributed length of time with rate λi. The problem is to assign the replacements from among the various possible types so as to minimize the expected total cost of providing an operative component for the entire life of the system. The principal result of this paper, generalizing previous work where T was assumed to have a degenerate or truncated exponential distribution, is that if F is an increasing failure rate function IFR the optimal replacement policy has a simple intuitive interval structure. An algorithm for finding the optimal policy is indicated. Some results are obtained for the case where component life distributions are not exponential.

On the Comparison of two Software Relibility Estimators

Two different estimators of the failure rate of software are compared. It is shown that one tends to be better when the number of errors is small to moderate and the other when the number of errors is large.