A Guide to Probability Theory And application

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A Simple Allocation Problem

The problem studied is that of how to allocate a fixed amount of some resource among various activities where, after a period of time, unused portions of the resource lose their value. Demand for the resource at each station is considered to be random. The problem is a special case of one usually referred to as the distribution of the effort. The question of determining the optimal amount of resource is also discussed.

Asymptotic Optimal Policies for the Stochastic Sequential Assignment Problem

Certain sets of numbers {a in }, i = 0,..., n, n = 1, 2,..., are known characterize an optimal sequential assignment policy. In this paper the limiting behavior as n -> \infty of the a in 's is studied.

Computing optimal sequential allocation rules in clinical trials

The problem of assigning one of several treatments in clinical trials is formulated as a discounted bandit problem that was studied by Gittins and Jones. The problem involves comparison of certain state dependent indices A recent characterization of the index is used to calculate more efficiently the values of these indices. 1 Introduction. We consider the well known problem of optimal allocation of treatments in clinical trials A simple version of the problem is as follows There are several possible treatments for a given disease. When a particular treatment n is used it is either effective with unknown probability θ or not effective wit

On the Maintenance of Systems Composed of Highly Reliable Components

We consider the dynamic repair allocation problem for a general multi-component system that is maintained by a limited number of repairmen. Component functioning and repair times are exponentially distributed random variables with known parameters. At most one repairman may be assigned to a failed component and it is possible to reassign a repairman from one failed component to another instantaneously. The objective is to determine repair allocation policies that maximize a measure of performance of the system such as the expected discounted system operation time or the availability of the system. We consider systems composed of highly reliable, i.e., small failure rates, components and study asymptotic techniques for the determination of optimal policies. In the final section we find asymptotically optimal policies for the series, parallel, and a system composed of parallel subsystems connected in series.first passage times, Markov decision processes