Economics and Information Theory

(2003). Comparison Methods for Stochastic Models and Risks. Technometrics: Vol. 45, No. 4, pp. 370-371.

Comparison Methods for Stochastic Models and Risks

Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

https://link.springer.com/content/pdf/10.1057%2Fjors.1984.224.pdf

Comparison Methods for Queues and Other Stochastic Models

The principal aim of this chapter is to familiarize the reader with the fact that the conditional distribution of the signal can be viewed as a stochastic process with values in the space of probability measures. While it is true that this chapter sets the scene for the subsequent chapters, it can be skipped by those readers whose interests are biased towards the applied aspects of the subject. The gist of the chapter can be summarized by the following.

The Stochastic Process π

We omit Gompertz’ method of fitting, which the reader will find in Makeham, below. Also omitted are Gompertz’ mortality tables, and his discussion of life expectancy and annuities under his Law. The hyperbolic logarithms in Art. 5 are the natural logs; we are not able to follow completely his integration, which is by Newton's method of fluxions.

On the Nature of the Function Expressive of the Law of Human Mortality

We consider the problem of allocating k active spares to n components of a series system in order to optimize its lifetime. Under the hypotheses that lifetimes of n components are identically distributed with distribution function F(-), lifetimes of k spares are identically distributed with distribution function G( ), lifetimes of components and spares are independently distributed, and that ln(G (x))/ ln(F(x)) is increasing hut, we show that the strategy of balanced allocation of spares optimizes the failure rate function of the system. Furthermore, under the hypotheses that lifetimes of n components are stochastically ordered, lifetimes of k spares are identically distributed, and that lifetimes of components and spares are independently distributed, we show that the strategy of balanced allocation of spares is superior to the strategy of allocating a larger number of components to stronger components. For coherent systems consisting of n identical components with n identical redundant (spare) components, we compare strategies of component and system redundancies under the criteria of reversed failure rate and likelihood ratio orderings. When spares and original components do not necessarily match in their life distributions, we provide a sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under reversed failure rate ordering. As a consequence, we show that, for r-out-of-rc systems, the strategy of component redundancy is superior to the strategy of system redundancy under the criterion of reversed failure rate ordering. When spares and original components match in their life distributions, we provide a necessary and sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under the likelihood ratio ordering. As a consequence, we show that, for r-out-of-n systems, with spares and original components matching in their life distributions, the strategy of component redundancy is superior to the strategy of system redundancy under the likelihood ratio ordering.

/pdf/optimal-allocation-of-active-spares-in-series-systems-and-1c1o13isu3.pdf

Optimal allocation of active spares in series systems and comparison of component and system redundancies

We present sufficient conditions for log-concavity and log-convexity of the residual life and the inactivity time. We also make stochastic comparisons on the residual life and the inactivity time in terms of the failure rate order, the mean residual life order and the usual stochastic order. These results strengthen some conclusions in Li and Zuo [11] and Yue and Cao [19] .

Stochastic Properties of Residual Life and Inactivity Time at a Random Time

By assuming that a random variable X possesses an aging property, we provide conditions under which the corresponding weighted version X 1, with weight function w 1(·), would also possess this aging property. Similarly, by assuming that two random variables X and Y are ordered with respect to a stochastic order, we provide conditions under which the corresponding weighted versions X 1 and Y 2, with weight functions w 1(·) and w 2(·), respectively, preserve this stochastic ordering. We also point out fallacies in the similar results claimed by Jain et al. (1989) and Bartoszewicz and Skolimowska (2006) and correct them.

Nitin Gupta

Papers

Optimal allocation of active spares in series systems and comparison of component and system redundancies

Stochastic Properties of Residual Life and Inactivity Time at a Random Time

Preservation of Some Aging Properties and Stochastic Orders by Weighted Distributions

Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components

Stochastic comparisons in systems with Frèchet distributed components