Extensions of A simple model :with applications in reliability, extinction of specizs, inventory depletion and URN sampling

Abstract

This paper is devoted to the study of che following tsodel:A series-parallel system consists of (k + 1) subsystem C₀C₁ ,…,C_k, , also called cut sets. Cut sec C₁ has n_i. com­ponents arranged in parallel, i = 0,1,…,k. Jo two cut sets have a component in comeon. This model was introduced and studied by El-Heweihi, Proschan, and Sethuraiaan (1978) under the assumption that component lifelengths are continuous, indepdent, and identically distributed random variables. They obtained several equivalent expressions for the probability that a specified cut set C₀, say, fails first. These expressions aere then used to derive qualitative properties of this probability, such as monotonicity, Schur-concavity, etc.

In this paper we obtain extensions of these results, tinder -he same assumptions ve study the probability that a. specified cut set C₀say, fails in the r^th place, r = l,2,…,k. This probability Is shown to retain most of the interesting qualitative features enjoyed in the special case r = 1. We then assure* that component lifelengths are identically distributed within a cut set, but allow then to vary a=ong cut sets. Under this more general assumption we derive expressions for and obtain properties of the probability that C_o fails in the r^th place

This generalization of the model of El-Heveihi, Proschan, and Sethuranan (1978), also has applications in the study of reliabilityextinction of species, inventory depletion, urn sampling, among others.

Keywords:

• reliability
• extinction of species
• inventor
• sampling
• schvr functions
• cut sets
• series-parallel system
• functions decreasing in transposition