This paper introduces a simple index that helps to assess the degree of aging or rejuvenation of repairable systems and non-repairable systems (components). The index ranges from –1 to 1. It is negative for the class of decreasing failure rate distributions and point processes with decreasing ROCOF and is positive for the increasing failure rate distributions and point processes with increasing...

A deeper observation is that a Markov chain X starts afresh from its value at some random times called stopping times. Generally, a stopping time is a random time τ that is non-anticipative. That is, we can tell that τ ≤ n from {X0, X1, . . . , Xn}, for any n ≥ 0. A simple example is the first hitting time TA of a set A ⊂ X . Another simple example is TA + 5. A simple counterexample is TA − 1. ...

A Cox process NCox directed by a stationary random measure ξ has second moment var NCox(0, t] = E(ξ(0, t]) + var ξ(0, t], where by stationarity E(ξ(0, t]) = (const.)t = E(NCox(0, t]), so long-range dependence (LRD) properties of NCox coincide with LRD properties of the random measure ξ. When ξ(A) = ∫ AνJ(u)du is determined by a density that depends on rate parameters νi (i∈ X) and the current s...

The question of the serial dependence of successive REM-NREM sleep cycles was examined. The experiments were performed in two different settings: 309 sleep episodes of 11 healthy young sleepers (age range, 20-36 years) were recorded under entrained conditions in the sleep laboratory; 5 of these subjects also slept in an isolation unit (underground apartment) with free-running sleep-wake cycles ...

This study describes the application of a renewal process to a very old legacy system working in the banking environment. The term renewal process refers to a combination of reverse engineering and restoration processes. The latter, similar to restructuring but taking account of the semantics of the components to be restructured, is better explained in the paper. Judged from the maintainer’s po...

We review the theory of renewal reward processes, which describes renewal processes that have some cost or reward associated with each cycle. We present a new simplified proof of the renewal reward theorem that mimics the proof of the elementary renewal theorem and avoids the technicalities in the proof that is presented in most textbooks. Moreover, we mention briefly the extension of the theor...