Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks

2022

Symmetry

Self Cite

This article is a study of vector-valued renewal-reward processes on Rd. The jumps of the process are assumed to be independent and identically distributed nonnegative random vectors with mutually dependent components, each of which may be either discrete or continuous (or a mixture of discrete and continuous components). Each component of the process has a fixed threshold. Operational calculus techniques and symmetries with respect to permutations are used to find a general result for the probability of an arbitrary weak ordering of threshold crossings. The analytic and numerical tractability of the result are demonstrated by an application to the reliability of stochastic networks and some other special cases. Results are shown to agree with empirical probabilities generated through simulation of the process.

Section: Reliability Theorymentioning

confidence: 99%

On the Exiting Patterns of Multivariate Renewal-Reward Processes with an Application to Stochastic Networks

2022

Symmetry

Self Cite

First Passage Analysis in a Queue with State Dependent Vacations

Dshalalow

2022

Axioms

This paper deals with a single-server queue where the server goes on maintenance when the queue is exhausted. Initially, the maintenance time is fixed by deterministic or random number T. However, during server’s absence, customers are screened by a dispatcher who estimates his service times based on his needs. According to these estimates, the dispatcher shortens server’s maintenance time and as the result the server returns earlier than planned. Upon server’s return, if there are not enough customers waiting (under the N-Policy), the server rests and then resumes his service. At first, the input and service are general. We then prove a necessary and sufficient condition for a simple linear dependence between server’s absence time (including his rest) and the number of waiting customers. It turns out that the input must be (marked) Poisson. We use fluctuation and semi-regenerative analyses (previously established and embellished in our past work) to obtain explicit formulas for server’s return time and the queue length, both with discrete and continuous time parameter. We then dedicate an entire section to related control problems including the determination of the optimal T-value. We also support our tractable formulas with many numerical examples and validate our results by simulation.

Fluctuation Analysis of a Soft-Extreme Shock Reliability Model

Dshalalow

2022

Mathematics

In this paper, we deal with a mixed reliability system decaying from natural wear, occasional soft and hard shocks that eventually lead the system to failure. The aging process alone is linear and it is escalated through soft shocks such that they lead to the system’s soft failure when the combined damage exceeds a threshold M. The other threat is that posed by occasional hard shocks. When the total number of them identified as critical (each critical shock exceeds a fixed threshold H) reaches N, the system becomes disabled. With N=1, a critical shock is extreme. The arrival stream of shocks is a renewal process marked by soft and hard shocks. We establish a formula for a closed form functional containing system’s time-to-failure, the state of the system upon its failure, and other useful statistical characteristics of the system using and embellishing fluctuation analysis and operational calculus. Special cases provide tame expressions that are computed and validated by simulation.