Well-Posedness of Gaver’s Parallel System Attended by a Cold Standby Unit and a  Repairman with Multiple Vacations

Haji, Abdukerim; Yunus, Bilikiz

doi:10.4236/jamp.2015.37101

Cited by 3 publications

(1 citation statement)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But they did not discuss the existence of the dynamic solution and the asymptotic stability of the dynamic solution. In [5], we proved the well-posedness and the existence of a unique positive dynamic solution of the system by using 0 Csemigroup theory of linear operators from [6] and [7]. In this paper, we prove that the dynamic solution converging to its static solution in the sense of the norm using the stochastic matrix and irreducibility of the corresponding semigroup, thus we obtain the asymptotic stability of the dynamic solution of this system.…”

Section: Introductionmentioning

confidence: 93%

Asymptotic Stability of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations

Haji¹

2015

WJET

Self Cite

View full text Add to dashboard Cite

We investigate Gaver's parallel system attended by a cold standby unit and a repairman with multiple vacations. By analysing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we prove that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution of the system.

show abstract

Section: Introductionmentioning

confidence: 93%

Asymptotic Stability of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations

Haji¹

2015

WJET

Self Cite

View full text Add to dashboard Cite

show abstract

Stochastic behavior of parallel system with expert repair and maintenance

Gitanjali¹,

Malik

2018

Life Cycle Reliab Saf Eng

View full text Add to dashboard Cite

Alternative Transient and Steady State Analysis of Some Gaver’s Parallel Systems

Khurodze,

Kakubava,

Kublashvili

et al. 2025

Int. j. math. eng. manag. sci.

View full text Add to dashboard Cite

This paper presents one of the most interesting generalizations of Gaver’s basic two-unit parallel system sustained by a cold standby unit and attended by a repairman with multiple vacations. The system was studied using the supplementary variables technique, like many other similar semi-Markov systems. D.P. Gaver, Jr. was the first to apply this method for constructing and studying reliability models, and since then it has been then widely used by other researchers to study various reliability problems. As a result, non-classical boundary value problem of mathematical physics with nonlocal boundary conditions has been obtained. Until now, a solution to this problem was obtained in terms of Laplace transforms. Naturally, the most significant part of the problem is a system of partial differential equations (Kolmogorov forward equations). In this study, we demonstrate that Kolmogorov equations are redundant and we can solve the problem by avoiding the necessity of using them. We present here a novel, purely probabilistic approach. The results are formulated as rigorous mathematical statements, offering a significant simplification in the reliability analysis of stochastic systems. Our findings show that this novel approach can be applied to study both semi-Markov and some non semi-Markov models where the supplementary variables technique is used.

show abstract

Well-Posedness of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations

Abstract: We investigate Gaver's parallel system attended by a cold standby unit and a repairman with multiple vacations. By using C 0-semigroup theory of linear operators in the functional analysis, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

Cited by 3 publications

References 7 publications

Asymptotic Stability of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations

Asymptotic Stability of Gaver’s Parallel System Attended by a Cold Standby Unit and a Repairman with Multiple Vacations

Stochastic behavior of parallel system with expert repair and maintenance

Alternative Transient and Steady State Analysis of Some Gaver’s Parallel Systems

Contact Info

Product

Resources

About