Journal ArticleDOI
Some Theoretical Properties of the Geometric and α-Series Processes
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TLDR
In this paper, it was shown that the increasing geometric process grows at most logarithmically in time while the decreasing geometric process is almost certain to have a time of explosion.Abstract:
The geometric process has been proposed as a simple model for use in reliability. Recently, the α-series process was proposed as a complementary model which can be used in situations where the geometric process is inappropriate. In this article, we show that the increasing geometric process grows at most logarithmically in time while the decreasing geometric process is almost certain to have a time of explosion. The α-series process grows either as a polynomial in time or exponentially in time. We also show that, unlike most renewal processes, the geometric process does not satisfy a central limit theorem, while the α-series process does.read more
Citations
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A replacement policy for a repairable system with its repairman having multiple vacations
Jishen Jia,Shaomin Wu +1 more
TL;DR: This paper considers a replacement policy for a repairable system with a repairman, who can have multiple vacations, and optimises replacement policy using geometric processes.
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An extended geometric process repair model for a cold standby repairable system with imperfect delayed repair
Yuan Lin Zhang,Guan Jun Wang +1 more
TL;DR: In this article, an extended geometric process repair model (EGPRM) was proposed to generalize the GPRM, and a repair-replacement problem for a cold standby repairable system consisting of two dissimilar components and one repairman was studied.
Journal ArticleDOI
An optimal replacement policy for a degenerative system with two-types of failure states
Guo Qing Cheng,Ling Li +1 more
TL;DR: A new general monotone process model for the degenerative system is introduced which is a generalization of the @a-series process model which aims to determine an optimal replacement policy such that the average cost rate is minimized.
Journal ArticleDOI
Geometric process repair model for an unreliable production system with an intermediate buffer
TL;DR: In this paper, an unreliable production system consisting of two machines (M1 and M2) is considered, in which M1 produces a single product type to satisfy a constant and continuous demand of M2 and it is subjected to random failures.
Journal ArticleDOI
Geometric-Like Processes: An Overview and Some Reliability Applications
TL;DR: An overview of geometric and the related geometric-like processes (GLP) is provided, including a brief review of the geometric process and some basic definitions, facts and references for geometric- like processes.
References
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Book
Probability and Measure
TL;DR: In this paper, the convergence of distributions is considered in the context of conditional probability, i.e., random variables and expected values, and the probability of a given distribution converging to a certain value.
BookDOI
Continuous-Time Markov Chains
TL;DR: In this paper, the steady state probability vector for a continuous-time Markov chain is found, where the transition probability is defined as the probability that the chain makes transitions into any given state at the same time as it makes transitions out of that same state.
Journal ArticleDOI
A note on the optimal replacement problem
TL;DR: In this article, a new repair replacement model for a deteriorating system is proposed, in which the successive survival times of the system form a geometric process and are stochastically nonincreasing, whereas the consecutive repair times after failure also constitute a geometrically non-decreasing process, and the explicit expressions of the long run average costs per unit time under these two kinds of replacement policy are calculated.
Journal ArticleDOI
Geometric processes and replacement problem
TL;DR: In this paper, the authors introduced the geometric process which is a sequence of independent nonnegative random variables, such that the distribution function of a random variable X n is F (a n−1 −1 normalized x), wherea is a positive constant, and the explicit expressions of the long-run average costs per unit time under each replacement policy are calculated, and therefore the corresponding optimal replacement policies can be found analytically or numerically.