Optimum replacement of a system subject to shocks

1986 ◽  
Vol 23 (1) ◽  
pp. 107-114 ◽  
Author(s):  
Mohamed Abdel-Hameed
Keyword(s):  
Poisson Process ◽  
Sufficient Conditions ◽  
Sample Path ◽  
Cost Structure ◽  
Homogeneous Poisson Process ◽  
Jump Size ◽  
Shock Process ◽  
Finite Period ◽  
Periodic Replacement ◽  
Non Homogeneous Poisson Process

A system is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement time for the system. Boland and Proschan [4] consider periodic replacement of the system and give sufficient conditions for the existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [3] establish similar results assuming that cost structure is time dependent, still requiring that the shock process is a non-homogeneous Poisson process. We show via a sample path argument that the results of [3] and [4] hold for any counting process whose jump size is of one unit magnitude.

Optimum replacement of a system subject to shocks

1986 ◽  
Vol 23 (01) ◽  
pp. 107-114 ◽  
Author(s):  
Mohamed Abdel-Hameed
Keyword(s):  
Poisson Process ◽  
Sufficient Conditions ◽  
Sample Path ◽  
Cost Structure ◽  
Homogeneous Poisson Process ◽  
Jump Size ◽  
Shock Process ◽  
Finite Period ◽  
Periodic Replacement ◽  
Non Homogeneous Poisson Process

A system is subject to shocks. Each shock weakens the system and makes it more expensive to run. It is desirable to determine a replacement time for the system. Boland and Proschan [4] consider periodic replacement of the system and give sufficient conditions for the existence of an optimal finite period, assuming that the shock process is a non-homogeneous Poisson process and the cost structure does not depend on time. Block et al. [3] establish similar results assuming that cost structure is time dependent, still requiring that the shock process is a non-homogeneous Poisson process. We show via a sample path argument that the results of [3] and [4] hold for any counting process whose jump size is of one unit magnitude.

Weak limit results for the extremes of a class of shot noise processes

1995 ◽  
Vol 32 (03) ◽  
pp. 707-726 ◽  
Author(s):  
Patrick Homble ◽  
William P. McCormick
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Impulse Response Function ◽  
Weak Limit ◽  
Intensity Function ◽  
Important Class ◽  
Homogeneous Poisson Process ◽  
Periodic Intensity ◽  
Non Homogeneous Poisson Process ◽  
Over Time

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

Order statistics, Poisson processes and repairable systems

1976 ◽  
Vol 13 (03) ◽  
pp. 519-529 ◽  
Author(s):  
Douglas R. Miller
Keyword(s):  
Order Statistics ◽  
Point Processes ◽  
Failure Time ◽  
Sufficient Conditions ◽  
Renewal Processes ◽  
Repairable Systems ◽  
Homogeneous Poisson Process ◽  
Necessary And Sufficient ◽  
Non Homogeneous Poisson Process ◽  
Weibull Process

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.

An optimal parking problem

1982 ◽  
Vol 19 (4) ◽  
pp. 803-814 ◽  
Author(s):  
Mitsushi Tamari
Keyword(s):  
Poisson Process ◽  
Decision Maker ◽  
Optimal Stopping ◽  
Renewal Process ◽  
A Priori ◽  
Expected Loss ◽  
Homogeneous Poisson Process ◽  
Optimal Stopping Rule ◽  
Non Homogeneous Poisson Process ◽  
Parking Place

The decision-maker drives a car along a straight highway towards his destination and looks for a parking place. When he finds a parking place, he can either park there and walk the distance to his destination or continue driving. Parking places are assumed to occur in accordance with a Poisson process along the highway. The decision-maker does not know the distance Y to his destination exactly in advance. Only an a priori distribution is assumed for Y and cases of typically important distribution are examined. When we take as loss the distance the decision-maker must walk and wish to minimize the expected loss, the optimal stopping rule and the minimum expected loss are obtained. In Section 3 a generalization to the cases of a non-homogeneous Poisson process and a renewal process is considered.

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