scholarly journals On history-dependent mixed shock models

2021 ◽  
pp. 1-18
Author(s):  
Dheeraj Goyal ◽  
Maxim Finkelstein ◽  
Nil Kamal Hazra
Keyword(s):  
Poisson Process ◽  
Failure Rate ◽  
Survival Function ◽  
Shock Model ◽  
Homogeneous Poisson Process ◽  
Proposed Model ◽  
Mean Lifetime ◽  
Polya Process ◽  
Pólya Process ◽  
Non Homogeneous Poisson Process

In this paper, we consider a history-dependent mixed shock model which is a combination of the history-dependent extreme shock model and the history-dependent $\delta$ -shock model. We assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as the particular cases. For the defined survival model, we derive the corresponding survival function, the mean lifetime and the failure rate. Further, we study the asymptotic and monotonicity properties of the failure rate. Finally, some applications of the proposed model have also been included with relevant numerical examples.

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.

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.

Poisson limits for a clustering model of strauss

1977 ◽  
Vol 14 (04) ◽  
pp. 776-784 ◽  
Author(s):  
Roy Saunders ◽  
Gerald M. Funk
Keyword(s):  
Poisson Process ◽  
Random Variable ◽  
Homogeneous Poisson Process ◽  
Clustering Model ◽  
Simulation Results ◽  
Non Homogeneous Poisson Process

In this article we present a limiting result for the random variable Yn (r) which arises in a clustering model of Strauss (1975). The result is that under some sparseness-of-points conditions the process {Yn (r): 0 ≦ r ≦ r ∞} converges weakly to a non-homogeneous Poisson process {Y(r): 0 ≦ r ≦ r ∞} when n → ∞. Simulation results are given to indicate the accuracy of the approximation when n is moderate and applications of the limiting result to tests for clustering are discussed.

Optimal dispatching of a poisson process

1969 ◽  
Vol 6 (03) ◽  
pp. 692-699 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Time Lag ◽  
Stopping Time ◽  
Processing Plant ◽  
Homogeneous Poisson Process ◽  
Batch Arrivals ◽  
Optimal Dispatch ◽  
Non Homogeneous Poisson Process

Items arrive at a processing plant at a Poisson rate λ. At time T, all items are dispatched from the system. An intermediate dispatch time is to be chosen to minimize the total wait of all items. It is shown that if the dispatch time must be chosen at time 0 then T/2 not only minimizes the expected total wait but it also maximizes the probability that the total wait is less than a for every a > 0. If the intermediate dispatch time is allowed to be a (random) stopping time, then it is shown that the policy which dispatches at time t iff N(t) > λ(T – t) is optimal, where N(t) denotes the number of items present at time t. The distribution of the optimal dispatch time and the optimal expected total wait are determined. A generalization to the case of a non-homogeneous Poisson process, a time lag, and batch arrivals is given. Finally, the case where the process goes on indefinitely and any number of dispatches are allowed (at a cost K per dispatch) is considered.

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