An age-dependent counting process generated from a renewal process

1988 ◽  
Vol 20 (4) ◽  
pp. 739-755 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Renewal Process ◽  
Counting Process ◽  
Intensity Function ◽  
Sufficient Condition ◽  
Minimal Repair ◽  
Homogeneous Poisson Process ◽  
Stochastic Convexity ◽  
Age Dependent ◽  
Non Homogeneous Poisson Process ◽  
Age Process

Let {N(t)} be a renewal process having the associated age process {X(t)}. Of interest is the counting process {M(t)} characterized by a non-homogeneous Poisson process with age-dependent intensity function λ (X(t)). The trivariate process {Y(t) = [M(t), N(t), X(t)]} is analyzed obtaining its Laplace transform generating function explicitly. Based on this result, asymptotic behavior of {S(t) = cM(t) + dN(t)} as t → ∞ is discussed. Furthermore, a sufficient condition is given under which {M(t), –N(t), X(t)} is stochastically monotone and associated. This condition also assures increasing stochastic convexity of {M(t)}. The usefulness of these results is demonstrated through an application to the age-dependent minimal repair problem.

An age-dependent counting process generated from a renewal process

1988 ◽  
Vol 20 (04) ◽  
pp. 739-755 ◽  
Author(s):  
Ushio Sumita ◽  
J. George Shanthikumar
Keyword(s):  
Renewal Process ◽  
Counting Process ◽  
Intensity Function ◽  
Sufficient Condition ◽  
Minimal Repair ◽  
Homogeneous Poisson Process ◽  
Stochastic Convexity ◽  
Age Dependent ◽  
Non Homogeneous Poisson Process ◽  
Age Process

Let {N(t)} be a renewal process having the associated age process {X(t)}. Of interest is the counting process {M(t)} characterized by a non-homogeneous Poisson process with age-dependent intensity function λ (X(t)). The trivariate process {Y(t) = [M(t), N(t), X(t)]} is analyzed obtaining its Laplace transform generating function explicitly. Based on this result, asymptotic behavior of {S(t) = cM(t) + dN(t)} as t → ∞ is discussed. Furthermore, a sufficient condition is given under which {M(t), –N(t), X(t)} is stochastically monotone and associated. This condition also assures increasing stochastic convexity of {M(t)}. The usefulness of these results is demonstrated through an application to the age-dependent minimal repair problem.

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.

Age-dependent minimal repair

1985 ◽  
Vol 22 (02) ◽  
pp. 370-385 ◽  
Author(s):  
Henry W. Block ◽  
Wagner S. Borges ◽  
Thomas H. Savits
Keyword(s):  
Continuous Function ◽  
Stochastic Model ◽  
Renewal Process ◽  
Failure Time ◽  
Minimal Repair ◽  
Life Distribution ◽  
Complete Repair ◽  
Age Dependent ◽  
Image Position

A stochastic model is developed to describe the operation in time of the following maintained system setting. A piece of equipment is put in operation at time 0. Each time it fails, a maintenance action is taken which, with probability p(t), is a complete repair or, with probability q(t)=1– p(t), is a minimal repair, where t is the age of the equipment in use at the failure time. It is assumed that complete repair restores the equipment to its good as new condition, that minimal repair restores the equipment to its condition just prior to failure and that both maintenance actions take negligible time. If the equipment's life distribution F is a continuous function, the successive complete repair times are shown to be a renewal process with interarrival distribution for t ≧ 0. Preservation and monotone properties of the model extending the results of Brown and Proschan (1983) are obtained.

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

1995 ◽  
Vol 32 (3) ◽  
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.

scholarly journals A Bayesian Model of COVID-19 Cases Based on the Gompertz Curve

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 228
Author(s):  
Ángel Berihuete ◽  
Marta Sánchez-Sánchez ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Monte Carlo ◽  
Prior Distribution ◽  
Bayesian Model ◽  
Real Data ◽  
Intensity Function ◽  
Specific Region ◽  
Mcmc Methods ◽  
Homogeneous Poisson Process ◽  
Gompertz Curve ◽  
Non Homogeneous Poisson Process

The COVID-19 pandemic has highlighted the need for finding mathematical models to forecast the evolution of the contagious disease and evaluate the success of particular policies in reducing infections. In this work, we perform Bayesian inference for a non-homogeneous Poisson process with an intensity function based on the Gompertz curve. We discuss the prior distribution of the parameter and we generate samples from the posterior distribution by using Markov Chain Monte Carlo (MCMC) methods. Finally, we illustrate our method analyzing real data associated with COVID-19 in a specific region located at the south of Spain.

The capacity of an intersection with non-stationary traffic

1976 ◽  
Vol 13 (02) ◽  
pp. 418-422
Author(s):  
Helmut Wegmann
Keyword(s):  
Road Traffic ◽  
Intensity Function ◽  
Major Road ◽  
Homogeneous Case ◽  
Homogeneous Poisson Process ◽  
Periodic Intensity ◽  
Minor Road ◽  
Traffic Stream ◽  
A Minor ◽  
Non Homogeneous Poisson Process

The average number of vehicles being able to enter an intersection per time unit from a minor road with a stop or yield sign — the capacity of the intersection — depends on the density of the traffic stream on the major road. In case the time-process of the major road traffic at the intersection is a non-homogeneous Poisson process with a periodic intensity function the capacity is calculated and compared with the capacity in the homogeneous case.

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