scholarly journals Nonhomogeneous Poisson Process and Compound Poisson Process in the Modelling of Random Processes Related to Road Accidents

2019 ◽  
Vol 26 (1) ◽  
pp. 39-46
Author(s):  
Franciszek Grabski
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Statistical Distribution ◽  
Nonhomogeneous Poisson Process ◽  
Model Parameters ◽  
Time Interval ◽  
Road Accidents ◽  
Compound Poisson ◽  
The Road ◽  
The Given

Abstract The stochastic processes theory provides concepts and theorems that allow building probabilistic models concerning accidents. So called counting process can be applied for modelling the number of the road, sea and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The new theoretical results regarding compound Poisson process are presented in the paper. A nonhomogeneous Poisson process and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number and number of injured and killed people in the Polish road. To estimate model parameters were used data coming from the annual reports of the Polish police [9, 10]. Constructed models allowed anticipating number of accidents at any time interval with a length of h and the accident consequences. We obtained the expected value of fatalities or injured and the corresponding standard deviation in the given time interval. The statistical distribution of fatalities number in a single accident and statistical distribution of injured people number and also probability distribution of fatalities or injured number in a single accident are computed. It seems that the presented examples explain basic concepts and results discussed in the paper.

scholarly journals Nonhomogeneus Generalisations of Poisson Process in the Modeling of Random Processes Related to Road Accidents

2020 ◽  
Vol 222 (3) ◽  
pp. 29-42
Author(s):  
Franciszek Grabski
Keyword(s):  
Poisson Process ◽  
Counting Process ◽  
Probabilistic Models ◽  
Compound Poisson Process ◽  
Annual Reports ◽  
Model Parameters ◽  
Road Accidents ◽  
Time Intervals ◽  
The Road ◽  
The Given

AbstractThe stochastic processes theory provides concepts, and theorems, which allow to build the probabilistic models concerning accidents. “Counting process” can be applied for modelling the number of road, sea, and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The nonhomogeneous Poisson process, and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number, and number of people injured and killed in Polish roads. To estimate model parameters were used data coming from the annual reports of the Polish police.

An Approach to Obsolescence Forecasting based on Hidden Markov Model and Compound Poisson Process

2019 ◽  
Vol 01 (02) ◽  
Author(s):  
Yosra Grichi ◽  
Yvan Beauregard ◽  
Thien-My Dao
Keyword(s):  
Life Cycle ◽  
Poisson Process ◽  
Product Life Cycle ◽  
Hidden Markov ◽  
Rapid Development ◽  
Compound Poisson Process ◽  
Stochastic Approach ◽  
Efficient Solutions ◽  
Model Parameters ◽  
Compound Poisson

The popularity of electronic devices has sparked research to implement components that can achieve better performance and scalability. However, companies face significant challenges when they use systems with a long-life cycle, such as in avionics, which leads to obsolescence problems. Obsolescence can be driven by many factors, primary among which could be the rapid development of technologies that lead to a short life cycle of parts. Moreover, obsolescence problems can prove costly in terms of intermittent stock availability and unmet demand. Therefore, obsolescence forecasting appears to be one of the most efficient solutions. This paper presents a review of gaps in the actual approaches and proposes a method that can better forecast the product life cycle. The proposed approach will help companies to improve obsolescence forecasting and reduce its impact in the supply chain. The method introduces a stochastic approach to estimate the obsolescence life cycle through simulation of demand data using Markov chain and homogeneous compound Poisson process. This approach uses multiple states of the life cycle curve based on the change in demand rate and introduces hidden Markov theory to estimate the model parameters. Numerical results are provided to validate the proposed method. To examine the accuracy of this approach, the standard deviation (STD) of obsolescence time is calculated. The results showed that the life cycle curves of parts can be predicted with high accuracy.

scholarly journals Limit theorems for the fractional nonhomogeneous Poisson process

2019 ◽  
Vol 56 (01) ◽  
pp. 246-264 ◽  
Author(s):  
Nikolai Leonenko ◽  
Enrico Scalas ◽  
Mailan Trinh
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Poisson Process ◽  
Limit Theorems ◽  
Compound Poisson Process ◽  
Nonhomogeneous Poisson Process ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Compound Poisson ◽  
Finite Dimensional

AbstractThe fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. We propose a similar definition for the (nonhomogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.

Approximations for the single-product production-inventory problem with compound Poisson demand and service-level constraints

1984 ◽  
Vol 16 (2) ◽  
pp. 378-401 ◽  
Author(s):  
A. G. De kok ◽  
H. C. Tijms ◽  
F. A. Van der Duyn Schouten
Keyword(s):  
Poisson Process ◽  
Compound Poisson Process ◽  
Service Level ◽  
Inventory Level ◽  
Inventory Problem ◽  
Single Product ◽  
Compound Poisson ◽  
Poisson Demand ◽  
Production Inventory ◽  
Random Fluctuations

We consider a production-inventory problem in which the production rate can be continuously controlled in order to cope with random fluctuations in the demand. The demand process for a single product is a compound Poisson process. Excess demand is backlogged. Two production rates are available and the inventory level is continuously controlled by a switch-over rule characterized by two critical numbers. In accordance with common practice, we consider service measures such as the average number of stockouts per unit time and the fraction of demand to be met directly from stock on hand. The purpose of the paper is to derive practically useful approximations for the switch-over levels of the control rule such that a pre-specified value of the service level is achieved.

Sign in / Sign up

Export Citation Format

Share Document