Tests on the Intensity of a Poisson Process

1975 ◽  
Vol 4 (8) ◽  
pp. 777-782
Author(s):  
John Saw
Keyword(s):  
Poisson Process

Lectures on the Poisson Process

2017 ◽  
Author(s):  
Günter Last ◽  
Mathew Penrose
Keyword(s):  
Poisson Process

Applying the Anderson-Darling Test to Suicide Clusters

Crisis ◽  
2013 ◽  
Vol 34 (6) ◽  
pp. 434-437 ◽  
Author(s):  
Donald W. MacKenzie
Keyword(s):  
Poisson Process ◽  
Goodness Of Fit ◽  
Small Samples ◽  
Massachusetts Institute Of Technology ◽  
Homogeneous Poisson Process ◽  
Cornell University ◽  
The Difference ◽  
Suicide Clusters ◽  
Institute Of Technology ◽  
Anderson Darling

Background: Suicide clusters at Cornell University and the Massachusetts Institute of Technology (MIT) prompted popular and expert speculation of suicide contagion. However, some clustering is to be expected in any random process. Aim: This work tested whether suicide clusters at these two universities differed significantly from those expected under a homogeneous Poisson process, in which suicides occur randomly and independently of one another. Method: Suicide dates were collected for MIT and Cornell for 1990–2012. The Anderson-Darling statistic was used to test the goodness-of-fit of the intervals between suicides to distribution expected under the Poisson process. Results: Suicides at MIT were consistent with the homogeneous Poisson process, while those at Cornell showed clustering inconsistent with such a process (p = .05). Conclusions: The Anderson-Darling test provides a statistically powerful means to identify suicide clustering in small samples. Practitioners can use this method to test for clustering in relevant communities. The difference in clustering behavior between the two institutions suggests that more institutions should be studied to determine the prevalence of suicide clustering in universities and its causes.

On some characterizations of the poisson process

1989 ◽  
Vol 26 (01) ◽  
pp. 176-181
Author(s):  
Wen-Jang Huang
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Arrival Time ◽  
Random Variables ◽  
Poisson Processes

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.

scholarly journals Generalized fractional Poisson process and related stochastic dynamics

2020 ◽  
Vol 23 (3) ◽  
pp. 656-693 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Alejandro P. Riascos
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Counting Process ◽  
Stochastic Dynamics ◽  
Waiting Times ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Fractional Operators ◽  
Fractional Poisson Process ◽  
Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

scholarly journals A bivariate two-state Markov modulated Poisson process for failure modeling

2021 ◽  
Vol 208 ◽  
pp. 107318
Author(s):  
Yoel G. Yera ◽  
Rosa E. Lillo ◽  
Bo F. Nielsen ◽  
Pepa Ramírez-Cobo ◽  
Fabrizio Ruggeri
Keyword(s):  
Poisson Process ◽  
Failure Modeling ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated

On the rate of Poisson process approximation to a Bernoulli process

1997 ◽  
Vol 34 (4) ◽  
pp. 898-907 ◽  
Author(s):  
Aihua Xia
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Discrete Time ◽  
Wasserstein Distance ◽  
Stein’S Method ◽  
Upper Bounds ◽  
Stein's Method ◽  
Bernoulli Process ◽  
Logarithmic Factor ◽  
Process Approximation

This note gives the rate for a Wasserstein distance between the distribution of a Bernoulli process on discrete time and that of a Poisson process, using Stein's method and Palm theory. The result here highlights the possibility that the logarithmic factor involved in the upper bounds established by Barbour and Brown (1992) and Barbour et al. (1995) may be superfluous in the true Wasserstein distance between the distributions of a point process and a Poisson process.

scholarly journals A Poisson Process-Based Random Access Channel for 5G and Beyond Networks

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 508
Author(s):  
Alaa Omran Almagrabi ◽  
Rashid Ali ◽  
Daniyal Alghazzawi ◽  
Abdullah AlBarakati ◽  
Tahir Khurshaid
Keyword(s):  
Poisson Process ◽  
Random Access ◽  
Communication Technologies ◽  
Channel Access ◽  
Access Channel ◽  
Machine Type Communications ◽  
Equal Chance ◽  
Random Access Channel ◽  
Access Mechanisms

The 5th generation (5G) wireless networks propose to address a variety of usage scenarios, such as enhanced mobile broadband (eMBB), massive machine-type communications (mMTC), and ultra-reliable low-latency communications (URLLC). Due to the exponential increase in the user equipment (UE) devices of wireless communication technologies, 5G and beyond networks (B5G) expect to support far higher user density and far lower latency than currently deployed cellular technologies, like long-term evolution-Advanced (LTE-A). However, one of the critical challenges for B5G is finding a clever way for various channel access mechanisms to maintain dense UE deployments. Random access channel (RACH) is a mandatory procedure for the UEs to connect with the evolved node B (eNB). The performance of the RACH directly affects the performance of the entire network. Currently, RACH uses a uniform distribution-based (UD) random access to prevent a possible network collision among multiple UEs attempting to access channel resources. However, in a UD-based channel access, every UE has an equal chance to choose a similar contention preamble close to the expected value, which causes an increase in the collision among the UEs. Therefore, in this paper, we propose a Poisson process-based RACH (2PRACH) alternative to a UD-based RACH. A Poisson process-based distribution, such as exponential distribution, disperses the random preambles between two bounds in a Poisson point method, where random variables occur continuously and independently with a constant parametric rate. In this way, our proposed 2PRACH approach distributes the UEs in a probability distribution of a parametric collection. Simulation results show that the shift of RACH from UD-based channel access to a Poisson process-based distribution enhances the reliability and lowers the network’s latency.

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