Revision as of 06:29, 22 April 2013

Poisson Random Variables

In 1837, the Poisson Distribution was introduced by Siméon Denis Poisson[1]. It has the pmf

$ P(X=x)= \frac{\lambda^x e^{-\lambda}}{x!} $

This distribution models the probablility that a number of events, x, will occur within a given time period when the average rate of occurance of such events on the same time interval is $ \lambda $.

Applications of Poisson Random Variables

Poisson random variables have many applications. This arises from the fact that many events in nature can be modeled as Poisson processes. Following are some examples of modern applications of the Poisson random variable.

Optimization

Poisson random variables are often used to model scenarios used to generate cost functions in optimization problems. For example, the economic lot scheduling problem aims to optimize the production of a certain number of products on a certain number of machines given a certain demand. If this is handled as a deterministic problem, it has been shown to be np-hard by Hsu in his 1983 paper on the topic [2]. Löhndorf & Minner have used Poisson random variables to make the problem stochastic, but also more feasible [3]. It should be noted that Löhndorf & Minner used a stuttering Poisson process to model their problem. Stuttering Poisson proceses are more generalized cases of Poisson processes wherein events occur at time periods dictated by a Poisson process but the number of events occurring within these time periods followas a geometric distribution [4].

=

Questions or comments on this page? Please post them below. I will try to keep up with any posts (even though only two weeks are left).

Question 1: (text)

Question 2: (text)

Question 3: (text)

References:

[1]: Jerzy Letkowski. Applications of the Poisson probability distribution. 2012. Retrieved from http://www.aabri.com/SA12Manuscripts/SA12083.pdf

[2]: Wen-Lian Hsu. On the General Feasibility Test of Scheduling Lot Sizes for Several Products on One Machine. Management Science: Vol. 29, No. 1, Jan 1983. Retrieved from http://www.jstor.org/stable/2631168

[3]: Löhndorf & Minner. Simulation optimization for the stochastic economic lot scheduling problem. IIE Transactions: Volume 45, Issue 7, 2013. Retrieved from http://www.tandfonline.com/doi/full/10.1080/0740817X.2012.662310#tabModule

[4]: R. M. Adelson. Compound Poisson Distributions. OR: Vol. 17, No. 1, Mar 1966. Retrieved from http://www.jstor.org/stable/3007241

Back to ECE302 Spring 2013, Prof. Boutin