09-More on Poisson Process

09-More on Poisson Process - Lecture set #9 IE111, Fall...

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IE111, Fall 2009 Lecture 9: The Poisson Process Revisited Recall the Poisson process models a “purely random processes”, in which an arrival is equally likely at any time. The parameter of a PP is the rate A Poisson Process models an “arrival process”. For example: the arrival of emergency patients to a hospital, the arrival of neutrons to a Geiger counter, the arrival of customers to a store, the arrival of orders to a factory, the arrival of calls to a phone network, the arrival of job requests to a server, etc. When an arrival process is a Poisson process, then if we let the random variable X be the number of arrivals in a time period of length T, then X has a Poisson PMF. That is, the Poisson distribution models the number of arrivals in a period of length T time units. P X (x) = e - α T ( α T) x / x! for x in {0,1,2,. ....} Note that in a Poisson process, the parameter is α T. α is called the “rate” of the process. Example Calls arrive to a 1-800 number according to a Poisson process with rate α = 1.7 calls per minute. What is the probability that at least 10 calls arrive i the next 10 minutes? The parameter is λ = α T = (1.7 calls/min)(10 min) = 17 calls. P(X 10) = 1 - P(X 9) = 1 - 0.026 = 0.99974 (note that the 0.026 comes from the tables) What is the expected number of calls in an 8 hour shift? The parameter is λ = α T = (1.7 calls/min)(8 hours) = (1.7 calls/min)(480 min) = 816 calls. The mean of the Poisson Distribution is λ , thus we expect 816 calls per shift. (Make sure the times units of α and T match!) The behavior that characterizes a Poisson Process is: 1. On average we see α arrivals per time unit 2. An arrival is equally likely to occur at any time. It is this second property that characterizes a Poisson Process.
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09-More on Poisson Process - Lecture set #9 IE111, Fall...

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