This is Revision: 2.1 .
The Poisson Process is the model we use in Queuing theory for ``Random arrivals''. It is in many ways the simplest model we could use for arrivals into a queue system because it assumes that the probability of an arrival in a small interval of time depends only on the size of the interval, not on any history of the process to that time.
The main reference for these notes is [(Hillier and Lieberman)(1990)]. Other OR books, such as [(Taha)(1992)] and [(Winston)(1994)] also explain the model. Many other books on statistics and queue models deal with this important stochastic process.
Small time interval h. Prob(event in interval h) = lh
Prob{event in stream 1 is in h } = l1 h
Prob{event in stream 2 is in h } = l2 h
Prob{some event is in h}
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Merged stream acts like a Poisson process with rate
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Whenever an event (arrival) occurs we decide to divert it into A or B stream according to fixed probabilities, pA and pB.
Prob{A event in h}
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The A stream acts like a Poisson stream with a rate of events = pAl
P{0 events in interval [t, t+h] } = 1 - lh where lh << 1
put
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Since probability of an event in h is independent of any others:
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and as h ® 0
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integrate
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But note that if t ® 0, p0(t) ® 1 ( certain to have 0 events!)
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The probability of 2 events in [0, t] is
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This leads us to the Poisson distribution
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Prob{next event after 0 occurs in [t, t+h] }
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thus p.d.f.
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NOTE
Since no memory the zero point can be an event.
Thus time between events is also exponential, l.
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the average time between events = 1/l.
Prob{kth next event is in [t, t+h] }
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This is an Erlang distribution with mean = k /l and variance = k/l2. It has shape parameter of k.
The Erlang Distribution is important in queueing theory and simulation.
Named after A K Erlang, a Danish telephone engineer, considered to be the founder of queueing theory. He wrote papers in 1917-1918.
The Ek is a continuous rv with density function
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A special type of Gamma distribution with an integer shape parameter. 2 parameters control its distribution (the scale parameter, m and its shape parameter, k). Mean is 1/m and its variance 1/km2.
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The Erlang distribution gets narrower as k increases. As k tends to infinity the variance tends to zero. Thus the Erlang can also represent the ``distribution'' of a constant - a deterministic value.
An intermediate value of k gives a shape between exponential and constant.
The Erlang can be thought of as the convolution (that is the sum of) k independent exponential random variables (each with the same mean).
To get a mean of 1/m these k exponentials must each have mean 1/km.
The variance of each exponential is 1/k2m2. Thus variance of their sum is k/k2m2 = 1/km2.
It is an important distribution in queueing theory because