2.6: Summary

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Page ID: 75633

Robert Gallager
Massachusetts Institute of Technology via MIT OpenCourseWare

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We started the chapter with three equivalent definitions of a Poisson process—first as a renewal process with exponentially distributed inter-renewal intervals, second as a stationary and independent increment counting process with Poisson distributed arrivals in each interval, and third essentially as a limit of shrinking Bernoulli processes. We saw that each definition provided its own insights into the properties of the process. We emphasized the importance of the memoryless property of the exponential distribution, both as a useful tool in problem solving and as an underlying reason why the Poisson process is so simple.

We next showed that the sum of independent Poisson processes is again a Poisson process. We also showed that if the arrivals in a Poisson process are independently routed to different locations with some fixed probability assignment, then the arrivals at these locations form independent Poisson processes. This ability to view independent Poisson processes either independently or as a splitting of a combined process is a powerful technique for finding almost trivial solutions to many problems.

It was next shown that a non-homogeneous Poisson process could be viewed as a (homogeneous) Poisson process on a non-linear time scale. This allows all the properties of (homogeneous) Poisson properties to be applied directly to the non-homogeneous case. The simplest and most useful result from this is (2.34), showing that the number of arrivals in any interval has a Poisson PMF. This result was used to show that the number of customers in service at any given time \(\tau\) in an \(\mathrm{M} / \mathrm{G} / \infty\) queue has a Poisson PMF with a mean approaching \(\lambda\) times the expected service time in the limit as \(\tau \rightarrow \infty\).

Finally we looked at the distribution of arrival epochs conditional on n arrivals in the interval \((0, t]\). It was found that these arrival epochs had the same joint distribution as the order statistics of n uniform IID rv’s in \((0, t]\). By using symmetry and going back and forth between the uniform variables and the Poisson process arrivals, we found the distribution of the interarrival times, the arrival epochs, and various conditional distributions.