SEMINARS
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Fall 2009
STATISTICS
COLLOQUIUM
Friday, November 6, 2009
2:30-3:00—Refreshments, Yost 327
3:00-4:00—Talk, Yost 101
Barbara Margolius, PhD
Department of Mathematics
Cleveland State University
The Much Maligned Bessel Formula for the Transition Probabilities of the Simplest Interesting Queue
The M/M/1 queue in which customers arrive to a single server
according to a Poisson process and are served at an
exponential rate is often referred to in the literature
as the simplest interesting queue. We study it in hope of
learning something that we can apply to more complex processes.
In this talk, we extend some classical formulas for single
server queues to single server queues with time-varying rates.
Note that there are a great many formulas for the transient
distribution of the single server queue and most do not have
time-varying analogs because their derivation relies in some
fundamental way on the transition rates being constant. In
particular, we focus on the standard textbook formula for the
transient distribution of the number in the queue. This formula
involves an infinite sum of modified Bessel functions. It
appears in most standard textbooks on queueing theory, but it is
not well liked as formulas go. It has been called “most
disheartening”[1975], “somewhat daunting”[1989], “far from
convenient”[1965] and “difficult to evaluate or interpret”[2009].
We will discuss how to evaluate and interpret this formula and
related formulas and how they might be extended to more complex
systems.
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