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Unformatted text preview: oisson
λ✲ M/M/1 M/M/1
λ µ1 λ ✲ ✲ µ2 Consider tandem M/M/1 queues. Departures from
ﬁrst are Poisson with rate λ. Assume service times
at rates µ1 and µ2, independent from queue to queue
and independent of arrivals at each.
Arrivals at queue 2 are Poisson at rate λ by Burke
and are independent of service times at 2. Thus the
second queue is M/M/1.
The states of the two systems are independent and
the time of a customer in system 1 is independent of
that in 2.
11 Random walks
Def: Let {Xi; i ≥ 1} b e a sequence of I ID rv’s, and let
Sn = X1 + X2 + · · · + Xn for n ≥ 1. The integertime
stochastic process {Sn; n ≥ 1} is called a random walk,
or, speciﬁcally, the random walk based on {Xi; i ≥ 1}.
We are used to sums of I ID rv’s, but here the interest
is in the process. We ask such questions as:
1) Threshold crossing: For given α > 0, what is the
probability that Sn ≥ α for at least one n ≥ 1; what
is the smallest n for which this crossing happens; and
what is the overshoot Sn − α?
2) Two thresholds: For given α > 0, β < 0, what is the
probability that {Sn; n ≥ 1 crosses α b efore it crosses
β , and what is the n at which the ﬁrst such crossing
o ccurs?
12 These thresholdcrossing problems are important in
studying overﬂow in queues, errors in digital commu
nication systems, hypothesis testing, ruin and other
catastrophes, etc.
In many of the important applications, the relevant
probabilities are very small and the problems are known
as large deviation problems.
Moment generating functions and their use in upper
b ounds on these small probabilities are important here.
We start with a brief discussion of 3 s...
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This note was uploaded on 01/13/2012 for the course ELECTRICAL 6.262 taught by Professor Staff during the Fall '11 term at MIT.
 Fall '11
 staff

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