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Unformatted text preview: ) Pr {X > t} = exp(−∏π t2 )
√
b) E [X ] = 1/(2 ∏).
Exercise 2.27. This problem is intended to show that one can analyze the long term
behavior of queueing problems by using just notions of means and variances, but that such
analysis is awkward, justifying understanding the strong law of large numbers. Consider an
M/G/1 queue. The arrival process is Poisson with ∏ = 1. The expected service time, E [Y ],
is 1/2 and the variance of the service time is given to be 1.
a) Consider Sn , the time of the nth arrival, for n = 1012 . With high probability, Sn will lie
within 3 standard derivations of its mean. Find and compare this mean and the 3σ range.
b) Let Vn be the total amount of time during which the server is busy with these n arrivals
(i.e., the sum of 1012 service times). Find the mean and 3σ range of Vn .
c) Find the mean and 3σ range of In , the total amount of time the server is idle up until
Sn (take In as Sn − Vn , thus ignoring any service time after Sn ).
d) An idle period starts when the server completes a service and there are no waiting
arrivals; it ends on the next arrival. Find the mean and variance of an idle period. Are
successive idle periods IID?
e) Combine (c) and (d) to estimate the total number of idle periods up to time Sn . Use
this to estimate the total number of busy periods.
f ) Combine (e) and (b) to estimate the expected length of a busy period.
Exercise 2.28. The purpose of this problem is to illustrate that for an arrival process with
independent but not identically distributed interarrival intervals, X1 , X2 , . . . ,, the number
of arrivals N (t) in the interval (0, t] can be a defective rv. In other words, the ‘counting
process’ is not a stochastic process according to our deﬁnitions. This illustrates that it is
necessary to prove that the counting rv’s for a renewal process are actually rv’s .
a) Let the distribution function of the ith interarrival interval for an arrival process be
FXi (xi ) = 1 − exp(−αi xi ) for some ﬁxed α ∈ (0, 1). Let Sn = X1 + · · · Xn and show that
E [Sn ] = 1 − αn−1
.
1−α b) Sketch a ‘reasonable’ sample function for N (t).
2
c) Find σSn . 2.7. EXERCISES 91 d) Use the Chebyshev inequality on Pr {Sn ≥ t} to ﬁnd an upper bound on Pr {N (t) ≤ n}
that is smaller than 1 for all n and for large enough t. Use this to show that N (t) is defective
for large enough t. Chapter 3 RENEWAL PROCESSES
3.1 Introduction Recall that a renewal process is an arrival process in which the interarrival intervals are
positive,1 independent and identically distributed (IID) random variables (rv’s). Renewal
processes (since they are arrival processes) can be speciﬁed in three standard ways, ﬁrst,
by the joint distributions of the arrival epochs, second, by the joint distributions of the
interrarival times, and third, by the joint distributions of the counting process {N (t); t ≥ 0}
in which N (t) represents the number of arrivals to a system in the interval (0, t]. The
simplest characterization is through the interarrival times, since they are IID. Perhaps the
most useful characterization, however, is through the counting process. These processes
are called renewal processes because the process probabilistically starts over at each arrival
epoch, Sn = X1 + · · · + Xn . That is, if the nth arrival occurs at Sn = τ , then, counting
from Sn = τ , the j th subsequent arrival epoch is at Sn+j − Sn = Xn+1 + · · · + Xn+j . Thus,
given Sn = τ , {N (τ + t) − N (τ ); t ≥ 0} is a renewal counting process with IID interarrival
intervals of the same distribution as the original renewal process. Because of this renewal
property, we shall usually refer to arrivals as renewals.
The ma jor reason for studying renewal processes is that many complicated processes have
randomly occurring instants at which the system returns to a state probabilistically equivalent to the starting state. These embedded renewal epochs allow us to separate the long term
behavior of the process (which can be studied through renewal theory) from the behavior
of the actual process within a renewal period.
Example 3.1.1. Consider a G/G/m queue. The arrival counting process to a G/G/m
queue is a renewal counting process, {N (t); t ≥ 0}. Each arriving customer waits in the
queue until one of m identical servers is free to serve it. The service time required by each
1 Renewal processes are often deﬁned in a slightly more general way, allowing the interarrival intervals Xi
to include the possibility 1 > Pr {Xi = 0} > 0. All of the theorems in this chapter are valid under this more
general assumption, as can be veriﬁed by complicating the proofs somewhat. Allowing Pr {Xi = 0} > 0
allows multiple arrivals at the same instant, which makes it necessary to allow N (0) to take on positive
values, and appears to inhibit intuition about renewals. Exercise 3.2 shows how to view these more general
renewal processes while using the deﬁnition here, thus show...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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