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Unformatted text preview: equent results about timeaverages. The next topic is the
expected renewal rate, E [N (t)] /t. If the Laplace transform of the interrenewal density
is rational, E [N (t)] /t can be easily calculated. In general, the Wald equality shows that
limt→1 E [N (t)] /t = 1/X . Finally, Blackwell’s theorem shows that the renewal epochs
reach a steadystate as t → 1. The form of this steadystate depends on whether the interrenewal distribution is arithmetic (see (3.18)) or nonarithmetic (see (3.17) and (3.19)).
Sections 3.4 and 3.5 add a reward function R(t) to the underlying renewal process; R(t)
depends only on the interrenewal interval containing t. The timeaverage value of reward
exists with probability 1 and is equal to the expected reward over a renewal interval divided
by the expected length of an interrenewal interval. Under some minor restrictions imposed
by the key renewal theorem, we also found that, for nonarithmetic interrenewal distributions, limt→1 E [R(t)] is the same as the timeaverage value of reward. These general
results were applied to residual life, age, and duration, and were also used to derive and
understand Little’s theorem and the PollaczekKhinchin expression for the expected delay
in an M/G/1 queue.
Finally, all the results above were shown to apply to delayed renewal processes.
For further reading on renewal processes, see Feller,[9], Ross, [16], or Wolﬀ, [22]. Feller still
appears to be the best source for deep understanding of renewal processes, but Ross and
Wolﬀ are somewhat more accessible. 3.9 Exercises Exercise 3.1. The purpose of this exercise is to show that for an arbitrary renewal process,
the number of renewals in (0, t] is a random variable for each t > 0, i.e., to show that N (t),
for each t > 0, is an actual rv rather than a defective rv.
a) Let X1 , X2 , . . . , be a sequence of IID interrenewal rv’s . Let Sn = X1 + · · · + Xn be the
corresponding renewal epochs for each n ≥ 1. Assume that each Xi has a ﬁnite expectation
X > 0 and use the weak law of large numbers to show that limn→1 Pr {Sn < t} = 0.
b) Use part a) to show that limn→1 Pr {N ≥ n} = 0 and explain why this means that N (t)
is not defective.
c) Prove that N (t) is not defective without assuming that each Xi has a mean (but assuming
that Pr {X = 0} 6= 1.) Hint: Lower bound each Xi by a binary rv Yi (i.e., Xi (ω ) ≥ Yi (ω )
for each sample point ω ) and show that this implies that FXi (xi ) ≤ FYi (yi ). Be sure you
understand this strange reversal of inequality signs. 128 CHAPTER 3. RENEWAL PROCESSES Exercise 3.2. Let {Xi ; i ≥ 1} be the interrenewal intervals of a renewal process generalized to allow for interrenewal intervals of size 0 and let Pr {Xi = 0} =α, 0 < α < 1.
Let {Yi ; i ≥ 1} be the sequence of nonzero interarrival intervals. For example, if X1 =
x1 >0, X2 = 0, X3 = x3 >0, . . . , then Y1 =x1 , Y2 =x3 , . . . , .
a) Find the distribution function of each Yi in terms of that of the Xi .
b) Find the PMF of the number of arrivals of the generalized renewal process at each epoch
at which arrivals occur.
c) Explain how to view the generalized renewal process as an ordinary renewal process with
interrenewal intervals {Yi ; i ≥ 1} and bulk arrivals at each renewal epoch.
Exercise 3.3. Let {Xi ; i≥1} be the interrenewal intervals of a renewal process and assume
e
that E [Xi ] = 1. Let b > 0 be an arbitrary number and Xi be a truncated random variable
e
e
deﬁned by Xi = Xi if Xi ≤ b and Xi = b otherwise.
hi
e
a) Show that for any constant M > 0, there is a b suﬃciently large so that E Xi ≥ M .
e
e
b) Let {N (t); t≥0} be the renewal counting process with interrenewal intervals {Xi ; i ≥ 1}
e
and show that for all t > 0, N (t) ≥ N (t). c) Show that for all sample functions N (t, ω ), except a set of probability 0, N (t, ω )/t < 2/M
for all suﬃciently large t. Note: Since M is arbitrary, this means that lim N (t)/t = 0 with
probability 1.
Exercise 3.4. a) Let J be a stopping rule and In be the indicator random variable of the
P
event {J ≥ n}. Show that J = n≥1 In .
b) Show that I1 ≥ I2 ≥ I3 ≥ . . . , i.e., show that for each n > 1, In (ω ) ≥ In+1 (ω ) for each
ω ∈ ≠ (except perhaps for a set of probability 0).
Exercise 3.5. Is it true for a renewal process that:
a) N (t) < n if and only if Sn > t?
b) N (t) ≤ n if and only if Sn ≥ t?
c) N (t) > n if and only if Sn < t?
Exercise 3.6. Let {N (t); t ≥ 0} be a renewal counting process and let m(t) = E [N (t)]
be the expected number of arrivals up to and including time t. Let {Xi ; i ≥ 1} be the
interrenewal times and assume that FX (0) = 0.
a) For all x > 0 and t > x show that E [N (t)X1 =x] = E [N (t − x)] + 1.
Rt
b) Use part (a) to show that m(t) = FX (t) + 0 m(t − x)dFX (x) for t > 0. This equation
is the renewal equation derived diﬀerently in (3.6).
c) Suppose that X is an exponential random variable of parameter ∏. Evaluate Lm (s) from
(3.7); verify that the inverse La...
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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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