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Unformatted text preview: process, with IID interrenewal times, is delayed until
after the epoch of the ﬁrst renewal. What we shall discover is intuitive — both the time
average behavior and, in essence, the limiting ensemble behavior are not aﬀected by the
11 This is often called the PASTA property, standing for Poisson arrivals see timeaverages. This holds
with great generality, requiring only that timeaverages exist and that the state of the system at a given
time t is independent of future arrivals. 3.7. DELAYED RENEWAL PROCESSES 123 distribution of the ﬁrst renewal epoch. It might be somewhat surprising, however, to ﬁnd
that this irrelevance of the distribution of the ﬁrst renewal epoch holds even when the mean
of the ﬁrst renewal epoch is inﬁnite.
To be more precise, we let {Xi ; i≥1} be a set of independent nonnegative random variables.
X1 has some distribution function G(x), whereas {Xi ; i ≥ 2} are identically distributed with
some distribution function F (x). Typically, G(P 6= F (x), since if equality held, we would
x)
have an ordinary renewal process. Let Sn = n Xi be the nth renewal epoch and let
i=1
N (t) be the number of renewal epochs up to and including time t (i.e., N (t) ≥ n if and only
if Sn ≤ t). {N (t); t ≥ 0} is then called a delayed renewal counting process. The following
simple lemma follows from lemma 3.1.
Lemma 3.2. Let {N (t); t ≥ 0} be a delayed renewal counting process. Then limt→1 N (t) =
1 with probability 1 and limt→1 E [N (t)] = 1.
Proof: Conditioning on X1 = x, we can write N (t) = 1 + N 0 (t − x) where N 0 {t; t ≥ 0} is the
ordinary renewal counting process with interrenewal intervals X2 , X3 , . . . . From Lemma
3.1, limt→1 N 0 (t − x) = 1 with probability 1, and limt→1 E [N 0 (t − x)] = 1. Since this is
true for every ﬁnite x > 0, and X1 is ﬁnite with probability 1, the lemma is proven.
Theorem 3.9 (Strong Law for Delayed Renewal Processes). Let a delayed renewal
R1
process have mean interrenewal interval X 2 = x=0 [1 − F (x)] dx. Then
lim t→1 N (t)
1
=
t
X2 with probability 1. (3.68) Proof: As in the proof of Theorem 3.1, we have
SN (t)
SN (t)+1
t
≤
≤
.
N (t)
N (t)
N (t)
PN (t)
SN (t)
X1
n=2 Xn
lim
= lim
+ lim
t→1 N (t)
t→1 N (t)
t→1 N (t) − 1 (3.69) N (t) − 1
.
N (t) (3.70) From Lemma 3.2, N (t) approaches 1 as t → 1. Thus for any ﬁnite sample value of X1 ,
the ﬁrst limit on the right side of (3.70) approaches 0. Since X1 is a random variable, it
takes on a ﬁnite value with probability 1, so this ﬁrst term is 0 with probability 1 (note that
this does not require X1 to have a ﬁnite mean). The second term in (3.70) approaches X 2
with probability 1 by the strong law of large numbers. The same argument applies to the
right side of (3.69), so that limt→1 N t t) = X 2 with probabilty 1. Eq. (3.68) then follows.
(
A truncation argument, as in Exercise 3.3, shows that the theorem is still valid if X 2 = 1. Next we look at the elementary renewal theorem and Blackwell’s theorem for delayed renewal processes. To do this, we view a delayed renewal counting process {N (t); t ≥ 0} as an
ordinary renewal counting process that starts at a random nonnegative epoch X1 with some
distribution function G(t). Deﬁne No (t − X1 ) as the number of renewals that occur in the interval (X1 , t]. Conditional on any given sample value x for X1 , {No (t−x); t−x≥0} is an ordinary renewal counting process and thus, given X1 = x, limt→1 E [No (t − x)] /(t−x) = 1/X 2 . 124 CHAPTER 3. RENEWAL PROCESSES Since N (t) = 1 + No (t − X1 ) for t > X1 , we see that, conditional on X1 = x,
lim t→1 E [N (t)  X1 =x]
E [No (t − x)] t − x
1
= lim
=
.
t→1
t
t−x
t
X2 (3.71) Since this is true for every ﬁnite sample value x for X1 , we can take the expected value over
X1 to get the following theorem:
Theorem 3.10 (Elementary Delayed Renewal Theorem). For a delayed renewal process with E [Xi ] = X 2 for al l i ≥ 2,
lim t→1 E [N (t)]
1
=
..
t
X2 (3.72) The same approach gives us Blackwell’s theorem. Speciﬁcally, if {Xi ; i≥2} are nonarithmetic,
then, using Blackwell’s theorem for ordinary renewal processes, for any δ > 0,
lim t→1 E [No (t − x + δ ) − No (t − x)]
1
=
.
δ
X2 (3.73) Thus, conditional on any sample value X1 = x, limt→1 E [N (t+δ ) − N (t)  X1 =x] = δ /X 2 .
Taking the expected value over X1 gives us limt→1 E [N (t + δ ) − N (t)] = δ /X 2 . The case
in which {Xi ; i≥2} are arithmetic with span d is somewhat more complicated. If X1 is
arithmetic with span d (or a multiple of d), then the ﬁrst renewal epoch must be at some
multiple of d and d/X 2 gives the expected number of arrivals at time id in the limit as
i → 1. If X1 is nonarithmetic or arithmetic with a span other than a multiple of d, then
the eﬀect of the ﬁrst renewal epoch never dies out, since all subsequent renewals occur at
multiples of d from this ﬁrst epoch. This gives us the theorem:
Theorem 3.11 (Blackwell for Delayed Renewal). If {Xi ; i≥2} are nonarithmetic, then,
for al l δ > 0,
lim t→1 E [N (t + ...
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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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