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Unformatted text preview: f size δ = 2−j , the probability of one arrival in an
increment is ∏δ and that of no arrival is 1 − ∏δ , and thus (2.17) is satisﬁed, and in fact
the o(δ ) terms are exactly zero. For arbitrary sized increments, it is clear that disjoint
increments have independent arrivals. The increments are not quite stationary, since, for
example, an increment of size 2−j −1 might contain a time that is a multiple of 2−j or might
not, depending on its placement. However, for any ﬁxed increment of size δ , the number
of multiples of 2−j (i.e., the number of possible arrival points) is either bδ 2j c or 1 + bδ 2j c.
Thus in the limit j → 1, the increments are both stationary and independent.
For each j , the j th Bernoulli process has an associated Bernoulli counting process Nj (t) =
Pbt2j c
er arrivals up to time t and is a discrete rv with the binomial
i=1 Yi . This is the numb° of ¢
j
2j
PMF. That is, pNj (t) (n) = btn c q n (1 − q )bt2 c−n where q = ∏2−j . We now show that this
PMF approaches the Poisson PMF as j increases Theorem 2.4. Consider the sequence of shrinking Bernoul li processes with arrival probability ∏2−j and timeslot size 2−j . Then for every ﬁxed time t > 0 and ﬁxed number of
arrivals n, the counting PMF pNj (t) (n) approaches the Poisson PMF (of the same ∏) with
increasing j , i.e., lim pNj (t) (n) = pN (t) (n). j →1 (2.18) 70 CHAPTER 2. POISSON PROCESSES Proof: We ﬁrst rewrite the binomial PMF, for bt2j c variables with q = ∏2−j as
µ j ∂µ
∂n
bt2 c
∏2−j
lim pNj (t) (n) = lim
exp[bt2j c(ln(1 − ∏2−j )]
j →1
j →1
n
1 − ∏2−j
µ j ∂µ
∂n
bt2 c
∏2−j
= lim
exp(−∏t)
(2.19)
j →1
n
1 − ∏2−j
µ
∂n
bt2j c · bt2j −1c · · · bt2j −n+1c
∏2−j
= lim
exp(−∏t) (2.20)
j →1
n!
1 − ∏2−j
(∏t)n exp(−∏t)
=
.
(2.21)
n!
We used ln(1 − ∏2−j ) = −∏2−j + o(2−j ) in (2.19) and expanded the combinatorial term
≥ −j ¥
∏
in (2.20). In (2.21), we recognized that limj →1 bt2j − ic 1−2 2−j = ∏t for 0 ≤ i ≤ n − 1.
∏
Since the binomial PMF (scaled as above) has the Poisson PMF as a limit for each n, the
distribution function of Nj (t) also converges to the Poisson distribution function for each
t. In other words, for each t > 0, the counting random variables Nj (t) of the Bernoulli
processes converge in distribution to N (t) of the Poisson process. By Deﬁnition 2, then,6
this limiting process is a Poisson process.
With the same scaling, the distribution function of the geometric distribution converges to
the exponential distribution function,
j −1c lim (1 − ∏2−j )bt2 j →1 = exp(−∏t). Note that no matter how large j is, the corresponding shrunken Bernoulli process can have
arrivals only at the discrete times that are multiples of 2−j . Thus the interarrival times
and the arrival epochs are discrete rv’s and in no way approach the densities of the Poisson
process. The distribution functions of these rv’s, however quickly approach the distribution
functions of the corresponding Poisson rv’s. This is a good illustration of why it is sensible
to focus on distribution functions rather than PDF’s or PMF’s. 2.3 Combining and splitting Poisson processes Suppose that {N1 (t), t ≥ 0} and {N2 (t), t ≥ 0} are independent Poisson counting processes7
of rates ∏1 and ∏2 respectively. We want to look at the sum process where N (t) = N1 (t) +
6 Note that the counting process is one way of deﬁning the Poisson process, but this requires the joint
distributions of the counting variables, not just the marginal distributions. This is why Deﬁnition 2 requires
not only the Poisson distribution for each N (t) (i.e., each marginal distribution), but also the stationary
and independent increment properties. Exercise 2.5 gives an example for how the binomial distribution can
be satisﬁed without satisfying the discrete version of the independent increment property.
7
Two processes {N1 (t); t ≥ 0} and {N2 (t); t ≥ 0} are said to be independent if for all positive integers k and all sets of times t1 , . . . , tk , the random variables N1 (t1 ), . . . , N1 (tk ) are independent of
N2 (t1 ), . . . , N2 (tk ). Here it is enough to extend the independent increment property to independence between increments over the two processes; equivalently, one can require the interarrival intervals for one
process to be independent of the interarrivals for the other process. 2.3. COMBINING AND SPLITTING POISSON PROCESSES 71 N2 (t) for all t ≥ 0. In other words, {N (t), t ≥ 0} is the process consisting of all arrivals
to both process 1 and process 2. We shall show that {N (t), t ≥ 0} is a Poisson counting
process of rate ∏ = ∏1 + ∏2 . We show this in three diﬀerent ways, ﬁrst using Deﬁnition
3 of a Poisson process (since that is most natural for this problem), then using Deﬁnition
2, and ﬁnally Deﬁnition 1. We then draw some conclusions about the way in which each
approach is helpful. Since {N1 (t); t ≥ 0} and {N2 (t); t ≥ 0}...
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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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