Discrete-time stochastic processes

Poisson processes for the sum process thus arrivals

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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 time-slot 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.

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