# From the results above for branching processes we see

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Unformatted text preview: x + x2 = (1 x)2 i.e., a double root at x = 1. In general when µ = 1, the graph of is tangent to the diagonal (x, x) at x = 1. This slows down the convergence of ⇢n to 1 so that it no longer occurs exponentially fast. In more advanced treatments, it is shown that if the o↵spring distribution has mean 1 and variance 2 > 0, then P1 (Xn > 0) ⇠ 2 n2 This is not easy even for the case of binary branching, so we refer to reader to Section 1.9 of Athreya and Ney (1972) for a proof. We mention the result here because it allows us to see that the expected time for the process to die out P n P1 (T0 > n) = 1. If we modify the branching process, so that p(0, 1) = 1 then in the modiﬁed process If µ < 1, 0 is positive recurrent 59 1.11. CHAPTER SUMMARY If µ = 1, 0 is null recurrent If µ > 1, 0 is transient Our ﬁnal example gives an application of branching processes to queueing theory. Example 1.54. M/G/1 queue. We will not be able to explain the name of this example until we consider Poisson processes in Chapter 2. However, imagine a que...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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