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TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET
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Therefore,  G r* s a 2 1 b 1 2 1 s 2 =2 1 b : 2 s 2 2 1 b  11:19
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The Mellin transform p* s therefore has a singularity set made of simple poles sk b 4i 1 b kp =2, for k integer. As we saw in the tutorial section about the Mellin transform, this kind of set creates periodic uctuating terms in front of the polynomial expansion. This periodic uctuation is re ected in the asymptotics of pn , namely: pn  b a lv1  v1 v0 log 2 v1 v0 l 1      P 1 b G sk exp 4ikp log n n b O n b e : log 2 k
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In other words, we have proved that pn  P log n n b , where P is a periodic function, of period log 2= 1 b , whose Fourier coef cients are proportional to G sk . Since function P is not constant (indeed Fourier coef cients are all nonzero), we don't have lim inf P x lim sup P x . Figure 11.4 displays function r x computed for b 0:5, which causes uctuating polynomial coef cients. 11.3.3 A Multiplexer Queue Under an In nite Number of On=Off Sources
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In the previous section we considered on=off sources served by separate parallel queues. In this section we consider that all on=off sources forward their packet to a
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Fig. 11.4
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Density function r x , which leads to uctuating asymptotics with b 0:5:
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11.3 QUEUEING UNDER ON=OFF SOURCES
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single server called the mux server. We assume that the mux server has an in nite buffer and performs exponential service times with mean 1. This will model a router, a switch, or a multiplexer device in the network. We also assume that:  for every i, the ith on=off source has peak rate li ;  the li are all identical and equal to a given l > 1;  for every on=off source we have n0 e2 and n1 ei e2, for some sequence i i ei > 0. To make the analysis relevant we need the system to be stable, which implies that the mean workload of the queue must be strictly smaller than 1:
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Our target is still to give an asymptotic estimate of pn , which is the probability that the mux queue length is greater than n. P We x a parameter b < 1. In the sequel P suppose that the series we ei is convergent. We call Z s the Dirichlet series I es and we assume that Z s is j 1 j absolutely converent for all complex numbers s with real part strictly less than 1 b. For example, ej j1= b 1 . Note that in this case the Dirichlet series Z s z s= 1 b ; that is, it can be identi ed with the Riemann zeta function. Figure 11.5 displays the n1 (off=on rates) and n0 (on=off rates) parameters of 60 such on=off sources with b 0:5.
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Fig. 11.5 Transition rates of 60 on=off sources with b 0:5. Transition rates from on state to off state ni (on=off rates); transition rate from off state to on state di (off=on rates).
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TRAFFIC AND QUEUEING FROM AN UNBOUNDED SET
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P Theorem 11.3.4. If the Dirichlet series i es has a simple pole on s 1 b with i residue m, then the quantity pn has a polynomial lower bound with exactly determined coef cients. Proof. To simplify, we assume that the Dirichlet series can be continued to a vertical strip 1 b < R s < 1 b e. Instead of looking directly at the queue length on the mux server mode, we consider again the parallel queues of the previous section, where each source has its own server. It is clear that the sum of the length of the parallel queues turns out to be always smaller than the length of the queue on the mux server. Our aim is to show that the sum of the parallel queues has a polynomial tail of degree b. Let g z be the probability generating function of the sum ofQ queue the lengths of the multiserver mode. We have the obvious identity g z n qi z , i 1 where qi z is the probability generating function of the queue size of the server attached to the on=off source number i. Generating functions qi z are computed according to Theorem 11.3.1. Let zi be the equivalent of z1 in Theorem 11.3.1 for on=off source i. We have formally
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