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8.3 MODEL OF SELF-SIMILAR TRAFFIC
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In the case where the random process ck;j i  1, equations similar to (8.25) were found by Cox [2]. Now we give some results for an asymptotically self-similar process Y . Theorem 8.3.4. Process Y . . . ; Y 1 ; Y0 ; Y1 ; . . . de ned by Eq. (8.11), with nite mean m EYt < I and variance s2 Var Yt < I is asymptotically secondorder self-similar with parameter 0 < b < 1, if r k as k 3 I. Proof. Since r k $ const k b , as k 3 I, from Eq. (8.1) we have     m 1 P 1 1 m m i r i $ 2c1 s2 m2 b m2 Var Y0 s2 m 2 1 b 2 b i 1 c2 m2 b ; as m 3 I; 8:27
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E c0;1 i c0;1 i k I ft0;1 > i kg $ const k b
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8:26
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where c1 and c2 are some positive constants and 0 < b < 1. Substituting Eq. (8.27) into (8.3) we get r m k $ 1 c m k 1 2 b 2c2 mk 2 b c2 m k 1 2 b 2c2 m2 b 2 as m 3 I:
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1 k 1 2 b 2k 2 b k 1 2 b ; 2
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It means, according to the de nition (8.10), that process Y is asymptotically secondorder self-similar with parameter 0 < b < 1. j Corollary 8.3.5. If a random process ck;j i is a constant one, ck;j i  ck;j for all i 0; 1; 2; . . . , then a process Y . . . ; Y 1 ; Y0 ; Y1 ; . . . de ned by Eq. (8.11), with nite mean m EYt < I and variance s2 Var Yt < I, will be asymptotically second-order self-similar with parameter 0 < b < 1, if Prft0;1 > kgE c2 jt0;1 > k $ const k 1 b ; 0;1 or Prft0;1 kgE c2 jt0;1 k $ const k 2 b ; 0;1 as k 3 I: 8:29 as k 3 I; 8:28
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BOUNDS ON THE BUFFER OCCUPANCY PROBABILITY
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Proof. Since c0;1 i does not depend on i, from Eq. (8.19) we have r k
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I l P Prft0;1 > kgE c2 jt0;1 > k : 0;1 s2 i 0
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Substituting Eq. (8.28) in the above equation, we obtain r k $ const
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i k 1 b $ const k b ;
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as k 3 I:
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Then from Theorem 8.3.4 it immediately follows that Y is an asymptotically secondorder self-similar process. Statement (8.29) can be proved in the same way. j
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8.4 ASYMPTOTICAL BOUNDS FOR BUFFER OVERFLOW PROBABILITY In this section we consider the process Y de ned by Eq. (8.11) as the input traf c of a single server queueing system with constant server rate equal to C and in nite buffer size. Suppose process Y has nite mean m EYt < C < I and nite variance s2 Var Yt < I. We will consider the particular form of the process Y . Namely, we consider the case when the random process ci;j t  1. Let Prft0;1 ig $ c0 i 2 b 8:30
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as i 3 I, with 0 < b < 1. Then, according to Eq. (8.29) with c0;1  1, process Y will be asymptotically second-order self-similar with Hurst parameter H 1 b=2. Now we are interested in the queue length behavior. Let nt be the length of the queue at the moment t. Then we have nt max 0; nt 1 Yt C : 8:31
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We will estimate the probability Prfnt > zg, that is, the stationary probability to nd, at moment t, the length of the queue bigger than z, for large value of z. For any given z, let us split the process Yt into two processes Yt 1 and Yt 2 , that is, Yt Yt 1 Yt 2 ;
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