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9.4 SECOND-ORDER CORRELATIONS
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Stationary M =G=I processes being time-reversible, we have the representation qb st sup Stb ct; t 0; 1; . . . I for the steady-state buffer content qb with I
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Hereafter, by an M =G=I input process we mean its stationary version fb*; t 0; 1; . . .g, which is fully characterized by the pair l; G . Moreover, from t now on, we always assume the stability condition rin  lE s < c: 9:14
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SECOND-ORDER CORRELATIONS
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Before discussing the asymptotics associated with buffer over ow induced by M =G=I input processes, we make a slight detour to explore the correlation structure of such input processes. 9.4.1 Correlation Properties
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In view of Proposition 9.2.1, the stationary version fb*; t 0; 1; . . .g has a wellt de ned (auto)covariance function G : R 3 R, say, * G h  Cov b*; bt h ; t Proposition 9.4.1. We have h 0; 1; . . . : 9:16 t; h 0; 1; . . . : 9:15
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The rst equality in Eq. (9.16) is established in Cox and Isham [7] and the second equality follows readily from the de nition (9.9). From Eq. (9.16) we nd the autocorrelation function g : R 3 R of the M=G=I process l; s to be given by g h  G h ^ P s > h ; G 0 h 0; 1; . . . : 9:17
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Note that g 0 1 as we recall that P s > 0 1. 9.4.2 Inverting g
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Proposition 9.4.1 shows that the correlation structure of the stationary M =G=I ^ input process l; s is completely determined by the pmf of s (thus of s). It turns out
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BUFFER ASYMPTOTICS FOR M=G=I INPUT PROCESSES
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that the inverse is true as well. Indeed, Eqs. (9.9) and (9.17) together imply ^ ^ g h g h 1 P s > h P s > h 1 1 P s > h ; h 0; 1; . . . ; E s
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so that the mapping h 3 g h is necessarily decreasing and integer-convex. Taking into account the facts g 0 1 and P s > 0 1, we conclude from Eq. (9.18) (with h 0) that E s 1 1 g 1 9:19
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with g 1 < 1 necessarily by the niteness of E s . Combining Eqs. (9.18) and (9.19) we nd that P s > h Note also from Eq. (9.20) that E s
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I P h 0
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h 0; 1; . . . :
9:20
P s > h
1 limh3I g h 1 g 1
9:21
and Eq. (9.19) imposes limh3I g h 0. A moment of re ection readily leads to the following invertibility result. Proposition 9.4.2. An R -valued sequence fg h ; h 0; 1; . . .g is the autocorrelation function of the M=G=I process l; s with integrable s if and only if the corresponding mapping h 3 g h is decreasing and integer-convex with g 0 1 > g 1 and limh3I g h 0, in which case the pmf G of s is given by Eq. (9.20). 9.5 LONG-RANGE DEPENDENCE
The existence of positive correlations in the sequence fb*; t 0; 1; . . .g is clearly t apparent from Eq. (9.16). The strength of such positive correlations can be formalized in several ways, which we now describe; additional material is available in Cox [8] and we refer the reader to Tsybakov and Georganas [32] for a discussion of alternative de nitions. The sequence fb*; t 0; 1; . . .g is said to be short-range dependent (SRD) if t
I P h 0
G h < I:
9:22
9.5 LONG-RANGE DEPENDENCE
Otherwise, the sequence fb*; t 0; 1; . . .g is said to be long-range dependent t (LRD). Easy calculations using Eq. (9.16) readily lead to the following simple characterization. Proposition 9.5.1. We have l G h E s s 1 2 h 0 so that the process is SRD if and only if E s2 is nite. Interesting subclasses of LRD processes can further be identi ed through the notion of second-order self-similarity. To do so, we introduce the rvs b m  t P 1 m 1 b* ; m k 0 mt k m 1; 2; . . . ; t 0; 1; . . . : 9:24
9:23
For each m 1; . . . , the rvs fb m ; t 0; 1; . . .g form a (wide-sense) stationary t sequence with correlation structure de ned by G m h  Cov b m ; b m t t h and g m h  G m h ; G m 0 h 0; 1; . . . : 9:25
For each H > 0 consider the mapping gH : R 3 R given by gH h  1 jh 1j2H 2jhj2H jh 1j2H ; 2 h 0; 1; . . . : 9:26
We say that the sequence fb*; t 0; 1; . . .g is exactly (second-order) self-similar if t Var b m d2 m b ; t m 1; 2; . . . 9:27
for some constants d2 > 0 and 0 < b < 1, a requirement equivalent to G h d2 gH h ; h 0; 1; . . . ; 9:28
where H  1 b=2 is known as the Hurst parameter of the process. The parameter H being in the range (0.5, 1), the mapping gH is strictly decreasing and integerconvex, with gH 0 1, and behaves asymptotically as gH h $ H 2H 1 h2H 2 h 3 I : 9:29