A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL in .NET

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7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL
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nonnegative random variables independent of fXon ; Xn ; n ! 1g representing off periods and these have common distribution Foff . The means are mon I
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 Fon s ds;
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moff
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 Foff s ds;
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which are assumed nite and the sum of the means is m : mon moff . Using these random variables we generate an alternating renewal sequence characterized as follows. 1. The interarrival distribution is Fon Foff and the mean interarrival time is m mon moff . 2. The renewal times are  0;
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n P i 1
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 Xi Yi ; n ! 1 :
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Because of the niteness of the means, the renewal process has a stationary version:  D; D
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 Xi Yi ; n ! 1 :
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where D is a delay random variable satisfying P D > x I I
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P Xon Yoff > s ds m 1 Fon Foff s ds: m
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However, making the process stationary in this manner has the disadvantage that the initial delay period D does not decompose into an on and an off period the way subsequent inter-renewal periods do and the following procedure is preferable for generating the stationary alternating renewal process. De ne independent random 0 0 variables B; Xon ; Yoff , which are assumed independent of fXon ; Xn ; n ! 1g and fYoff ; Yn ; n ! 1g, by P B 1
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0 P Xon 0 P Yoff
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mon 1 P B 0 ; m I 1 Fon s 0 > x ds : 1 Fon x ; mon x I 1 Foff s 0 > x ds : 1 Foff x : moff x
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FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING
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The delay random variable D 0 is de ned by
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0 0 D 0 B Xon Yoff 1 B Yoff :
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This delayed renewal sequence   n P fSn ; n ! 0g : D 0 ; D 0 Xi Yi ; n ! 1
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is a stationary renewal process. 7.2.2 High Variability Induces Long-Range Dependence
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Consider the indicator process fZt g, which is 1 iff t is in an on period. Thus, for t ! D 0 ,  Zt and if 0 1; 0; if Sn t < Sn Xn 1 ; some n if Sn Xn 1 t < Sn 1 ; some n
t < D 0 we de ne ( Zt 1; 0; if B 1 and 0 otherwise:
0 t < Xon ;
A standard renewal argument gives the following result [22]. Proposition 7.2.1. fZt ; t ! 0g is strictly stationary and P Zt 1 mon : m
Conditional on Zt 1, the subsequent sequence of on=off periods is the same as seen from time 0 in the stationary process with B 1. It is easiest to express long-range dependence in terms of slow decay of covariance functions so we consider the second-order properties of the stationary process fZt g (See Heath et al. [22].) The basis for the next result is a renewal theory argument. Theorem 7.2.2. The covariance function g s Cov Zt ; Zt s
7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL
of the stationary process fZ t ; t ! 0g is   s mon moff 0  off s u Fon U du F g s m m 0   m m 0 on off Fon U 1 Foff s m m m m 1 s on 2 off z s o U dw ; m 0 m where U and z t
I P n 0
Fon Foff n
  Foff x Fon t x dx
0 mon Fon 1 Foff t 0 moff Foff 1 Fon t :
How do we analyze the asymptotic behavior of g as a function of s Note g s is of the form h i g s const lim z U v z U s
so we need rates of convergence in the key renewal theorem. This can be based on a theorem of Frenk [19] and is given in Heath et al. [22]. Theorem 7.2.3. Assume that there is an n ! 1 such that Fon Foff n is nonsingular. Suppose  Fon t t a L t ; t 3 I;
where 1 < a < 2 and L is slowly varying at in nity and assume   Foff t o Fon t ; Then g t $ m2 off t a 1 L t ; a 1 m3 t 3 I: t 3 I: