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7.2 A SINGLE CHANNEL ON/OFF COMMUNICATION MODEL
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The rst downgoing ladder epoch of f
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xi ; n ! 0g is  0 : 7:2
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 n P  N inf n > 0 : xi
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The tail behavior of the maximum in a (discrete) cycle is described next. See Asmussen [2] and Heath et al. [21]. Proposition 7.2.4. For the stable queueing process fX Sn g satisfying 1 Fon x x a L x ; a > 1; x 3 I;
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the maximum over a cycle has a distribution tail asymptotic to the tail of the on distribution and, in particular, " P
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 N W n 0
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#  X Sn > x $ P x1 > x E N  $ P 1 r X1 > x E N a  $ 1 r Fon x E N :
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Note that this tail is lighter than the tail of W or V from the previous section. Interestingly, the off distribution in uences this expression only through the factor  E N . From maxima over discrete cycles, it is relatively simple to derive tail behavior over a continuous time cycle: de ne C1 SN .  Corollary 7.2.5. Assume the contents process fX t g is stable and 1 Fon x x a L x ; a > 1; x 3 I:
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The distribution tail of the maximum of the contents process over one cycle is asymptotic to the tail of the on distribution; P C 1 W
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  X s > x $ 1 r a Fon x E N :
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Again note the minimal effect of Foff , which only affects the answer through the  multiplicative factor E N . Now the behavior up to an arbitrary time t is derived from the random number of cycles squeezed into 0; t and we have the following result from Heath et al. [21]. (Let Dr 0; I be the right continuous functions on 0; I with nite left limits; Dl 0; I are the left continuous functions with nite right limits.)
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FLUID QUEUES, ON=OFF PROCESSES, AND TELETRAFFIC MODELING
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Theorem 7.2.6.
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Assume fX t g stable and 1 Fon x x a L x ; a > 1; x 3 I:
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De ne the quantile function 1 b s 1 Fon  2 s :
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Let fYa t ; t > 0g be the extremal process [44] generated by Fa x expf x a g; so that P Ya t De ne the rescaled extremal process Sa t 1 r Y t : m1=a a x Fta x : x>0
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Then in Dr 0; I Dl 0; I , as u 3 I,   2  M u M u 2 ; A Sa ; Sa : b u b u In particular, we get for the rst passage process, as u 3 I, 1 Fon u t u A and lim P  1 r a 1 Fon L t L m  x P E 1 x 1 e x ; x > 0;
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2 Ya
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where E 1 is a unit exponential random variable. Also, as x 3 I, 1 Fon x E t x 3 m ; 1 r a
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so equivalently E t x $ m xa =L x : 1 r a
The approximation to E t x is only rst order and not expected to be dazzlingly good but to test this we did two modest simulations with Fon Pareto; a 1:5; r 0:53  the same Pareto or Foff constant off times equal to 3; The number of replications was 500 and the levels L were 2, 5, 10, 22, 46, 100, 215, 464. The simulation with the constant off times shows our approximation is surprisingly effective but, as expected, having variability in the off distribution makes the approximation less accurate. See Fig. 7.1. As a nal experiment, we decided to test the correctness of the intuition that a high level crossing by the content level was due to a single very long on period, rather than due to gradual buildup. We ran 1000 simulation runs of the system. Each simulation consisted of running the system until level L 64 was crossed and