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 for every x ! 0 and y ! 0 such that Foff y > 0. The class of distributions satisfying this condition includes Gamma distributions, distributions with compact support, and others. For related nite buffer results, see Jelenkovic [29] and Zwart [50]. 7.4 AN INFINITE SOURCE MODEL
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The dif culty in analyzing the J -channel on=off model of the previous section suggests letting J 3 I in order to achieve additional tractability. So suppose now there are an in nite number of nodes in a network. Sources turn on and initiate sessions at Poisson time points fGk ; k ! 1g. The rate of the Poisson process is l. The lengths of sessions are i.i.d. random variables fXn g, which have a common distribution Fon and assume E X1 mon . During a session, an active source transmits at rate 1 and whenever there is work, the single server works at rate r. An important quantity is N t number of sessions in progress at t I P 1 Gk <t Gk Xk
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the number of busy servers in the M =G=I queue: Traditional queueing theory recognizes N t as the number of busy trunklines in an in nite line telephone model. It is well known and easy to derive the following facts. 1. N t is Poisson distributed. At equilibrium the mean is lmon .
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2. At equilibrium, the stationary version of N has covariance function at lag s I   equal to s Fon u du. So if Fon is heavy tailed, there will be slow decay of the covariance function and long-range dependence will be present. Both facts follow readily from the fact that the point process with points f Gk ; Xk g is a two-dimensional Poisson process with mean measure l ds F dy . We de ne the contents process X by dX t N t dt r1 X t >0 dt so the input rate is random depending on the number of active sources at a given time. To ensure stability we assume long-term input rate lmon < r: For additional discussion of this model, see Boxma [8], Brichet et al. [10], Konstantopoulos and Lin [32], and Resnick and Rootzen [46]. A discrete time analog of this model is discussed in 9. 7.4.1 Activity Periods
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Following Jelenkovic and Lazar [30, 31], we de ne an activity period to be a busy period in the corresponding M=G=I queue so if at time 0 the initial conditions are that N 0 0 and a node turns on, then the length of the initial activity period is inf ft > 0 : N t 0g. Note that an activity period could end with nonzero content; it just depends on there being no active sources. The length of an activity period is a random variable whose distribution is known by means of a Laplace transform formula as given in Takacs [47] or Hall [20]. Based on this, we may derive another interesting quantity. Let n be the number of sessions in an activity period and de ne  p u P
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to be the probability that at least one of the sessions forming an activity period is longer than u. Then based on the Takacs formula [47] for the Laplace transform of the M =G=I busy period, we may derive the following (see Resnick and Samorodnitsky [41]): p u u
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This allows us (see Resnick and Samorodnitsky [41] for the proof ) to generalize a result of Jelenkovic and Lazar [28]. See also 10. Our result weakens assumptions on Fon and weakens the assumption r < 1. Theorem 7.4.1. Let I be the change in the buffer content during the activity period. Assume that the stability condition holds, and that lmon < r < 1 lm: 7:3
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Assume the session length distribution Fon has a regularly varying (or only dominatedly varying) tail with index a > 1. Then lim  P I > x  elmon : x 1 r lmon 7:4
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