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The modelling of Ethernet data and of signals that are heavy-tailed with infinite variance. (English) Zbl 1020.60082

Let U i be i.i.d. interrenewal times such that P(Uu)=u -α L U (u), α(1,2); let W i be i.i.d. rewards independent of {U i ,i=1,2,} with EW=0 and σ 2 =EW 2 < (FVR, finite variance rewards) or P(|W|w)=w -β L W (w), w>0 (IVR, infinite variance rewards) (L U , L W are slowly varying functions at infinity). The renewal reward process is defined as W(t)=W n if t belongs to the nth interrenewal interval, W * (t,M)= m=1 M 0 t W (m) (u)du, where W (m) are i.i.d. copies of W(t). The author considers the limit behavior of W * (Tt,M) as T, M. E.g. in the FVR case it is shown that if L U * is a slowly varying function such that x>0, L U * (u) -α L(u 1/α L U * (u)x)1 as u and

lim T M T α-1 (L U * (MT)) α =,

then

lim T W * (Tt,M) T (3-α)/2 M 1/2 (L U (T)) 1/2 =σ 0 B H (t)

(in distribution), where B H is a standard fractional Brownian motion. In other results the limit processes are the symmetric Lévy motion and a symmetric β-stable process. The processes W (m) are used to describe a centered load of one workstation m in the Ethernet local area network at time t. Then W * is the aggregated load.

MSC:
60K15Markov renewal processes
90B18Communication networks (optimization)