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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 \(P(U\geq u)=u^{-\alpha}L_U(u)\), \(\alpha\in(1,2)\); let \(W_i\) be i.i.d. rewards independent of \(\{U_i,\;i=1,2,\dots\}\) with \(E W=0\) and \(\sigma^2=E W^2<\infty\) (FVR, finite variance rewards) or \(P(|W|\geq w)=w^{-\beta}L_W(w)\), \(w>0\) (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 \(n\)th interrenewal interval, \(W^*(t,M)=\sum_{m=1}^M\int_0^t W^{(m)}(u) du\), where \(W^{(m)}\) are i.i.d. copies of \(W(t)\). The author considers the limit behavior of \(W^*(Tt,M)\) as \(T\to\infty\), \(M\to\infty\). E.g. in the FVR case it is shown that if \(L^*_U\) is a slowly varying function such that \(\forall x>0\), \(L^*_U(u)^{-\alpha}L(u^{1/\alpha}L^*_U(u)x)\to 1\) as \(u\to\infty\) and \[ \lim_{T\to\infty}{M\over T^{\alpha-1}}(L_U^*(MT))^\alpha =\infty, \] then \[ \lim_{T\to\infty}{W^*(Tt,M)\over T^{(3-\alpha)/2}M^{1/2}(L_U(T))^{1/2}}=\sigma_0 B_H(t) \] (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 \(\beta\)-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:

60K15 Markov renewal processes, semi-Markov processes
90B18 Communication networks in operations research
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