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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 $P(U\ge u)=u^{-\alpha}L_U(u)$, $\alpha\in(1,2)$; 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 $P(|W|\ge w)=w^{-\beta}L_W(w)$, $w>0$ (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.
Reviewer: R.E.Maiboroda (Kyïv)

60K15Markov renewal processes
90B18Communication networks (optimization)
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