## 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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### References:

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