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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 $P\left(U\ge u\right)={u}^{-\alpha }{L}_{U}\left(u\right)$, $\alpha \in \left(1,2\right)$; let ${W}_{i}$ be i.i.d. rewards independent of $\left\{{U}_{i},\phantom{\rule{4pt}{0ex}}i=1,2,\cdots \right\}$ with $EW=0$ and ${\sigma }^{2}=E{W}^{2}<\infty$ (FVR, finite variance rewards) or $P\left(|W|\ge w\right)={w}^{-\beta }{L}_{W}\left(w\right)$, $w>0$ (IVR, infinite variance rewards) (${L}_{U}$, ${L}_{W}$ are slowly varying functions at infinity). The renewal reward process is defined as $W\left(t\right)={W}_{n}$ if $t$ belongs to the $n$th interrenewal interval, ${W}^{*}\left(t,M\right)={\sum }_{m=1}^{M}{\int }_{0}^{t}{W}^{\left(m\right)}\left(u\right)du$, where ${W}^{\left(m\right)}$ are i.i.d. copies of $W\left(t\right)$. The author considers the limit behavior of ${W}^{*}\left(Tt,M\right)$ 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 $\forall x>0$, ${L}_{U}^{*}{\left(u\right)}^{-\alpha }L\left({u}^{1/\alpha }{L}_{U}^{*}\left(u\right)x\right)\to 1$ as $u\to \infty$ and

$\underset{T\to \infty }{lim}\frac{M}{{T}^{\alpha -1}}{\left({L}_{U}^{*}\left(MT\right)\right)}^{\alpha }=\infty ,$

then

$\underset{T\to \infty }{lim}\frac{{W}^{*}\left(Tt,M\right)}{{T}^{\left(3-\alpha \right)/2}{M}^{1/2}{\left({L}_{U}\left(T\right)\right)}^{1/2}}={\sigma }_{0}{B}_{H}\left(t\right)$

(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 ${W}^{\left(m\right)}$ 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 90B18 Communication networks (optimization)