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Function-indexed empirical processes based on an infinite source Poisson transmission stream. (English) Zbl 1259.60036
The paper considers an infinite source Poisson transmission process defined by $X(t) = \sum\limits_{l \in {\mathbb Z}} {{W_l}{{\text{1}}_{[{\Gamma _l} \leqslant t < {\Gamma _l} + {Y_l}]}}},\quad t \in ( - \infty ,\infty ),$ where the triples $$\{ ({\Gamma _l},{Y_l},{W_l}),l \in {\mathbb Z}\}$$ of session arrival times, durations and transmission rates satisfy the following assumptions:
(i) $$\{ {\Gamma _l} : l \in {\mathbb Z}\}$$ are the points of a homogeneous Poisson process on the real line;
(ii) $$\{ (Y,W),({Y_l},{W_l}) : l \in {\mathbb Z}\}$$ are i.i.d. random pairs with values in $$(0,\infty ) \times [0,\infty )$$ and independent of the arrival times, the random variable $$W$$ is positive with positive probability, and $$Y$$ has finite expectation and infinite variance;
(iii) there exists a measure $$\nu$$ on $$(0,\infty ] \times [0,\infty ]$$ such that $$\nu ((0,\infty ] \times [0,\infty ]) = 1$$ and, as $$n \to \infty$$, $$n{\operatorname{P}}(({Y \mathord{\left/ {\vphantom {Y {a(n),W) \in \cdot )\mathop \to \limits^v }}} \right. \kern-\nulldelimiterspace} {a(n),W) \in \cdot )\mathop \to \limits^v }}\nu$$, where $$\mathop \to \limits^v$$ denotes vague convergence on $$(0,\infty ] \times [0,\infty ]$$, and $$a$$ is the left continuous inverse of $${1 \mathord{\left/ {\vphantom {1 {\bar F}}} \right. \kern-\nulldelimiterspace} {\bar F}}$$ – here, $$F$$ is the distribution function of $$Y$$, and $$\bar F = 1 - F$$.
The authors study the large time behavior of the empirical process ${J_T}(\phi ) = \int_0^T {\phi ({X_h}(s))ds} ,$ where $$h > 0$$, $${X_h}(s) = \{ X(s + t) : 0 \leqslant t \leqslant h\}$$, and $$\phi$$ is a real valued function. The main result of the paper is stated as a functional central limit theorem in the Skorohod $${M_1}$$ topology.

MSC:
 60F17 Functional limit theorems; invariance principles 60K25 Queueing theory (aspects of probability theory)
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References:
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