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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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