Queueing approximation of suprema of spectrally positive Lévy process
Let \(X\) be a Lévy process with expectation zero, and let \(W = {\sup _{0 \leqslant t < \infty}}(X(t) - \beta t)\), where \(0 < \beta < \infty \). The random variable \(W\) appears in many areas of applied probability, such as queueing theory, risk theory. In the queueing theory it appears as a limit of appropriately normalized stationary waiting times in heavy traffic for some queues. This fact leads to studying the weak convergence \({\omega _n} = {\sup _{0 \leqslant t < \infty }}({X_n}(t) - {\beta _n}(t))\mathop \to \limits^D W\). It is known that this convergence is valid if: (I) \({X_n}\mathop \to \limits^D X\) in \(D[0,\infty )\) with Skorokhod \({J_1}\) topology, and \(X\) is stochastically continuous; (II) \({\beta _n}(t) \to \beta t\) for each \(t \geqslant 0\); and (IIIA) \({\lim _{m \to \infty }}\lim {\sup _{n \to \infty }}P({\sup _{t \geqslant m}}({X_n}(t) - {\beta _n}(t)) > \varepsilon ) = 0\) for each \(\varepsilon > 0\). In some cases condition (IIIA) was replaced by condition (III): \(\{ {\omega _n}\} \) is tight. The paper shows that, for a given Lévy measure \(\nu \) concentrated on \((0,\infty )\) such that \(\int_1^\infty x \nu (dx) < \infty \), there exists a sequence of M/GI/1 queueing systems in heavy traffic such that \({\omega _n}\mathop \to \limits^D W\) with Lévy process \(X\) having measure \(\nu \). Theorem 1 contains the construction of that sequence. As a consequence of Theorem 1, the authors get the fact that in the set of limiting distributions of stationary waiting times in heavy traffic, there are some convolutions of Mittag-Leffler distributions or distributions of suprema of Poisson processes, Compound Poisson processes, or Gamma processes with negative trends. It is shown also that condition (III) for GI/G/1 queues is not implied by conditions (I) and (II).
scientific article; zbMATH DE number 5702561
scientific article