Abstract
In this note, we prove that the speed of convergence of the workload of a Lévy-driven queue to the quasi-stationary distribution is of order 1/t. We identify also the Laplace transform of the measure giving this speed and provide some examples.
Highlights
In this paper, we consider a storage system with Lévy netput
Let X ≡ (X (t))t be a Lévy process, which is defined on the filtered space (, F, {Ft }t≥0, P) with the natural filtration that satisfies the usual assumptions of right continuity and completion
It is worth noting that ξ may be negative, since it pertains to the speed of convergence to the Yaglom limit
Summary
We consider a storage system with Lévy netput. In other words, the workload process {Q(t), t ≥ 0} is a spectrally one-sided Lévy process X (t) that is reflected at 0: Q(t) := X (t) − infs≤t (X (s))−,. Kyprianou [15] found the Laplace transform of the QS distribution for the workload process of the stable M/G/1 queue with service times that have a rational moment generating function. The speed of convergence (in total variation) to the QS distribution for population processes has been studied in [5]. A contribution of this paper lies in proving that the speed of convergence to the quasi-stationary distribution is surprisingly slow (of order 1/t). If we want to simulate the quasi-stationary distribution directly from its definition given in (3), the result stated in (3) shows that the speed of such a simulation is very slow. The main result in (3) contrasts the typical results derived for the regular stationary distribution of Markov processes where, in most of the cases, the rate is exponential.
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