engleski

Ergodicity of Levy-driven SDEs arising from multiclass many-server queues

We study the ergodic properties of a class of multidimensional piecewise Ornstein Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with bursty arrivals and/or asymptotically negligible service interruptions in the Hal n-Whitt regime as special cases. In these queueing models, the Ito equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Levy process, or (2) an anisotropic Levy process with independent one- dimensional symmetric alpha-stable components, or (3) an anisotropic Levy process as in (2) and a pure-jump Levy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) alpha- stable Levy process as a special case. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponenial and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence is polynomial, and we provide a sharp quantitative characterisation of the rate via matching upper and lower bounds.

piecewise Ornstein Uhlenbeck processes with jumps ; (sub)geometric ergodicity

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