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  • The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

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  • Offer Kella
  • Journal:
    Journal of Applied Probability / Volume 49 / Issue 3 / September 2012
    Published online by Cambridge University Press:
    04 February 2016, pp. 883-887
    Print publication:
    September 2012
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  • The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.