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Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models

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  • F. G. Badía

    (University of Zaragoza)

  • C. Sangüesa

    (University of Zaragoza)

Abstract

We consider a counting processes with independent inter-arrival times evaluated at a random end of observation time T, independent of the process. For instance, this situation can arise in a queueing model when we evaluate the number of arrivals after a random period which can depend on the process of service times. Provided that T has log-convex density, we give conditions for the inter-arrival times in the counting process so that the observed number of arrivals inherits this property. For exponential inter-arrival times (pure-birth processes) we provide necessary and sufficient conditions. As an application, we give conditions such that the stationary number of customers waiting in a queue is a log-convex random variable. We also study bounds in the approximation of log-convex discrete random variables by a geometric distribution.

Suggested Citation

  • F. G. Badía & C. Sangüesa, 2017. "Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 647-664, June.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:2:d:10.1007_s11009-016-9520-9
    DOI: 10.1007/s11009-016-9520-9
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    References listed on IDEAS

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