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

F. G. Badía () and C. Sangüesa ()
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F. G. Badía: University of Zaragoza
C. Sangüesa: University of Zaragoza

Methodology and Computing in Applied Probability, 2017, vol. 19, issue 2, 647-664

Abstract: 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.

Keywords: Log-convexity; Counting process; Queueing model; Stochastic order; 60E05; 60E15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s11009-016-9520-9

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