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On arrivals that see time averages: a martingale approach

Published online by Cambridge University Press:  14 July 2016

Benjamin Melamed  and
Ward Whitt
Benjamin Melamed*
Affiliation:
NEC Research Institute
Ward Whitt*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: NEC Research Institute, 4 Independence Way, Princeton, NJ 08540, USA.
∗∗Postal address: AT&T Bell Laboratories, Room 2C-178, Murray Hill, NJ 07974, USA.
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Abstract

This paper is a sequel to our previous paper investigating when arrivals see time averages (ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed process (assumed to be well defined) and basic properties of Riemann–Stieltjes integrals. Here similar results are obtained using the stochastic intensity associated with the martingale theory of point processes, as in Brémaud (1981). In the martingale framework, the ASTA result is almost an immediate consequence of the definition of a stochastic intensity. In a stationary framework, the results characterize the Palm distribution, but stationarity is not assumed here. Watanabe's (1964) martingale characterization of a Poisson process is also applied to establish a general version of anti–PASTA: if the points of the point process are appropriately generated by the observed process and the observed process is Markov with left-continuous sample paths, then ASTA implies that the point process must be Poisson.

Keywords

EMBEDDED PROCESSESPALM MEASUREPASTAQUEUESCUSTOMER AVERAGES AND TIME AVERAGESTHE ARRIVAL THEOREMPOINT PROCESSESSTOCHASTIC INTENSITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 2 , June 1990 , pp. 376 - 384
Copyright
Copyright © Applied Probability Trust 1990 

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