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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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References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Bremaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.CrossRefGoogle Scholar
Bremaud, P. (1989) Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability. Queueing Systems 5, 99–112 CrossRefGoogle Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1981) Queues and Point Processes. Akademie-Verlag, Berlin (and Wiley, Chichester, 1982).Google Scholar
Green, L. and Melamed, B. (1990) An anti-PASTA result for Markovian systems. Operat. Res. 38, 173–175.CrossRefGoogle Scholar
König, D. and Schmidt, V. (1980) Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes. J. Appl. Prob. 17, 753–767.CrossRefGoogle Scholar
König, D., Miyazawa, M. and Schmidt, V. (1983) On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions. J. Appl. Prob. 20, 860–871.Google Scholar
König, D., Schmidt, V. and Van Doorn, E. A. (1989) On the ‘PASTA’ property and a further relationship between customer and time averages in stationary queueing systems. Stochastic Models 5, 261–272.Google Scholar
Melamed, B. (1982) On Markov jump processes imbedded at jump epochs and their queueing-theoretic applications. Math. Operat. Res. 7, 111–128.Google Scholar
Melamed, B. and Whitt, W. (1990) On arrivals that see time averages. Operat. Res. 38, 156–172.Google Scholar
Miyazawa, M. (1982) Simple derivation of invariance relations and their applications. J. Appl. Prob. 19, 183–194.Google Scholar
Serfozo, R. F. (1989a) Poisson functionals of Markov processes and queueing networks. Adv. Appl. Prob. 21, 595–611.CrossRefGoogle Scholar
Serfozo, R. F. (1989b) Markov network processes: congestion-dependent routing and processing. Queueing Systems 5, 5–36.Google Scholar
Stidham, S. and El-Taha, M. (1989) Sample-path analysis of processes with imbedded point processes. Queueing Systems 5, 131–166.Google Scholar
Varaiya, P. and Walrand, J. (1981) Flows in queueing networks: a martingale approach. Math. Operat. Res. 6, 387–404.Google Scholar
Walrand, J. (1988) An Introduction to Queueing Networks, Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Watanabe, S. (1964) On discontinuous additive functionals and Levy measures of a Markov process. Japan J. Math. 34, 53–70.CrossRefGoogle Scholar
Wolff, R. W. (1982) Poisson arrivals see time averages. Operat. Res. 30, 223–231.CrossRefGoogle Scholar
Wolff, R. W. (1990) A note on PASTA and anti-PASTA for continuous-time Markov chains. Operat. Res. 38. To appear.CrossRefGoogle Scholar