Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T19:10:37.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Further results on ASTA for general stationary processes and related problems

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa  and
Ronald W. Wolff
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
Ronald W. Wolff*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Information Sciences, Science University of Tokyo, Noda-city, Chiba 278, Japan.
∗∗ Postal address: Department of IEOR, University of California, Berkeley, CA 94720, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the equivalence of state probabilities of a general stationary process at an arbitrary time and at embedded epochs of a given point process, which is called ASTA (Arrivals See Time Averages). By using an event-conditonal intensity, we give necessary and sufficient conditions for ASTA for a large class of state sets, which determines a state distribution. We do not need any additional assumptions except that the general process has left-hand limits at all points of time. Especially, for a stationary pure-jump process with a point process, ASTA is obtained for all state sets. As an application of those results, Anti-PASTA is obtained for a pure-jump Markov process and a certain class of GSMP (Generalized Semi-Markov Processes), where Anti-PASTA means that ASTA implies that the arrival process is Poisson.

Keywords

QUEUESPURE-JUMP PROCESSGSMPPASTAANTI-PASTACONDITIONAL INTENSITYPOINT PROCESSPALM DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 4 , December 1990 , pp. 792 - 804
Copyright
Copyright © Applied Probability Trust 1990 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Part of this work was done when R. W. Wolff visited the Science University of Tokyo in 1989.

References

Baccelli, F. and Bremaud, P. (1987) Palm Probabilities and Stationary Queueing Systems. Lecture Notes in Statistics 41, Springer-Verlag, New York.CrossRefGoogle Scholar
Bremaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.Google Scholar
Bremaud, P. (1989a) Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability. QUESTA 5, 99–113.Google Scholar
Bremaud, P. (1989b) Palm-Martingale calculus, event and time averages: The stationary and the non-stationary cases. Proc. Vth Vilnius Internat. Conf. Probability and Mathematical Statistics,Google Scholar
Bremaud, P. (1990) Events and time averages: a review. Second Symposium on Queueing Theory and Related Topics, Karpacz, Poland.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, New York.Google Scholar
Green, L. and Melamed, B. (1990) An anti-PASTA result for Markovian systems, Operat. Res. 30, 173–175.Google Scholar
König, D. and Schmidt, V. (1989) EPSTA: The coincidence of time-stationary and customer-stationary distributions. QUESTA 5, 247–263.Google Scholar
König, D. and Schmidt, V. (1990) Extended and conditional versions of the PASTA property. Adv. Appl. Prob. Google 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
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 Walrand, J. (1986) On the one-dimensional distributions of counting processes with stochastic intensities. Stochastics 19, 1–9.CrossRefGoogle Scholar
Melamed, B. and Whitt, W. (1990a) On arrivals that see time averages. Operat. Res. 38, 156–172.CrossRefGoogle Scholar
Melamed, B. and Whitt, W. (1990b) On arrivals that see time averages: A martingale approach. J. Appl. Prob. 27, 376–384.Google Scholar
Miyazawa, M. and Yamazaki, G. (1988) The basic equations for a supplemented GSMP and its applications to queues. J. Appl. Prob. 25, 565–578.Google Scholar
Schassberger, R. (1978) Insensitivity of steady-state distributions of generalized semi-Markov processes with speeds. Adv. Appl. Prob. 10, 836–851.Google Scholar
Stidham, S. Jr. and El-Taha, M. (1988) Sample-path analysis of processes with imbedded point processes. QUESTA. Google Scholar
Walrand, J. (1988) An Introduction to Queueing Networks. Prentice-Hall, New Jersey.Google 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, 176–177.Google Scholar