Further results on ASTA for general stationary processes and related problems

1990 ◽  
Vol 27 (4) ◽  
pp. 792-804 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Ronald W. Wolff
Keyword(s):  
Point Process ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Jump Process ◽  
Arrival Process ◽  
General Process ◽  
Left Hand ◽  
Time Averages ◽  
Semi Markov Processes ◽  
Necessary And Sufficient

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.

Further results on ASTA for general stationary processes and related problems

1990 ◽  
Vol 27 (04) ◽  
pp. 792-804 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Ronald W. Wolff
Keyword(s):  
Point Process ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Jump Process ◽  
Arrival Process ◽  
General Process ◽  
Left Hand ◽  
Time Averages ◽  
Semi Markov Processes ◽  
Necessary And Sufficient

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.

PRODUCT-FORM MARKOVIAN QUEUEING SYSTEMS WITH MULTIPLE RESOURCES

2019 ◽  
pp. 1-9 ◽  
Author(s):  
Valeriy Naumov ◽  
Konstantin Samouylov
Keyword(s):  
Queueing Systems ◽  
Sufficient Conditions ◽  
Jump Process ◽  
Product Form ◽  
Temporary Increase ◽  
Stationary Probability Distribution ◽  
Markov Jump ◽  
Multiple Resources ◽  
Resource Requirements ◽  
Necessary And Sufficient

In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.

Event and time averages: a review

1992 ◽  
Vol 24 (2) ◽  
pp. 377-411 ◽  
Author(s):  
Pierre Brémaud ◽  
Raghavan Kannurpatti ◽  
Ravi Mazumdar
Keyword(s):  
Continuous Time ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Process Models ◽  
Stationary Case ◽  
Unified Framework ◽  
Martingale Approach ◽  
Point Process Models ◽  
Time Averages ◽  
Necessary And Sufficient

This article reviews results related to event and time averages (EATA) for point process models, including PASTA, ASTA and ANTIPASTA under general hypotheses. In particular, the results for the stationary case relating the Palm and martingale approach are reviewed. The non-stationary case is discussed in the martingale framework where minimal conditions for ASTA generalizing earlier work are presented in a unified framework for the discrete- and continuous-time cases. In addition, necessary and sufficient conditions for ASTA to hold in the stationary case are discussed in the case even when stochastic intensities may not exist and a short proof of the ANTIPASTA results known to date are given.

A note on exceedances and rare events of non-stationary sequences

1993 ◽  
Vol 30 (04) ◽  
pp. 877-888 ◽  
Author(s):  
J. Hüsler
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Rare Events ◽  
Sufficient Conditions ◽  
Stationary Sequence ◽  
Poisson Processes ◽  
Regularity Conditions ◽  
Mixing Conditions ◽  
Necessary And Sufficient ◽  
Point Process Of Exceedances

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

Processes with new better than used first-passage times

1984 ◽  
Vol 16 (03) ◽  
pp. 667-686 ◽  
Author(s):  
J. G. Shanthikumar
Keyword(s):  
Point Process ◽  
Markov Processes ◽  
Probability Space ◽  
Sufficient Conditions ◽  
General Point ◽  
First Passage ◽  
Semi Markov Processes ◽  
New Better Than Used ◽  
Better Than ◽  
Passage Times

Let with Z(0) = 0 be a random process under investigation and N be a point process associated with Z. Both Z and N are defined on the same probability space. Let with R 0 = 0 denote the consecutive positions of points of N on the half-line . In this paper we present sufficient conditions under which (Z, R) is a new better than used (NBU) process and give several examples of NBU processes satisfying these conditions. In particular we consider the processes in which N is a renewal and a general point process. The NBU property of some semi-Markov processes is also presented.

Characterizing pure loss GI/G/1 queues with renewal output

1974 ◽  
Vol 75 (1) ◽  
pp. 103-107 ◽  
Author(s):  
D. J. Daley
Keyword(s):  
Poisson Process ◽  
Service Time ◽  
Renewal Process ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arrival Process ◽  
Renewal Function ◽  
Service Distribution ◽  
Necessary And Sufficient

SummaryIn a pure loss GI/G/1 queueing system, necessary and sufficient conditions are given for the output to be a renewal process. These conditions involve dependence between the service distribution and the renewal function of the arrival process: for example, if pr {service time < ξ} = 0 for some ξ > 0, then it is sufficient for the renewal function to be that of a quasi-Poisson process with index ξ.

scholarly journals Weak convergence of first-rare-event times for semi-Markov processes

2020 ◽  
Author(s):  
AISDL
Keyword(s):  
Weak Convergence ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Rare Event ◽  
Necessary And Sufficient Conditions ◽  
Finite Set ◽  
In Series ◽  
Event Times ◽  
Semi Markov Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions for weak convergence of first-rareevent times for semi-Markov processes with finite set of states in series of schemes are obtained.

Event and time averages: a review

1992 ◽  
Vol 24 (02) ◽  
pp. 377-411 ◽  
Author(s):  
Pierre Brémaud ◽  
Raghavan Kannurpatti ◽  
Ravi Mazumdar
Keyword(s):  
Continuous Time ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Process Models ◽  
Stationary Case ◽  
Unified Framework ◽  
Martingale Approach ◽  
Point Process Models ◽  
Time Averages ◽  
Necessary And Sufficient

This article reviews results related to event and time averages (EATA) for point process models, including PASTA, ASTA and ANTIPASTA under general hypotheses. In particular, the results for the stationary case relating the Palm and martingale approach are reviewed. The non-stationary case is discussed in the martingale framework where minimal conditions for ASTA generalizing earlier work are presented in a unified framework for the discrete- and continuous-time cases. In addition, necessary and sufficient conditions for ASTA to hold in the stationary case are discussed in the case even when stochastic intensities may not exist and a short proof of the ANTIPASTA results known to date are given.

High-Level exceedances of regenerative and semi-stationary processes

1980 ◽  
Vol 17 (02) ◽  
pp. 423-431 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Point Processes ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Poisson Processes ◽  
Partial Sums ◽  
Stationary Processes ◽  
Special Cases ◽  
Cumulative Amount ◽  
High Level ◽  
Necessary And Sufficient

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

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