EPSTA: The coincidence of time-stationary and customer-stationary distributions

In the first part of this paper we present an overview of relationships between time- and customer-stationary distributions of queueing processes. These have been proved by using the properties of random marked point processes, stochastic processes with embedded point processes, Palm distributions and an intensity conservation principle. In the second part a necessary and sufficient condition is established for the coincidence of the two types of stationary distributions, using conditional intensities. We also formulate the property of EPSTA that includes PASTA and ASTA as particular cases. A further result concerns the conditional EPSTA property. Applications to particular queueing systems are considered.

[1]  V. Schmidt,et al.  The stationary waiting time process in single-server priority queues with general low priority arrival process , 1984 .

[2]  V. Schmidt,et al.  Queues and Point Processes , 1983 .

[3]  Ronald W. Wolff,et al.  Poisson Arrivals See Time Averages , 1982, Oper. Res..

[4]  V. Schmidt.  Stationary processes with piecewise monotonous trajectories , 1983 .

[5]  V. Schmidt,et al.  On a single-server queue with multiple transaction , 1985 .

[6]  Pierre Brémaud,et al.  Characteristics of queueing systems observed at events and the connection between stochastic intensity and palm probability , 1989, Queueing Syst. Theory Appl..

[7]  Ward Whitt,et al.  On Arrivals That See Time Averages , 1990, Oper. Res..

[8]  Erik A. van Doorn,et al.  Conditional PASTA , 1988 .

[9]  G. F. Newell,et al.  Introduction to the Theory of Queues. , 1963 .

[10]  Dieter König.  Methods for Proving Relationships between Stationary Characteristics of Queueing Systems with Point Processes , 1980, J. Inf. Process. Cybern..

[11]  E. A. van Doorn,et al.  On the “pasta” property and a further relationship between customer and time averages in stationary queueing systems , 1989 .

[12]  Muhammad El-Taha,et al.  Sample-path analysis of processes with imbedded point processes , 1989, Queueing Syst. Theory Appl..

[13]  Jacob Cohen,et al.  On up- and downcrossings , 1977, Journal of Applied Probability.

[14]  D. König,et al.  Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes , 1980 .

[15]  W. Whitt,et al.  On arrivals that see time averages : a martingale approach , 1990 .

[16]  Masakiyo Miyazawa,et al.  On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions , 1983 .

[17]  V. Schmidt,et al.  RELATIONSHIPS BETWEEN TIME/CUSTOMER STATIONARY CHARACTERISTICS OF TANDEM QUEUES ATTENDED BY A SINGLE SERVER , 1984 .

[18]  P. Franken,et al.  Queues and Point Processes , 1983 .

[19]  J. Grandell.  Doubly stochastic Poisson processes , 1976 .

[20]  Ralph L. Disney,et al.  Stationary queue-length and waiting-time distributions in single-server feedback queues , 1984 .

[21]  Dieter König,et al.  Stationary queue-length characteristics in queues with delayed feedback , 1985, Journal of Applied Probability.