Conditional intensities and coincidence properties of stochastic processes with embedded point processes. (English) Zbl 0776.60061
Let \(X(t)\), \(t\in R\), be a continuous-time stochastic process with embedded point processes \(T_ n\), that has the conditional intensity \(\lambda_ A\) at a fixed point in time \(t\) given that \(X(t)\) takes a value from the set \(A\) of state space. There are given weak sufficient conditions on \(A\) for the existence of \(\lambda_ A\). The EPSTA (embedded points see time averages) and GEPSTA (conditional EPSTA) properties of PMP (process with embedded marked point process) are characterized by the invariance property of \(\lambda_ A\). For Markovian network processes the result is given without using any Markov-type assumption. The characterization theorem is proved for PMP with MUSTA (moving units see time averages for the unmoved units) property, too.
Reviewer: N.Kordzakhia (Tbilisi)
MSC:
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
Keywords:
conditional intensities; queueing processes; point processes; embedded marked point process; Markovian network processesReferences:
| [1] | Asmussen, S., Ladder heights and the Markov-modulated M/GI/1 queue, Stochastic Process. Appl., 37, 313-326 (1991) · Zbl 0734.60091 |
| [2] | Brémaud, P., Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability, Queueing Syst., 5, 99-112 (1989) · Zbl 0686.60096 |
| [3] | Brémaud, P.; Kannurpatti, R.; Mazundar, R., Event and time averages: A review, TIMS/ORSA Conference (1991), Monterrey |
| [4] | Daley, D. J.; Vere-Jones, D., An Introduction to the Theory of Point Processes (1988), Springer: Springer New York · Zbl 0657.60069 |
| [5] | Disney, R. L.; König, D.; Schmidt, V., Stationary queue-length and waiting-time distributions in single-server feedback queues, Adv. Appl. Probab., 16, 437-446 (1984) · Zbl 0536.60090 |
| [6] | Franken, P.; König, D.; Arndt, U.; Schmidt, V., Queues and Point Processes (1982), Wiley: Wiley New York · Zbl 0505.60058 |
| [7] | Kelly, F. P., Reversibility and Stochastic Networks (1979), Wiley: Wiley New York · Zbl 0422.60001 |
| [8] | König, D., Methods for proving relationships between stationary characteristics of queueing systems with point processes, J. Inform. Process. Cybern., 16, 521-543 (1980) · Zbl 0477.60088 |
| [9] | König, D.; Miyazawa, M.; Schmidt, V., On the identification of Poisson arrivals in queues with coinciding time-stationary and customer-stationary state distributions, J. Appl. Probab., 20, 860-871 (1983) · Zbl 0527.60082 |
| [10] | König, D.; Schmidt, V., Imbedded and non-imbedded stationary characteristics of queueing systems with varying service rate and point processes, J. Appl. Probab., 17, 753-767 (1980) · Zbl 0435.60094 |
| [11] | König, D.; Schmidt, V., EPSTA: The coincidence of time-stationary and customer-stationary distributions, Queueing Syst., 5, 247-264 (1989) · Zbl 0681.60099 |
| [12] | König, D.; Schmidt, V., Extended and conditional versions of the PASTA property, Adv. Appl. Probab., 22, 510-512 (1990) · Zbl 0702.60086 |
| [13] | König, D.; Schmidt, V., Random Point Processes. An Introduction with Examples of Applications (1992), Teubner: Teubner Stuttgart, [In German.] · Zbl 0778.60037 |
| [14] | König, D.; Schmidt, V.; van Doorn, E. A., On the PASTA property and a further relationship between customer and time averages in stationary queueing systems, Stochastic Models, 5, 261-272 (1989) · Zbl 0671.60096 |
| [15] | Melamed, B.; Whitt, W., On arrivals that see time averages, Oper. Res., 38, 156-172 (1990) · Zbl 0702.60093 |
| [16] | Melamed, B.; Whitt, W., On arrivals that see time averages: A martingale approach, J. Appl. Probab., 27, 376-384 (1990) · Zbl 0713.60098 |
| [17] | Miyazawa, M.; Wolff, R. W., Further results on ASTA for general stationary processes and related problems, J. Appl. Probab., 28, 792-804 (1990) · Zbl 0726.60048 |
| [18] | Regeterschot, G. J.K.; van Doorn, E. A., Conditional PASTA, Oper. Res. Lett., 7, 229-232 (1988) · Zbl 0659.60128 |
| [19] | Serfozo, R. F., Markovian network processes: Congestion-dependent routing and processing, Queueing Syst., 5, 5-36 (1989) · Zbl 0714.60082 |
| [20] | Stidham, S.; El Taha, M., Sample-path analysis of processes with embedded point processes, Queueing Syst., 5, 131-166 (1989) · Zbl 0682.60083 |
| [21] | Walrand, J., An Introduction to Queueing Networks (1988), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0854.60090 |
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