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Stationary queue-length characteristics in queues with delayed feedback

Published online by Cambridge University Press:  14 July 2016

Dieter König  and
Volker Schmidt
Dieter König*
Affiliation:
Mining Academy, Freiberg
Volker Schmidt*
Affiliation:
Mining Academy, Freiberg
*
∗Postal address: Sektion Mathematik, Bergakademie Freiberg, DDR-9200 Freiberg (Sachs), GDR.
∗Postal address: Sektion Mathematik, Bergakademie Freiberg, DDR-9200 Freiberg (Sachs), GDR.
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Abstract

A class of two-node queueing networks with general stationary ergodic governing sequence is considered. This means that, in particular, a non-Poissonian arrival process and dependent service times, as well as a non-Bernoulli feedback mechanism are admitted. A mixing condition ensures that the limiting distributions of the number of customers in the nodes observed in continuous time as well as at certain embedded epochs can be expressed by the Palm distributions of appropriately chosen marked point processes. This gives the possibility of connecting the classical concept of embedding with a general point-process approach. Furthermore, it leads to simple proofs of relationships between the limiting distributions. An example is given to illustrate how these relationships can be used to derive explicit formulas for various stationary queueing characteristics.

Keywords

QUEUEING NETWORKSDELAYED FEEDBACKQUEUE LENGTHEMBEDDINGLIMIT DISTRIBUTIONSRELATIONSHIPSPOINT PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 394 - 407
Copyright
Copyright © Applied Probability Trust 1985 

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References

[1] Disney, R. L., König, D. and Schmidt, V. (1984) Stationary queue length and waiting time distributions in single-server feedback queues. Adv. Appl. Prob. 16, 437–446.Google Scholar
[2] Foley, R. D. and Disney, R. L. (1983) Queues with delayed feedback. Adv. Appl. Prob. 15, 162–182.Google Scholar
[3] Franken, P., König, O., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, Chichester.Google Scholar
[4] Jackson, J. R. (1957) Networks of waiting lines. Operat. Res. 5, 518–521.Google Scholar
[5] König, D. (1980) Methods for proving relationships between stationary characteristics of queueing systems with point processes. Elektron. Informationsverarbeit. Kybernet. 16, 521–543.Google Scholar
[6] König, D., Matthes, K. and Nawrotzki, K. (1967) Verallgemeinerungen der Erlangschen und Engsetschen Formeln (Eine Methode in der Bedienungstheorie). Akademie-Verlag, Berlin.Google Scholar
[7] 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.Google Scholar
[8] König, D. and Schmidt, V. (1985) Convergence theorems for single-server feedback queues generated by a general class of marked point processes (in Russian). Teorija Verojatnost. i Primenen 30.Google Scholar
[9] König, D. and Schmidt, V. (1984) Relationships between various stationary characteristics in feedback queues generated by a general class of marked point processes (in Russian). Izvest. Akad. Nauk SSSR, Techn. Kibernet. No. 2, 110–114.Google Scholar
[10] Loynes, R. M. (1962) The stability of a queue with nonindependent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[11] Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
[12] Matthes, K. and Nawrotzki, K. (1962) Ergodizitätseigenschaften rekurrenter Ereignisse. II. Math. Nachr. 24, 245–253.CrossRefGoogle Scholar
[13] Miyazawa, M. (1977) Time and customer processes in queues with stationary inputs. J. Appl. Prob. 14, 349–357.Google Scholar
[14] Wirth, K.-D. (1984) Some remarks on feedback queues. Elektron. Informationsverarbeit. Kybernet. 20, 55–64.Google Scholar