Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-02T22:15:19.219Z Has data issue: false hasContentIssue false
  • English
  • Français

On joint queue-length characteristics in infinite-server tandem queues with heavy traffic

Published online by Cambridge University Press:  01 July 2016

Volker Schmidt
Volker Schmidt*
Affiliation:
Mining Academy of Freiberg
*
∗Postal address: Bergakademie Freiberg, Sektion Mathematik, Bernhard-von-Cotta-Str. 2, DDR-9200 Freiberg, GDR.
Get access Rights & Permissions [Opens in a new window]

Abstract

For m infinite-server queues with Poisson input which are connected in a series, a simple proof is given of a formula derived in [3] for the generating function of the joint customer-stationary distribution of the successive numbers of customers a randomly chosen customer finds at his arrival epochs at two queues of the system. In this connection, a shot-noise representation of the queue-length characteristics under consideration is used. Moreover, using this representation, corresonding asymptotic formulas are derived for infinite-server tandem queues with general high-density renewal input.

Keywords

QUEUEING NETWORKCORRELATION COEFFICIENTHIGH DENSITYLIMIT THEOREM
Type
Research Article
Information
Advances in Applied Probability , Volume 19 , Issue 2 , June 1987 , pp. 474 - 486
Copyright
Copyright © Applied Probability Trust 1987 

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.)

References

1. Borovkov, A. A. (1967) On limit laws for service processes in multi-channel systems (in Russian). Sib. Mat. Zhurnal 8, 983–1004.Google Scholar
2. Borovkov, A. A. (1980) Asymptotic Methods in Queueing Theory (in Russian). Nauka, Moscow.Google Scholar
3. Boxma, O. J. (1984) M/G/8 tandem queues. Stoch. Proc. Appl. 18, 153–164.Google Scholar
4. Franken, P. and Kerstan, J. (1968) Queueing systems with infinitely many servers (in German). In Operations Research and Mathematical Statistics I, ed. Bunke, O., Akademie-Verlag, Berlin, 67–76.Google Scholar
5. Franken, P., König, D., Arndt, U. and Schmidt, V. (1982) Queues and Point Processes. Wiley, Chichester.Google Scholar
6. Gnedenko, B. V., König, D. et al. (1984) Handbook of Queueing Theory (in German), Vol. II. Akademie-Verlag, Berlin.Google Scholar
7. Heinrich, L. and Schmidt, V. (1985) Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.Google Scholar
8. Iglehart, D. L. (1973) Weak convergence of compound stochastic processes. Stoch. Proc. Appl. 1, 11–31.Google Scholar
9. König, D., Matthes, K. and Nawrotzki, K. (1967) Generalizations of the Erlang and Engset Formulas (A Method in Queueing Theory) (in German). Akademie-Verlag, Berlin.Google Scholar
10. Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
11. Mechata, K. M. and Deivamoney Selvam, D. (1984) Covariance structure of infinite server queues in tandem. Opsearch 21, 172–178.Google Scholar
12. Schmidt, V. (1984) On shot noise processes induced by stationary marked point processes. J. Inf. Proc. Cybernet. 20, 397–406.Google Scholar
13. Schmidt, V. (1985) Qualitative and asymptotic properties of shot-noise fields: A point process approach. Proc. 53rd Sci. Session on Stochastics , Academy of Sciences of GDR, Inst. of Mathematics, Berlin, 1–56.Google Scholar
14. Vainshtein, A. D. and Kreinin, A. Ja. (1983) Some characteristics of tandem queues with infinite number of channels (in Russian). Proc. Seminar on Stability Problems for Stochastic Models , VNIISI (Institute for System Analysis), Moscow, 24–36.Google Scholar
15. Vere-Jones, D. (1968) Some applications of probability generating functionals to the study of input-output streams. J. R. Statist. Soc. B30, 321–333.Google Scholar
16. Whitt, W. (1982) On the heavy-traffic limit theorem for GI/G/8 queues. Adv. Appl. Prob. 14, 171–190.Google Scholar
17. Whitt, W. (1984) Departures from a queue with many busy servers. Math. Operat. Res. 9, 534–544.Google Scholar