Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T05:43:43.278Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic analysis of queueing systems with identical service

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

F. I. Karpelevitch  and
A. Ya. Kreinin
F. I. Karpelevitch*
Affiliation:
Moscow Institute of Railway Transport Engineers
A. Ya. Kreinin*
Affiliation:
University of Toronto
*
∗Postal address: Moscow Institute of Railway Transport Engineers, 129851 Moscow, 3-ya Mytishchinskaya 10, Russia.
∗∗Postal address: Computer Systems Research Institute, University of Toronto, 6 King's College Road, Toronto, Ontario, Canada M5S 1A1. (Present address: Algorithmics Inc., 822 Richmond Street West, Toronto, Ontario, Canada M6J 1C9.)
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a heavy traffic regime in queueing systems with identical service. These systems belong to the class of multi-phase systems with dependent service times in different service nodes. We study the limit behaviour of the waiting time vector in heavy traffic. Both transient behaviour and the stationary regime are considered. Our analysis is based on the conception of ‘approximated functionals', which appeared to be fruitful in weak convergence theory of stochastic processes.

Keywords

ARRIVAL PROCESSHEAVY TRAFFIC LIMITSIDENTICAL SERVICEJOINT STATIONARY DISTRIBUTIONSQUEUEING SYSTEMSREFLECTED DIFFUSION PROCESSESSOJOURN TIMESWAITING TIMESWEAK CONVERGENCE

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 267 - 281
Copyright
Copyright © Applied Probability Trust 1996 

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] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Borovkov, A. A. (1972) Weak convergence of functionals of random sequences and processes defined on the hole axis. Proc. Stecklov Math. Inst. 128, 41–65.Google Scholar
[3] Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer, Berlin.CrossRefGoogle Scholar
[4] Borovkov, A. A. (1984) Asymptotic Methods in Queueing Theory. Wiley, New York.Google Scholar
[5] Borovkov, A. A. (1986) Limit theorems for queueing networks. Theory Prob. Appl. 31, 413–427.Google Scholar
[6] Boxma, O. J. (1979) On a tandem queueing model with identical service times at both counters I, II. Adv. Appl. Prob. 11, 619–659.Google Scholar
[7] Glynn, P. W. and Whitt, W. (1991) A new view of the heavy-traffic limit theorems for infinite-server queues. Adv. Appl. Prob. 23, 188–209.Google Scholar
[8] Glynn, P. W. and Whitt, W. (1991) Departures from many queues in series. Ann. Appl. Prob. 1, 546–572.Google Scholar
[9] Grigelionis, B. and Miculavichus, R. (1984) Diffusion approximation in queueing theory. Fundamentals of Teletraffic Theory. (Proc. Third Int. Seminar on Teletraffic Theory), pp. 147–158.Google Scholar
[10] Harrison, J. M. (1973) The heavy traffic approximation for single server queues in series. J. Appl. Prob. 10, 613–629.CrossRefGoogle Scholar
[11] Harrison, J. M. (1978) The diffusion approximation for tandem queues in heavy traffic. Adv. Appl. Prob. 10, 886–905.CrossRefGoogle Scholar
[12] Harrison, J. M. and Reiman, M. I. (1981) On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math. 41, 345–361.Google Scholar
[13] Harrison, J. M. and Williams, R. J. (1987) Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22, 77–115.Google Scholar
[14] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic I. Adv. Appl. Prob. 2, 150–175.Google Scholar
[15] Iglehart, D. L. and Whitt, W. (1970) Multiple channel queues in heavy traffic II: Sequences, networks and batches. Adv. Appl. Prob. 2, 355–364.Google Scholar
[16] Karpelevitch, F. I. and Kreinin, A. Ya. (1981) Two-phase queueing system GI/G/1/8 ? G/1/8 under heavy traffic conditions. Theory Prob. Appl. 26, 293–313.Google Scholar
[17] Karpelevitch, F. I. and Kreinin, A. Ya. (1990) Asymptotic analysis of telegraphic message switching systems. Prob. Inf. Trans. 26, 261–274.Google Scholar
[18] Karpelevitch, F. I. and Kreinin, A. Ya. (1992) Joint distributions in Poissonian tandem queues. Queueing Systems 12, 274–286.Google Scholar
[19] Kelly, F. P. (1984) An asymptotic analysis of blocking. Modelling and Performance Evaluation Methodology, pp. 3–20, Springer, Berlin.Google Scholar
[20] Klimov, G. P. (1970) Several solved and unsolved problems of the service by queues in series (in Russian). Izv. AN USSR, Ser. Tech. Kibern. 6, 88–92.Google Scholar
[21] Krichagina, E. V., Liptzer, R. Sh. and Pukhalsky, A. A. (1988) The diffusion approximation for queues with input flow, depending on a queue state and general service. Theory Prob. Appl. 33, 124–135.Google Scholar
[22] Kogan, Ya. A. and Pukhalsky, A. A. (1988) Tandem queues with finite intermediate waiting room and blocking in heavy traffic. Prob. Control Int. Theory 17, 3–13.Google Scholar
[23] Lind Val, T. (1973) Weak convergence of probability measures and random functions in the functional space D([0, 8)). J. Appl. Prob. 10, 109–121.CrossRefGoogle Scholar
[24] Liptzer, R. Sh. and Shiryaev, A. N. (1989) Theory of Martingales. Kluwer, Boston.CrossRefGoogle Scholar
[25] Macarichev, A. V. (1984) Analysis of a tandem queueing system with identical service times at both counters for various service disciplines. Fundamentals of Teletraffic Theory. Proc. Third Int. Seminar on Teletraffic Theory, pp. 298–301.Google Scholar
[26] Prohorov, Yu. V. (1956) Convergence of random processes and limit theorems in probability theory. Theory Prob. Appl. 1, 157–214.Google Scholar
[27] Reiman, M. I. (1984) Open queueing networks in heavy traffic. Math. Operat Res. 9, 441–458.Google Scholar
[28] Reiman, M. I. (1988) A multiclass feedback queue in heavy traffic. Adv. Appl. Prob. 20, 179–207.Google Scholar
[29] Reiman, M. I. and Simon, B. (1990) A network of priority queues in heavy traffic: one bottleneck station. Queueing Systems 6, 33–58.Google Scholar
[30] Skorohod, A. V. (1965) Studies in the Theory of Random Processes. Addison-Wesley, New York.Google Scholar
[31] Szczotka, W. (1992) Weak convergence under mapping. Proc. Math. Statist. 13, 39–58.Google Scholar
[32] Szczotka, W. and Kelly, F. P. (1990) Asymptotic stationarity of queues in series and the heavy traffic approximation. Ann. Prob. 18, 1232–1248.Google Scholar
[33] Vinogradov, O. P. (1983) Tandem queue with identical service (in Russian). Math. Notes 33, 141–146.Google Scholar
[34] Vinogradov, O. P. (1985) Queues in series with identical service in heavy traffic (in Russian). Iz. AN USSR, ser. Tech. Kibern. 6, 29–32.Google Scholar
[35] Wentzell, A. D. (1975) Course on Theory of Random Processes. Nauka, Moscow.Google Scholar
[36] Whitt, W. (1974) Heavy traffic limit theorems for queues: a survey. Math. Methods in Queueing Theory. (Lect. Notes in Econ. Math. Syst). 98, 307–350.Google Scholar
[37] Whitt, W. (1971) Weak convergence theorems for priority queues: preemptive resume discipline. J. Appl. Prob. 8, 79–94.CrossRefGoogle Scholar