Abstract
The joint queue length process in polling systems with and without switchover times is studied. If the service discipline in each queue satisfies a certain property it is shown that the joint queue length process at polling instants of a fixed queue is a multitype branching process (MTBP) with immigration. In the case of polling models with switchover times, it turns out that we are dealing with an MTBP with immigration in each state, whereas in the case of polling models without switchover times we are dealing with an MTBP with immigration in state zero. The theory of MTBPs leads to expressions for the generating function of the joint queue length process at polling instants. Sufficient conditions for ergodicity and moment calculations are also given.
Similar content being viewed by others
References
K.B. Athreya and P.E. Ney,Branching Processes (Springer-Verlag, Berlin, 1972).
J.E. Baker and I. Rubin, Polling with a general service order table, IEEE Trans. Commun. 35 (1987) 283–288.
O.J. Boxma, H. Levy and U. Yechiali, Cyclic reservation schemes for efficient operation of multiple queue single server systems, to appear in Ann. Oper. Res.
S.W. Fuhrmann, Performance analysis of a class of cyclic schedules, Bell Laboratories technical memorandum 81-59531-1 (March 1981).
S.W. Fuhrmann, A decomposition result for a class of polling models, IBM Research Report, Zürich (May 1991).
S.W. Fuhrmann and R.B. Cooper, Stochastic decompositions in the M/G/1 queue with generalized vacations, Oper. Res. 33 (1985) 1117–1129.
N. Kaplan, The multitype Galton-Watson process with immigration, Ann. Prob. 1 (1973) 947–953.
L. Kleinrock and H. Levy, The analysis of random polling systems, Oper. Res. 36 (1988) 716–732.
A.G. Konheim and H. Levy, Efficient analysis of polling systems, in:Proc. INFOCOM'92, Florence, Italy, May 1992.
H. Levy, Optimization of polling systems: The fractional exhaustive service method, Technical report, Department of Computer Science, Tel Aviv University (1988).
H. Levy, Analysis of cyclic polling systems with binomial gated service, in:Performance of Distributed and Parallel Systems, eds. T. Hasegawa, H. Takagi, Y. Takahashi (North-Holland, Amsterdam, 1989) pp. 127–139.
H. Levy and M. Sidi, Polling systems with correlated arrivals, IEEE Infocom 1989 (1989) 907–913.
A.G. Pakes, A branching process with a state dependent immigration component, Adv. Appl. Prob. 3 (1971) 301–314.
M.P. Quine, The multitype Galton-Watson process with immigration, J. Appl. Prob. 7 (1970) 411–422.
R. Ramaswamy and L.D. Servi, The busy period of theM/G/1 vacation model with a Bernoulli schedule, Commun. Stat.-Stoch. Models 4 (1988) 507–521.
J.A.C. Resing, Asymptotic results in feedback systems, Ph.D. Thesis, Technical University Delft (1990).
E. Seneta,Non-negative Matrices and Markov Chains, 2nd ed. (Springer-Verlag, New York, 1981).
M. Sidi and H. Levy, Customers routing in polling systems, in:Performance 1990, eds. P.J.B. King, I. Mitrani and R.J. Pooley (North-Holland, Amsterdam, 1990) pp. 319–331.
H. Tkagi,Analysis of Polling Systems (MIT Press, Cambridge, MA, 1986).
H. Takagi, Analysis of polling systems with a mixture of exhaustive and gated service disciplines, J. Oper. Res. Soc. Jpn. 32 (1989) 450–161.
H. Takagi, Queueing analysis of polling models: an update,Stochastic Analysis of Computer and Communications Systems, ed. H. Takagi (Elsevier Science Publ., Amsterdam, 1990) pp. 267–318.
Author information
Authors and Affiliations
Additional information
This work was done while the author was at the Centre for Mathematics and Computer Science (CWI) in Amsterdam, The Netherlands.
Rights and permissions
About this article
Cite this article
Resing, J.A.C. Polling systems and multitype branching processes. Queueing Syst 13, 409–426 (1993). https://doi.org/10.1007/BF01149263
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01149263