Abstract
In this paper we present a unified analysis of the BMAP/G/1 cyclic polling model and its application to the gated and exhaustive service disciplines as examples. The applied methodology is based on the separation of the analysis into service discipline independent and dependent parts. New expressions are derived for the vector-generating function of the stationary number of customers and for its mean in terms of vector quantities depending on the service discipline. They are valid for a broad class of service disciplines and both for zero- and nonzero-switchover-times polling models.
We present the service discipline specific solution for the nonzero-switchover-times model with gated and exhaustive service disciplines. We set up the governing equations of the system by using Kronecker product notation. They can be numerically solved by means of a system of linear equations. The resulting vectors are used to compute the service discipline specific vector quantities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Takagi, H.: Analysis of Polling Systems. MIT Press, Cambridge (1986)
Lucantoni, D.L.: New results on the single server queue with a batch Markovian arrival process. Stoch. Models 7, 1–46 (1991)
Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Applications. Dekker, New York (1989)
Kasahara, S., Takine, T., Takahashi, Y., Hasegawa, T.: MAP/G/1 queues under N-policy with and without vacations. J. Oper. Res. Soc. Jpn. 39(2), 188–212 (1996)
Chang, S.H., Takine, T., Chae, K.C., Lee, H.W.: A unified queue length formula for BMAP/G/1 queue with generalized vacations. Stoch. Models 18, 369–386 (2002)
Chang, S.H., Takine, T.: Factorization and stochastic decomposition properties in bulk queues with generalized vacations. Queueing Syst. 50, 165–183 (2005)
Banik, A.D., Gupta, U.C., Pathak, S.S.: BMAP/G/1/N queue with vacations and limited service discipline. Appl. Math. Comput. 180, 707–721 (2006)
Takine, T.: The nonpreemptive priority MAP/G/1 queue. Oper. Res. 47, 917–927 (1999)
Takine, T., Takahashi, Y.: On the relationship between queue lengths at a random instant and at a departure in the stationary queue with BMAP arrivals. Stoch. Models 14, 601–610 (1998)
Nishimura, S.: A spectral method for a nonpreemptive priority BMAP/G/1/N queue. Stoch. Models 21, 579–597 (2005)
Borst, S.C., Boxma, O.J.: Polling models with and without switchover times. Oper. Res. 45, 536–543 (1997)
Eisenberg, M.: Queues with periodic service and changeover time. Oper. Res. 20, 440–451 (1972)
Saffer, Zs., Telek, M.: Stability of periodic polling system with BMAP arrivals. Europ. J. Oper. Res. 197(1), 188–195 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is partially supported by the OTKA K61709 grant.
Rights and permissions
About this article
Cite this article
Saffer, Z., Telek, M. Unified analysis of BMAP/G/1 cyclic polling models. Queueing Syst 64, 69–102 (2010). https://doi.org/10.1007/s11134-009-9136-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-009-9136-7