Cyclic-type Polling Models with Preparation Times

N. Perel, J. L. Dorsman, M. Vlasiou

Abstract

We consider a system consisting of a server serving in sequence a fixed number of stations. At each station there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queuing networks, to an extension of polling systems, and surprisingly to random graphs. We are interested in the waiting time of the server. The waiting time of the server satisfies a Lindley-type equation of a non-standard form. We give a sufficient condition for the existence of a limiting waiting time distribution in the general case, and assuming preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We provide detailed computations for a special case and extensive numerical results investigating the effect of the system’s parameters to the performance of the server.

References

  1. Asmussen, S. (2003). Applied Probability and Queues. Springer Verlag.
  2. Asmussen, S. and Sigman, K. (1996). Monotone stochastic recursions and their duals. Probability in the Engineering and Informational Sciences, 10(1):1-20.
  3. Boon, M. A. A., Van der Mei, R. D., and Winands, E. M. M. (2011). Applications of polling systems. Surveys in Operations Research and Management Science, 16(2):67-82.
  4. Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. Wiley Series in Probability and Statistics. John Wiley & Sons Ltd., Chichester.
  5. Boxma, O. J. and Groenendijk, W. P. (1988). Two queues with alternating service and switching times. In Boxma, O. J. and Syski, R., editors, Queueing Theory and its Applications (Liber Amicorum for J. W. Cohen), pages 261-282. Amsterdam: North-Holland.
  6. Cohen, J. W. (1982). The Single Server Queue. NorthHolland Publishing Co., Amsterdam.
  7. Eisenberg, M. (1979). Two queues with alternating service. SIAM Journal on Applied Mathematics, 36:287-303.
  8. Franks, G., Al-Omari, T., Woodside, M., Das, O., and Derisavi, S. (2009). Enhanced modeling and solution of layered queuing networks. IEEE Transactions on Software Engineering, 35:148-161.
  9. Gamarnik, D., Nowicki, T., and Swirszcz, G. (2006). Maximum weight independent sets and matchings in sparse random graphs. Exact results using the local weak convergence method. Random Structures & Algorithms, 28(1):76-106.
  10. Kalashnikov, V. (2002). Stability bounds for queueing models in terms of weighted metrics. In Suhov, Y., editor, Analytic Methods in Applied Probability, volume 207 of American Mathematical Society Translations Ser. 2, pages 77-90. American Mathematical Society, Providence, RI.
  11. Lindley, D. V. (1952). The theory of queues with a single server. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 48, pages 277- 289.
  12. Litvak, N. and Vlasiou, M. (2010). A survey on performance analysis of warehouse carousel systems. Statistica Neerlandica, 64(4):401-447.
  13. McGinnis, L. F., Han, M. H., and White, J. A. (1986). Analysis of rotary rack operations. In White, J., editor, Proceedings of the 7th International Conference on Automation in Warehousing, pages 165-171, San Francisco, California. Springer.
  14. Park, B. C., Park, J. Y., and Foley, R. D. (2003). Carousel system performance. Journal of Applied Probability, 40(3):602-612.
  15. Resing, J. A. C. (1993). Polling systems and multitype branching processes. Queueing Systems. Theory and Applications, 13(4):409-426.
  16. Takagi, H. (1986). Analysis of polling systems. MIT press.
  17. Tijms, H. C. (1994). Stochastic Models: an Algorithmic Approach. Wiley, Chichester.
  18. van der Mei, R. D. (2007). Towards a unifying theory on branching-type polling systems in heavy traffic. Queueing Systems. Theory and Applications, 57(1):29-46.
  19. van Vuuren, M. and Winands, E. M. M. (2007). Iterative approximation of k-limited polling systems. Queueing Systems. Theory and Applications, 55(3):161-178.
  20. Vlasiou, M. (2006). Lindley-type Recursions. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands.
  21. Yechiali, U. (1993). Analysis and control of polling systems. In Donatiello, L. and Nelson, R., editors, Performance Evaluation of Computer and Communication Systems, Joint Tutorial Papers of Performance 7893 and Sigmetrics 7893, volume 729, pages 630-650, London, UK. Springer Berlin / Heidelberg.
Download


Paper Citation


in Harvard Style

Perel N., Dorsman J. and Vlasiou M. (2013). Cyclic-type Polling Models with Preparation Times . In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES, ISBN 978-989-8565-40-2, pages 14-23. DOI: 10.5220/0004206500140023


in Bibtex Style

@conference{icores13,
author={N. Perel and J. L. Dorsman and M. Vlasiou},
title={Cyclic-type Polling Models with Preparation Times},
booktitle={Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},
year={2013},
pages={14-23},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0004206500140023},
isbn={978-989-8565-40-2},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,
TI - Cyclic-type Polling Models with Preparation Times
SN - 978-989-8565-40-2
AU - Perel N.
AU - Dorsman J.
AU - Vlasiou M.
PY - 2013
SP - 14
EP - 23
DO - 10.5220/0004206500140023