Skip to main content

Modeling and optimization of buffers and servers in finite queueing networks

Abstract

The joint buffer and server optimization problem (BCAP) is a non-linear optimization problem with integer decision variables that optimizes the numbers of buffers and servers such that the resulting throughput is greater than a pre-defined threshold throughput. This work presents a detailed review of the current literature that addresses allocation problems, particularly the BCAP, and a quite effective methodology for solving this problem, which consists of a combination of approximate methods and the Powell algorithm, a derivative-free optimization algorithm. The methodology was applied to networks of queues in the basic topologies series, split, and merge, producing very encouraging results that pointed at robust and homogeneous solutions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. Almeida, M.A.C., Cruz, F.R.B.: A note on Bayesian estimation of traffic intensity in single-server Markovian queues. Commun. Stat. Simul. Comput. 47(9), 2577–2586 (2018)

    Article  Google Scholar 

  2. Almeida, M.A.C., Cruz, F.R.B., Oliveira, F.L.P., de Souza, G.: Bias correction for estimation of performance measures of a Markovian queue. Oper. Res. (to appear) (2017). https://doi.org/10.1007/s12351-017-0351-4. (Available on line 08 Oct 2017)

    Google Scholar 

  3. Andriansyah, R., van Woensel, T., Cruz, F.R.B., Duczmal, L.: Performance optimization of open zero-buffer multi-server queueing networks. Comput. Oper. Res. 37(8), 1472–1487 (2010)

    Article  Google Scholar 

  4. Balsamo, S., de Nitto, Personé V., Onvural, R.: Analysis of Queueing Networks with Blocking. Kluwer Academic Publishers, Dordrecht (2001)

    Book  Google Scholar 

  5. Conway, R., Maxwell, W.L., McClain, J.O., Thomas, L.J.: The role of work-in-process inventories in serial production lines. Oper. Res. 36, 229–241 (1988)

    Article  Google Scholar 

  6. Cruz, F.R.B., MacGregor Smith, J., Medeiros, R.O.: An \(M/G/C/C\) state dependent network simulation model. Comput. Oper. Res. 32(4), 919–941 (2005)

    Article  Google Scholar 

  7. Cruz, F.R.B., Duarte, A.R., van Woensel, T.: Buffer allocation in general single-server queueing networks. Comput. Oper. Res. 35(11), 3581–3598 (2008)

    Article  Google Scholar 

  8. Cruz, F.R.B., Oliveira, P.C., Duczmal, L.: State-dependent stochastic mobility model in mobile communication networks. Simul. Model. Pract. Theory 18(3), 348–365 (2010)

    Article  Google Scholar 

  9. Cruz, F.R.B., Kendall, G., While, L., Duarte, A.R., Brito, N.L.C.: Throughput maximization of queueing networks with simultaneous minimization of service rates and buffers. Math. Probl. Eng. 2012, 19 (2012)

    Article  Google Scholar 

  10. Cruz, F.R.B., Quinino, R.C., Ho, L.L.: Bayesian estimation of traffic intensity based on queue length in a multi-server \(M/M/s\) queue. Commun. Stat. Simul. Comput. 46(9), 7319–7331 (2017)

    Article  Google Scholar 

  11. Cruz, F.R.B., Almeida, M.A.C., D’Angelo, M.F.S.V., van Woensel, T.: Traffic intensity estimation in finite Markovian queueing systems. Math. Probl. Eng. 2018, 15 (2018)

    Article  Google Scholar 

  12. Cruz, F.R.B., Duarte, A.R., Souza, G.L.: Multi-objective performance improvements of general finite single-server queueing networks. J. Heuristics 24(5), 757–781 (2018)

    Article  Google Scholar 

  13. Cruz, F.R.B., Santos, M.A.C., Oliveira, F.L.P., Quinino, R.C.: Estimation in a general bulk-arrival markovian multi-server finite queue. Oper. Res. (to appear) (2018). https://doi.org/10.1007/s12351-018-0433-y. (Available on line 06 Oct 2018)

    Google Scholar 

  14. Dallery, Y., Gershwin, S.B.: Manufacturing flow line systems: a review of models and analytical results. Queueing Syst. 12(1–2), 3–94 (1992)

    Article  Google Scholar 

  15. Dallery, Y., Stecke, K.E.: On the optimal allocation of servers and workloads in closed queueing networks. Oper. Res. 38(4), 694–703 (1990)

    Article  Google Scholar 

  16. Daskalaki, S., MacGregor Smith, J.: Combining routing and buffer allocation problems in series-parallel queueing networks. Ann. Oper. Res. 125(1–4), 47–68 (2004)

    Article  Google Scholar 

  17. Dorda, M., Teichmann, D.: Modelling of freight trains classification using queueing system subject to breakdowns. Math. Probl. Eng. 2013 (Article ID 307652), 11 (2013)

  18. Hall, N.G., Sriskandarajah, C.: A survey of machine scheduling problems with blocking and no-wait in process. Oper. Res. 44(3), 510–525 (1996)

    Article  Google Scholar 

  19. Harris, J.H., Powell, S.G.: An algorithm for optimal buffer placement in reliable serial lines. IIE Trans. 31, 287–302 (1999)

    Google Scholar 

  20. Hillier, F.S., So, K.C.: The effect of the coefficient of variation of operation times on the allocation of storage space in production line systems. IIE Trans. 23(2), 198–206 (1991)

    Article  Google Scholar 

  21. Jackson, J.R.: Networks of waiting lines. Oper. Res. 5(4), 518–521 (1957)

    Article  Google Scholar 

  22. Johnson, D.S., Lenstra, J.K., Rinnooy Kan, A.H.G.: The complexity of the network design problem. Networks 8(4), 279–285 (1978)

    Article  Google Scholar 

  23. Kendall, D.G.: Stochastic processes occurring in the theory of queues and their analysis by the method of embedded Markov chains. Ann. Math. Stat. 24, 338–354 (1953)

    Article  Google Scholar 

  24. Kerbache, L., MacGregor Smith, J.: The generalized expansion method for open finite queueing networks. Eur. J. Oper. Res. 32, 448–461 (1987)

    Article  Google Scholar 

  25. Kerbache, L., MacGregor Smith, J.: Asymptotic behavior of the expansion method for open finite queueing networks. Comput. Oper. Res. 15(2), 157–169 (1988)

    Article  Google Scholar 

  26. Kose, S.Y., Kilincci, O.: Hybrid approach for buffer allocation in open serial production lines. Comput. Oper. Res. 60, 67–78 (2015)

    Article  Google Scholar 

  27. Kuehn, P.: Approximate analysis of general queuing networks by decomposition. IEEE Trans. Commun. 27(1), 113–126 (1979)

    Article  Google Scholar 

  28. MacGregor Smith, J.: Optimal workload allocation in closed queueing networks with state dependent queues. Ann. Oper. Res. 231(1), 157–183 (2015)

    Article  Google Scholar 

  29. MacGregor Smith, J., Barnes, R.: Optimal server allocation in closed finite queueing networks. Flex. Serv. Manuf. J. 27(1), 58–85 (2015)

    Article  Google Scholar 

  30. MacGregor Smith, J., Cruz, F.R.B.: The buffer allocation problem for general finite buffer queueing networks. IIE Trans. 37(4), 343–365 (2005)

    Article  Google Scholar 

  31. MacGregor Smith, J., Daskalaki, S.: Buffer space allocation in automated assembly lines. Oper. Res. 36(2), 343–358 (1988)

    Article  Google Scholar 

  32. MacGregor Smith, J., Cruz, F.R.B., van Woensel, T.: Optimal server allocation in general, finite, multi-server queueing networks. Appl. Stoch. Mod. Bus. Ind. 26(6), 705–736 (2010)

    Article  Google Scholar 

  33. Nahas, N.: Buffer allocation and preventive maintenance optimization in unreliable production lines. J. Intell. Manuf. 28(1), 85–93 (2017)

    Article  Google Scholar 

  34. Perros, H.G.: Queueing networks with blocking: a bibliography. ACM Sigmet 12, 8–12 (1984)

    Article  Google Scholar 

  35. Reiser, M., Kobayashi, H.: Accuracy of the diffusion approximation for some queuing systems. IBM J. Res. Dev. 18(2), 110–124 (1974)

    Article  Google Scholar 

  36. Springer, M.C., Makens, P.K.: Queueing models for performance analysis: selection of single station models. Eur. J. Oper. Res. 58(1), 123–145 (1992)

    Article  Google Scholar 

  37. Suri, R.: An overview of evaluative models for flexible manufacturing systems. Ann. Oper. Res. 3, 13–21 (1985)

    Article  Google Scholar 

  38. Whitt, W.: Open and closed models for networks of queues. AT & T Bell Lab. Tech. J. 63(9), 1911–1979 (1984)

    Article  Google Scholar 

  39. van Woensel, T., Cruz, F.R.B.: Optimal routing in general finite multi-server queueing networks. PLoS ONE 9(7), e102075 (2014)

    Article  Google Scholar 

  40. van Woensel, T., Andriansyah, R., Cruz, F.R.B., MacGregor Smith, J., Kerbache, L.: Buffer and server allocation in general multi-server queueing networks. Int. Trans. Oper. Res. 17(2), 257–286 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

This research has been partially funded by the Brazilian agencies CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico of the Ministry for Science and Technology), under grants 300825/2016-1 and 305515/2018-7, and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais), under grants CEX-PPM-00564-17 and PPM-00321-18.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to F. R. B. Cruz.

Ethics declarations

Conflict of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Martins, H.S.R., Cruz, F.R.B., Duarte, A.R. et al. Modeling and optimization of buffers and servers in finite queueing networks. OPSEARCH 56, 123–150 (2019). https://doi.org/10.1007/s12597-019-00362-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-019-00362-7

Keywords

  • Buffer and server allocation
  • Finite queues
  • Queueing networks
  • Generalized expansion method