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Stability and performance of greedy server systems

A review and open problems

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Abstract

Consider a queueing system in which arriving customers are placed on a circle and wait for service. A traveling server moves at constant speed on the circle, stopping at the location of the customers until service completion. The server is greedy: always moving in the direction of the nearest customer. Coffman and Gilbert conjectured that this system is stable if the traffic intensity is smaller than 1; however, a proof or counterexample remains unknown. In this review, we present a picture of the current state of this conjecture and suggest new related open problems.

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References

  1. Altman, E., Foss, S.: Polling on a space with general arrival and service distribution. Oper. Res. Lett. 20, 187–194 (1997)

    Article  Google Scholar 

  2. Altman, E., Levy, H.: Queueing in space. Adv. Appl. Probab. 26, 1095–1116 (1994)

    Article  Google Scholar 

  3. Bertsimas, D.J., van Ryzin, G.: A stochastic and dynamic vehicle routing problem in the Euclidean plane. Oper. Res. 39, 601–615 (1993)

    Article  Google Scholar 

  4. Borst, S.C.: Polling Systems. CWI Tract, Amsterdam (1996)

    Google Scholar 

  5. Coffman, E.G., Gilbert, E.N.: A continuous polling system with constant service times. IEEE Trans. Inf. Theory 32, 584–591 (1986)

    Article  Google Scholar 

  6. Coffman, E.G., Gilbert, E.N.: Polling and greedy servers on line. Queueing Syst. 2, 115–145 (1987)

    Article  Google Scholar 

  7. Coffman, E.G., Stolyar, A.: Continuous polling on graphs. Probab. Eng. Inf. Sci. 7, 209–226 (1993)

    Article  Google Scholar 

  8. Eliazar, I.: The snowblower problem. Queueing Syst. 45, 357–380 (2003)

    Article  Google Scholar 

  9. Eliazar, I.: From polling to snowplowing. Queueing Syst. 51, 115–133 (2005)

    Article  Google Scholar 

  10. Foss, S., Last, G.: Stability of polling systems with exhaustive service policies and state-dependent routing. Ann. Appl. Probab. 6, 116–137 (1996)

    Article  Google Scholar 

  11. Foss, S., Last, G.: On the stability of greedy polling systems with general service policies. Probab. Eng. Inf. Sci. 12, 49–68 (1998)

    Article  Google Scholar 

  12. Fuhrmann, S.W., Cooper, R.B.: Applications of the decomposition principle in M/G/1 vacation models to two continuum cyclic models. AT&T Bell Lab. Tech. J. 64, 1091–1098 (1985)

    Google Scholar 

  13. Harel, A., Stulman, A.: Polling, greedy and horizon servers on a circle. Oper. Res. 43, 177–186 (1995)

    Article  Google Scholar 

  14. Knuth, D.E.: The Art of Computer Programming, vol. 3. Addison-Wesley, Reading (1973)

    Google Scholar 

  15. Kroese, D.P.: Heavy-traffic analysis for continuous polling models. J. Appl. Probab. 34, 720–732 (1997)

    Article  Google Scholar 

  16. Kroese, D.P., Schmidt, V.: A continuous polling system with general service times. Ann. Appl. Probab. 2, 906–927 (1992)

    Article  Google Scholar 

  17. Kroese, D.P., Schmidt, V.: Queueing systems on a circle. J. Math. Methods Oper. Res. 37, 303–331 (1993)

    Article  Google Scholar 

  18. Kroese, D.P., Schmidt, V.: Single server queues with spatially distributed arrivals. Queueing Syst. 17, 317–345 (1994)

    Article  Google Scholar 

  19. Kroese, D.P., Schmidt, V.: Light-traffic analysis for queues with spatially distributed arrivals. Math. Oper. Res. 21, 137–157 (1996)

    Article  Google Scholar 

  20. Kurkova, I.A., Menshikov, M.V.: Greedy algorithm, z 1 case. Markov Process. Relat. Fields 3(2), 243 (1997)

    Google Scholar 

  21. Leskelä, L., Unger, F.: Stability of spatial polling system with greedy myopic server. Ann. Oper. Res. (2010). doi:10.1007/s10479-010-0762-6. http://arXiv.org/abs/0908.4585

    Google Scholar 

  22. Robert, Ph.: The evolution of a spatial stochastic network. Stoch. Proc. Appl. (2010). doi:10.1016/j.spa.2010.03.013. http://arXiv.org/abs/0908.3256

    Google Scholar 

  23. Schassberger, R.: Stability of polling networks with state-dependent server routing. Probab. Eng. Inf. Sci. 9, 539–550 (1995)

    Article  Google Scholar 

  24. Takagi, H.: Analysis of Polling Systems. MIT Press, Cambridge (1986)

    Google Scholar 

  25. Takagi, H.: Queueing analysis of Pollin systems: An update. In: Takagi, H. (ed.) Stochastic Analysis of Computer and Communication Systems, pp. 267–318. Elsevier Science, Amsterdam (1990)

    Google Scholar 

  26. Takagi, H.: Queueing analysis of polling models: Progress in 1990–1994. In: Dshalalow, J.H. (ed.) Frontiers in Queueing: Models and Applications in Science and Engineering, pp. 119–146. CRC Press, Boca Raton (1997)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of Queensland, Brisbane, Australia

    Leonardo Rojas-Nandayapa & Dirk P. Kroese

  2. School of Mathematics and Computer Sciences, Heriot-Watt University, Edinburgh, UK

    Sergey Foss

Authors
  1. Leonardo Rojas-Nandayapa
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  2. Sergey Foss
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  3. Dirk P. Kroese
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Correspondence to Leonardo Rojas-Nandayapa.

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Rojas-Nandayapa, L., Foss, S. & Kroese, D.P. Stability and performance of greedy server systems. Queueing Syst 68, 221–227 (2011). https://doi.org/10.1007/s11134-011-9235-0

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  • DOI: https://doi.org/10.1007/s11134-011-9235-0

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