Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Single-server queues with spatially distributed arrivals

Abstract

Consider a queueing system where customers arrive at a circle according to a homogeneous Poisson process. After choosing their positions on the circle, according to a uniform distribution, they wait for a single server who travels on the circle. The server's movement is modelled by a Brownian motion with drift. Whenever the server encounters a customer, he stops and serves this customer. The service times are independent, but arbitrarily distributed. The model generalizes the continuous cyclic polling system (the diffusion coefficient of the Brownian motion is zero in this case) and can be interpreted as a continuous version of a Markov polling system. Using Tweedie's lemma for positive recurrence of Markov chains with general state space, we show that the system is stable if and only if the traffic intensity is less than one. Moreover, we derive a stochastic decomposition result which leads to equilibrium equations for the stationary configuration of customers on the circle. Steady-state performance characteristics are determined, in particular the expected number of customers in the system as seen by a travelling server and at an arbitrary point in time.

This is a preview of subscription content, log in to check access.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. [1]

    E. Altman and S. Foss, Polling on a graph with general arrival and service time distribution, Preprint, INRIA Centre Sophia Antipolis (1993)

  2. [2]

    E. Altman, P. Konstantopoulos and Z. Liu, Stability, monotonicity and invariant quantities in general polling systems, Queueing Syst. 11 (1992) 35–57.

  3. [3]

    E. Altman and U. Yechiali, Cyclic Bernoulli polling, Zeits. Oper. Res. 38 (1993) 55–76.

  4. [4]

    E. Altman and H. Levy, Queueing in space, Preprint 4-93, Rutgers University, New Brunswick, NJ (1993).

  5. [5]

    S. Asmussen,Applied Probability and Queues (Wiley, Chichester, 1987).

  6. [6]

    K.B. Athreya and P. Ney, A new approach to the limit theory of recurrent Markov chains, Trans. Amer. Math. Soc. 245 (1978) 493–501.

  7. [7]

    I. Bardhan and K. Sigman, Rate conservation law for stationary semimartingales, Prob. Eng. Inf. Sci. 7 (1993) 1–17.

  8. [8]

    H.C.P. Berbee,Random Walks with Stationary Increments and Renewal Theory, Mathematical Centre Tracts 112 (Mathematisch Centrum, Amsterdam, 1979).

  9. [9]

    D.J. Bertsimas and G. van Ryzin, A stochastic and dynamic vehicle routing problem in the Euclidean plane, Oper. Res. 39 (1991) 601–615.

  10. [10]

    A. Borovkov and R. Schassberger, Ergodicity of a polling network, Stoch. Proc. Appl. 50 (1994) 253–262.

  11. [11]

    O.J. Boxma and W.P. Groenendijk, Pseudo-conservation laws in cyclic service systems, J. Appl. Prob. 24 (1987) 949–964.

  12. [12]

    O.J. Boxma and J.A. Weststrate, Waiting times in polling systems with Markovian server routing, in:Messung, Modellierung und Bewertung von Rechnersystemen und Netzen, eds. G. Siege and J.S. Lie, (Springer, Berlin, 1989) pp. 89–104.

  13. [13]

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

  14. [14]

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

  15. [15]

    R.B. Cooper and G. Murray, Queues served in cyclic order, Bell Syst. Techn. J. 48 (1969) 675–689.

  16. [16]

    D.J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes (Springer, New York, 1980).

  17. [17]

    M. Eisenberg, Queues with periodic service and changeover times, Oper. Res. 20 (1972) 440–451.

  18. [18]

    P. Franken, D. König, U. Arndt and V. Schmidt,Queues and Point Processes (Wiley, Chichester, 1982).

  19. [19]

    C. Fricker and M.R. Jaibi, Monotonicity and stability of periodic polling models, Queueing Syst. 15 (1994) 211–238.

  20. [20]

    S.W. Fuhrmann, Symmetric queues served in cyclic order, Oper. Res. Lett 4 (1985) 139–144.

  21. [21]

    S.W. Fuhrmann and R.B. Cooper, Stochastic decomposition in theM/G/1 queue with generalized vacations, Oper. Res. 33 (1985) 1117–1129.

  22. [22]

    S.W. Fuhrmann and R.B. Cooper, Application of decomposition principle in M/G/1 vacation model to two continuum cyclic queueing models — especially token-ring LANs, AT&T Techn. J. 64 (1985) 1091–1098.

  23. [23]

    L. Georgiadis and W. Szpankowski, Stability of token passing rings, Queueing Syst. 11 (1992) 7–33.

  24. [24]

    S. Karlin and H.M. Taylor,A Second Course in Stochastic Processes (Academic Press, New York, 1981).

  25. [25]

    J. Keilson and L.D. Servi, Oscillating random walk models forGI/G1 vacation systems with Bernoulli schedules, J. Appl. Prob. 23 (1986) 790–802.

  26. [26]

    L. Kleinrock and H. Levy, The analysis of random polling systems, Oper. Res. 36 (1988) 716–732.

  27. [27]

    D. König and V. Schmidt,Random Point Processes (Teubner, Stuttgart, 1992; in German).

  28. [28]

    A.G. Koheim and B. Meister, Waiting lines and times in a system with polling, J. ACM 21 (1974) 470–490.

  29. [29]

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

  30. [30]

    D.P. Kroese and V. Schmidt, Queueing systems on a circle, Zeits. Oper. Res. 37 (1993) 303–331.

  31. [31]

    D.P. Kroese and V. Schmidt, Light-traffic analysis for queues with spatially distributed arrivals, Math. Oper. Res., to appear.

  32. [32]

    P.J. Kühn, Multiqueue systems with nonexhaustive cyclic service, Bell Syst. Techn. J. 58 (1979) 671–698.

  33. [33]

    T. Lindvall,Lectures on the Coupling Method (Wiley, New York, 1992).

  34. [34]

    K. Matthes, J. Kerstan and J. Mecke,Infinitely Divisible Point Processes (Wiley, Chichester, 1978).

  35. [35]

    R. Mazumdar, V. Badrinath, F. Guillemin and C. Rosenberg, On pathwise rate conservation for a class of semi-martingales, Stoch. Proc. Appl. 47 (1993) 119–130.

  36. [36]

    M. Miyazawa, Rate conservation laws: A survey, Queueing Syst. 15 (1994) 1–58.

  37. [37]

    E. Nummelin, A splitting technique for Harris recurrent Markov chains, Zeits. Wahrscheinlichkeitstheorie verw. Geb. 43 (1978) 309–318.

  38. [38]

    J.A.C. Resing, Polling systems and multitype branching processes, Queueing Syst. 13 (1993) 409–426.

  39. [39]

    R. Schassberger,Queues (Springer, Wien, 1973).

  40. [40]

    K. Sigman and R.W. Wolff, A review of regenerative processes, SIAM Rev. 35 (1993) 269–288.

  41. [41]

    F.M. Spieksma and R.L. Tweedie, Strengthening ergodicity to geometric ergodicity for Markov chains, Stoch. Models 10 (1994) 45–74.

  42. [42]

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

  43. [43]

    Tedijanto, Exact results for the cyclic service queue with a Bernoulli schedule, Perf. Eval. 11 (1990) 107–115.

  44. [44]

    H. Thorisson, Construction of a stationary regenerative process, Stoch. Proc. Appl. 42 (1992) 237–253.

  45. [45]

    R.L. Tweedie, Criteria for classifying general Markov chains, Adv. Appl. Prob. 8 (1976) 737–771.

  46. [46]

    R.L. Tweedie, The existence of moments for stationary Markov chains, J. Appl. Prob. 20 (1983) 191–196.

  47. [47]

    R.W. Wolff, Work conserving priorities, J. Appl. Prob. 7 (1970) 327–337.

  48. [48]

    R.W. Wolff, Poisson arrivals see time averages, Oper. Res. 30 (1982) 223–231.

  49. [49]

    U. Yechiali, Optimal dynamic control of polling systems, in:Queueing Performance and Control in ATM, eds. J.W. Cohen and C.D. Pack (North-Holland, 1991) pp. 205–217.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kroese, D.P., Schmidt, V. Single-server queues with spatially distributed arrivals. Queueing Syst 17, 317–345 (1994). https://doi.org/10.1007/BF01158698

Download citation

Keywords

  • Single-server queue
  • spatially distributed arrival points
  • travelling server
  • Brownian motion
  • embedded Markov chain
  • stability
  • Tweedie's lemma
  • regenerative processes
  • stochastic decomposition
  • equilibrium equations
  • mean queue length