Skip to main content
Log in

Tandem queues with subexponential service times and finite buffers

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We focus on tandem queues with subexponential service time distributions. We assume that number of customers in front of the first station is infinite and there is infinite room for finished customers after the last station but the size of the buffer between two consecutive stations is finite. Using (max, +) linear recursions, we investigate the tail asymptotics of transient response times and waiting times under both communication blocking and manufacturing blocking schemes. We also discuss under which conditions these results can be generalized to the tail asymptotics of stationary response times and waiting times.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Altiok, T.M., Stidham, S.: A note on transfer lines with unreliable machines, random processing times, and finite buffers. IIE Trans. 14(2), 125–127 (1982)

    Google Scholar 

  2. Asmussen, S.: Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8(2), 354–374 (1998)

    Article  Google Scholar 

  3. Asmussen, S., Møller, J.R.: Tail asymptotics for M/G/1 type queueing processes with subexponential increments. Queueing Syst. 33(1), 153–176 (1999)

    Article  Google Scholar 

  4. Asmussen, S., Henriksen, L.F., Klüppelberg, C.: Large claims approximations for risk processes in a Markovian environment. Stoch. Process. Appl. 54, 29–43 (1994)

    Article  Google Scholar 

  5. Asmussen, S., Klüppelberg, C., Sigman, K.: Sampling at subexponential times, with queueing applications. Stoch. Process. Appl. 79(2), 265–286 (1999)

    Article  Google Scholar 

  6. Ayhan, H., Kim, J.K.: A general class of closed fork and join queues with subexponential service times. Stoch. Mod. 23(4), 523–535 (2007)

    Article  Google Scholar 

  7. Ayhan, H., Palmowski, Z., Schlegel, S.: Cyclic queueing networks with subexponential service times. J. Appl. Probab. 41(3), 791–801 (2004)

    Article  Google Scholar 

  8. Baccelli, F., Foss, S.: Moments and tails in monotone-separable stochastic networks. Ann. Appl. Probab. 14(2), 612–650 (2004)

    Article  Google Scholar 

  9. Baccelli, F., Cohen, G.J., Olsder, G., Quadrat, J.-P.: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley Series on Probability and Mathematical Statistics. Wiley, New York (1992)

    Google Scholar 

  10. Baccelli, F., Schlegel, S., Schmidt, V.: Asymptotics of stochastic networks with subexponential service times. Queueing Syst. 33(1), 205–232 (1999)

    Article  Google Scholar 

  11. Baccelli, F., Lelarge, M., Foss, S.: Asymptotics of subexponential max plus networks: the stochastic event graph case. Queueing Syst. 46(1), 75–96 (2004)

    Article  Google Scholar 

  12. Baccelli, F., Foss, S., Lelarge, M.: Tails in generalized Jackson networks with subexponential service-time distributions. J. Appl. Probab. 42(2), 513–530 (2005)

    Article  Google Scholar 

  13. Borst, S., Boxma, O., Jelenković, P.: Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows. Queueing Syst. 43(4), 273–306 (2003)

    Article  Google Scholar 

  14. Brandwajn, A., Jow, Y.L.L.: An approximation method for tandem queues with blocking. Oper. Res. 36(1), 73–83 (1988)

    Article  Google Scholar 

  15. Chistyakov, V.P.: A theorem on sums of independent, positive random variables and its applications to branching processes. Theory Probab. Appl. 9, 640–648 (1964)

    Article  Google Scholar 

  16. Cline, D.B.H.: Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72, 529–557 (1986)

    Article  Google Scholar 

  17. Dieker, A.B., Lelarge, M.: Tails for (max, plus) recursions under subexponentiality. Queueing Syst. 53(4), 213–230 (2006)

    Article  Google Scholar 

  18. Embrechts, P., Goldie, C.M.: On convolution tails. Stoch. Process. Appl. 13, 263–278 (1982)

    Article  Google Scholar 

  19. Embrechts, P., Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insur. Math. Econ. 1(1), 55–72 (1982)

    Article  Google Scholar 

  20. Foss, S., Korshunov, D.: Heavy tails in multi-server queue. Queueing Syst. 52(1), 31–48 (2006)

    Article  Google Scholar 

  21. Foss, S., Konstantopoulos, T., Zachary, S.: Discrete and continuous time modulated random walks with heavy-tailed increments. J. Theor. Probab. 20(3), 581–612 (2007)

    Article  Google Scholar 

  22. Huang, T., Sigman, K.: Steady-state asymptotics for tandem, split-match and other feed-forward queues with heavy tailed service. Queueing Syst. 33(1), 233–259 (1999)

    Article  Google Scholar 

  23. Jelenković, P.R., Lazar, A.A.: Subexponential asymptotics of a Markov-modulated random walk with queueing applications. J. Appl. Probab. 35(2), 325–347 (1998)

    Article  Google Scholar 

  24. Lelarge, M.: Packet reordering in networks with heavy-tailed delays. Math. Methods Oper. Res. 67(2), 341–371 (2008)

    Article  Google Scholar 

  25. Martin, J.B.: Large tandem queueing networks with blocking. Queueing Syst. 41, 45–72 (2002)

    Article  Google Scholar 

  26. Miyoshi, N.: On the subexponential properties in stationary single-server queues: a palm-martingale approach. Adv. Appl. Probab. 36(3), 872–892 (2004)

    Article  Google Scholar 

  27. Pakes, A.G.: On the tails of waiting-time distributions. J. Appl. Probab. 12(3), 555–564 (1975)

    Article  Google Scholar 

  28. Perros, H.G., Altiok, T.: Approximate analysis of open networks of queues with blocking: tandem configurations. IEEE Trans. Softw. Eng. 12(3), 450–461 (1986)

    Google Scholar 

  29. Pitman, E.J.G.: Subexponential distribution functions. J. Aust. Math. Soc. A 29, 337–347 (1980)

    Article  Google Scholar 

  30. Takine, T.: Subexponential asymptotics of the waiting time distribution in a single-server queue with multiple Markovian arrival streams. Stoch. Mod. 17(4), 429–448 (2001)

    Article  Google Scholar 

  31. Whitt, W.: The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution. Queueing Syst. 36(1), 71–87 (2000)

    Article  Google Scholar 

  32. Willekens, E., Teugels, J.L.: Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time. Queueing Syst. 10(4), 295–311 (1992)

    Article  Google Scholar 

  33. Xia, C.H., Liu, Z., Squillante, M.S., Zhang, L.: Lower bounds for LRD/GI/1 queues with subexponential service times. Probab. Eng. Inf. Sci. 18(01), 87–101 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hayriye Ayhan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kim, JK., Ayhan, H. Tandem queues with subexponential service times and finite buffers. Queueing Syst 66, 195–209 (2010). https://doi.org/10.1007/s11134-010-9182-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11134-010-9182-1

Keywords

Mathematics Subject Classification (2000)

Navigation