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Regularly varying tails in a queue with discrete autoregressive arrivals of order p

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Abstract

We consider a discrete time single server queueing system where the service time of a customer is one slot, and the arrival process is governed by a discrete autoregressive process of order p (DAR(p)). For this queueing system, we investigate the tail behavior of the queue size and the waiting time distributions. Specifically, we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β−1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β.

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References

  1. Addie, R.G., Zukerman, M.: Performance evaluation of a single server autoregressive queue. Aust. Telecommun. Res. 28(1), 25–32 (1994)

    Google Scholar 

  2. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  3. Choi, B.D., Kim, B., Hwang, G.U., Kim, J.-K.: The analysis of a multi server queue fed by discrete time autoregressive process of order 1. Oper. Res. Lett. 32(1), 85–93 (2004)

    Article  Google Scholar 

  4. Dehnert, M., Helm, W.E., Hutt, M.-T.: A discrete autoregressive process as a model for short-range correlations in DNA sequences. Physica A 327, 535–553 (2003)

    Article  Google Scholar 

  5. Dehnert, M., Helm, W.E., Hutt, M.-T.: Information theory reveals large-scale synchronisation of statistical correlations in eukaryote genomes. Gene 345, 81–90 (2005)

    Article  Google Scholar 

  6. Elwalid, A., Heyman, D., Laksman, T.V., Mitra, D., Weiss, A.: Fundamental bounds and approximations for ATM multiplexers with applications to video teleconferencing. IEEE J. Sel. Areas Commun. 13(6), 1004–1016 (1995)

    Article  Google Scholar 

  7. Finch, P.D., Pearce, C.: A second look at a queueing system with moving average input process. J. Aust. Math. Soc. 5, 100–106 (1965)

    Google Scholar 

  8. He, J., Sohraby, K.: An extended combinatorial analysis framework for discrete-time queueing systems with general sources. IEEE/ACM Trans. Netw. 11(1), 95–110 (2003)

    Article  Google Scholar 

  9. Heyman, D., Tabatabai, A., Lakshman, T.V.: Statistical analysis and simulation study of video teleconferencing traffic in ATM networks. IEEE Trans. Circuits Syst. Video Technol. 2(1), 49–59 (1992)

    Article  Google Scholar 

  10. Hwang, G.U., Choi, B.D., Kim, J.-K.: The waiting time analysis of a discrete time queue with arrivals as an autoregressive process of order 1. J. Appl. Probab. 39(3), 619–629 (2002)

    Article  Google Scholar 

  11. Hwang, G.U., Sohraby, K.: On the exact analysis of a discrete-time queueing system with autoregressive inputs. Queueing Syst. 43(1–2), 29–41 (2003)

    Article  Google Scholar 

  12. Jacobs, P.A., Lewis, P.A.W.: Discrete time series generated by mixtures. III: Autoregressive processes (DAR(p)). Technical Report NPS55-78-022, Naval Postgraduate School (1978)

  13. Jacobs, P.A., Lewis, P.A.W.: Stationary discrete autoregressive-moving average time series generated by mixtures. J. Time Ser. Anal. 4(1), 19–36 (1983)

    Google Scholar 

  14. Kim, B., Chang, Y., Kim, Y.C., Choi, B.D.: A queueing system with discrete autoregressive arrivals. Perform. Eval. 64, 148–161 (2007)

    Article  Google Scholar 

  15. Kim, B., Sohraby, K.: Tail behavior of the queue size and waiting time in a queue with discrete autoregressive arrivals. Adv. Appl. Probab. 38, 1116–1131 (2006)

    Article  Google Scholar 

  16. Makowski, A., Melamed, B., Whitt, W.: On averages seen by arrivals in discrete time. Proc. IEEE Conf. Dec. Control 28, 1084–1086 (1989)

    Article  Google Scholar 

  17. McKenzie, E.: Discrete variate time series. In: Shanbhag, D.N., Rao, C.R. (eds.) Handbook of Statistics 21, Stochastic Processes: Modelling and Simulation, pp. 573–606. North-Holland, Amsterdam (2003)

    Google Scholar 

  18. Pearce, C.: A queueing system with general moving average input and negative exponential service time. J. Aust. Math. Soc. 6, 223–236 (1966)

    Article  Google Scholar 

  19. Ryu, B.K., Elwalid, A.: The importance of long-range dependence of VBR video traffic in ATM traffic engineering: Myths and Realities, ACM SIGCOMM, Stanford, CA, pp. 3–14 (1996)

  20. Sigman, K.: Appendix: A primer on heavy-tailed distributions. Queueing Syst. 33, 261–275 (1999)

    Article  Google Scholar 

  21. Widjaja, I., Elwalid, A.: Performance issues in VC-merge capable switches for IP over ATM networks. Proc. IEEE INFOCOM’98 1, 372–380 (1998)

    Google Scholar 

  22. Widjaja, I., Elwalid, A.: Performance issues in VC-merge capable switches for multiprotocol label switching. IEEE JSAC 17(6), 1178–1189 (1999)

    Google Scholar 

  23. Xiong, Y., Bruneel, H.: Buffer contents and delay for statistical multiplexers with fixed-length packet-train arrivals. Perform. Eval. 17(1), 31–42 (1993)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics Education, Chungbuk National University, 12, Gaeshin-dong, Heungduk-ku, Cheongju, Chungbuk, 361-763, Republic of Korea

    Jeongsim Kim

  2. Department of Mathematics and Telecommunication Mathematics Research Center, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul, 136-701, Republic of Korea

    Bara Kim

Authors
  1. Jeongsim Kim
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  2. Bara Kim
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Correspondence to Bara Kim.

Additional information

This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).

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Kim, J., Kim, B. Regularly varying tails in a queue with discrete autoregressive arrivals of order p . Queueing Syst 56, 93–102 (2007). https://doi.org/10.1007/s11134-007-9035-8

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  • DOI: https://doi.org/10.1007/s11134-007-9035-8

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