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The intercept term of the asymptotic variance curve for some queueing output processes

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  • Hautphenne, Sophie
  • Kerner, Yoav
  • Nazarathy, Yoni
  • Taylor, Peter

Abstract

We consider the output processes of some elementary queueing models such as the M/M/1/K queue and the M/G/1 queue. An important performance measure for these counting processes is their variance curve v(t), which gives the variance of the number of customers in the time interval [0, t]. Recent work has revealed some non-trivial properties dealing with the asymptotic rate at which the variance curve grows. In this paper we add to these results by finding explicit expressions for the intercept term of the linear asymptote. For M/M/1/K queues our results are based on the deviation matrix of the generator. It turns out that by viewing output processes as Markovian Point Processes and considering the deviation matrix, one can obtain explicit expressions for the intercept term, together with some further insight regarding the BRAVO (Balancing Reduces Asymptotic Variance of Outputs) effect. For M/G/1 queues our results are based on a classic transform of D. J. Daley. In this case we represent the intercept term of the variance curve in terms of the first three moments of the service time distribution. In addition we shed light on a conjecture of Daley, dealing with characterization of stationary M/M/1 queues within the class of stationary M/G/1 queues, based on the variance curve.

Suggested Citation

  • Hautphenne, Sophie & Kerner, Yoav & Nazarathy, Yoni & Taylor, Peter, 2015. "The intercept term of the asymptotic variance curve for some queueing output processes," European Journal of Operational Research, Elsevier, vol. 242(2), pages 455-464.
    • Handle: RePEc:eee:ejores:v:242:y:2015:i:2:p:455-464
    DOI: 10.1016/j.ejor.2014.10.051
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    References listed on IDEAS

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    1. Tan, Bar[iota]s, 1999. "Variance of the output as a function of time: Production line dynamics," European Journal of Operational Research, Elsevier, vol. 117(3), pages 470-484, September.
    2. Tan, Baris, 1997. "Variance of the throughput of an N-station production line with no intermediate buffers and time dependent failures," European Journal of Operational Research, Elsevier, vol. 101(3), pages 560-576, September.
    3. Kevin B. Hendricks, 1992. "The Output Processes of Serial Production Lines of Exponential Machines with Finite Buffers," Operations Research, INFORMS, vol. 40(6), pages 1139-1147, December.
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    5. Daley, D. J., 1975. "Further second-order properties of certain single-server queueing systems," Stochastic Processes and their Applications, Elsevier, vol. 3(2), pages 185-191, April.
    6. Brown, Mark & Solomon, Herbert, 1975. "A second-order approximation for the variance of a renewal reward process," Stochastic Processes and their Applications, Elsevier, vol. 3(3), pages 301-314, July.
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    Cited by:

    1. Peter Braunsteins & Sophie Hautphenne & Peter G. Taylor, 2016. "The roles of coupling and the deviation matrix in determining the value of capacity in M/M/1/C queues," Queueing Systems: Theory and Applications, Springer, vol. 83(1), pages 157-179, June.
    2. Sarat Babu Moka & Yoni Nazarathy & Werner Scheinhardt, 2023. "Diffusion parameters of flows in stable multi-class queueing networks," Queueing Systems: Theory and Applications, Springer, vol. 103(3), pages 313-346, April.
    3. Romero-Silva, Rodrigo & Shaaban, Sabry & Marsillac, Erika & Hurtado, Margarita, 2018. "Exploiting the characteristics of serial queues to reduce the mean and variance of flow time using combined priority rules," International Journal of Production Economics, Elsevier, vol. 196(C), pages 211-225.

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