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Busy periods in M/M/∞ systems with heterogeneous servers

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Abstract

This note gives a solution for the problem of finding the probability density and probability distribution functions of the N-busy-period length for the M/M/∞ system where the servers are not necessarily the same. A solution in case of the same servers was done in [3].

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References

  1. M. Brown and S.M. Ross, Some results for infinite server Poisson queues, J. Appl. Prob. 6 (1969) 604–611.

    Google Scholar 

  2. B.V. Gnedenko and I.N. Kovalenko, Introduction to Queueing Theory (Israel Program for Scientific Translation, Jerusalem, 1968).

  3. F. Guillemin and D. Pinchon, Continued fraction analysis of the duration of an excursion in an M/M/∞ system, J. Appl. Prob. 35 (1998) 165–183.

    Google Scholar 

  4. F. Guillemin and D. Pinchon, Excursions of birth and death processes, orthogonal polynomials, and continued fractions, J. Appl. Prob. 36 (1999) 752–770.

    Google Scholar 

  5. F. Guillemin, G. Rubino, B. Sericola and A. Simonian, Transient analysis of statistical multiplexing of data connections on an ATM link, in: Proc. ITC'15 (Washington, DC, 1997).

  6. F. Guillemin and A. Simonian, Transient characteristics of an M/M/∞ system, J. Appl. Prob. 27 (1995) 862–888.

    Google Scholar 

  7. M. Ismail, J. Letessier, and G. Valent, Linear birth and death models and associated Laguerre and Meixner polynomials, J. Approx. Theory 55 (1988) 337–348.

    Google Scholar 

  8. L. Kleinrock, Queueing Systems, Vol. 1 (Wiley, New York, 1975).

  9. G.A. Korn and T.M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968).

    Google Scholar 

  10. J.A. Morrison, L.A. Shepp and C.J. Van Wyk, A queueing analysis of hashing with lazy deletion, SIAM J. Comput. 16(6) (1987) 1155–1164.

    Google Scholar 

  11. J. Preater, M/M/∞ transience revisited, J. Appl. Prob. 34 (1997) 1061–1067.

    Google Scholar 

  12. J. Preater, A perpetuity and the M/M/∞ ranked server system, J. Appl. Prob. 34 (1997) 508–513.

    Google Scholar 

  13. A.F. Siegel, Asymptotic coverage distributions on the circle, Ann. Probab. 7 (1979) 651–661.

    Google Scholar 

  14. W. Stadje, The busy period of the queueing system M/G/∞, J. Appl. Prob. 22 (1985) 697–704.

    Google Scholar 

  15. H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation Vol. 1 (North-Holland, Amsterdam, 1991).

  16. B. Tsybakov, Busy periods in system with heterogeneous servers or channels, Problems of Information Transmission 41(3) (2005) 280–295.

    Google Scholar 

  17. B.S. Tsybakov, Probability of heavy traffic period in Third generation CDMA mobile communication, Mobile Networks and Applications 6 (2000) 463–470.

    Google Scholar 

  18. R.W. Wolff, Stochastic Modeling and the Theory of Queues (Prentice Hall, New Jersey, 1989).

    Google Scholar 

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Correspondence to Boris Tsybakov.

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AMS Subject Classification 60K25 68M20

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Tsybakov, B. Busy periods in M/M/∞ systems with heterogeneous servers. Queueing Syst 52, 153–156 (2006). https://doi.org/10.1007/s11134-006-4435-8

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  • DOI: https://doi.org/10.1007/s11134-006-4435-8

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