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Busy period distribution in state-dependent queues

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Abstract

We consider a state-dependent M/M/1 queue in which the arrival rate is a function of the instantaneous unfinished work (work backlog) in the system, and the customer's exponential service time distribution is allowed to depend on the unfinished work in the system at the instant that customer arrived. We obtain asymptotic approximations to both the busy period distributions as well as the residual busy period distribution. Our approximations are valid for systems with a rapid arrival rate and small mean service times.

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References

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

    Google Scholar 

  2. C.M. Bender and S.A. Orszag,Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

    Google Scholar 

  3. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Asymptotic analysis of a state-dependent M/G/1 queueing system, SIAM J. Appl. Math. 46(1986)483.

    Google Scholar 

  4. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Distribution of the maximum buffer content during a busy period for state-dependent M/G/1 queues, Comm. Statist.: Stochastic Models 3 (2) (1987).

  5. J.A. Morrison, Asymptotic analysis of the waiting-time distribution for a large closed processor-sharing system, SIAM J. Appl. Math. 46(1986)140.

    Google Scholar 

  6. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, A Markov-modulated M/G/1 queue I: stationary distribution, Queueing Systems 1(1987)355.

    Google Scholar 

  7. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, A Markov-modulated M/G/1 queue II: busy period and time for buffer overflow, Queueing Systems 1(1987)375.

    Google Scholar 

  8. C. Knessl, B.J. Matkowsky, Z. Schuss and C. Tier, Busy period distribution in statedependent queues, Appl. Math. Tech. Rep. No. 8529, North-western University (1986).

  9. W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York, 1966).

    Google Scholar 

  10. M. Abramowitz and I.A. Stegun,Handbook of Mathematical Functions (Dover Publ., New York, 1964).

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Sciences and Applied Mathematics, Northwestern University, 60201, Evanston, Illinois, USA

    C. Knessl & B. J. Matkowsky

  2. Department of Mathematics, Tel-Aviv University, Tel-Aviv, Israel

    Z. Schuss

  3. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 60680, Chicago, Illinois, USA

    C. Tier

Authors
  1. C. Knessl
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  2. B. J. Matkowsky
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  3. Z. Schuss
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  4. C. Tier
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Additional information

This research was supported in part by NSF Grants DMS-84-06110 and DMS-86-20267 and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences. C. Knessl was partially supported by an I.B.M. Graduate Fellowship.

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Knessl, C., Matkowsky, B.J., Schuss, Z. et al. Busy period distribution in state-dependent queues. Queueing Syst 2, 285–305 (1987). https://doi.org/10.1007/BF01158903

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  • DOI: https://doi.org/10.1007/BF01158903

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