Advertisement

Further Results for the Queue G/G/1

  • N. U. Prabhu
Part of the Applications of Mathematics book series (SMAP, volume 15)

Abstract

In the preceding chapter we derived some basic results concerning ladder processes associated with a random walk, and applied them to the queueing systems M/M/1, G/M/1, and M/G/1. In these cases we were able to derive the various distributions by using the special properties of the random walk. We now proceed with the general discussion and derive the transforms of these distributions, establish the Wiener-Hopf factorization (stated earlier without proof) and derive further results for the single-server queue.

Keywords

Random Walk Service Time Busy Period Idle Period Server Queue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen, Sparre E. (1953a): On sums of symmetrically dependent random variables. Skan. Aktuar. 36, 123–138.Google Scholar
  2. Andersen, Sparre E. (1953b): On the fluctuations of sums of random variables I. Math. Scand. 1, 263–285.MathSciNetzbMATHGoogle Scholar
  3. Andersen, Sparre E. (1954): On the fluctuations of sums of random variables II. Math. Scand. 2, 195–223.MathSciNetzbMATHGoogle Scholar
  4. Baxter, Glen (1958): An operator identity, Pacific J. Math. 8, 649–663.MathSciNetzbMATHGoogle Scholar
  5. Beněs, V. E. (1957): On queues with Poisson arrivals. Ann. Math. Statist. 28, 670–677.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Blackwell, D. (1953): Extension of a renewal theorem. Pacific J. Math. 3, 315–320.MathSciNetzbMATHGoogle Scholar
  7. Erdös, P. and Kac, M. (1946): On certain limit theorems of the theory of probability. Bull Amer. Math. Soc. 52, 292–302.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Feller, W. (1959): On combinatorial methods in fluctuation theory. The Harold Cramér Volume, 75–91, John Wiley, New York.Google Scholar
  9. Feller, W. (1971): An Introduction to Probability Theory and Its Applications, Volume 2, 3rd edition. John Wiley, New York.zbMATHGoogle Scholar
  10. Heyde, C. C. (1967): A limit theorem for random walks with drift. J. Appl. Prob. 4, 144–150.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Kemperman, J. H. B. (1961): The First Passage Problem for a Stationary Markov Chain. University of Chicago Press, Chicago.Google Scholar
  12. Kendall, D. G. (1957): Some problems in the theory of dams. J. Roy. Stat. Soc, B19, 207–212.MathSciNetGoogle Scholar
  13. Krein, M. G. (1958): Integral equations on a half-line with kernel depending upon the difference of the arguments. Uspekhi Mat. Nauk 13, 3–120 [Amer. Math. Soc. Translations Series 2, 22, 163–288].Google Scholar
  14. Lindley, D. V. (1952): Theory of queues with a single server. Proc. Comb. Phil Soc. 48, 277–289.MathSciNetCrossRefGoogle Scholar
  15. Prabhu, N. U. (1970): Limit theorems for the single server queue with traffic intensity one: J. Appl Prob. 7, 227–233.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Smith, W. L. (1953): On the distribution of queueing times. Proc. Camb. Phil Soc. 49, 449–461.zbMATHCrossRefGoogle Scholar
  17. Spitzer, F. (1956): A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82, 323–339.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Spitzer, F, (1957): The Wiener-Hopf equation whose kernel is a probability density. Duke Math. J. 24, 327–344.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Spitzer, F. (1960a): The Wiener-Hopf equation whose kernel is a probability density II. Duke Math. J. 27, 363–372.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Spitzer, F. (1960b): A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94, 150–160.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1980

Authors and Affiliations

  • N. U. Prabhu
    • 1
  1. 1.School of Operations ResearchCornell UniversityIthacaUSA

Personalised recommendations