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Limit theorems for multiphase queueing systems

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Abstract

One gives a limit theorem for the joint distribution of the stationary waiting times of customers in the queues of a multiphase queueing system, functioning in a heavy traffic regime. One proves that the joint distribution function of the waiting times is a solution of a problem with a directional derivative for an elliptic differential equation in a polyhedral angle.

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Literature cited

  1. T. Suzuki and M. Maruta, “Inequalities for two bulk queues and a tandem queue,” Mem. Defense Acad. Jpn., No. 10, 81–102 (1970/71).

    Google Scholar 

  2. D. Stoyan, “Inequalities for multiserver queues and a tandem queue,” Zastos. Mat.,15, 415–419 (1976/77).

    Google Scholar 

  3. J. M. Harrison, “The diffusion approximation for tandem queues in heavy traffic,” Adv. Appl. Probab.,10, No. 4, 886–905 (1978).

    Google Scholar 

  4. F. I. Karpelevich and A. Ya. Kreinin, “Two-phase queueing system under heavy traffic conditions,” Teor. Veroyatn. Primen.,26, No. 2, 302–320 (1981).

    Google Scholar 

  5. A. A. Borovkov, Stochastic Processes in Queueing Theory, Springer, New York (1976).

    Google Scholar 

  6. A. Ya. Khinchin, Asymptotic Laws of Probability Theory [in Russian], GONTI, Moscow (1936).

    Google Scholar 

  7. E. M. Landis, Second-Order Elliptic and Parabolic Equations [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  8. A. V. Bitsadze, Boundary Value Problems for Second Order Elliptic Equations, North-Holland, Amsterdam (1968).

    Google Scholar 

  9. A. A. Borovkov, “The convergence of distributions of functionals of random sequences and processes that are given on the entire axis,” Trudy Mat. Inst. Akad. Nauk SSSR,128, 41–65 (1972).

    Google Scholar 

  10. V. G. Maz'ya and B. P. Paneyakh, “Degenerate elliptic pseudodifferential operators and the problem with oblique derivative,” Tr. Mosk. Mat. Obshch.,31, 237–295 (1974).

    Google Scholar 

  11. V. G. Maz'ya and B. A. Plamenevskii, “On the oblique derivative problem in a domain with a piecewise-smooth boundary,” Funkts. Anal. Prilozhen.,5, No. 3, 102–103 (1971).

    Google Scholar 

  12. V. G. Maz'ya and B. A. Plamenevskii, “On boundary-value problems for a second-order elliptic equation in a domain with edges,” Vestn. Leningr. Univ. Mat. Mekh. Astron., No. 1, Issue 1, 102–108 (1975).

    Google Scholar 

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Authors
  1. F. I. Karpelevich
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  2. A. Ya. Kreinin
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Additional information

Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 212–229, 1986.

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Karpelevich, F.I., Kreinin, A.Y. Limit theorems for multiphase queueing systems. J Math Sci 38, 2288–2298 (1987). https://doi.org/10.1007/BF01093830

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  • DOI: https://doi.org/10.1007/BF01093830

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