Abstract
We consider a single-line service system with a Palm arrival rate and exponential service time, with n-1 places in the queue. Let τn be the moment of first loss of a customer. It is assumed that\(\alpha _0 = \int_0^\infty {e^{ - t_{dF (t) \to 0} } } \), where F(t) is the distribution function of the time interval between successive arrivals of customers. We shall study the class of limiting distributions of the quantity τnδ (α0), where δ (α0) is some normalizing factor. We shall obtain conditions for which Pτn/Mτn<→ 1 −e−t.
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O. P. Vinogradov, “The problem of the maximum length of a queue, and its uses,” Teoriya Veroyatn. i Ee Primen.,13, No. 2, 366–375 (1968).
B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables [in Russian], Moscow (1949).
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Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 541–546, May, 1968.
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Vinogradov, O.P. Limiting distribution for the moment of first loss of a customer in a single-line service system with a limited number of positions in the queue. Mathematical Notes of the Academy of Sciences of the USSR 3, 345–348 (1968). https://doi.org/10.1007/BF01150987
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DOI: https://doi.org/10.1007/BF01150987