Abstract
An M/G/1 retrial queue with batch arrivals is studied. The queue length K μ is decomposed into the sum of two independent random variables. One corresponds to the queue length K ∞ of a standard M/G/1 batch arrival queue, and another is compound-Poisson distributed. In the case of the distribution of the batch size being light-tailed, the tail asymptotics of K μ are investigated through the relation between K ∞ and its service times.
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References
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
Asmussen, S., Klüppelberg, C., Sigman, K.: Sampling at subexponential times, with queueing applications. Stoch. Process. Appl. 79, 265–286 (1999)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Cohen, J.W.: On a single-server queue with group arrivals. J. Appl. Probab. 13, 619–622 (1976)
Cline, D.B.H.: Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields 72, 529–557 (1986)
Embrechts, P., Goldie, C.M., Veraverbeke, N.: Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheor. Verw. Geb. 49, 335–347 (1979)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin, (1999)
Falin, G.I.: On a multiclass batch arrival retrial queue. Adv. Appl. Probab. 20, 483–487 (1988)
Falin, G.I.: A survey of retrial queues. Queueing Syst. 7, 127–167 (1990)
Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman & Hall, London (1997)
Gaver, D.P. Jr.: Imbedded Markov chain analysis of a waiting-line process in continuous time. Ann. Math. Stat. 30, 698–720 (1959)
Haji, R., Newell, G.F.: A relation between stationary queue and waiting time distributions. J. Appl. Probab. 8, 617–620 (1971)
Klüppelberg, C.: Subexponential distributions and characterizations of related classes. Probab. Theory Relat. Fields 82, 259–269 (1989)
Klüppelberg, C.: Asymptotic ordering of distribution functions on convolution semigroup. Semigroup Forum 40, 77–92 (1990)
Klukarni, V.G.: Expected waiting times in a multiclass batch arrival retrial queue. J. Appl. Probab. 23, 144–154 (1986)
Li, Q.-L.: Constructive Computation in Stochastic Models with Applications: The RG-Factorization. Tsinghua University Press/Springer, Beijing/Berlin (2010)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)
Shang, W., Liu, L., Li, Q.-L.: Tail asymptotics for the queue length in an M/G/1 retrial queue. Queueing Syst. 52, 193–198 (2006)
Shimura, T., Watanabe, T.: Infinite divisibility and generalized subexponentiality. Bernoulli 11, 445–469 (2005)
van Ommeren, J.C.W.: Exponential expansion for the tail of the waiting-time probability in the single-server queue with batch arrivals. Adv. Appl. Probab. 20, 880–895 (1988)
Yang, T., Templeton, J.G.C.: A survey on retrial queues. Queueing Syst. 2, 201–233 (1987)
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Yamamuro, K. The queue length in an M/G/1 batch arrival retrial queue. Queueing Syst 70, 187–205 (2012). https://doi.org/10.1007/s11134-011-9268-4
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DOI: https://doi.org/10.1007/s11134-011-9268-4