Abstract
We consider two classes of multiserver retrial queueing systems. For the first class, the rate of retrial requests depends on the number of customers on the orbit, and for the second class, the rate is constant. The input flow is assumed to be a regenerative one and the service time has an arbitrary distribution. Based on the synchronization of the input flow and an auxiliary service process, we establish necessary and sufficient stability conditions for models of both classes.
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References
L. G. Afanaseva and E. E. Bashtova, “Coupling method for asymptotic analysis of queue with regenerative input and unreliable server,” Queueing Syst., 76, No. 2, 125–147 (2014).
L. G. Afanaseva and A. B. Tkachenko, “Multichannel queueing systems with regenerative input flow,” Theory Probab. Its Appl., 58, No. 2, 174–192 (2014).
S. R. Artalejo, “Stationary analysis of the characteristics of theM|M|2 queue with constant repeated attempts,” Opsearch, 33, 83–95 (1996).
S. R. Artalejo, A. Gómez-Corral, and M. F. Neuts, “Analysis of multiserver queues with constant retrial rate,” Eur. J. Oper. Res., 135, 569–581 (2001).
B. D. Choi, J. W. Shin, and W. C. Ahn, “Retrial queues with collision arising from unslotted CSMA—CD protocol,” Queueing Syst., 11, 335–356 (1992).
B. D. Choi, K. H. Rhee, and K. K. Park, “The M|G|1 retrial queue with retrial rate control policy,” Probab. Eng. Inform. Sci., 7, 29–46 (1993).
B. D. Choi, K. H. Rhee, K. K. Park, and C. E. M. Pearce, “An M|M|1 retrial queue with control policy and general retrial times,” Queueing Syst., 14, 175–292 (1993).
S. W. Cohen, “Basic problems of telephone traffic theory and influence of repeated calls,” Philips Telecommun. Rev., 18, 49–100 (1957).
G. I. Falin and J. R. Artalejo, “Approximation for multiserver queues with bulking/retrial discipline,” OR Spectrum, 17, 239–244 (1995).
G. I. Falin and S. G. C. Templeton, Retrial Queues, Chapman & Hall, London (1997).
G. Fayolle, “A simple telephone exchange with delayed feedback,” in: O. S. Boxma, S. W. Cohen, and H. C. Tijms, eds., Teletraffic Analysis in Computer Performance Evaluation, Elsevier, Amsterdam (1986).
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley (1950).
A. Gómez-Corral and M. F. Ramalhoto, “On the stationary distribution of Markovian process arising in the theory of multiserver retrial queueing systems,” Math. Comput. Model., 30, 141–158 (1999).
J. Grandell, Doubly Stochastic Poisson Process, Springer, Berlin (1976).
M. F. Neuts, Structured Stochastic Matrices of M|G|1 Type and Their Applications, Marcel Dekker, New York (1989).
B. A. Sevastyanov, “An ergodic theorem for Markov processes and its application to telephone systems with refusals,” Theory Probab. Its Appl., 2, No. 1, 104–112 (1957).
H. Thorisson, Coupling, Stationarity and Regeneration, Springer, New York (2000).
R. I. Wilkinson, “Theories for toll traffic engineering in the USA,” Bell Syst. Tech. J., 35, 421–514 (1950).
A. Zeifman, Y. Satin, E. Morozov, R. Nekrasova, A. Gorshenin, “On the ergodicity bounds for a constant retrial rate queueing model,” in: 8th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT) (2016), pp. 269–272.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 3, pp. 5–18, 2018.
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Afanaseva, L.G. Stability Conditions for Retrial Queueing Systems with Regenerative Input Flow. J Math Sci 254, 446–455 (2021). https://doi.org/10.1007/s10958-021-05317-2
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DOI: https://doi.org/10.1007/s10958-021-05317-2