×

Averaging methods for transient regimes in overloading retrial queueing systems. (English) Zbl 1042.60528

Summary: A new approach is suggested for studying transient and stable regimes in overloading retrial queueing systems. This approach is based on limit theorems of averaging principle and diffusion approximation types for so-called switching processes. Two models of retrial queueing systems of the types \((\overline M/\overline G/\overline 1)w.r\) (multidimensional Poisson input flow, one server with general service times, retrial system) and \(M/M/m/w.r\) (\(m\) servers with exponential service) are considered in the case when the intensity of calls that reapply for the service tends to zero. For the number of reapplying calls, functional limit theorems of averaging principle and diffusion approximation types are proved.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Yang, T.; Templeton, J. G.C., A survey on retrial queues, Queueing Systems, 2, 203-233 (1987) · Zbl 0658.60124
[2] Falin, G. I., A survey of retrial queues, Queueing Systems, 7, 127-168 (1990) · Zbl 0709.60097
[3] Kulkarni, V. G.; Liang, H. M., Retrial queues revisited, (Dshalalow, J. H., Frontiers in Queueing Models and Applications in Science and Engineering (1997), CRC Press), 19-34 · Zbl 0871.60074
[4] Neuts, M. F.; Ramalhoto, M. F., A service model in which the server is required to search for customers, Journal of Applied Probability, 21, 157-166 (1984) · Zbl 0531.60089
[5] Falin, G. I.; Artalejo, J. R.; Martin, M., On the single server retrial queue with priority customers, Queueing Systems, 14, 439-455 (1993) · Zbl 0790.60076
[6] Artalejo, J. R., A queueing system with returning customers and waiting line, Operations Research Letters, 17, 191-199 (1995) · Zbl 0836.90072
[7] Falin, G. I., Estimation of retrial rate in a retrial queue, Queueing Systems, 19, 231-246 (1995) · Zbl 0836.60101
[8] Falin, G. I.; Artalejo, J. R., Approximation for multiserver queues with balking/retrial discipline, OR Spektrum, 17, 239-244 (1995) · Zbl 0843.90046
[9] Martin, M.; Artalejo, J. R., Analysis of an M/G/1 queue with two types of impatient units, Advances in Applied Probability, 27, 840-861 (1995) · Zbl 0829.60085
[10] Artalejo, J. R.; Falin, G. I., On the orbit characteristics of the M/G/1 retrial queue, Naval Research Logistics, 43, 1147-1161 (1996) · Zbl 0859.60088
[11] Artalejo, J. R., Analysis of an M/G/1 queue with constant repeated attempts and server vacations, Computer and Operations Research, 24, 493-504 (1997) · Zbl 0882.90048
[12] Anisimov, V. V.; Atadzhanov, Kh. L., Asymptotic analysis of highly-reliable systems with repeated call, Issledov. Oper, i ASU, 37, 32-36 (1991), (in Russian) · Zbl 0764.60038
[13] Anisimov, V. V.; Atadzhanov, Kh. L., Diffusion approximation of systems with repeated calls, Theory of Probab. and Math. Statistics, 44, 3-8 (1991) · Zbl 0764.60038
[14] Anisimov, V. V.; Atadzhanov, Kh. L., Diffusion approximation of systems with repeated calls and unreliable server, Journal of Mathematical Sciences, 72, 3032-3034 (1994) · Zbl 0850.90015
[15] Falin, G. I.; Templeton, J. G.C., Retrial Queues (1997), Chapman and Hall · Zbl 0944.60005
[16] Buslenko, N. P.; Kalashnikov, V. V.; Kovalenko, I. N., Lectures on the Theory of Complex Systems (1973), Sov. Radio: Sov. Radio Moscow, (in Russian)
[17] Basharin, G. P.; Bocharov, P. P.; Kogan, Ja. A., Analysis of Queues in Computing Networks (1989), Nauka: Nauka Moscow, (in Russian) · Zbl 0708.68005
[18] Anisimov, V. V., Random Processes with Discrete Component, Limit Theorems (1988), Kiev University, (in Russian) · Zbl 0686.60092
[19] Anisimov, V. V.; Lebedev, E. A., Stochastic Queueing Networks. Markov Models (1992), Kiev University, (in Russian)
[20] Chen, H.; Mandelbaum, A., Hierarchical modelling of stochastic networks, (Yao, D. D., Stochastic Modeling and Analysis of Manufacturing Systems (1994)), 107-131, Part I, II
[21] Harrison, J. M., Balanced fluid models of multiclass queueing network: A heavy traffic conjecture, (Stochastic Networks. Stochastic Networks, IMA Volumes in Mathematics and its Appl., Volume 71 (1995)), 1-20 · Zbl 0838.90045
[22] Harrison, J. M.; Williams, R. J., A multiclass closed queueing network with unconventional heavy traffic behavior, Ann. Appl. Probab., 6, 1, 1-47 (1996) · Zbl 0865.60078
[23] Anisimov, V. V., Switching processes, Cybernetics, 13, 4, 590-595 (1977)
[24] Anisimov, V. V., Limit theorems for switching processes and their applications, Cybernetics, 14, 6, 917-929 (1978) · Zbl 0447.60075
[25] Anisimov, V. V., Vor. Publ. Ser., Aarhus Univ., 40, 235-262 (1992) · Zbl 0821.60044
[26] Anisimov, V. V., Switching processes: Averaging principle, diffusion approximation and applications, (Acta Applicandae Mathematicae, 40 (1995), Kluwer: Kluwer The Netherlands), 95-141 · Zbl 0827.60022
[27] Anisimov, V. V., Asymptotic analysis of switching queueing systems in conditions of low and heavy loading, (Matrix-Analytic Methods in Stochastic Models. Matrix-Analytic Methods in Stochastic Models, Lecture Notes in Pure and Appl. Mathem. Series, Volume 183 (1996), Marcel Dekker), 241-260 · Zbl 0872.60073
[28] Ez̆ov, I. I.; Skorokhod, A. V., Markov processes which are homogeneous in the second component, Theor. Probab. Appl., 14, 679-692 (1969) · Zbl 0196.20003
[29] Anisimov, V. V., Asymptotic consolidation of the states of random processes, Cybernetics, 9, 3, 494-504 (1973)
[30] Gikhman, I. I.; Skorokhod, A. V., Theory of Random Processes II (1973), Nauka: Nauka Moscow, (in Russian) · Zbl 0298.60024
[31] Hersh, R., Random evolutions: Survey of results and problems, Rocky Mount. J. Math., 4, 3, 443-475 (1974) · Zbl 0366.60005
[32] Kurtz, T., A limit theorem for perturbed operator semigroups with applications to random evolutions, J. Funct. Anal., 12, 55-67 (1973) · Zbl 0246.47053
[33] Pinsky, M., Random Evolutions, (Lecture Notes in Math., Volume 451 (1975), Springer-Verlag: Springer-Verlag New York), 89-100
[34] Korolyuk, V. S.; Swishchuk, A. V., Random Evolutions (1994), Kluwer Academic · Zbl 0813.60083
[35] Anisimov, V. V.; Zakusilo, O. K.; Dontchenko, V. S., The Elements of Queueing Theory and Asymptotic Analysis of Systems, ((1987), Visca Scola: Visca Scola Kiev), 248, (in Russian)
[36] Anisimov, V. V.; Aliev, A. O., Limit theorems for recurrent processes of semi-Markov type, Theor. Probab. and Math. Statist., 41, 7-13 (1990) · Zbl 0733.60102
[37] Anisimov, V. V., Averaging principle for switching recurrent sequences, Theor. Probab. and Math. Statist., 45, 1-8 (1991) · Zbl 0834.60041
[38] Anisimov, V. V., Averaging principle for switching processes, Theory Probab. and Math. Statist., 46, 1-10 (1992) · Zbl 0835.60018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.