For citation:
Lapatin I. L., Nazarov A. A. Output process of the M|GI|1 is an asymptotical renewal process. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 1, pp. 100-110. DOI: 10.18500/1816-9791-2021-21-1-100-110, EDN: TMICEG
Output process of the M|GI|1 is an asymptotical renewal process
Most of the studies on models with retrials are devoted to the research of the number of applications in the system or in the source of repeated calls using asymptotic and numerical approaches or simulation. Although one of the main characteristics that determines the quality of the communication system is the number of applications served by the system per unit of time. Information on the characteristics of the output processes is of great practical interest, since the output process of one system may be incoming to another. The results of the study of the outgoing flows of queuing networks are widely used in the modeling of computer systems, in the design of data transmission networks and in the analysis of complex multi-stage production processes. In this paper, we have considered a single server system with redial, the input of which receives a stationary Poisson process. The service time in considered system is a random value with an arbitrary distribution function B(x). If the customer enters the system and finds the server busy, it instantly joins the orbit and carries out a random delay there during an exponentially distributed time. The object of study is the output process of this system. The output is characterized by the probability distribution of the number of customers that have completed service for time t. We have provided the study using asymptotic analysis method under low rate of retrials limit condition. We have shown in the paper that the output of retrial queue M|GI|1 is an asymptotical renewal process. Moreover, the lengths of the intervals in output process are the sum of an exponential random value with the parameter lambda + kappa and a random variable with the distribution function B(x). The results of a numerical experiment show that the probability distributions of the number of served customers in the system are practically the same for significantly different distribution laws B(x) of service time if the service times have the same first two moments.
- Gnedenko B. V., Kovalenko I. N. Vvedenie v teoriyu massovogo obsluzchivaniya [Introduction to Queuing Theory]. Moscow, LKI, 2007. 400 p. (in Russian).
- Wilkinson R. I. Theories for toll traffic engineering in the USA. The Bell System Technical Journal, 1956, vol. 35, no. 2, pp. 421–507.
- Artalejo J. R., Gomez-Corral A. Retrial Queueing Systems: A Computational Approach. Berlin, Springer, 2008. 318 p.
- Falin G., Templeton J. Retrial Queues. London, Chapman and Hall, 1997. 320 p.
- Artalejo J. R., Phung-Duc T. Single server retrial queues with two way communication. Applied Mathematical Modelling, 2013, vol. 37, no. 4, pp. 1811–1822. https://doi.org/10.1016/j.apm.2012.04.022
- Nazarov A., Paul S., Gudkova I. Asymptotic analysis of Markovian retrial queue with two-way communication under low rate of retrials condition. Proceedings 31st European Conference on Modelling and Simulation. Budapest, Hungary, ECMS, 2017, pp. 687–693. https://doi.org/10.7148/2017-0687
- Burke P. J. The output of queueing systems. Operations Research, 1956, vol. 4, iss. 6, pp. 629–753. https://doi.org/10.1287/opre.4.6.699
- Reich E. Waiting times when queues are in tandem. The Annals of Mathematical Statistics, 1957, vol. 28, no. 3, pp. 768–773.
- Finch P. D. The output process of the queueing system M|G|1. Journal of the Royal Statistical Society: Series B (Methodological), 1959, vol. 21, iss. 2, pp. 375–380. https://doi.org/10.1111/j.2517-6161.1959.tb00344.x
- Projdakova Е. V., Fedotkin М. А. Control of output flows in the system with cyclic servicing and readjustments. Automation and Remote Control, 2008, vol. 69, no. 6, pp. 993– 1002. https://doi.org/10.1134/S000511790806009X
- Green D. Departure processes from MAP/PH/1 queues. Thesis (Ph. D.), University of Adelaide, Department of Applied Mathematics, 1999. 12 p.
- Lapatin I. L. Issledovanie vykhodiashchikh potokov modelei massovogo obsluzhivaniia s neogranichennym chislom priborov [Investigation of output process queuing models with an unlimited number of devices]. Diss. Cand. Sci. (Phys. and Math.), Tomsk State University. Tomsk, 2012. 138 p. (in Russian).
- Lapatin I. L., Nazarov A. A. Investigation of output process RQ system M/M/1 in the asymptotic condition of a large delay in orbit. Distributed computer and communication networks: control, computation, communications (DCCN-2018): Procedings of the XXI International Scientific Conference. Moscow, Izdatel’stvo RUDN, 2018, pp. 246–252 (in Russian).
- Lopuchova S. V., Nazarov A. A. Research of general independent process. Tomsk State University Journal of Control and Computer Science, 2007, no. 1, pp. 67–76 (in Russian).
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