Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Nazarov A. A., Paul S. V., Lizyura O. D. Heavy outgoing call asymptotics for MMPP|M|1 retrial queue with two way communication and multiple types of outgoing calls. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 1, pp. 111-124. DOI: 10.18500/1816-9791-2021-21-1-111-124, EDN: AHILEU

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.03.2021
Full text:
(downloads: 1605)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
519.872
EDN: 
AHILEU

Heavy outgoing call asymptotics for MMPP|M|1 retrial queue with two way communication and multiple types of outgoing calls

Autors: 
Nazarov Anatoly A., Tomsk State University
Paul Svetlana V., Tomsk State University
Lizyura Olga D., Tomsk State University
Abstract: 

In this paper, we consider a single server retrial queue MMPP|M|1 with two way communication and multiple types of outgoing calls. Calls received by the system occupy the device for operating, if it is free, or are sent to orbit, where they make a random delay before the next attempt to occupy the device. The duration of the delay has an exponential distribution. The main issue of this model is an existence of various types of outgoing calls in the system. The intensity of outgoing calls is different for different types of outgoing calls. The operating time of the outgoing calls also differs depending on the type and is exponential random variable, the parameters of which in the general case do not coincide. The device generates calls from the outside only when it does not operate the calls received from the flow. We use asymptotic analysis methods under two limit conditions: high rate of outgoing calls and low rate of serving outgoing calls. The aim of the current research is to derive an asymptotic stationary probability distribution of the number of incoming calls in the system that arrived from the flow, without taking into account the outgoing call if it is operated on the device. In this paper, we obtain asymptotic characteristic function under aforementioned limit conditions. In the limiting condition of high intensity of outgoing calls, the asymptotic characteristic function of the number of incoming calls in a system with repeated calls and multiple types of outgoing calls is a characteristic function of a Gaussian random variable. The type of the asymptotic characteristic function of the number of incoming calls in the system under study in the limiting condition of long-term operation of the outgoing calls is uniquely determined.

References: 
  1. Artalejo J. R., Gomez-Corral A. Retrial Queueing Systems. Berlin, Springer, 2008. 320 p.
  2. Falin G., Templeton, J. Retrial Queues. London, CRC Press, 1997. 320 p.
  3. Bhulai S., Koole G. A queueing model for call blending in call centers. IEEE Transactions on Automatic Control, 2003, vol. 48, no. 8, pp. 1434–1438. https://doi.org/10.1109/TAC.2003.815038
  4. Aguir S., Karaesmen F., Ak¸sin Z., Chauvet F. The impact of retrials on call center performance. OR Spectrum, 2004, vol. 26, no. 3, pp. 353–376. https://doi.org/10.1007/s00291-004-0165-7
  5. Morozov E., Phung-Duc T. Regenerative Analysis of Two-Way Communication OrbitQueue with General Service Time. In: Y. Takahashi, T. Phung-Duc, S. Wittevrongel, W. Yue, eds. Queueing Theory and Network Applications. QTNA 2018. Lecture Notes in Computer Science, vol. 10932. Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-93736-6_2
  6. Sakurai H., Phung-Duc T. Scaling limits for single server retrial queues with twoway communication. Annals of Operations Research, 2016, no. 247, pp. 229–256. https://doi.org/10.1007/s10479-015-1874-9
  7. Dragieva V., Phung-Duc T. Two-way communication M/M/1/1 queue with server-orbit interaction and feedback of outgoing retrial calls. In: A. Dudin, A. Nazarov, A. Kirpichnikov, eds. Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2017. Communications in Computer and Information Science, vol. 800. Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-68069-9_20
  8. Sakurai H., Phung-Duc T. Two-way communication retrial queues with multiple types of outgoing calls. TOP, 2015, vol. 23, pp. 466–492. https://doi.org/10.1007/s11750-014-0349-5
  9. Nazarov A. A., Paul S. V., 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. Netherlands, 2017, pp. 678–693.
  10. Nazarov A. A., Phung-Duc T., Paul S. V. Heavy Outgoing Call Asymptotics for MMPP/M/1/1 Retrial Queue with Two-Way Communication. In: A. Dudin, A. Nazarov, A. Kirpichnikov, eds. Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2017. Communications in Computer and Information Science, vol. 800. Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-68069-9_3
Received: 
11.11.2019
Accepted: 
20.02.2020
Published: 
01.03.2021
Short text (in English):
(downloads: 156)