Skip to main content
Log in

On the total amount of resources occupied by serviced customers

Automation and Remote Control Aims and scope Submit manuscript

Abstract

We consider a model of a multi-server queueing system with losses caused by lack of resources necessary to service claims. A claim accepted for servicing occupies a random amount of resources of several types with given distribution functions. Random vectors that define the requirements of claims for resources are independent of the processes of customer arrivals and servicing, mutually independent, and identically distributed. Under the assumptions of a Poisson arrival process and exponential service times, we analytically find the joint distribution of the number of customers in the system and the vector of amounts of resources occupied by them. We show sample computations that illustrate an application of the model to analyzing the characteristics of a videoconferencing service in an LTE wireless network.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, J.G., Buzzi, S., Wan Choi, et al., What Will 5G Be?, IEEE J. Selected Areas Commun., 2014, vol. 32, no. 6, pp. 1065–1082.

    Article  Google Scholar 

  2. Galinina, O., Andreev, S.D., Gerasimenko, M., et al., Capturing Spatial Randomness of Heterogeneous Cellular/WLAN Deployments with Dynamic Traffic, IEEE J. Selected Areas Commun., 2014, vol. 32, no. 6, pp. 1083–1099.

    Article  Google Scholar 

  3. Buturlin, I.A., Gaidamaka, Y.V., and Samuylov, A.K., Utility Function Maximization Problems for Two Cross-layer Optimization Algorithms in OFDM Wireless Networks, Proc. 4th Int. Congress on Ultra Modern Telecommunications and Control Syst. ICUMT, St. Petersburg, 2012, pp. 63–65.

    Google Scholar 

  4. Hongseok, K. and De Veciana, G., Leveraging Dynamic Spare Capacity in Wireless Systems to Conserve Mobile Terminals’ Energy, IEEE/ACM Trans. Networking, 2010, vol. 18, no. 3, pp. 802–815.

    Article  Google Scholar 

  5. Romm, E.L. and Skitovich, V.V., On Certain Generalization of Problem of Erlang, Autom. Remote Control, 1971, vol. 32, no. 6, pp. 1000–1003.

    MATH  Google Scholar 

  6. Tikhonenko, O.M., Determination of the Characteristics of Queueing Systems with Limited Memory, Autom. Remote Control, 1997, vol. 58, no. 6, pp. 969–973.

    MathSciNet  MATH  Google Scholar 

  7. Tikhonenko, O.M. and Klimovich, K.G., Analysis of Queuing Systems for Random-Length Arrivals with Limited Cumulative Volume, Probl. Peredachi Inform., 2001, vol. 37, no. 1, pp. 78–88.

    MathSciNet  MATH  Google Scholar 

  8. Tikhonenko, O.M., Generalized Erlang Problem for Queueing Systems with Bounded Total Size, Probl. Peredachi Inform., 2005, vol. 41, no. 3, pp. 64–75.

    MathSciNet  Google Scholar 

  9. Gimpelson, L.A., Analysis of Mixtures of Wide- and Narrow-Band Traffic, IEEE Trans. Commun. Technol., 1965, vol. 13, no. 3, pp. 258–266.

    Article  Google Scholar 

  10. Kelly, F.P., Loss Networks, Ann. Appl. Probab., 1991, no. 1, pp. 319–378.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ross, K.W., Multiservice Loss Models for Broadband Telecommunication Networks, New York: Springer-Verlag, 1995.

    Book  MATH  Google Scholar 

  12. Basharin, G.P., Samouylov, K.E., Yarkina, N.V., and Gudkova I.A., A New Stage in Mathematical Teletraffic Theory, Autom. Remote Control, 2009, vol. 70, no. 12, pp. 1954–1964.

    Article  MathSciNet  MATH  Google Scholar 

  13. Naumov, V.A. and Samuilov, K.E., On Modeling Queueing Systems with Multiple Resources, Vestn. Ross. Univ. Druzhby Narodov, Ser. Mat. Informatika. Fiz., 2014, no. 3, pp. 60–64.

    Google Scholar 

  14. Naumov, V., Samouylov, K., Sopin, E., et al., Two Approaches to Analyzing Dynamic Cellular Networks with Limited Resources, 6th Int. Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), St. Petersburg, 2014, pp. 585–588.

    Google Scholar 

  15. Korolyuk, V.S. and Turbin, A.F., Protsessy markovskogo vosstanovleniya v zadachakh nadezhnosti sistem (Markov Renewal Processes in Problems of System Reliability), Kiev: Naukova Dumka, 1982.

    MATH  Google Scholar 

  16. ITU-T Recommendation H.264: Advanced Video Coding for Generic Audiovisual Services, Geneva, February 2014.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. A. Naumov.

Additional information

Original Russian Text © V.A. Naumov, K.E. Samuilov, A.K. Samuilov, 2016, published in Avtomatika i Telemekhanika, 2016, No. 8, pp. 125–135.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naumov, V.A., Samuilov, K.E. & Samuilov, A.K. On the total amount of resources occupied by serviced customers. Autom Remote Control 77, 1419–1427 (2016). https://doi.org/10.1134/S0005117916080087

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117916080087

Navigation