DOI QR코드

DOI QR Code

Queueing System Operating in Random Environment as a Model of a Cell Operation

Kim, Chesoong;Dudin, Alexander;Dudina, Olga;Kim, Jiseung

  • Received : 2016.03.03
  • Accepted : 2016.05.20
  • Published : 2016.06.30

Abstract

We consider a multi-server queueing system without buffer and with two types of customers as a model of operation of a mobile network cell. Customers arrive at the system in the marked Markovian arrival flow. The service times of customers are exponentially distributed with parameters depending on the type of customer. A part of the available servers is reserved exclusively for service of first type customers. Customers who do not receive service upon arrival, can make repeated attempts. The system operation is influenced by random factors, leading to a change of the system parameters, including the total number of servers and the number of reserved servers. The behavior of the system is described by the multi-dimensional Markov chain. The generator of this Markov chain is constructed and the ergodicity condition is derived. Formulas for computation of the main performance measures of the system based on the stationary distribution of the Markov chain are derived. Numerical examples are presented.

Keywords

Multi-Server Queueing;Multi-Dimensional Markov Chain;Random Environment;Cell Operations

References

  1. Buchholz, P., Kemper, P., and Kriege, J. (2010), Multiclass Markovian arrival processes and their parameter fitting, Performance Evaluation, 67, 1092-1106. https://doi.org/10.1016/j.peva.2010.08.006
  2. Choi, B. D., Melikov, A., and Velibekov, A. (2008), A simple numerical approximation of joint probabilities of calls in service and calls in the retrial group in a picocell, Applied Computational Mathematics, 7, 21-30.
  3. Cordeiro, J. D. and Kharoufeh, J. P. (2012), The unreliable M/M/1 retrial queue in a random environment, Stochastic Models, 28, 29-48. https://doi.org/10.1080/15326349.2011.614478
  4. Do, T.V. (2011), Solution for a retrial queueing problem in cellular networks with the Fractional Guard Channel policy, Mathematical and Computer Modelling, 53, 2059-2066. https://doi.org/10.1016/j.mcm.2010.05.011
  5. Dudin, A. N. and Nazarov, A. A. (2015), The MMAP/M/R/0 queueing system with reservation of servers operating in a random environment, Problems of Information Transmission, 51, 274-283.
  6. Dudin, A., Kim, C. S., Dudin, S., and Dudina, O. (2015), Priority Retrial Queueing Model Operating in Random Environment with Varying Number and Reservation of Servers, Applied Mathematics and Computations, 269, 674-690. https://doi.org/10.1016/j.amc.2015.08.005
  7. Graham, A. (1981), Kronecker products and matrix calculus with applications. Cichester: Ellis Horwood.
  8. He, Q. M. (1996), Queues with marked calls, Advances in Applied Probability, 28, 567-587. https://doi.org/10.1017/S000186780004862X
  9. Kim, C. S., Dudin, A. N., Klimenok, V. I., and Khramova, V. V. (2009), Erlang loss queueing system with batch arrivals operating in a random environment, Computers and Operations Research, 36, 674-967. https://doi.org/10.1016/j.cor.2007.10.022
  10. Kim, C. S., Klimenok, V. I., and Dudin, A. N. (2014), Optimization of Guard Channel Policy in Cellular Mobile Networks with Account of Retrials, Computers and Operation Research, 43, 181-190. https://doi.org/10.1016/j.cor.2013.09.005
  11. Kim, C. S., Klimenok, V., Mushko, V., and Dudin, A. (2010), The BMAP/PH/N retrial queueing system operating in Markovian random environment, Computers and Operations Research, 37, 1228-1237. https://doi.org/10.1016/j.cor.2009.09.008
  12. Klimenok, V. I. and Dudin, A.N. (2006), Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory, Queueing Systems, 54, 245-259. https://doi.org/10.1007/s11134-006-0300-z
  13. Tran-Gia, P. and Mandjes, M. (1997), Modeling of customer retrial phenomenon in cellular mobile networks, IEEE Journal on Selected Areas in Communications, 15, 1406-1414. https://doi.org/10.1109/49.634781
  14. Wu, J., Liu, Z., and Yang, G. (2001), Analysis of the finite source MAP/PH/N retrial G-queue operating in a random environment, Applied Mathematical Modelling, 35, 1184-1193.
  15. Yang, G., Yao, L.-G., and Ouyang, Z.-S. (2013), The MAP/PH/N retrial queue in a random environment. Acta Mathematicae Applicatae Sinica, 29, 725-738. https://doi.org/10.1007/s10255-013-0251-1
  16. Zhou, Z. and Zhu, Y. (2013), Optimization of the ($MAP_,\;MAP_2)/(PH_1,\;PH_2)/N$ retrial queue model of wireless cellular networks with channel allocation, Computers and Electrical Engineering, 39, 1637-1649. https://doi.org/10.1016/j.compeleceng.2012.08.004

Cited by

  1. Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks vol.78, pp.8, 2017, https://doi.org/10.1134/S000511791708001X

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)