Abstract
In the paper we consider a GI/GI/\(\infty \) queuing system with n types of customers under the assumptions that customers arrive at the queue according to a renewal process and occupy random resource amounts, which are independent of their service times. Since, in general, the analytical solution of the corresponding Kolmogorov differential equations is not available, we focus on the amount of resources occupied by each class of customers under the assumption of infinitely growing arrival rate, and derive its first and second-order asymptotic approximations. In more detail, we show that the n-dimensional probability distribution of the total resource amount is asymptotically n-dimensional Gaussian, and we verify the accuracy of the asymptotics (in terms of Kolmogorov distance) by means of discrete event simulation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alexander, M., Anatoly, N.: Asymptotic analysis of the infinite-server queueing system with high-rate semi-markov arrivals. In: 2014 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), pp. 507–513, October 2014
Alexander, M., Anatoly, N.: Queueing network with high-rate arrivals. Eur. J. Oper. Res. 254(1), 161–168 (2016)
Ali, T., Winfried, G., Javad, T.: Optimal policies of M(t)/M/c/c queues with two different levels of servers. 249, 10 (2015)
Efrosinin, D., Feichtenschlager, M.: Optimal control of M(t)/M/K queues with homogeneous and heterogeneous servers. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds.) DCCN 2016. CCIS, vol. 678, pp. 132–144. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-51917-3_13
Fedorova, E.: The second order asymptotic analysis under heavy load condition for retrial queueing system MMPP/M/1. In: Dudin, A., Nazarov, A., Yakupov, R. (eds.) ITMM 2015. CCIS, vol. 564, pp. 344–357. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25861-4_29
Lisovskaya, E., Moiseeva, S., Pagano, M.: On the total customers’ capacity in multi-server queues. In: Dudin, A., Nazarov, A., Kirpichnikov, A. (eds.) ITMM 2017. CCIS, vol. 800, pp. 56–67. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68069-9_5
Ekaterina, L., Svetlana, M., Michele, P., Viktoriya, P.: Study of the MMPP/GI/\(\infty \) queueing system with random customers’ capacities. Inf. Appl. 11(4), 109–117 (2017)
Pankratova, E., Farkhadov, M., Gelenbe, E.: Research of heterogeneous queueing system SM|M\(^{(n)}|\infty \). In: Dudin, A., Nazarov, A., Kirpichnikov, A. (eds.) ITMM 2017. CCIS, vol. 800, pp. 122–132. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68069-9_10
Pankratova, E., Moiseeva, S.: Queueing system MAP/M/\(\infty \) with n types of customers. In: Dudin, A., Nazarov, A., Yakupov, R., Gortsev, A. (eds.) ITMM 2014. CCIS, vol. 487, pp. 356–366. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-13671-4_41
Kumar, V., Gupta, A.K., Kumar, S., Kumar, V.: Mobility management in heterogeneous wireless networks based on IEEE 802.21 framework. In: 2015 International Conference on Advances in Computer Engineering and Applications, pp. 930–935, March 2015
Oleg, T., Wojciech, K.: Queueing system with processor sharing and limited memory under control of the AQM mechanism. Autom. Remote Control 76(10), 1784–1796 (2015)
Valeriy, N., Konstantin, S.: Analysis of multi-resource loss system with state-dependent arrival and service rates. Probab. Eng. Inf. Sci. 31(4), 413–419 (2017)
Valeriy, N., Konstantin, S., Eduard, S., Sergey, A.: Two approaches to analyzing dynamic cellular networks with limited resources. In: 2014 6th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), pp. 485–488, October 2014
Valeriy, N., Konstantin, S., Nataliya, Y., Eduard, S., Sergey, A., Andrey, S.: LTE performance analysis using queuing systems with finite resources and random requirements. In: 2015 7th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), pp. 100–103, October 2015
Wang, R., Jouini, O., Benjaafar, S.: Service systems with finite and heterogeneous customer arrivals. Manuf. Serv. Oper. Manag. 16(3), 365–380 (2014)
Borodakiy, V.Y., Samouylov, K.E., Gudkova, I.A., Markova, E.V.: Analyzing mean bit rate of multicast video conference in lte network with adaptive radio admission control scheme\(^{*}\). J. Math. Sci. 218(3), 257–268 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Lisovskaya, E., Moiseeva, S., Pagano, M. (2018). Multiclass GI/GI/\(\infty \) Queueing Systems with Random Resource Requirements. In: Dudin, A., Nazarov, A., Moiseev, A. (eds) Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM WRQ 2018 2018. Communications in Computer and Information Science, vol 912. Springer, Cham. https://doi.org/10.1007/978-3-319-97595-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-97595-5_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-97594-8
Online ISBN: 978-3-319-97595-5
eBook Packages: Computer ScienceComputer Science (R0)