Извините, этот техт доступен только в “Американский Английский”. For the sake of viewer convenience, the content is shown below in the alternative language. You may click the link to switch the active language.
QUEUE ANALYSIS IN THE G/G/1 SYSTEM BASED ON HYPEREXPONENTIAL DISTRIBUTIONS
Marina A. Buranova, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia, buranova-ma@psuti.ru
Abstract
For real-time applications, the change in packet delay, which is usually called delay jitter, is one of the most important parameters of quality of service in modern infocommunication networks when processing multimedia streams, along with the delay and the probability of packet loss. This article discusses approaches to assessing jitter in modern infocommunication networks when processing non-Poissonian traffic in the G/G/1 system. To approximate the system of an arbitrary queue G/G/1, a model based on hyperexponential distributions of the type H2/H2/1 was used. As a processed stream, the implementation of IP network traffic with the absence of mutual correlations between the time intervals between packets and the times of packet processing is used. An important task in this case is to determine the parameters of hyperexponential distributions for the time intervals between packets and packet processing times. The paper presents two methods for determining the parameters of hyperexponential distributions: using the method of moments and using the EM-algorithm. The paper analyzes the influence of the network load factor on the jitter value when using two models for determining the parameters of hyperexponential distributions. This analysis showed a slight increase in jitter with increasing network load both in the case of using the method of moments and in the case of the EM algorithm. It has been determined that the estimates of the jitter values in the case of applying the method of moments and the EM algorithm are quite close in value. The main result of the work is to obtain simple calculation formulas for the analysis of jitter in the G/G/1 system using hyperexponential distributions.
Keywords: jitter, queuing system, hyperexponential distribution, EM-algorithm, method of moments.
References
1. Sheluhin O.I., Tenyakshev A.M., Osin A.V. (2003). Fraktalnie processi v telekommunikaciyah [Fractal Processes in Telecommunications]. Edited by. O.I. Sheluhina. Radiotehnika. 480 p. (in Russian)
2. Taggu M.S. (1988). Self-similar processes. In S. Kotz and N. Johnson, editors, Encyclopedia of Statistical Sciences. Wiley, New York, v. 8, pp. 352-357.
3. Feldmann, A., & Whitt, W. (1997). Fitting Mixtures of Exponentials to Long-Tail Distributions to Analyze Network Performance Models. Proceedings IEEE INFOCOM’97, pp. 1096-1104.
4. Andersen A.T., Jensen A., Nielsen B.F. (1995), Modelling and performance study of packet-traffic with self-similar characteristics over several time-scales with Markovian arrival processes (MAP). Twelfth Nordic Teletraffic Seminar, NTS 12, pp. 269-283.
5. Lucantoni D.M. (1993). The ÂÌÀÐ/G/1 queue: a tutorial. Models and Techniques for Performance Evaluation of Computer and Communication Systems, Springer, New York, pp. 330-358.
6. Neuts M.F. (1989). Structured stochastic matrices of M/G/l type and their applications. Marcel Dekker, New York, 512 p.
7. Asmussen, S., Koole G. (1993). Marked point processes as limits of Markovian arrival streams. J. Appl. Probab, vol. 30, no. 2, pp. 365-372.
8. Dbira H., Girard A., Sanso B. (2016). Calculation of packet jitter for non-poisson traffic. Annals of telecommunications, vol. 71, issue 5-6, pp. 223-237.
9. Kartashevsky V.G., Buranova M.A. (2019). Modelirovanie dzhittera paketov pri peredache po mul’tiservisnoj seti. [Modeling of jitter packets when transmitting over a multiservice network]. Informacionnye tekhnologii i telekommunikacii [Information technologies and telecommunications], vol. 17, no 1, pp. 34-40. (in Russian)
10. Kartashevskiy I., Buranova M. (2019). Calculation of Packet Jitter for Correlated Traffic. International Conference on «Internet of Things, Smart Spaces, and Next Generation Networks and Systems. NEW2AN 2019», vol. 11660, pp. 610-620. (Lecture Notes in Computer Science, Springer, Cham). DOI: 10.1007 / 978-3-030-30859-9_53.
11. L. Kleinrock (1975). Queueing Systems, Vol. I: Theory, Wiley, New York, NY, 432 p.
12. Keilson, J., Machihara, F. (1985). Hyperexponential waiting time structure in hyperexponential system. Journal of the Operation Society of Japan, vol. 28, no. 3, pp. 242-250. DOI: 10.15807/jorsj.28.242.
13. Demichelis C, Chimento P. (2000). IP Packet Delay Variation Metric for IP Performance Metrics (IPPM). Institution IETF, RFC 33934, 21 p. DOI: 10.17487/RFC3393.
14. Internet protocol data communication service IP packet transfer and availability performance parameters. ITU-T Recommendation Y.1540, 2002, 33 p.
15. Kartashevsky V.G., Buranova M.A. (2019). Modelirovanie dzhittera paketov pri peredache po mul’tiservisnoj seti [Modeling of jitter packets when transmitting over a multiservice network]. Informacionnye tekhnologii i telekommunikacii [Information technologies and telecommunications], vol. 17, no. 1, pp. 34-40 (in Russian)
16. Buranova M.A., Ergasheva D.R., Kartashevskiy V.G. (2019). Using the EM-algorithm to Approximate the Distribution of a Mixture by Hyperexponents. 2019 International Conference on Engineering and Telecommunication (EnT) Dolgoprudny, Russia, pp. 1-4. DOI: 10.1109/EnT47717.2019.9030551.
17. Tarasov V.N., Kartashevskij I.V. (2014). Opredelenie srednego vremeni ozhidaniya trebovanij v upravlyaemoj sisteme massovogo obsluzhivaniya N2/N2/1 [Determination of the average waiting time for requirements in a managed mass service system]. Sistemy upravleniya i informacionnye tekhnologii [Control systems and information technologies], no. 3(57), pp. 92-96. (In Russian)
18. Tarasov V.N., Gorelov G.A., Ushakov Yu.A. (2014). Vosstanovlenie momentnyh harakteri-stik raspredeleniya intervalov mezhdu paketami vhodyashchego trafika [Recovery of moment characteristics of the distribution of intervals between packets of incoming traffic]. Infokommunikacionnie tehnologii [Infocommunication technologies], no. 2, pp. 40-44. (In Russian)
19. Day N.E. (1969). Estimating the components of a mixture of normal distributions. Biometrika, vol. 56, no. 3, pp. 463-474.
20. Korolev V.Yu. (2007). EM-algoritm ego modifikacii i ih primenenie k zadache razdeleniya smesej veroyatnostnyh raspredelenij. Teoreticheskij obzor [EM-algorithm for its modification and their application to the problem of separation of mixtures of probability distributions. Theoretical review]. Moscow: IPI RAN, 94 p. (In Russian)
21. Voroncov K.V. Matematicheskie metody obucheniya po precedentam (teoriya obucheniya mashin) [Mathematical teaching methods by precedents (machine learning theory)], Available at: http://www.machinelearning.ru/wiki/images/6/6d/Voron-ML-1.pdf. (In Russian)
22. Baird, SR.: Estimating mixtures of exponential distributions using maximum likelihood and the EM algorithm to improve simulation of telecommunication networks, Available at: https://open.library.ubc.ca/collections/ubctheses/831/items/1.0090805.
Information about author:
Buranova Marina, Associate Professor of Information Security Department, PhD in Technical Sciences, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia