Matthew N.O. Sadiku · Sarhan M. Musa Performance Analysis of Computer Networks Performance Analysis of Computer Networks Matthew N.O Sadiku • Sarhan M Musa Performance Analysis of Computer Networks Matthew N.O Sadiku Roy G Perry College of Engineering Prairie View A&M University Prairie View, TX, USA Sarhan M Musa Roy G Perry College of Engineering Prairie View A&M University Prairie View, TX, USA ISBN 978-3-319-01645-0 ISBN 978-3-319-01646-7 (eBook) DOI 10.1007/978-3-319-01646-7 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013947166 © Springer International Publishing Switzerland 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To my late dad, Solomon, late mom, Ayisat, and my wife, Kikelomo To my late father, Mahmoud, mother, Fatmeh, and my wife, Lama Preface Modeling and performance analysis play an important role in the design of computer communication systems Models are tools for designers to study a system before it is actually implemented Performance evaluation of models of computer networks during the architecture design, development, and implementation stages provides means to assess critical issues and components It gives the designer the freedom and flexibility to adjust various parameters of the network in the planning rather than in the operational phase The major goal of the book is to present a concise introduction to the performance evaluation of computer communication networks The book begins by providing the necessary background in probability theory, random variables, and stochastic processes It introduces queueing theory and simulation as the major tools analysts have at their disposal It presents performance analysis on local, metropolitan, and wide area networks as well as on wireless networks It concludes with a brief introduction to self-similarity The book is designed for a one-semester course for senior-year undergraduate and graduate engineering students The prerequisite for taking the course is a background knowledge of probability theory and data communication in general The book can be used in giving short seminars on performance evaluation It may also serve as a fingertip reference for engineers developing communication networks, managers involved in systems planning, and researchers and instructors of computer communication networks We owe a debt of appreciation to Prairie View A&M University for providing the environment to develop our ideas We would like to acknowledge the support of the departmental head, Dr John O Attia, and college dean, Dr Kendall Harris Special thanks are due to Dr Sadiku’s graduate student, Nana Ampah, for carefully going through the entire manuscript (Nana has graduated now with his doctoral degree.) Dr Sadiku would like to thank his daughter, Ann, for helping in many ways especially with the figures Without the constant support and prayers of our families, this project would not have been possible Prairie View, TX, USA Prairie View, TX, USA Matthew N.O Sadiku Sarhan M Musa vii Contents Performance Measures 1.1 Computer Communication Networks 1.2 Techniques for Performance Analysis 1.3 Performance Measures References 1 Probability and Random Variables 2.1 Probability Fundamentals 2.1.1 Simple Probability 2.1.2 Joint Probability 2.1.3 Conditional Probability 2.1.4 Statistical Independence 2.2 Random Variables 2.2.1 Cumulative Distribution Function 2.2.2 Probability Density Function 2.2.3 Joint Distribution 2.3 Operations on Random Variables 2.3.1 Expectations and Moments 2.3.2 Variance 2.3.3 Multivariate Expectations 2.3.4 Covariance and Correlation 2.4 Discrete Probability Models 2.4.1 Bernoulli Distribution 2.4.2 Binomial Distribution 2.4.3 Geometric Distribution 2.4.4 Poisson Distribution 2.5 Continuous Probability Models 2.5.1 Uniform Distribution 2.5.2 Exponential Distribution 2.5.3 Erlang Distribution 5 8 11 12 13 14 20 20 22 22 23 28 28 29 30 31 33 33 34 34 ix Problems 263   dP P ¼ rP À dt K (10.41) where P is population size, K is capacity, and t is time Setting x ¼ P/K in Eq (10.41) gives dx ¼ rxð1 À xÞ dt (10.42) Logistic map is a discrete representation of Eq (10.42) and is written as recurrence relation as follows: xnþ1 ¼ rxn ð1 À xn Þ (10.43) This equation has been used to obtain self-similar time sequence which could be used for traffic generation for wireless network systems [19] Values of r in the range 3.50 < r < 3.88 and < x0 < 0.5 have been used 10.6 Summary Studies of both Ethernet traffic and variable bit rate (VBR) video have demonstrated that these traffics exhibit self-similarity A self-similar phenomenon displays the same or similar statistical properties when viewed at different times scales Pareto distribution is a heavy-tailed distribution with infinite variance and is used in modeling self-similar traffic The most common method of generating self-similar traffic is to simulate several sources that generate constant traffic and then multiplex then with ON/OFF method using heavy-tailed distribution such as Pareto We analytically modeled the performance of a single server queue with almost self-similar input traffic and exponentially distributed service times Logistic map for self-similar traffic generation is used for wireless network OPNET can be used to simulate the network traffic’s self-similarity [20] Problems 10.1 (a) Explain the concept of self-similarity (b) What is a self-similar process? 10.2 Show that the Brownian motion process B(t) with parameter H ¼ 1/2 is self-similar Hint: Prove that B(t) satisfy conditions in Eqs (10.1) to (10.3) 10.3 Show that the Eq (10.14) is valid and that the variance of Pareto distribution is infinite 264 10 Self-Similarity of Network Traffic 10.4 If X is a random variable with a Pareto distribution with parameters α and δ, then show that the random variable Y ¼ ln (X/δ) has an exponential distribution with parameter α 10.5 Evaluate and plot σ in Eq (10.24) for < ρ < 0.2 with r ¼ 0.01 References W E Leland et al., “On the self-similar nature of Ethernet traffic,” Computer Communications Review, vol 23, Oct 1993, pp 183-193 –, “On the self-similar nature of Ethernet traffic (extended version),” IEEE/ACM Transactions on Networking, vol 5, no 6, Dec 1997, pp 835-846 M E Crovella and A Bestavros, “Self-similarity in World Wide Web traffic: Evidence and possible causes,” IEEE/ACM Transactions on Networking, vol 5, no 6, Dec 1997, pp 835-846 C D Cairano-Gilfedder and R G Cleggg, “A decade of internet research—advances in models and practices,” BT Technology Journal, vol 23, no 4, Oct 2005, pp 115-128 B Tsybakov and N D Georganas, “On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution,” IEEE/ACM Transactions on Networking, vol 5, no 3, June 1997, pp 397-409 W Stallings, High-Speed Networks and Internets: Performance and Quality of Service Upper Saddle, NJ: Prentice Hall, 2nd ed., 2002, pp 219-247 W Jiangto and Y Geng, “An intelligent method for real-time detection of DDOS attack based on fuzzy logic,” Journal of Electronics (China), vol 25, no 4, July 2008, pp 511-518 D Kouvatsos (ed.), Performance Evaluation and Applications of ATM Networks Boston, MA: Kluwer Academic Publishers, 2000, pp 355-386 A Ost, Performance of Communication Systems New York: Springer Verlag, 2001, pp 171-177 10 K Park and W Willinger (eds.), Self-similar Network Traffic and Performance Evaluation New York: John Wiley & Sons, 2000 11 J.M Pitts and J A Schormans, Introduction to IP and ATM Design and Performance Chichester, UK: John Wiley & Sons, 2000, pp 287-298 12 Z Harpantidou and M Paterakis, “Random multiple access of broadcast channels with Pareto distributed packet interarrival times,” IEEE Personal Communications, vol 5, no 2, April 1998, pp 48-55 13 Z Hadzi-Velkov and L Gavrilovska, “Performance of the IEEE 802.11 wireless LANs under influence of hidden terminals and Pareto distributed packet traffic,” Proceedings of IEEE International Conference on Personal Wireless Communication, 1999, pp 221-225 14 W Willinger et al., “Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level,” IEEE/ACM Transactions on Networking, vol 5, no 1, 1997, pp 71-86 15 A R Prasad, B Stavrov, and F C Schoute, “Generation and testing of self-similar traffic in ATM networks,” IEEE International Conference on Personal Wireless Communications, 1996, pp 200-205 16 N Bhatnagar, “Model of a queue with almost self-similar or fractal-like traffic,” Proc IEEE GLOBECOM ‘97, 1997, pp 1424-1428 17 E Y Peterson and P M Ulanov, “Methods for simulation of self-similar traffic in computer networks,” Automatic Control and Computer Science, vol 36, no 6, 2002, pp 62-69 18 M S Taqqu, “The modeling of Ethernet data and of signals that are heavy-tailed with infinite variance.” Scandinavian Journal of Statistics, vol 29, 2002, pp 273-295 References 265 19 R Yeryomin and E Petersons, “Generating self-similar traffic for wireless network simulation,” Proc of Baltic Congress of Future Internet and Communications, 2011, pp 218-220 20 Y Fei et al., “An intrusion alarming system based on self-similarity of network traffic,” Wuhan University Journal of Natural Sciences (WUJNS), vol 10, no 1, 2005, pp 169-173 Appendix A: Derivation for M/G/1 Queue In this appendix, we apply the method of z-transform or generating functions to find the waiting time of the M/G/1 model The probability of having k arrivals during the service time t is ð pk ¼ ð pðkÞdH ðtÞ ¼ 0 ðλtÞk Àλt e dH ðtÞ k! (A.1) where H(t) is the service time distribution Let N be the number of customers present in the system and Q be the number of customers in the queue Let the probability that an arriving customer finds j other customers present be Πj ¼ ProbðN ¼ jÞ, j ¼ 0, 1, 2, Á Á Á (A.2) It can be shown using the theorem of total probability and the equilibrium imbedded-Markov-chain that Πj ¼ pj Π0 þ jþ1 X pjÀiþ1 Πi , j ¼ 0, 1, 2, Á Á Á (A.3) i¼1 We define the probability-generating functions gð z Þ ¼ X Πj zj (A.4a) pj z j (A.4b) j¼0 hð z Þ ¼ X j¼0 M.N.O Sadiku and S.M Musa, Performance Analysis of Computer Networks, DOI 10.1007/978-3-319-01646-7, © Springer International Publishing Switzerland 2013 267 268 Appendix A: Derivation for M/G/1 Queue Substituting (Eq A.4a) into (Eq A.3) results in gð z Þ ¼ ðz À 1ÞhðzÞ Π0 z À hð z Þ (A.5) The normalization equation X Πj ¼ (A.6) j¼0 implies that g(1) ¼ With a single application of L’Hopital’s rule, we find Π0 ¼ À ρ (A.7) where ρ ¼ λ/μ ¼ λτ If we define η(s) as the Laplace-Stieltjes transform of the service-time distribution function H(t), ð ηðsÞ ¼ eÀst dH ðtÞ (A.8) Substitution of (Eq A.1) into (Eq A.4b) yields hðzÞ ¼ ηðλ À λzÞ (A.9) and substitution of (Eq A.7) and (Eq A.9) into (Eq A.5) leads to gðzÞ ¼ ðz À 1Þηðλ À λzÞ ð À ρÞ z À ηðλ À λzÞ (A.10) Differentiating this and applying L’Hopital rule twice, we obtain   ρ2 σ2 1þ þρ g ð1Þ ¼ 2ð À ρÞ τ (A.11) The mean values of the number of customers in the system and queue are respectively given by Eð N Þ ¼ X jΠj ¼ g ð1Þ (A.12a) j¼0 EðQÞ ¼ EðNÞ À ρ (A.12b) Appendix A: Derivation for M/G/1 Queue 269 By applying Little’s theorem, the mean value of the response time is Eð N Þ ρτ @ σ2A ¼ 1þ þτ Eð T Þ ¼ λ 2ð À ρÞ τ ¼ Eð W Þ þ τ Thus we obtain the mean waiting time as   EðQÞ ρτ σ2 ¼ 1þ Eð W Þ ¼ λ 2ð À ρÞ τ which is Pollaczek-Khintchine formula (A.13) Appendix B: Useful Formulas n X i¼1 n X n i2 ¼ ðn þ 1Þð2n þ 1Þ i¼1 n X n i ¼ ð n þ 1Þ " i ¼ i¼1 n X X , 1Àx jxj < xn ¼ xk , 1Àx jxj < n¼k k X xn ¼ x À xkþ1 , 1Àx x 6¼ xn ¼ À xkþ1 , 1Àx x 6¼ n¼1 k X n¼0 X nxn ¼ n¼1 n¼1 À nx ¼ x n2 ðn þ 1Þ2 xn ¼ n¼1 n ¼ i i¼1 X k X #2 x ð1 À xÞ2 , jxj < Á À xk À kxk ð1 À xÞ ð1 À xÞ2 , x 6¼ M.N.O Sadiku and S.M Musa, Performance Analysis of Computer Networks, DOI 10.1007/978-3-319-01646-7, © Springer International Publishing Switzerland 2013 271 272 Appendix B: Useful Formulas X n2 x n ¼ n¼1 X x ð1 þ x Þ ð1 À xÞ3 nðn þ 1Þxn ¼ n¼1 2x ð1 À x Þ3 X ðn þ kÞ! n k! x ¼ , n! ð1 À xÞkþ1 n¼0 X xn ¼ ex , n! n¼0 X n¼0 X xð2nÀ1Þ ex À eÀx ¼ , ð2n À 1Þ! n¼1 n xÀn ¼ n   X n k¼1 k jxj < 1, k ! À1