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1 38 6 Vol. 38 No ADVANCES IN MECHANICS Nov. 25, 2008 *,,, /,.,,,., ( ),., ( ),.,,, 3 ( )., ( ).,.,,,,, 1,,. [1,2],,, [3 8].,.,, (autonomous system, AS)., AS.,, AS. : [9 13] [14 18] Faloutsos [19],., Barabasi Albert [9], [10]., Yook [11]. Zhou [12] AS, :.,, AS., Chen [13].,..,,.,, [14,15].,,.,,,,. : , : (G-YF51) tamwm@eie.polyu.edu.hk

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7 6 : ( ) 1:., i j.,,. ( T c ), T c = t e t s = T c. 2:. i j,,,. 6, j 2, t (j) 2,.,, T c < T c., T c = t (j) 2 t s. 3:. ( i K i j K j ).,, T c = T c.,. i, i j. 5 : (1) λ i : i. (2) λ = 1 N λ i : N i=1, N ; (3) 1/µ:,. (4) 1/θ i : i, i. (5) c l : l ( )., : (1) i t, 1/λ [24,26,27] i, f(t) = λ i e λit (7), f( ) (pdf).,., i, λ i = αn i (8) α > 0,. (scale-free user network, SFUN),, λ i. (fullyconnected user network, FCUN),,,.

8 , ( N = ) 4 ( M = 4), , 5 ( n = 5) α = 1/500., λ, (2), T c f(t c ) = µe µtc (9), 1/µ 4 min, A = 2 λn µm = 200 E (10) (3) (, τ m ).,,. τ m, 1/θ [28] i pdf f(τ m ) = θ i e θiτm (11), 3. 1,, (zero mobility, ZM), [27]. 2, (identical mobility, IM), θ i = 1 N N i=1 θ i = θ, i = 1, 2,..., N., (power-law mobility, PLM), ( ), ( )., θ i, (3) p b2 :.. (4) p b = p b1 + p b2 :.. 6, (FCUN) (SFUN) (ZM) (IM) (PLM). 6.1,, c 1 = c 2 = c 3 = c 4 =. IM PLM FCUN SFUN, 7 1/ θ.,,,. f(θ i ) θ ν i (12) ν > 0., θ, θ = 1/120 ( )., : (1) Ā:. ( 1min). (2) p b1 :.. 7 IM PLM FCUN SFUN, 8.,.,,,,,

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10 ,,,. FCUN,,,., SFUN, FCUN,. 11 FCUN SFUN, 12., p b1 p b2. p b.,,, p b1.,, p b2.,, p b1., p b2., p b p b1, p b2.,. FCUN SFUN 12(c),, SFUN( ) FCUN. SFUN ( 11),, SFUN FCUN ( 10). 12 p b p b1 p b2

11 6 : 825, λ. 30 min, θ=1 / 30. FCUN SFUN, , λ = 0.008, SFUN FCUN. IM PLM 13.,,.,,. IM,,. PLM,,,,,, λ,, (FCUN) (SFUN),. FCUN.,, SFUN., 3, (ZM) (IM) (PLM).,., SFUN FCUN, SFUN,.,.,,.,,.,,,.,,, ( ). 1 Erdos P, Rényi A. On the evolution of random graphs. Publ Math Inst Hung Acad Sci, 1959, 5: Bollobas B. Random Graphs. London: Academic, Milgram S. The small world problem. Psychology Today, 1967, 2: Watts D J, Strogatz S H. Collective dynamics of smallworld networks. Nature, 1998, 393: Kleinberg J M. Navigation in a small world. Nature, 2000, 406: Strogatz S H. Exploring complex networks. Nature, 2001, 410: Boots M, Sasaki A. Small worlds and the evolution of virulence: infection occurs locally and at a distance. Proc R Soc Lond, 1999, B266(1432): Wang X F, Chen G. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans on Circuits Systems I, 2002, 49: Baraási A-L, Albert R. Emergence of scaling in random networks. Science, 1999, 286: Albert R, Baraási A-L. Topology of evolving networks: local events and universality. Physical Review Letters, 2000, 85: Yook S, Jeong H, Barabási A L. Modeling the internet s large-scale topology. Proceedings of the National Academy of Sciences, 2002, 99(21): Zhou S, Mondraón R J. Accurately modeling the internet topology. Physical Review E, 2004, 70: Chen G, Fan Z P, Li X. Modeling the complex internet topology. In: Complex Dynamics in Communication Networks. New York: Springer-Verlag, Zhao L, Park K, Lai Y C. Attack vulnerability of scale-free networks due to cascading breakdown. Physical Review E, 2004, 70: Zhao L, Lai Y C, Park K, Ye N. Onset of traffic congestion in complex networks. Physical Review E, 2005, 71: Moura A P S. Fermi-Dirac statistics and traffic in complex networks. Physical Review E, 2005, 71: Eisler Z, Kertész J. Random walks on complex networks with inhomogeneous impact. Physical Review E, 2005, 71:

12 Wang W X, Wang B H, Yin C Y, et al. Traffic dynamics based on local routing protocol on a scale-free network. Physical Review E, 2006, 73: Faloutsos M, Faloutsos P, Faloutsos C. On power-law relationships of the internet topology. Computer Communication Review, 1999, 29(4): Xia Y, Tse C K, Lau F C M, et al. Traffic congestion analysis in complex networks. In: Proceedings of IEEE International Symposium on Circuits and Systems , USA IEEE CAS Society, Kos Island, Greece, Arrowsmith D, Bernardo M, Sorrentino F. Communication models with distributed transmission rates and buffer sizes. In: Proceedings of IEEE International Symposium on Circuits and Systems , USA IEEE CAS Society, Kos Island, Greece, Zheng X, Lau F C M, Tse C K. Study of LDPC codes built on scale-free networks. In: Proceedings of International Symposium on Nonlinear Theory and its Applications, Japan: Research Society of Nonlinear Theory and its Applications , Bologna, Italy, ,,.., : ,, Bellamy J C. Digital Telephony. New York: Wiley, Gross D, Harris C M. Fundamentals of Queuing Theory. Chapter 1. New York: John Wiley & Sons, Aiello W, Chung F, Lu L. A random graph model for massive graphs. In: Proceedings of 32nd Annual ACM Symposium on Theory of Computing. ACM Special Interest Group on Algorithms and Computation Theory Portland, USA, Xia Y X, Tse C K, Tam W M, et al. Scale-free user network approach to telephone network traffic analysis. Physical Review E, 2005, 72: Hung H N, Lee P C, Lin Y B, Peng N F. Modeling channel assignment of small-scale cellular networks. IEEE Trans. on Wireless Communications, 2005, 4(2): STUDY ON MOBILE COMMUNICATIONS WITH A COMPLEX UNDERLYING USER NETWORK * TAM Waiman LAU Chungming TSE Chikong Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong, China Abstract In the traditional study of mobile cellular systems, all users are assumed to have the same behaviour: they have the same probability of making/receiving a call, and they will move around the network with identical mobility. Moreover, the underlying user network is assumed to be fully connected, implying that all users know each other and they will communicate with each other with the same probability. In a practical environment, each user has a different list of acquaintances including relatives, friends and colleagues, with whom the user is most likely to make contact, and the size of the list varies with individual users. In addition, different users will acquire different levels of mobility, depending on various factors such as job or age. To evaluate the performance of a mobile cellular system more realistically, we model the user network with a scale-free network in which the number of acquaintances of a user follows a power-law distribution. Moreover, the mobile system is studied under three user mobility distributions: zero mobility, identical mobility and power-law mobility. Finally, the system performance indicators, such as the carried traffic and the blocking probabilities, are evaluated based on computer simulations, which show that both the user network configuration and the mobility of users affect the performance of the mobile system. Keywords complex network, fully-connected network, scale-free network, mobile cellular system, blocking probability, mobility The project supported by the Research Foundation of Hong Kong Polytechnic University (G-YF51) tamwm@eie.polyu.edu.hk

第一章数学系的历史沿革第一节数学系的渊源和机构变革情况 1949 年 6 月, 邸耀宗厉瑞康在太原市北郊上兰村原进山中学的废墟上筹建兵工职业学校,1950 年改为兵工高级职

第一章数学系的历史沿革第一节数学系的渊源和机构变革情况 1949 年 6 月, 邸耀宗厉瑞康在太原市北郊上兰村原进山中学的废墟上筹建兵工职业学校,1950 年改为兵工高级职前言中北大学数学系是伴随着学校的发展, 在几代人的努力下不断发展壮大起来的从 1953 年太原机械工业学校数学教研组到 2006 年理学院数学系, 其间凝聚了中北几代数学人的汗水 2011 年, 正值中北大学建校 70 周年校庆, 我们认为