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If Qh is a SFBM of the multifractional function h, then Qh is the overall H lderian of exponent H for all 0 < H < H (I), on any compact interval J I Thus, in terms of regularity, the SFBM is a satisfactory model We will see, in section 65, that we can completely identify the multifractional function: moments and amplitudes of the jumps 6432 Generalized multifractional Brownian motion Let us now outline the work of [AYA 99] The authors propose to circumvent the low frequency problems encountered within the de nition of MBM, by replacing the multifractional function h with a sequence of regular functions hn , whose limit, which will play the role of the multifractional function, can be very irregular Let us rst specify the technical conditions relating to the sequence (hn )n N DEFINITION 69 A function h is said to be locally H lderian of exponent r and of constant c > 0 on if, for all t1 and every t2 , such that, |t1 t2 | 1, we have: |h(t1 ) h(t2 )| c|t1 t2 |r

Such a function will be called (r, c) H lderian We can consequently de ne the multifractional sequences which generalize the multifractional functions for the GMBM DEFINITION 610 We will call a multifractional sequence a sequence (hn )n N of H lderian functions (r, cn ) with values in an interval [a, b] ]0, 1[ and we will call its lower limit a generalized multifractional function (GMF): h(t) = lim inf hn (t)

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[113] DARPA, Information Survivability Conference and Exposition, DISCEX 2001, Anaheim, CA, June 2001 [114] R Chadha, Y-H Cheng, J Chiang, G Levin, S-W Li, and A Poylisher, Policy-Based Mobile Ad Hoc Network Management, Policy 2004, Yorktown Heights, NY, June 2004 [115] R Chadha, Y-H Cheng, J Chiang, G Levin, S-W Li, and A Poylisher, Policy-Based Mobile Ad Hoc Network Management for DRAMA, Milcom, 2004 [116] R Chadha, Y-H Cheng, J Chiang, G Levin, S-W Li, A Poylisher, L LaVergn, and S Newan, Scalable Policy Management for Ad Hoc Networks, Milcom, 2005 [117] S Herzog, J Boyle, R Cohen, D Durham, R Rajan and A Sastry, COPS Usage for RSVP, IETF RFC 2749, January 2000 [118] R Braden, Resource ReSerVation Protocol (RSVP) Version 1 Functional Speci cation, IETF RFC 2205, September 1997 [119] K Chan, J Seligson, D Durham, S Gai, K McCloghrie, S Herzog, F Reichmeyer, R Yavatkar, and A Smith, COPS Usage for Policy Provisioning (COPS-PR), IETF RFC 3084, March 2001 [120] M Wahl, T Howes, and S Kille, Lightweight Directory Access Protocol (v3), IETF RFC 2251, December 1997 [121] M Blaze, J Feigenbaum, and J Lacy, Decentralized Trust Management, Proceedings of the 17th IEEE Symposium on Security and Privacy, IEEE Computer Society, New York, 1996, pp 164 173, [122] M Blaze, J Feigenbaum, J Ioannidis, and A Keromytis, The KeyNote Trust-Management System Version 2, IETF RFC 2704, September 1999 [123] A D Keromytis, S Ioannidis, M Greenwald, and J Smith, The STRONGMAN Architecture, DISCEX 20003, Washington, DC, April 2003 [124] J Burns, A Cheng, P Gurung, S Rajagopalan, P Rao, D Rosenbluth, A V Surendran, and D M Martin Jr, Automatic Management of Network Security Policy, DISCEX 2001, California, June 2001 [125] T Markham and C Payne, Security at the Network Edge: A Distributed Firewall Architecture, DISCEX 20001, California, June 2001 [126] Y B Ko, and N Vaidya, Location-aided Routing (LAR) in Mobile Ad hoc Networks, in Proceedings of MobiCom, 1998, pp 66 75 [127] S Basagni, I Chlamtac, V Syrotiuk, and B Woodward, A Distance Routing Effect Algorithm for Mobility (DREAM), in Proceedings of MOBICOM, Dallas, TX, October 1998, pp 76 84 [128] Karp, B and Kung, HT, GPSR: Greedy Perimeter Stateless Routing for Wireless Networks, in Proceedings of MobiCom 2000, pp 243 254 [129] J C Navas and T Imielinski, Geographic Addressing and Routing, in MOBICOM, Budapest, 26 30 September 1997 [130] D Johnson and D Maltz, Dynamic Source Routing in Ad Hoc Wireless Networks, Mobile Computing, 1996, pp 153 181 [131] V Park and M Corson, A Highly Adaptive Distributed Routing Algorithm for Mobile Wireless Networks, Proc Infocom, 1997 [132] C Perkins and E Royer, Ad-Hoc On-Demand Distance Vector Routing, in Proceedings of Second IEEE Workshop Mobile Computing Systems and Applications, February 1999, pp 90 100 [133] T Camp, J Boleng, B Williams, L Wilcox, and W Navidi, Performance Comparison of Two Location Based Routing Protocols for Ad Hoc Networks, Proc IEEE Infocom, 2002.

if (hn )n N veri es the following properties: for all and all t0 , there exists n0 (t0 , ) and h0 (t0 , ) > 0 such that, for all n > n0 and, |h| < h0 we have: hn (t0 + h) > h(t0 ) for all t, we have h(t) < r and cn = O(n) In the preceding de nition, it is essential that the generalized multifractional function is a limit when the index n tends towards + ; we will see that this translates the high frequency portion of the information contained in the multifractional sequence In addition, the GMF set contains very irregular functions like, for example, 0 < a < b < 1: t b + (a b)1F (t)

bl D l u(t)

corresponding, by Laplace transform, to the symbol: H(s) =

(734)

.

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