= dv 3V (r + a 1) 3 r 3 f(r) = 1. = ( (r + r 2

Transcription

1 Random Waypoint Model in n-dimensional Space Esa Hyytiä and Joma Vitamo Netwoking Laboatoy, Helsinki Univesity of Technology, Finland Abstact The andom waypoint model (RWP) is one of the most widely used mobility models in pefomance analysis of mobile wieless netwoks. In this pape we extend the pevious wok by deiving an analytical fomula fo the stationay distibution of a node moving accoding to a RWP model in n-dimensional space. Key wods: mobility modeling, andom waypoint model, ad hoc netwoking Intoduction Random waypoint model (RWP) is one of the most widely used mobility models in pefomance analysis of wieless ad hoc netwoks. In the taditional RWP model in R, the path of the node is defined by a sequence of andom waypoints, P, P,..., placed andomly using a unifom distibution in some convex domain D R. At time t = the node is placed at some point P D, eithe andomly using, e.g., unifom distibution o at some fixed stating point. Then the node moves at constant speed v along a line towads the next waypoint P. Once the node eaches waypoint P.O. Box 3, FIN 5 HUT, Finland tel , fax , . esa.hyytia@iki.fi This wok has been suppoted by the Academy of Finland (gant n:o 7454) and the Finnish Defence Foces Technical Reseach Cente. P it takes a new heading towads the next waypoint P etc. Each line segment between two waypoints is efeed to as a leg and its length is denoted by l. The RWP model was oiginally poposed in [] and has since then been studied extensively. The stationay node distibution in RWP model (in plane) has been studied, e.g., in [3 6]. In [4] Bettstette et al. pesent a method to deive an appoximate fomula fo the stationay node distibution in two-dimensional unit squae. Using this analysis as a stating point we deived in [] an exact fomula fo the node distibution in R.In[],we also analysed the extension of the model the nodes pause fo a andom time at the waypoints. In this pape we extend ou ealie wok in [] by consideing the RWP model in an n-dimensional convex set. Using a technique simila to that in [] we deive an analytical fomula fo the stationay node distibution in this case. The Pepint submitted to Elsevie Science 9 Novembe 4

2 thee-dimensional RWP pocess seves as an elementay model fo, e.g. mobile uses in an office building o a shopping cente. Additionally, it may be a useful mobility model fo eithe aibone o undewate objects. D dv da a da Spatial Node Distibution in R 3 Fig.. The integal in Eq. (3) is equal to times the volume of the shaded domain [,4]. x 3 x Fo claity,we fist conside the natual extension of RWP model to the thee-dimensional space and then genealize the esults to an n- dimensional space. θ φ x Let V denote the volume of the convex domain D R 3, l an abitay leg and l the mean length of a leg. Futhemoe, let f() denote the pdf of the node location at. Similaly as in [,4], we stat by consideing the length of the intesection of an abitay leg l and a diffeential volume element dv at, denoted by l dv. Note that the mean length of the intesection coesponds to the faction of time the node spends in dv duing a single leg. In paticula, efeing to Fig. we infe, f() = E[ l dv ], () l dv E[ l dv ] = E[ l dv ] d 3. () V D Let a = a (, ) denote the distance fom to the bounday of the domain in a given diection and a the distance fom to the bounday in the opposite diection. Fom Fig. we deduce that dv = da and da/da = /( + a ). Consequently, Fig.. Spheical Coodinates. E[ l dv ]= l(, ) dv d 3 V D = 3V (( + a ) da da) = ( ( + 3V a ) 3 3 ) da = dv 3V ( + a ) 3 3. (3) Substituting Eqs. () and (3) back into Eq. () yields, f() = ( + a ) 3 3 d 3 3lV D The volume of 3-dimensional cone is /3 height aea of the base, and the shaded domain coesponds to the diffeence between two cones.

3 a x 3 θ a x x Fig. 3. Due to symmety in a unit sphee one can conside points (,,) fo which a and a ae a function of and θ. = 3lV = lv d a d ( ( + a ) 3 3) d ( (a + a ) 4 (a 4 + a4 )). Moe specifically, using spheical coodinates we get (see Fig. ) f() = dθ sin θ dφ H(,θ,φ). (4) 6lV H(,θ,φ)=(a + a ) 4 (a 4 + a4 ), and the distances a and a depend on the shape of the paticula domain D. 3 Spatial Node Distibution in R n The same pocedue can be genealized to n dimensions. The volume of an n-dimensional cone is V n (h) = h A n, h is the heightanda coesponds to the aea of the base. Let D R n be a convex set with volume V. Then it is easy to see that the stationay distibution of the RWP pocess in D is given by f() = ( + a ) n n d n, nlv D n which can be expessed as f() = n(n +)V l d H(, ). (5) H(, ) = (a + a ) n+ (a n+ + a n+ ). The mean length of leg l is obtained fom the nomalization condition f() d n =, l = n(n +)V d n d H(, ). (6) D 4 Examples Example : RWP model in plane: Fo n = the geneal expession (5) yields f() = 6lV = lv (a + a ) 3 (a 3 + a3 ) dθ a a (a + a ) dθ, which is identical to the equation deived in []. Example : Unit spheeinr 3 : Due to the symmety, the pdf is a function of distance = only and without loss of genealitywe can conside the point =(,,). Fom Fig. 3 one 3

4 immediately obtains that a (, θ) = sin θ cos θ, a (, θ) = sin θ + cos θ. (7) Using the 3-dimensional expession (4) we get f() = 3 sin θ a a (a +3a a +a ) dθ. 6π l Witing h(, θ) =sinθ a a (a +3a a +a ) =sinθ ( )( cos θ) and h() = we get h(, θ) dθ = ( ) 3 4π h() d = 9π 35. Thus the mean length of leg l and the pdf of the node location at ae l = , f() = 35 88π ( ). In the spheically symmetic case, the pdf of the andom vaiable =, denoted by f d (), is f d () =4π f(), f d () = 35 7 ( ). The cumulative pdf fo the unit sphee is illustated in Fig. 4, each section epesents a pobability mass of.. Fig. 4. Stationay node distibution of the RWP pocess in the unit sphee. Each section epesents a pobability mass of.. Example 3: Unit hypesphee inr n : The aea and volume of the n-dimensional unit hypesphee ae [7] A n = πn/ Γ(n/), and V n = π n/ Γ(n/+). Again, due to the symmety, the pdf is a function of distance only and we can abitaily take =(,...,,). Eq. (7) holds still with θ being the angle between and the x n -axis. Diffeential suface aea of the hypesphee between θ and θ + dθ is A n sin n θdθ. Hence, we have with f() = h() = A n n(n +)V n l h(), sin n θ H(, θ) dθ H(, θ) = ( (a + a ) n+ (a n+ + a n+ ) ). In Fig. 5 the pdf of the node location at, f(), and the pdf of the distance fom the 4

5 n=5 n=4 n=3 n= n= n=3 (a) n=5 n= (b) Fig. 5. Pdf of the node location (left) and distance fom the oigin (ight) in an n-dimensional unit hypesphee, n =, 3, 4, 5. oigin, f d (), ae depicted fo dimensions n =, 3, 4, 5. Themaximum value of the pdf f() attained at the cente of the hypesphee inceases as the dimension n inceases. Futhemoe, fom the Fig. 5(b) it can be noted that as the dimensions incease the pobability mass of the andom vaiable shifts towads the suface, =. Finally, the mean length of leg given by Eq. (6) can be witten fo the unit hypesphee as A n A n l n = n(n +)Vn n h() d. Fo n =,...,5 we obtain the well-known esults [8,9]: l = 3.667, l 4 = π.4, l = 8 45π.95, l 5 = l 3 = , Refeences [] E. Hyytiä, P. Lassila, L. Nieminen, J. Vitamo, Spatial node distibution in the andom waypoint mobility model, submitted fo publication (Ap. 4). [] D. B. Johnson, D. A. Maltz, Dynamic souce outing in ad hoc wieless netwoks, in: T. Imielinski, H. Koth (Eds.), Mobile Computing, Vol. 353, Kluwe Academic Publishes, 996, pp. 53 8, chapte 5. [3] G. Resta, P. Santi, An analysis of the node spatial distibution of the andom waypoint model fo Ad Hoc netwoks, in: Poc. ACM Wokshop on Pinciples of Mobile Computing (POMC), Toulouse, Fance,. [4] C. Bettstette, G. Resta, P. Santi, The node distibution of the andom waypoint mobility model fo wieless ad hoc netwoks, IEEE Tans. Mobile Computing (3) (3). [5] W. Navidi, T. Camp, Stationay distibutions fo the andom waypoint mobility model, IEEE Tans. Mobile Computing 3 () (4). [6] C. Bettstette, H. Hatenstein, X. Péez- Costa, Stochastic popeties of the andom waypoint mobility model, ACM/Kluwe Wieless Netwoks (5). [7] E. W. Weisstein, Hypesphee, MathWold A Wolfam Web Resouce. wold.wolfam.com/hypesphee.html [8] E. W. Weisstein, Ball line picking, MathWold A Wolfam Web Resouce. wold.wolfam.com/balllinepicking.html [9] L. A. Santaló, Integal geomety and geometic pobability, Addison-Wesley,