On Some Aspects of Random Walks for. Modelling Mobility in a Communication Network* E.R. Barnes

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1 ESL-P-783 Noveiber 1977 O Some Aspects of Radom Walks for Modellig Mobility i a Commuicatio Network* S.K. Leug-Ya-Cheog E.R. Bares This ote examies a two-dimesioal symmetric radom walk model of mobility for termials i a commuicatio etwork. A stochastic process associated with the locatio of a termial is defied. For a certai locatio fidig scheme, the mea time betwee termial trasmissios is derived. We also give a first-order aalysis of the trade-off betwee the amout of locatio related data to be trasmitted per uit time ad the accuracy about the termials' positios. * This work was supported by ARPA uder Grat ONR/N C-1183 t Massachusetts Istitute of Techology I.B.M. Thomas J. Watso Research Ceter

2 -2-1. Itroductio This paper examies some aspects of a stochastic process which might be used to model the mobility of users i a mobile commuicatio etwork. Specifically, we model the motio of a user as a two-dimesioal symmetric radom walk. This might be a appropriate model for a patrol car i a city. We assume that there is a cotroller who eeds to be kept iformed of the locatios of the various users to withi some tolerace. Here we will require that each user otify the cotroller of its exact locatio wheever it hits a square boudary of a certai size cetered o the poit it occupied (Poit C i figure 1) the last time it commuicated its locatio to the cotroller. IC ' - _ User iforms cotroller of its locatio as soo as it hits this square boudary Figure 1 Sice the cotroller kows poit C, the user eed oly trasmit a idex which idicates the poit at which it hits the square boudary. I the followig sectios, we will derive expressio for (1) the expected umber of steps or time a user takes to reach the boudary startig from the ceter C ad (2) the probabilities with which the user hits the differet poit o the boudary ad the correspodig etropy. It is clear

3 -3- that the smaller the size of the square boudary, the better iformed is the cotroller of the locatio of the user. O the other had, oe would surmise that the user will have to trasmit more data back. This trade-off is discussed i sectio IV. II. Expected time betwee user trasmissios. I this sectio, we derive a expressio for the expected time for a two-dimesioal symmetric radom walk startig from the ceter of a square boudary to reach some poit o the boudary. For coveiece, we cosider the square to have legth r ad subdivide it ito 2 small squares as show i figure 2 for the case =4. We assume to be a eve iteger. Tr Tr 3Tr T 7[ 4 4 Figure 2 Let us deote the expected time to hit the boudary startig from the poit (x,y) by tx ' IT 2 T (-l) 'i.y Of course x ad y take o values i the set {0, ' 'TI We are primarily iterested i t 1 /2, 7 1 /2 eve though it will be fairly easy to write expressios for t for arbitrary x ad y. x,y Startig from (x,y), the user ca move to (x+h,y), (x-h,y), (x,y+h) or (x,y-h) where h = f/ ad each motio occurs with equal probability 1/4. The coditioal expected time to hit the boudary assumig the first step takes the '

4 -4- user to (x',y') is t,,+l. This argumet shows that t satisfies the differece equatio t - t + -t + 1 t(la) txy 4 tx-h,y 4 x+h,y 4 tx,y-h 4 x,y+h (la) where 0 < x < IT, 0 <y < 'T with the boudary coditios t = t = t = 0. (lb) O,y r,y x,o x,tr I order to solve this system of equatios, we first cosider the eigevalue problem 4Xt t +t + t + t (3) x,y x-h,y x+h,y x,y-h x,y+h associated with the homogeeous part of (1). It should be observed that as x ad y rage over the set {, 2 ( ) }( 3 ) represets a (-1)2 (-1) matrix eigevalue problem. The eigevalues ad eigevectors of this system are give i [1, p.289]. The eigevalues are X, = (cos ph + cos qh), p,q = 1,2,...,-1 (4) pq 2 ad the correspodig eigevectors are U,1 p,q Ul,-l u2,1 u2,-1 _ -l,-1

5 -5- where u = si prh si qsh, p,q = 1,2,...,-1. This ca be verified by rts settig t y=si px si qy ad otig that 2 si px cos ph = si p(x+h) + si p(x-h). (6) We ow show that the eigevectors defied by (5) are mutually orthogoal, that is, -1 D si prh si qsh si p'rh si q'sh = 0 (7) r,s=l if p' y p or q' # q D = si pr - si pr -. I si qs - si q's - (8) r=-l s=l Thus to show that the eigevectors are orthogoal it suffices to show that if p' $ p, -1 r=l si pr - si p'r -= 0. (9) -1-1 I sipr- si p'r- ff= I [cos(p-p ')r - cos(p+p')r - (10) rl2 From [2, p.78, Eq. (418)], -1 r=l cos r(p-p') - = cos (p-p') si cosec (p-p) (11) =cos (p-p') [ si (p-p') cos (p-p') - cos (p-p') si (p-p') 2 cosec (p-p') T. 2 V2' (12) =-cos (p-p') -. 2

6 -6- I (13) we have used the fact that p' $ p. Similarly, -1 r=l cos r(p+p') - - cos 2 (p+p) (14) Fially, substitutig (13) ad (14) i (10) yields -1 r=l si pr si p'- = [ -cos (p-p') + cos (p+p') ] (15) [ {-1- cos(p-p')t} + {1 + cos(p+p')7} ] (16) = [ - 2 si pf si p'7] = 0 (17) It is ow fairly easy to obtai the solutio of (1) i terms of the eigevectors U. Let T deote the (-l)2 dimesioal vector P,q th,h th,2h T = th,(-l)h t2h,h t2h,(-l)h t(l)h(-l)h Sice the U p,q are liearly idepedet, we ca write -i U (18) p,q=l pq pq

7 -7- for some costats ~ to be determied. Similarly we ca write the (-l)2 P,q vector 1, all of whose compoets are oes, as -l *= EC ap U. (19) pq=l pq p,q Takig the dot product of both sides of (19) with U.. ad recallig the fact that the eigevectors U p,q are orthogoal, we obtai (20) i j = i,jui,j2 (20) -1 where IUi,jl 2 i si irh si jsh. (21) r,s=l It ca be verified [2, p.82] that Ui jl 2 = 2 /4. Also, U..= I si irh si jsh (22) r,s=l 1 i j co j ~ = 1 [1-(-1) i ][1-(-l) i] cot it cot 2f (23) I (23) we have used the fact [2, p. 7 8 ] that -i Y si irh = [1-(-1) i cot (24) 2 2 r=l Thus from (20) we ca write [--i j ictr2 o j725 [1-(-1) J[1-(-1)J] cot - 2 cot 2 aij= _2 (25)

8 -8- Now, if we deote the matrix associated with the homogeeous part of (1) by A, the solvig (1) is equivalet to solvig AT = IT - 1, i.e. BT = -1 (26) 2 2 where B = A - I ad I is the (-l) X(-l) idetity matrix. Sice BU = AU - U = ( -1)U (27) p,q P,q P,q P,q P,q the eigevalues of B are (X. -1) ad its eigevectors are U. From (26) pq p,q ad (18) we obtai Therefore -l > -l BT = I p (X -1)U = -1 = -_ a U. (28) p,q=l,q,q p,q=l pq h -1 P,q _Pq (29) -2[1-(-1) P ] [1-(-l)q] cot pr cot q r 2 2 (30) (cos p + cos q 7 - ) Fially, substitutig (30) ito (18) yields si pt si gr q si p I si q27 si -l ~-2 1 -(-1) P ] [ 1 -)- 1 )q ] cot p i cot q T si q(-l)7 p,q=l 2 (cos + cos -2) 2 q2 si p2 si si p( 1)T q (-1) sr -31)

9 -9- Equatio (31) gives us a expressio for the expected umber of steps a symmetric two-dimesioal radom walk takes to hit a square boudary of size startig from some poit iside the boudary. Let us deote by t the expected time t/2,/2 to hit the boudary startig from the ceter. The - -l -8 cot P cot q- si P si ( t I (32) pq=l1 (cos P! + cos - 2) Figure 3 shows t for values of ragig from 2 to 20. III. Probabilities of boudary poits Let p deote the probability that the radom walk will first hit xy * * the square boudary at poit (x,y ) give that it starts from the poit (x,y). I this sectio we will derive a expressio for P/2, /2 Usig a argumet similar to that leadig to (1), we see that Px,y satisfies the differece equatio * 1 * 1 * 1 * 1 * = (33a) Px,y 4 Px-h,y 4 Px+h,y 4 Px,y-h 4 Px,y+h where 0 < x < 7, 0 < y < 7 with the boudary coditios * * * * * P 0,y P T~ r~y P, x,0 Px x,ti = 0 except that Px*,y* = (33b)

10 -10- For otatioal coveiece, let us deote the poit iside the boudary immediately adjacet to (x*, y*) by {x', y'). Also let P deote the (-l) 2 dimesioal vector h,h Ph,2h P = Ph, (-l)h P2h,h (-l)h,(-l)h The we ca write (33) as AP = IP - - e (34) where A is the same matrix as i the previous sectio ad e is the (-l) dimesioal vector with 1 i the positio correspodig to x',y' ad O's elsewhere. Because of the symmetry, there is o loss of geerality i assumig that x = X ad < y* < T I this case, x' = (-l), y' = y* 2 - h ad the o-zero etry i ex,, is the [(-2)(-l) + Z]-th etry where = y Proceedig i a maer completely aalogous to the method of the previous sectio, ad settig

11 -l 4 -x', Y1 ap U (35) e x,, y ' q pq P,q we fid that si [i(-l)h] si jkh (36) i, j 2 Also lettig -l P = I U (37) p,q=l pq p,q we fid that B = p,q = _ 2 si[p(-l)h] si qgh (38) p -1 (cos ph + cos qh -2) P,q Thus, si pt si q ' q si p si q2 -l 2 si p(-l) - si qq. P = si si q(-l) 2 7r T p,q=l [2 - (cos - + cos q s si P21T si si P i q(-l)si (39) From (39) we ca deduce that -l 2 si p(-l) T si q si si P/2,ir/2 pcq~l 2E2 (cos p,q=l 2 p [2 -(cos + cos -) I

12 -12- Usig (40), the etropy H() i bits associated with the radom variable idicatig the boudary poit the radom walk first hits was plotted as a fuctio of i figure 4. For a square boudary of size, there are 4(-1) boudary poits which could first be hit. If we assume that all these poits are equi-probable, the resultig etropy H () = 2 + log 2 (-1) is clearly a upperboud o H(). It was foud umerically that for eve positive itegers less tha or equal to 20, the differece betwee H () max ad H() was mootomically icreasig but did ot exceed The maximum percetage error was below 3%. IV. Discussio I order to give a rough idicatio of the trade-off betwee the size,, of the square ad the amout of iformatio to be trasmitted back to the cotroller every time uit, a plot of H()/t* is show i figure 5. From a practical viewpoit, the trade-off betwee [2 + log 2 (-l)]/t* ad should be examied. Fially, we ote that the symmetric model assumed here may be refied i may ways. For example, a more appropriate model of a patrol car might exclude the possibility of the car makig a U-tur. Ufortuately, such models seem to be hard to aalyze. t Note that we are eglectig the fact that there is depedece betwee the time of hittig the boudary ad the boudary poit which is first hit. A more accurate aalysis should take this ito accout.

13 t ate agaist boudar size. Figure 5. Data rate agaist boudary size.

14 t 80 t O so Figure 3. Expected time to hit boudary agaist size of boudary.

15 t H() _ 1_ I - 0 i I I I Figure 4. H() agaist boudary size.

16 -16- REFERENCES [1]. L. Fox, Numerical Solutio of Ordiary ad Pastrial Differetial Equatios, Pergamo Press, [2]. L.B.W. Jolley, Summatio of Series, Dover, 1961.

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?

(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m? MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle