t,
Figure D. 1 : T im e d om ain rep resentation o f contention p ro b lem described in Section 4.3 .1 .3 .
The sym bols in figure D. 1 are explained below:
• event 1: ordinary location change request is issued at a local exchange outside the
recoimected child domain due to a re-registration by a user that was previously registered at a
local exchange inside the reconnected child domain,
• event 2: flagged location change request issued by reconnected child node,
• C : tim e at w hich a management re-configuration occurs,
• f, : tim e at w hich event 1 occurs,
• event 1 has to occur at least m sec before event 2 {i.e. event 2 cannot occur within
• event 2 has to occur within f^ .
The values o f At^ and Af^ depend on each specific situation. In the exam ples o f figures 4.5 and
4.6, Af^ represents the time it takes for the location change request to travel from the local
exchange where it was generated to the new parent node m inus the tim e it takes for the flagged
location change request to travel from reconnected child to new parent node (/. e. so that location
change request arrives at new parent node before flagged packet). And Ar^ is the time it takes for
the location change request to travel from new parent node to reconnected child, goin g through
the direct link from old-parent node (/. e. so that flagged packet is issued before location change
We want to calculate the probability o f the contention problem occurring after a management re
configuration. The situation can be summarized as follows: given that a m anagement re
configuration occurs at , w e want to find the probability that event 1 takes place at least
m sec before event 2 and that event 2 occurs w ithin . Therefore w e need to calculate the rate at
w hich event 1 occurs, w e also need to calculate the probability o f event 2 not occurring w ithin
Af^ and occurring w ithin Af^.
The probability that event 2 occurs w ithin a certain tim e interval Ai = r, - is given by:
(D.l)
where w (i) is the number o f outdated database entries at reconnected child node at tim e i . If r
is the rate at w hich the reconnected child node issues flagged location change requests, then.
m
m .
Hence, the probability that event 2 does not occur w ithin Ai^ is given by:
and the probability that event 2 occurs w ithin A t , is given by:
rAi. / \
P i ~ 7TT ’ (D .5)
4 ^ a )
where - r ( t ^ - r A t ^ .
Let's now calculate the rate at w hich event 1 occurs. T he symbols that w ill be used in the
calculation are the following:
• N = number o f country cells,
• n = Yj^ = c e ll’s population at steady state,
• - fraction o f c e ll’s population that m oves internally and fraction that m oves to other
cells, respectively.
Hence,
• w hich people m ove w ithin their cell,
• w hich people m ove to other cells.
The values o f and / ^ , depend on the adopted population model. If it is assumed that people
can m ove to any o f the N country cells with equal probability, then:
J o u , J ^
where / is the total fraction o f the c e ll’s population that m oves (either internally or to other
cells).
If, however, the population model is based on the difihision movem ent, in w hich people are
allow ed to m ove only to one o f the c e ll’s four nearest neighbours, then and can be
defined as input parameters. This is the current model adopted by the sim ulation, the full
description is given in Section 6.1.1. The first population model, in w hich people m ove in hops to
distant cells, represents the worst case scenario, the second m odel, based on the diffusion
m ovem ent, is a more realistic one, where people m ove most locally. The calculation w ill be carried out for the worst case scenario so that it gives the m aximum probability o f the contention
problem occurring.
At steady state, the rate at w hich people arrive at a given cell is:
= n k (D .6)
At
where
k
is defined asAt ■
The probability that a user is listed as one o f the outdated entries in the reconnected child node's
database is Hence, the rate at w hich people that are listed in the reconnected child node's
m(t ) m(t )k
= , N ^ O (D .7)
c N
If h is the number o f local exchanges that are not inside the reconnected child node domain, then
the rate at w hich people (that are listed in reconnected child's database as outdated entries) arrive
at a cell outside the reconnected child domain is given by;
( t )hk
^ ’ N ^ O (D .8)
N
The expression above has to be placed within the problem tim e frame, hence w e are interested in
the number o f outdated entries at reconnected child node's database at tim e :
m ( t , ) h k
T, N ^ O (D .9)
where m { t ^ = - A/^ ) . If = 0 and if w e allow to slide along the tim e axis then:
= t ^ + A t ^ : . A t ^ + (D.IO)
and the expression for A, becomes:
A ,W = - ( D . l l )
Therefore, w e have:
• the probability that event 2 does not occur w ithin At^
r ( t + A t A m
Pi {t ) = 1 , 0 < t <---At^ , m ^ ^ 0 (D. 12)
r
the probability that event 2 occurs w ithin At^ :
th e rate at w h ic h e v e n t 1 o c c u r s a c c o r d in g to th e p r o b le m tim e fram e:
- rt^hk
N ( D M )
As p, and are independent from each other and as event 1 is independent from event 2, the
three expressions can be m ultiplied together in order to obtain the rate at which the contention
problem occurs: (D .15) = M r. {m^ - r t ) h k N (D .16) - r ( t + Ar^ )y V
0 < r < ^ - ( A r ^ + A r , ) , A ^ O, m^ ^ 0
If w e now integrate (r) over a tim e interval A r , w e w ill obtain the number o f contentions that
m ight occur w ithin this AT :
I ^2 {t)dt
(DAI)
1- ^(r + A r J rAr. m„ - r X (m„ - r t ) h k V N dt (D .18) h k 5(Ar) = — Ar^ ( A r )'+ M r , o < A r < ^ - ( A r , + A r J , A r ^ o , m , (D .19)Equation (D. 19) is a polynom ial o f the second order:
s - a t + b t + c where, h k • a = Ar. A: 2m. , _ h k