The sign alling load for each node according to its hierarcliical level k is given below. Only moves
tliat represent location area crossings are taken into account.
A: = 0 signalling com ing from users:
signalling com ing from parent:
total signalling: ^la - 1 n f N,„ At Af (A. 14) (A. 15) (A. 16)
0 < & < r o o t => signalling com ing from children: a*
signalling com ing from parent:
total signalling: = a' A:,. Af (A^/o la J nf_ At CA17) (A. 18) (A. 19) At
k - r o o t signalling com ing from children: = {n,^ ' ) —
(total signalling)
(A .20)
(A .21)
L et’s then analyse those equations. Equation (A. 16) shows that the total signalling load in each
local exchange is dependent on the location area boundary crossing rate. The dependency on
is due to the adopted population model (that assumes the users have equal probability o f m oving to
any o f the location areas). However, as a constant value. Note that
equations (A. 16) and (A. 10) are the same as VLRs in the GSM strategy and the local exchanges in the R EM US strategy process the same amount o f signalling, i.e. the signalling for the location area they are responsible for. As the system is expanded by the addition o f more local exchanges, if the number o f users being served by each local exchange is kept constant then the signalling load on the local exchange is not affected.
For 0 < & < ro o t and k = r o o t, the total signalling load depends on the total number o f location
areas, Nia, and on the number o f location areas under the node’s domain, a * . Tliis is due to the adopted random population model. The objective o f REM US hierarchical structure is to explore tlie locality o f the requests so that high level nodes are rarely involved in the transactions, this objective cannot be achieved for the random population model.
Equal ion (A. 19) shows that the signalling for 0 < k <root is proportional to the number o f local
exchanges under the node’s domain. The main dependencies are: k (hierarciucal level), a
(number o f children per node) and (number o f location areas). For a fixed number o f location
areas and a constant number o f children per node, the signalling increases with the hierarchical level because higher level nodes are responsible for a liigher number o f local exchanges (see figure A .2). An important point to note is that the signalling load on a given level tends to the constmU
(maximum) value 2a * a s N ,^ ( i.e . the proportion o f requests that cross the boundarv o f the node’s domain becom es dominant). Hence, as the network is expmided the signalling load on a given parent node does not change if the number o f location areas under its domain is kept constant. This characteristic makes the system highly scaleable.
^ 1.5 XI k ( h i e r a r c h i c a l le v e l) - s i g n a l l i n g lo a d - u p p e r lim it Figure A .2: V ariation o f a cc o rd in g to k, fo r 0 < k < ro o t .
In equation (A.21), for k = r o o t , the signalling is proportional to the number o f location
areas, N ^ . As previously discussed, this is due to the adopted population model where random
user movement is assumed. The objective is to build the network topology so that the root o f the network is rarely involved in the requests.
In tlie worst case scenario, in which people move randomly with equal probability o f the destination being any o f the location areas, the locality o f the transactions cannot be explored and REMUS strategy cannot be optimized. Nevertheless, it is a scaleable system, in that tlie system can
be expanded without increasing the signalling load at all nodes (for 0 < & < root there is an upper limit for the signalling load) except the root. The root node is particularly affected because, for this population model, it has to deal with all inter location area transactions that are not within the domains o f its cliildren. Tliis causes the signalling load across the system to become dependent on the number o f location areas. It is important to emphasise that tlie dependency o f the signalling load across the network on Nia is not due to the system ’s strategy but due to the adopted population model. Any strategy would depend on the number o f location areas when exposed to this population model.
A . 2 .2 C a s e 2: D i f f u s i o n - b a s e d P o p u l a t i o n M o d e l
A.2.2.1 Binary Tree
Again \vc consider n users per local exchange where a fraction / o f them move out o f their location area in a single time interval. This means that the signalling information that must be dealt with by the parent enters the local exchange from below at a rate nf. Let’s consider a binaiv tree mapped to a one-dimensional connin' IxN, in which n f 2 people move to each of the two neighbouring location areas. The first hierarchical layer above the local exchanges receives 2 n f signalling requests from its children. One half o f this information is to be passed back to the children and the remainder passed up to parent. This means the next parent also recci\ es 2 n f signalling requests from its children and this is true of any node at any layer o f the network. O f the n f signalling requests that are passed down to the children the load is equally split between the two. Tlierefore the child receives a load o f nf/2 from its parent. Because o f the hierarchical nature o f the network, this extrapolates to a load o f nf/4 from the grand parent and a load o f nf/H from the great grand parent. This procedure is repeated till the root o f the tree is reached. From this we can calculate the load at the k‘‘‘ le\ el o f the tree, that is obtained from all nodes that have a parent, as
{ r o o t - k ) j
I n f + n f ' ^ — , { ) < k < r o o t (A. 22)
1=1 2
The root is a special case in which it has no parent and therefore passes all the signalling information it receives to its children. Tliis means that the child receives n f signalling requests rather then np2. In order to include this we have to consider what fraction o f the e.xtra nf/2
signalling requests are present at a given level k. This is simply
n f
cÉ -rr (A .23)
Combining equations (A .22) and (A .23) w e obtain the signalling load at level k as
( r o o l - k ) J = 2/7/ + + 77/ ^ . 0 < k < root ( A. 24 ) ^ 1=1 But. ( r o o l - k ) , j i r o o t k ) n f ^ — = /7/ 2 (A .25) (Proof: l - ~ X < 1 180
[ r o o t- k ) ^ 2 ^ { r o o l-k ) ^
^
^ 2' 2^(rool-k)/=1
4 ) ^ 1 ^ 2 I 1
/ J 2 ' / 2 ^ { r o o t - k ) f ^ i r o o t - k ) ^ ^ { r o o l - k )
Equation (A. 24) then becom es
x,^ = 2 n f + n f = 3 n f , 0 < k < r o o t (A .26)
For the local exchanges hierarchical level (k = 0), the signalling load is given by
y . f { r o o l - k ) .
For the root, as it does not have a parent, the only contribution to the signalhng load com es from its children,
^ r o o , = ^ ^ f ^ k = root (A. 28)
N ote that the signalling for high level nodes ( 0 < & < root) is independent o f the hierarchical level
k. This result is due to the fact that the network topology (binary tree) is mapped onto a on e dim ensional country. Hence the number o f location area borders external to the node’s domain is the same for all high level nodes although their domain area increases for higher A;. If a two- dim ensional country is used, this is no longer the case, because as tlie node’s domain increases w hen m oving to higher layers its external borders are also expanded and hence the signalling becom es dependent on hierarchical layer k. This w ill be seen in the next subsection in w hich the general case for a two-dim ensional country is discussed.
N evertheless, the above calculations w ere able to demonstrate REM US design strategy, in w hich the signalling is distributed throughout the network avoiding the creation o f bottlenecks. This sim ple calculation has shown that the signalling load in the REM US network is kept constant even if the system is expanded indefinitely.
A. 2.2.2 G eneral C ase
L et’s try then to generalize the analysis in the previous subsection for any uniform n-nary tree. The result can be appUed to countries o f any dim ension because its based on tlie percentage o f the n ode’s domain borders that represents external borders at its parent’s domain area. Again we
sin gle time interval. The signalling load for tlie local exchanges is then given by the same expression as before as it is independent o f the adopted population model,
= 2 n f, A - 0 (A .29)
T his is again due to new users entering tlie corresponding location area and cancellations due to users that m oved out. This is based on tlie fact tliat the system is at steady state and hence the number o f users entering the location area is equal to the number o f users leaving it. The signalling load equation for a node at hierarchical level k is given below.
= "((/mt )k + )* ), 0 < & < ro o t (A. 30)
W here f„ t represents the fraction o f the population attached to one o f the location areas under the n od e’s domain that m oves without crossing its boundary - i.e. m ovem ent internal to the node’s domain. And fout represents the fraction o f the population that leaves the node’s domain.
If,
bi„t = average fraction o f location area border internal to node’s domain,
bext = average fraction o f location area border external to node’s domain,
a = number o f children per node.
Then,
(A .31)
i f
out) k -^ext^{.fout)k-\ (A. 3 2)But,
f (A .33)
Therefore,
(A .34)
Equation (A .30) then becom es
For the root, all m ovem ent is internal to its domain since it encom passes all location areas. Hence
L ,- , =
^
^ = ^00/
(A.36)
As it can be seen from equations (A.29), (A .35) and (A .36), the signalling load main dependencies are: the number o f children per parent node, a {i.e. network topology) and the fraction o f location area borders internal and external to node’s domain, and bext {i e. shape and size o f location areas and population model). Therefore, if those parameters, together with the number o f users per location area and their movem ent rate, are kept constant, the network can be expanded indefinitely w ithout affecting the network signalling load.
A .2 .2 .3 Q u atern ary T ree
If w e consider a quaternary tree and square location areas, then
^mt
a = 4
A nd the general equations (A .35) and (A .36) becom e
x ^ = 3 n /2 * 0 < k < r o o t (A .37)
t = (Au38)
A s in the results for the binary tree, the signalling is constant and does not change as the network is expanded (there is no dependency on the number o f local exchanges in tlie system). It is important to note, though, that the signalling load increases with hierarchical level k. Consequently, as the network is expanded, the root node is shifted to higher hierarchical levels, and hence its load increases. This is due to the fact that the external borders o f a node’s domain becom e larger with
hierarchical level k. The population model assum es the crossing rate is proportional to the border length, causing the switch throughput dependency on the hierarchical level. However, as discussed in Chapter 3, the objective is to design the system so that it can dynamically adapt its topology to the usage pattern. This causes the nodes domain to be established in such a way that most transactions are confined to it, eliminating the dependency o f the switch tliroughput on the hierarchical level.