A. 1.1 General case
L e t’s a n a ly s e th e e ase o f a d istrib u te d d a ta b a se w h e re every n e tw o rk re so u rc e lias a k n o w n fixed
d a ta so u rc e tliat n e v e r m oves. S p ecifically , th e lo c a tio n o f th is d ata so u rce is fix e d w ith in o n e o f
th e n o d es in tlie sy stem (th e h o m e d atab a se). T h is d ata so u rce c o n ta in s th e n e tw o rk reso u rces
c u rre n t lo c a tio n m a k in g it p o ssib le to ro u te q u e rie s to them . I f th e n e tw o rk re s o u rc e m o v es to
a n o th e r lo catio n , th e n etw o rk sig n als b a c k th e lo catio n eluuige to th e h o m e d atab a se.
A sim p lifie d m o d el o f su ch a n etw o rk can be re g a rd e d as a sy stem c o n sistin g o f N n odes. T o each
n o d e w e a tta c h n u sers. T liis m e a n s th at each n o d e h as n p eo p le u sin g it as a h o m e d atab a se. In
a d d itio n to tliis w e c o n s id e r the w o rst e ase scen ario , i.e. w h e n from th e n p e o p le c u rre n tly a tta c h e d
to a g iv e n n o d e, n / N p eo p le h av e th e ir h o m e d a ta b a se at n o d e A',. I f w e a s s u m e p e rio d ic b o u n d a rs
c o n d itio n s , so th a t e v e n n o d e h as th e sam e n u m b e r o f n e ig h b o u rs to all sid es, it is p o ssib le to fin d
th e a v e ra g e n u m b e r o f n o d es th e sig n a llin g h as to trav erse d u e to th e n eed to re tu rn to th e fix ed
d a ta source.
If,
N - to tal n u m b e r o f n o d es in th e sy stem ,
d = lattice d im e n s io n (it in d ic a te s tlie n etw o rk d e g re e o f c o n n e c tiv ity ),
/7„ ni, = n u m b e r o f sites / h o p s aw ay,
D = a v e ra g e d is ta n c e b e tw e e n tw o nodes. T h e n , D is g iv e n by. — % . 2
.
D = ' V /7 / + --- 777,/ ( A .l) ^ M A, j L 2 173,V
F o r d = 2, th e n u m b e r o f sites at a giv en d ista n c e / fo rm s a d ia m o n d o f 4/ sites for / < . F or
f y ^
/ > th e n u m b e r o f sites d e c re a se s w ith in c re a sin g / an d is g iv en by - / j . H en ce the
a v e ra g e d is ta n c e b e tw e e n tw o n o d es can be c a lc u la te d by th e follow in g ex p ressio n .
N
/ J = ± V 4 , = + - y1 4 l , V % -/ / ( A 2)
T h e se c o n d p art o f e q u a tio n (A .2) can be e x p a n d e d as follow s:
Af/2
1 y 4fA'^-,)=T ywT-- yr
2 2 2 ( /I 3) By m a k in g j = i . th e ab o v e e x p re s sio n b eco m e s. 7 = 1 y + - 1/^ A/M r I/A 7 + - (A .4 )± y A ' ) ^ y + A y y _ ± y / _ ± y A L _ ± ÿ , v
A' 4 ^ 2 A' 4 ^ TV 4 ^ 4 TV 4 A/X A A//: 2 2 (A 5) at/2 j _ A 4 _ ^ _ j _ 4 . 2 4 T V ^ A / /V 2 2 Af 2 4 (A 6) A/% 4 4 , 2 N 7=1 (A .7 )S u b stitu tin g e x p re s sio n (A .7) in to e q u a tio n (A .2),
A / % A / %
/=0
(A 8)
D = N ’ (A.9)
Hence, for a two-dimensional grid, the average distance between two nodes (assuming periodic boundary conditions) is proportional to the square root o f the total number o f nodes in the system.
The scalability o f such a system is compromised by the dependency on as it cmi be seen from the plot on figure A. 1. The scalability problem is due to the fact that the signalling has always to return to a fixed home base node, causing the signalling across the network to be proportional to
the average distance between nodes, D, and consequently. .
60 0 600 4 0 0 2 0 0 100 0 0 2 0 0 0 0 0 4 0 0 0 0 0 6 0 0 0 0 0 8 0 0 0 0 0 1 0 0 0000 num ber o r n o d e s. N Figure A .l :( N " ^ ) /2 plot.
A. 1.2 Signalling Load for the GSM Strategy (HLR/VLR)
The GSM strategy for mobility management adopts the approach described in the previous Section. In the GSM strategy, users have a fixed data source, the Home Location Register (HLR). Every' location area crossing has to be reported back to the HLR and call requests need to be forwarded to tlie HLR to retrieve information about the current VLR the called user is registered with. This represents a centralization in the system and causes the signalling across the network to be proportional to the average distance between HLRs and VLRs, i.e. the average number of nodes/switches tlie signalling has to travel from HLRs to VLRs. The signalling load analysis is given below. Only m oves that cross location area boundaries are o f interest.
If.
^HLR = number o f HLRs,
= number o f location areas,
= location area boundary crossing rate,
Npop = total nmnber of people registered in the system.
It is assumed that each HLR lias A users using it as their home database. The average VLR
signalling load due to location area crossings, i.e. the axerage number of VLR updates per unit time, is given by,
(A. 10)
The expression above is due to new registrations as users enter tlie corresponding location area and cancellations from users tliat left that location area. The average number o f updates to the HLR
databases {i.e. HLR incoming signalling load), , is then given by.
11)
^ HIM
The expression shows the signalling load on the HLR themselves and how it can be alleviated by
the introduction o f more HLRs {i.e. )• What this expression does not show is the
amount o f signalling crossing tlie network that eannnot be optimized due to the fact that the protocol has a centralized strategy, that means tlie HLR represents a fixed data source, and the relationship between the user’s unique identifier and the HLR is static. Therefore we need to be able to evaluate the amount of signalling generated due to updates and requests for data retneval having to be reported back to a fixed network point, the HLR. Let’s then calculate the total amount o f signalling across the network due to packets interchanged between HLRs and VLRs. Assuming people move to any o f the location areas with equal probability, it can be said that at steady state
each VLR has people from each HLR. The total signalling load across the system is
then equal to.
'lolal ~ ^HlJt^ HIMX 2 X D - X N x 2 x D - 2 N / / a (A. 12)
Where D is the average distance between HLRs and VLRs, and hence it is the average number of nodes through which the signalling needs to propagate. Tlie factor 2 above is introduced to account for HLR incom ing and outgoing packets. The calculation for D was given in the previous Section, equation (A .9) gives the expression for D for the two-dimensional case. The final expression for
the total signalling load across the network for the two-dimensional case is then given by.
The main dependencies are (number o f location areas), (location area boundary crossing
rate) and N (number o f routing switches between HLRs and VLRs). The scalability problem is to tlie fact tliat all location area crossings have to be reported to the respective HLR, causing the
signalling load across the network to be proportional to and .