In Chapter 4, Sections 4 .3 .1.1 and 4.3.1.2, the different procedures for database restoration after a network reconfiguration are discussed. It was pointed out the possibility o f simultaneous error
detection in case inconsistencies are present in both parts o f the data path being restored, i.e. below the reconnected child and between reconnected child and old parent, and the consequent potential for clash between the two recovery m echanism s that are initiated. For the case o f restoration on demand, there is potential for clash between the flagged location change request travelling from reconnected child to old parent and call request travelling downwards. For the case o f immediate restoration, the potential for clash is between the flagged location change request travelling from reconnected child to old parent and the stamp packet travelling downwards. It is important then to calculate the probability o f such problem arising to check if it requires the introduction o f special m echanism s in the protocol. The follow in g notation w ill be used in the calculations;
= total number o f errors present in the network,
= number o f errors per node at hierarchical level k,
= number o f database entries at a node at hierarchical level k,
Ic = reconnected ch ild ’s hierarchical level,
lop = old parent node’s hierarchical level,
I root = root’s hierarchical level,
^HNk^ number o f high level (parent) nodes under the domain o f a node at hierarchical level k,
a = number o f child nodes per parent node (a uniform network tree is assumed, in w hich all parent nodes have a children).
The probability o f finding an error in the data path below the reconnected child is given by,
P a
=
P\ P \ ) P l P \ y ^ ~ P l ) p ^ P \ ) f ^ ~ P l ) - ^ ~ Pr , - \ ) Pn( ^ 0
Where p\, P 2 , P s , . . . P n are the probability o f finding an error at tlie first, second, third, .., n*^ hierarchical level below the reconnected child. This is a conditional probability in w hich the probability o f finding an error at the first hierarchical level is p\, and given that no error was found
until tlie n* level (local exchange level) is reached. Equation (C. 1) can be written in a more concise form ; n fc -i P a ^ P x k = l V (C.2)
In order to sim plify the above expression, let’s now assum e that the probability o f finding an error
at a node at any liierarchical level is the same - i.e. = g , for all k.
k = l 7=1 k - l V ^ = 2 But, x - x (C.3) = ^ . * < 1 ( C 4 ) 1=2 ^ ^ Therefore, P , = l - ( l - g ) " (C .5)
W here n is the number o f hierarchical levels below the reconnected child, hence n = 1^ and equation (C.5) can be rewritten,
p . = I - ( l - g ) ' - (C.6)
A s it can be seen from the above expression, as - > o o , ( l- g ) '" - ^ 0 (since ( l - g ) < l ) and
hence ^ \ . So, as the data path becom es longer, the probability o f detecting an error tends to
1. That means that the higher the reconnected ch ild ’s liierarchical level, the higher the probability
o f finding an error in the data path below it. H owever, the lower the probability g o f finding an error at any one node the more slow ly probability pa w ill tend to 1.
N ow w e need to calculate the probability o f finding an error in the data patli from reconnected child to old parent node. Because the packet is emitted by the reconnected child, this node w ill not detect an error. The data path from new parent node till the node that is the root o f the sub-tree encom passing both new and old parents (this node w ill be called com m on parent from now on)
does not contain the information. The flagged location change request is building this part o f the data path, hence, no error w ill be detected tliere. From the com m on parent to the old parent the information should be present and the flagged location change request has the task o f deleting it. It
is in this stretch o f the data patli that an error can be found. There are two cases to consider depending on the node tlie isolated child reconnects to, if it is under the old parent’s domain or not. If the isolated child reconnects to a node under the old parent’s domain, tlien the only node tliat can
be found containing an inconsistency is the old parent itself, as it is tlie only node in tlie flagged packet patli tliat should contain the required information. We need then to calculate the probability
Pb o f the isolated child node reconnecting to a node under tlie old parent’s domain and multiply it
by the probability o f finding an error at the old parent node = g ) - The probability pb is
given by the ratio o f prospective new parents imder old parent’s domain (excluding the isolated child and its domain) to the total number o f prospective new parent’s in the system (again excluding the isolated child, its domain and the old parent itself). A ssum ing that every parent node (also referred to as high level node) is a prospective new parent, then pb is given by,
^ + 0 _ ^HNl^p + l )
^ H N t o t a l + 1 “ fiN lc + ^ ^ H N l o t a l 0
The parameters and refer to the number o f high level nodes under the domain o f the
old parent and the domain o f the reconnected child, respectively. The number, o f high level
(parent) nodes under the domain o f a node at hierarchical level k is given by the follow ing expression,
*-i a'' - a
~ ' k > 2 (C.8)
1=1
Equation (C.7) then becom es.
In the second case, in which the isolated child node reconnects to a node outside the old parent’s domain, the probability o f detecting an error w ill depend on the distance between old parent and com m on parent nodes as this is the stretch o f data path that contains the required information. We have then to calculate tlie probability pc o f the com m on parent node being at a particular hierarchical level. If j is the distance between the old parent and the com m on parent hierarchical levels, then pc is given by.
Tlie above expression has to be multiplied by the probability pd o f finding an error in tlie data patli betw een old parent and com m on parent. As for tlie case o f the data patli below the reconnected child, this probability is given by equation (C.5), where n is now tlie distance between old parent and com m on parent, i.e. n = j.
( C . l l )
The probability, p ’, o f finding an error in the data patli between reconnected cliild and old parent is
then equal to.
P ' ^ P b S + ' ^ { p c j P d j ) (C .1 2 )
j=i
A ssum ing that the error detection in the path below the reconnected child and in the path between reconnected child and old parent represent independent events, then the final expression for the probability o f simultaneous error detection is given by.
P = P a ^ P (C .1 3 )
P b g + Y . i P c j P d j )
7=1
(C.14)
(C .1 5 )
Expanding the summation in the above equation w e obtain.
a — I z ( l - g ) - l
(C .1 6 )
Therefore, the main dependencies are:
g = —^ = fraction o f database entries that are corrupted.
Ic = reconnected ch ild ’s hierarchical level,
lop = old parent’s hierarchical level,
I root ^ root’s hierarchical level,