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h : height of the tree structure
d : degree of role group
g
If d equals to d for all i then G equals to d . In general, the roles are included in
not all of role groups. Thus, an unnecessary role group creation can be avoided for
determining the proper value of h . If the roles are not grouped,
s needs to be trans-
d members. From the viewpoint of the reliable delivery, a role specifica-
tion certificate at level l of the tree structure has to be delivered to
l
mitted to
W
(
l
)
=
d
h
l
s needs to be transmitted to d members. Thus, it
receivers. If the roles are grouped,
Wl
()
=
d
has to be delivered to
receivers. Let M( l ) be the frequency of the trans-
mission of a role specification certificate
s in order to be successfully delivered to
all W ( l ) receivers.
The probability that one of these W ( l ) receivers (say w ) will not receive the updated
role specification if it is transmitted once is equal to the probability of packet loss, p ,
for that receiver. Let M be the frequency of role specification transmissions neces-
sary for receiver w to successfully receive the role specification certificate. Since all
the packet loss events for receiver w , including replicated packet and retransmissions,
are mutually independent,
M
is geometrically distributed as in [14]. Thus,
w
m
PM
[
≤=−
m
]
1
p
,
m
1
(1)
w
EM
[
]
=−
/(1)
p
(2)
Equation (1) represents the probability that the role specification certificate is deliv-
ered successfully within m packet transmissions. Equation (2) represents the expected
number of packet transmission. Since lost packet events at different receivers are
independent each other, the probability
w
PM l
[()
m
]
that all the W ( l ) receivers will
receive the packet within m transmissions is as shown in Equation (3).
()
Wl
mWl
()
pM l
[()
≤=
m
]
PM
[
≤=−
m
] 1
p
)
(3)
w
w
=
1
The expected frequency of the role specification packet transmission can be computed
as following:
mWl
1( )
EM l
[
( )]
=
PM l
[
( )
m
]
=
(1
(1
p
)
)
(4)
m
=
1
m
=
1
We can compute F ( l ) numerically using Equation (4) by truncating the summation
when the m th value falls below the threshold.
4 Performance Evaluation
From Equation (1) through (4), we can measure the expected number of role specifi-
cation packet transmission, E [ M ( l )], for the performance comparison. For each given
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