Information Technology Reference
In-Depth Information
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