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λ
0
λ
0
λ
1
λ
1
λ
2
λ
2
0,2,0
0,2,0
1,2,0
1,2,0
2,2,0
2,2,0
3,2,0
3,2,0
µ
1
µ
1
µ
2
µ
2
2
2
ω
ω
2
2
ω
ω
λ
1
λ
1
λ
2
λ
2
µ
1
µ
1
1,2,1
1,2,1
2,2,1
2,2,1
3,2,1
3,2,1
µ
2
µ
2
2
2
ω
ω
2
2
ω
ω
λ
1
λ
1
λ
2
λ
2
λ
3
λ
3
µ
1
µ
1
1,3,0
1,3,0
2,3,0
2,3,0
3,3,0
3,3,0
4,3,0
4,3,0
µ
3
µ
3
µ
2
µ
2
2
2
ω
ω
2
ω
ω
λ
2
λ
2
λ
3
λ
3
µ
2
µ
2
2,3,1
2,3,1
3,3,1
3,3,1
4,3,1
4,3,1
µ
3
µ
3
µ
2
µ
2
2
2
ω
ω
2
2
ω
ω
λ
2
λ
2
λ
3
λ
3
λ
4
λ
4
2,4,0
2,4,0
3,4,0
3,4,0
4,4,0
4,4,0
5,4,0
5,4,0
µ
3
µ
3
µ
4
µ
4
: state with
h
servers failed,
k
level of redundancy and detection process
d
h
,
k,d
h
,
k,d
λ
h
λ
h
: aggregate server failure rate with
h
server failed
µ
h
: aggregate repair rate with
h
server failed
ω
: mean detection rate
Figure 13.4
A Markov chain for PRT with
K
min
=
2,
K
max
=
4 and Erlang-2 detection time
it analytically to obtain the first passage time directly. For larger values, the resulting solutions
are very complex and again we make use of Maple to obtain numerical solutions.
For the sake of verification, we have also developed a simulation program to measure the
systemMTTF. For large values of
K
min
and
K
max
, the simulation time required is extraordinary
long. But for smaller values of
K
min
and
K
max
the simulation time is manageable and the
simulation results do confirm the correctness of the numerical results obtained from Maple.
13.5 Performance Evaluation
Using the reliability models in Sections 13.4 and 13.5, we answer in this section the question
of how much bandwidth overhead can be saved by PRT under the constraint that the system
reliability is at least as good as FEC. Table 13.1 summarizes the system parameters used in
computing the following numerical results.
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