SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

87
or
λµ 00 = 1 +λµ 10
λµ 01 = λµ 11
(λ + τ)µ 10 = τµ 00
(λ + τ)µ 10 = 1 +τµ 01
−1 = −λµ 00 +λµ 10
0= −λµ 01 λµ 11
0= τµ 00 −(λ + τ)µ 10
−1 = τµ 01 −(λ + τ)µ 11
The solution is
[ ]
µ00 µ 01
=
µ 10 µ 11
[ λ+τ
λ
τ
2 λ
1
1
λ 2 λ
]
Therefore, E[TTF]= λ+τ
λ 2
+ 1 λ = 2λ+τ
λ 2 .
System A has the largest E[TTF].
System E[TTF]
A 111,333.33
B 105,473.68
C 100,200.00
D 86,358.28
16. (a) Let H i represent the holding time on a visit to state i.
Let F N|L=i represent the cdf implied by the ith row of P, the transition matrix of
the embedded Markov chain.
1. n ← 0
T 0 ← 0
S 0 ← s
2. i ← S n
3. T n+1 ← T n + FH −1
i
(random())
4. S n+1 ← S n + F −1
N|L=i (random())
5. n ← n +1
goto 2
(b) Replace step 3 with
3. T n+1 ← T n +E[H i ]
(c) No clocks need to be maintained. When we use 16(b), we only need to generate
Markov-chain transitions and we eliminate variability due to the holding times.