SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

59
Pr{≤ 1 defective} = a + b + c +13d
= p 12
11(π 1 p 2 11 + π 2 p 21 p 11 + π 1 p 11 p 12
+13π 1 p 12 p 21 )
= p 12
11(π 1 (p 2 11 + p 11 p 12 +13p 12 p 21 )+
π 2 p 21 p 11 )
For (6.11) this ≈ 0.9169
For 16(a) this ≈ 0.1299
Notice that the Markov-chain model gives a lower probability of accepting a good
process, and a higher probability of accepting a bad process.
17. (a) M = {1, 2, 3, 4} = {A, B, C, D} is the location of the AGV
n represents the number of trips
(b) p (5)
24 ≈ 0.1934
(c) π 4 =3/16 = 0.1875
⎛
P = ⎜
⎝
0 1/2 1/2 0
1/3 0 1/3 1/3
1/3 1/3 0 1/3
1/3 1/3 1/3 0
⎞
⎟
⎠
18. (a)
(2, 000, 000)p 12 +(2, 000, 000)p 22 +(2, 000, 000)p 32 =2, 107, 800
(b) If “today” is 1994, then 16 years have passed
(2, 000, 000)(p (16)
11 + p (16)
21 + p (16)
31 ) ≈ 2, 019, 566
19. (a) M = {0, 1, 2,...,k} is the number of prisoners that remain in the prison
n is the number of months
Let
( i
b(i, l) = p
l)
l (1 − p) i−l
which is the probability l prisoners are paroled if there are i in prison.
Then P has the following form