SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

109
= (1− ρ 1 )(1 − ρ 2 )
∞∑
i=0
ρ i 1
1
1 − ρ 2
1 1
= (1− ρ 1 )(1 − ρ 2 )
1 − ρ 1 1 − ρ 2
= 1
23. Let B be a Bernoulli random variable that takes the value 1 with probability r. The
event {B =1} indicates that a product fails inspection.
The only change occurs in system event e 3 .
e 3 ()
(complete inspection)
S 2,n+1 ← S 2,n − 1
(one fewer product at inspection)
if {S 2,n+1 > 0} then
(if another product then start)
C 3 ← T n+1 + FZ −1 (random()) (set clock for completion)
endif
B ← FB
−1 (random())
if {B =1} then
(product fails inspection)
S 1,n+1 ← S 1,n + 1
(one more product at repair)
if {S 1,n+1 =1} then
(if only one product then start)
C 2 ← T n+1 + FX −1 (random()) (set clock for completion)
endif
endif
24. The balance equations are
rate in = rate out
(1 − r)µ (2) π (0,1) = λπ (0,0)
λπ (i−1,0) + rµ (2) π (i−1,1) +(1− r)µ (2) π (i,1) = (λ + µ (1) )π (i,0) ,i>0
µ (1) π (1,j−1) +(1− r)µ (2) π (0,j+1) = (λ + µ (2) )π (0,j) ,j>0
λπ (i−1,j) + µ (1) π (i+1,j−1) + rµ (2) π (i−1,j+1) +(1− r)µ (2) π (i,j+1) = δπ (i,j) ,i,j>0
where δ = λ + µ (1) + µ (2) .
The steady-state probabilities are