SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

102 CHAPTER 8. QUEUEING PROCESSES
(c)
(d)
(e)
(f)
20
∑
l q = (j − 3)π j ≈ 6.20 jobs waiting
j=4
3π 0 +2π 1 + π 2 +0(π 3 + π 4 + ···+ π 20 ) ≈ 0.055 idle computers
π 0 ≈ 0.02 or 2% of the time
(d)/3 ≈ 0.018 or 1.8% of the time
13. No answer provided.
14. (a) For the M/M/1/c, letρ = λ/µ.
Therefore,
d j =
{
(λ/µ) j , j =0, 1, 2,...,c
0, j = c +1,c+2,...
π 0 =
( c∑
) −1
d i =
i=0
1
∑ ci=0
ρ i
π j = π 0 d j = π 0 ρ j ,j=0, 1,...,c
(b) For the M/M/c/c
⎧
⎨ (λ/µ) j
, j =0, 1,...,c
j!
d j =
⎩
0, j = c +1,c+2,...
Therefore,
π 0 =
( c ∑
i=0
d i
) −1
=
1
∑ ci=0
(λ/µ) i /i!
(λ/µ) j
π j = π 0 d j = π 0 , j =0, 1,...,c
j!