SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

100 CHAPTER 8. QUEUEING PROCESSES
(e) Let the reneging rate be β =1/5 call/minute = 12 calls/hour for customers on
hold.
with µ = 20.
µ i =
{
iµ, i =1, 2
2µ +(i − 2)β, i =3, 4,...
(f) l q = ∑ ∞
j=3 (j − 2)π j
Therefore, we need the π j for j ≥ 3.
⎧
⎨
d j =
⎩
(λ/µ) j
j!
, j =0, 1, 2
2µ 2 ∏ j
i=3
λ j
(2µ+(i−2)β),
j =3, 4,...
1
π 0 = ∑ ∞j=0
dj ≈ 1
∑ 20
j=0 dj ≈ 1
2.77 ≈ 0.36
since ∑ n
j=0 d j does not change in the second decimal place after n ≥ 20.
Therefore
20
∑
l q ≈ (j − 2)π j ≈ 0.137 calls on hold
j=3
11. (a) M = {0, 1, 2,...,k+ m} is the number of users connected or in the wait queue.
λ i = λ, i =0, 1,...
{
iµ, i =1, 2,...,k
µ i =
kµ +(i − k)γ, i = k +1,k+2,...,k+ m
(b) l q = ∑ m
j=k+1 (j − k)π j
(c) λπ k+m (60 minutes/hour)
(d) The quantities in (b) and (c) are certainly relevant. Also w q , the expected time
spent in the hold queue.
12. We approximate the system as an M/M/3/20/20 with τ = 1 program/minute, and µ
= 4 programs/minute
λ i =
µ i =
{
(20 − i)τ, i =0, 1,...,19
0, i =20, 21,...
{
iµ, i =1, 2
3µ, i =3, 4,...,20