SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

98 CHAPTER 8. QUEUEING PROCESSES
8. (a) We approximate each system as an M/M/1 queue.
One-person system
λ = 24 cars/hour
µ = 30 cars/hours
w q =2/15 ≈ 0.13 hour
w = w q +1/µ =1/6 ≈ 0.17 hour
Two-person system
µ = 48 cars/hour
w q =1/48 ≈ 0.02 hour
w = w q +1/µ =1/24 ≈ 0.04 hour
(b) Need π 0 for each case
One person π 0 =1/5
Two person π 0 =1/2
(c) The fraction of time that the intercom is blocked is
∞∑
η = π j = 1− π 0 − π 1 − π 2
j=3
2∑
= 1− (1 − ρ)ρ j
j=0
One person η ≈ 0.51
Two person η ≈ 0.13
So in the one-person system the intercom is blocked over half the time.
(d) Now we model the system as an M/M/1/6 queue.
• One-person system
π 6 ≈ 0.066
so the lost-customer rate is λπ 6 ≈ 1.6 customers/hour.
• Two-person system
π 6 ≈ 0.008
so λπ 6 ≈ 0.2 customer/hour.
9. This is one possible approach to the problem:
Since the user population is relatively large, treat the arrival process as Poisson with
rate
λ = (10000)(0.7) = 7000 calls/day
≈ 292 calls/hour
≈ 4.86 calls/minute