1. 24 February 2019
Model Antrian M/D/1

2. Steady-State Parameters of M/D/1 Queue
l 1 l r2
L = = r
m 2 m(m- l) (1 – r)
1 1 l rm-1
W = = m
m 2 m(m- l) 2(1 – r)
1 l 1 rm-1
Wq = =
2 m(m- l) 2 (1 – r)
1 l r2
Lq = =
2 m(m- l) 2 (1 – r)
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3. Example 1
Arrivals to an airport are all directed to the same runway. At a certain time of the day, these arrivals are Poisson distributed at a rate of 30 per hour. The time to land an aircraft is a constant 90 seconds. Determine Lq, Wq, L and W for this airport. In this case l= 0.5 per minute, and 1/m = 1.5 minutes, or m = 2/3 per minute.
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4. Example 1 (cont.) The runway utilization is
r = l / m = (1/2) / (2/3) = 3/4
The steady-state parameters are given by
Lq = {(3/4) 2} / {2 (1 - 3/4)}
= 9 / 8 = aircraft
Wq = Lq / l = (9/8) / (1/2) = 2.25 minutes
W = Wq + 1 / m = = 3.75 minutes
L = Lq + l / m = = aircraft
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5. Example 2
ABC Car Wash is an automated car wash. Each customer deposits four quarters in a coin slot, drives the car into the auto-washer, and waits while the car is automatically washed. Cars arrive at an average rate of 20 cars per hour (Poisson). The service time is exactly 2 minutes.
24 February 2019
Note that this is actually an M/D/1 model (a special case of M/G/1)
l = 20 cars per hour
1/m = (2 min) / (60 min/hr) = 1/30 hours
s = 0