24 February 2019 Model Antrian M/D/1.

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24 February 2019 Model Antrian M/D/1

Steady-State Parameters of M/D/1 Queue l 1 l2 1 r2 L = --- + --- ---------- = r + --- --------- m 2 m(m- l) 2 (1 – r) 1 1 l 1 rm-1 W = --- + --- ---------- = m-1 + --- -------- m 2 m(m- l) 2(1 – r) 1 l 1 rm-1 Wq = --- -------- = --- -------- 2 m(m- l) 2 (1 – r) 1 l 1 r2 Lq = --- --------- = --- ------- 2 m(m- l) 2 (1 – r) 24 February 2019 30 16

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. 24 February 2019 30 13

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 = 1.125 aircraft Wq = Lq / l = (9/8) / (1/2) = 2.25 minutes W = Wq + 1 / m = 2.25 + 1.5 = 3.75 minutes L = Lq + l / m = 1.125 + 0.75 = 1.875 aircraft 24 February 2019 30 13

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