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Probability, Statistics and Queueing

For a server that can be treated as an M/M/1 queue, the arrival rate increased with the average service time unchanged, and the utilization rose from 0.8 to 0.9. By what factor did the average waiting time Wq (the time spent waiting in the queue) change? Which of the following is closest?

  1. A1.125 times
  2. B2 times
  3. C2.25 times
Show answer and explanation

Answer: C. 2.25 times

Wq = ρ/(1 − ρ) × Ts, and Ts is unchanged, so the ratio of ρ/(1 − ρ) is needed. At ρ = 0.8 it is 0.8/0.2 = 4, and at ρ = 0.9 it is 0.9/0.1 = 9, so the factor is 9 ÷ 4 = 2.25. As the utilization approaches 1, 1 − ρ approaches 0, so a small increase in utilization makes the waiting time grow sharply.

  • A. This is 0.9 ÷ 0.8 = 1.125, assuming the waiting time is proportional to the utilization.
  • B. This is the ratio of the average response time W = Ts/(1 − ρ), 0.2 ÷ 0.1 = 2, not the ratio of the waiting time, which excludes the processing time.
  • C. Correct. (0.9/0.1) ÷ (0.8/0.2) = 9 ÷ 4 = 2.25 times.
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