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Probability, Statistics and Queueing

Requests arrive at a service window at an average of 12 per hour, and each takes an average of 4 minutes to handle. Treating it as an M/M/1 queue, which of the following is the utilization of the window?

  1. A0.8
  2. B1.25
  3. C3
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Answer: A. 0.8

The utilization ρ is the product of the arrival rate λ and the average service time Ts, and it expresses the fraction of time the window is busy. With matching units, 4 minutes = 1/15 hour, so ρ = 12 × 1/15 = 0.8. Using the service rate μ = 15 per hour, ρ = λ/μ = 12/15 gives the same result.

  • A. Correct. 12 per hour × 4 minutes ÷ 60 minutes = 0.8.
  • B. This is 15 ÷ 12 = 1.25, with the numerator and denominator of λ/μ reversed. If the utilization were 1 or more, the queue would grow without limit.
  • C. This is 12 ÷ 4 = 3, calculated without matching the time units.
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More Probability, Statistics and Queueing questions

  1. Requests arrive at a server at an average of 3 per minute, and each takes an average of 15 seconds to process. Treating the server as an M/M/1 queue, which of the following is the average time from the arrival of a request until its processing is finished (the average response time)?
  2. A web server holds an average of 12 requests, counting those being processed and those waiting. Requests arrive at 40 per second and are processed at the same rate (steady state). Which of the following is the average time a request spends in the server, found from Little's law?
  3. 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?
  4. There are two service windows of the same speed. In scheme A, each customer picks one of the windows with equal probability on arrival and joins that window's own queue. In scheme B, customers join a single queue and are served by whichever window becomes free first. Arrivals are random, and the total arrival rate and service speed are the same in both schemes. Which of the following correctly compares the average waiting time?
  5. A server with an average service time of 0.2 seconds is treated as an M/M/1 queue. To keep the average response time (waiting time plus processing time) at or below 1.0 second, which of the following is closest to the maximum arrival rate that can be accepted?