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GATE 2024 ME – Question 35

Operations Research · Simple queuing models · 1 mark · Multiple choice

A queueing system has one single server workstation that admits an infinitely long queue. The rate of arrival of jobs to the queueing system follows the Poisson distribution with a mean of 5 jobs/hour. The service time of the server is exponentially distributed with a mean of 6 minutes. In steady state operation of the queueing system, the probability that the server is not busy at any point in time is

  1. 0.20
  2. 0.17
  3. 0.50
  4. 0.83

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Correct answer: (C) 0.50

Explanation

This is an M/M/1 queue with $\lambda = 5$ per hour and $\mu = \frac{60}{6} = 10$ per hour. The utilisation is $\rho = \frac{\lambda}{\mu} = 0.5$, and the probability that the server is idle is $1 - \rho = 0.5$.