Arrival rate at a M|M|1 queuing system is 0.11 per minute, while service rate is 0.33 per minute. Then, which of the following statements is true ? (I) The probability that a customer has to wait is 0.67. (II) The probability of waiting for more than 10 minutes is 0.03.
Only (II) is true
\(\lambda = 0.11\), \(\mu = 0.33\), \(\rho = \frac{\lambda}{\mu} = \frac{1}{3}\)
P(customer has to wait) \(= P(n \ge 1) = \rho = 0.33 \ne 0.67\): (I) is false.
Waiting time in queue: \(P(W_q > t) = \rho\, e^{-(\mu - \lambda)t}\)
\(P(W_q > 10) = \frac{1}{3}e^{-0.22 \times 10} = \frac{1}{3}e^{-2.2} = \frac{0.1108}{3} \approx 0.037 \approx 0.03\): (II) holds.
Hence, Only (II) is true.
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