Discussion
In usual notations of queueing systems, which of the following relationships is not true ?
Answer: D
The relationship that is not true is \(L_{s} = L_{q} + 1/\lambda\).
The correct relation is \(L_{s} = L_{q} + \lambda/\mu\): the mean number in the system exceeds the mean number in the queue by the mean number in service, which for a single server is the utilisation \(\rho = \lambda/\mu\). It follows from multiplying \(W_{s} = W_{q} + 1/\mu\) by \(\lambda\) and applying Little's law.
The other three are true. \(W_{s} = W_{q} + 1/\mu\) says time in system is waiting time plus mean service time. \(L_{s} = \lambda W_{s}\) and \(L_{q} = \lambda W_{q}\) are Little's law for the system and for the queue.
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