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The impact of utilization rate of a queueing system on its average queue time

I do not agree with the assertion that "both high and low utilization rate mean we are going to have a long average queue time". If $\rho$ is low (close to 0), $E[QT]$ is close to 0. Low ...
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If $x=\min\{f(\mathbf{a}),1-\epsilon\}$, how can we model and partition $x$?

Start with the constraints $$x \le f(\mathbf{a})$$ and $$x \le 1-\epsilon.$$ If the nature of the problem is that larger values of $x$ are always preferable in objective terms to smaller values of $x$,...
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