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\section{Analytical Treatment}  In this section we introduce the mathematical formalism of the M/M/c queue and summarize the main results. For the complete discussion of the problem, the reader should refer to \cite{allen1978queueing}. \cite{allen1978queueing} - page 274 and following.  \\ The number of customers in the system is a discrete random variable that evolves according to a \textit{continuous time Markov chain}, with the following transition probabilities:  \begin{itemize}  \item probability $p_{up}$ that in the time interval $\Delta t$ the number of customers in the system increases by one unit is $p_{up} = \lambda \Delta t$; 
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