2 Simulations

2.1 Assumptions and notations

In the old ancient times, when queuing theorists [2] were involved in designing computers from scratch, any hypothesis that alleviates hand computations was unavoidable and, therefore, every queuing system was seen as "quite M/M/1". Nowadays, computers are ubiquitous and using M/M models has to be avoided -unless required by the very nature of the phenomenon.

In what follows, the service process is assumed by $n$ identical servers. Any service is independent from anything else, like if its duration was be picked at random at arrival time, with pdf $S$. The total capacity of service is $\mu=n/\operatorname{E}_{B}\left(t\right)$. Customers are arriving one at a time and inter-arrivals are assumed to be iid (independent and identically distributed), with pdf $A$, flow $\lambda=1/\operatorname{E}_{A}\left(t\right)$ and $\rho=\lambda/\mu<1$.

Simulations have been conducted using $Ga$ (Gamma) servers with parameters $a=2.42$ and $b=77.17$ (and therefore mean = $1/\mu_{1}=a\, b$, variance $a\, b^{2}$ and $svc\left(B\right)=1/a$), while arrival flows are either M ( $svc\left(A\right)=1$) or Ga ( $svc\left(A\right)=1.25$). The load factor is either $\rho=0.933$ or $\rho=0.973$.

2.2 Batch-Mean Method

Each presented results has been obtained by simulating $K\, N=400\times50\,000$ events, inducing the simulation of around $T=1\,000\,000$ customers. Splitting this simulation into $K$ batches is useful for dealing with rounding errors in great additions, and also allows parallelization when using a suitable random generator[3]. Moreover, when long range dependence can be neglected, this division into batches can be used to estimate the sd of the general estimator from the experimental value of the sd of the $K$ partial estimators [4].