# Documentation

## Model Multiple Servers

### Blocks that Model Multiple Servers

You can use the N-Server and Infinite Server blocks to model a bank of identical servers operating in parallel. The N-Server block lets you specify the number of servers using a parameter, while the Infinite Server block models a bank of infinitely many servers.

To model multiple servers that are not identical to each other, you must use multiple blocks. For example, to model a pair of servers whose service times do not share the same distribution, use a pair of Single Server blocks rather than a single N-Server block. The example in Select the First Available Server illustrates the use of multiple Single Server blocks with a switch.

### Model an M/M/5 Queuing System

The example below (open model) shows a system with infinite storage capacity and five identical servers. In the notation, the M stands for Markovian; M/M/5 means that the system has exponentially distributed interarrival and service times, and five servers.

The plot below shows the waiting time in the queuing system.

You can compare the empirical values shown in the plot with the theoretical value, E[S], of the mean system time for an M/M/m queuing system with an arrival rate of λ=1/2 and a service rate of μ=1/5. Using expressions in [2], the computation is as follows.

$\begin{array}{l}\rho =\frac{\lambda }{m\mu }=\frac{\left(1/2\right)}{5\left(1/5\right)}=\frac{1}{2}\hfill \\ {\pi }_{0}={\left[1+\sum _{n=1}^{m-1}\frac{{\left(m\rho \right)}^{n}}{n!}+\frac{{\left(m\rho \right)}^{m}}{m!}\frac{1}{1-\rho }\right]}^{-1}\approx 0.0801\hfill \\ E\left[S\right]=\frac{1}{\mu }+\frac{1}{\mu }\frac{{\left(m\rho \right)}^{m}}{m!}\frac{{\pi }_{0}}{m{\left(1-\rho \right)}^{2}}\approx 5.26\hfill \end{array}$

Zooming in the plot shows that the empirical value is close to 5.26.