# Multiple Server Model Calculator

Instructions: You can use this Multiple Server Model Calculator, by providing the arrival rate per time period $$(\lambda)$$, the service rate per time period $$(\mu)$$, and the number of servers $$(s)$$ using the form below:

Arrival Rate per time period $$(\lambda)$$ =
Service Rate per time period $$(\mu)$$ =
Number of Servers $$(s)$$ =
Time period unit =

#### Multiple Server Model Calculator

More about the Multiple Server Model for you to have a better understanding of what this calculator will provide you. The Multiple Server Model (or usually known as M/M/s server discipline) occurs in the setting of a waiting line in which there is one or more servers, the customers are supposed to arrive at a random rate that is specified as a Poisson distribution for a given time period (or the inter-arrival times are exponentially distributed), and the service times are exponentially distributed. The main parameters of a waiting line are:

$\text{Probability of no units in the system } = P_0 = \displaystyle \frac{1}{\displaystyle \sum_{n=0}^{s-1} \frac{1}{n!} \left(\frac{\lambda}{\mu}\right) + \frac{1}{s!} \left(\frac{\lambda}{\mu}\right)^s \frac{s\mu}{s\mu - \lambda}}$ $\text{Average Number of Units in the System } = L_s = \frac{\lambda \mu (\lambda/\mu)^s}{(s-1)!(s\mu - \lambda)^2} P_0 + \frac{\lambda}{\mu}$ $\text{Average Number of Units in the Queue } = L_q = L_s - \frac{\lambda}{\mu}$ $\text{Average Time a unit spend in the System } = W_s = \frac{ \mu (\lambda/\mu)^s}{(s-1)!(s\mu - \lambda)^2} P_0 + \frac{1}{\mu}$ $\text{Average Time a unit spend in the Queue } = W_q = W_s - \frac{1}{\mu}$ $\text{Utilization Factor } = \rho = \frac{\lambda}{\mu}$

Other common waiting line model is the single server model, M/M/1, and as we go making different assumptions about number of lines, servers and channels, we can arrive to fairly complex waiting line models.

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