 # Single Server Model Calculator

Instructions: You can use this Single Server Model Calculator, by providing the arrival rate per time period $$(\lambda)$$, and the service rate per time period $$(\mu)$$, using the form below: Arrival Rate per time period $$(\lambda)$$ = Service Rate per time period $$(\mu)$$ = Time period unit =

#### Single Server Model Calculator

More about the Single Server Model for you to have a better understanding of what this calculator will provide you. The Single Server Model (or usually known as M/M/1 server discipline) occurs in the setting of a waiting line in which there is only one server, 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:

$\text{Average Number of Units in the System } = L_s = \frac{\lambda}{\mu - \lambda}$ $\text{Average Number of Units in the Queue } = L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}$ $\text{Average Time a unit spend in the System } = W_s = \frac{1}{\mu - \lambda}$ $\text{Average Time a unit spend in the Queue } = W_q = \frac{\lambda}{\mu (\mu - \lambda)}$ $\text{Utilization Factor } = \rho = \frac{\lambda}{\mu}$ $\text{Probability of no units in the system } = P_0 = 1 - \frac{\lambda}{\mu}$

Other common waiting line model is the multiple server model, M/M/s, 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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