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.

In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us.

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