# Single Server Model Calculator

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Instructions:
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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:

## Single Server Model Calculator

More about the
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Single Server Model
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for you to have a better understanding of what this calculator will provide you.

### What is a single server model about?

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.

### Single Server Model Parameters

The main parameters of a waiting line are:

\[ \text{Average Number of Units in the System } = L_s = \frac{\lambda}{\mu - \lambda}\] \[ \text{Average Number of Units in the Queue } = L_q = \frac{\lambda^2}{\mu(\mu - \lambda)}\] \[ \text{Average Time a unit spend in the System } = W_s = \frac{1}{\mu - \lambda}\] \[ \text{Average Time a unit spend in the Queue } = W_q = \frac{\lambda}{\mu (\mu - \lambda)}\] \[ \text{Utilization Factor } = \rho = \frac{\lambda}{\mu}\] \[ \text{Probability of no units in the system } = P_0 = 1 - \frac{\lambda}{\mu}\]### Are there other Waiting Line Models

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.

Also you can use this single period model calculator to get all the steps of the calculation in the case of a normally distributed demand, though the idea of shortage and overage apply as well for discrete distributions.