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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