Engset calculation

In queueing theory, the Engset formula is used to determine the blocking probability of an M/M/c/c/N queue (in Kendall's notation).

The formula is named after its developer, T. O. Engset.

Consider a fleet of ${\displaystyle c}$ vehicles and ${\displaystyle N}$ operators. Operators enter the system randomly to request the use of a vehicle. If no vehicles are available, a requesting operator is "blocked" (i.e., the operator leaves without a vehicle). The owner of the fleet would like to pick ${\displaystyle c}$ small so as to minimize costs, but large enough to ensure that the blocking probability is tolerable.

Let

Then, the probability of blocking is given by

By rearranging terms, one can rewrite the above formula as

where ${\displaystyle {}_{2}F_{1}}$ is the Gaussian Hypergeometric function.

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