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 vehicles and 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 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 [1]
By rearranging terms, one can rewrite the above formula as [2]
where is the Gaussian Hypergeometric function.
There are several recursions [3] that can be used to compute in a numerically stable manner.
Alternatively, any numerical package that supports the hypergeometric function can be used. Some examples are given below.
fromscipy.specialimporthyp2f1P=1.0/hyp2f1(1,-c,N-c,-1.0/(Lambda*h))MATLAB with the Symbolic Math Toolbox
P=1/hypergeom([1,-c],N-c,-1/(Lambda*h))In practice, it is often the case that the source arrival rate is unknown (or hard to estimate) while , the offered traffic per-source, is known. In this case, one can substitute the relationship
between the source arrival rate and blocking probability into the Engset formula to arrive at the fixed point equation
where
While the above removes the unknown from the formula, it introduces an additional point of complexity: we can no longer compute the blocking probability directly, and must use an iterative method instead. While a fixed-point iteration is tempting, it has been shown that such an iteration is sometimes divergent when applied to . [2] Alternatively, it is possible to use one of bisection or Newton's method, for which an open source implementation is available.
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