Engset Formula

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.

Example application

Consider a fleet of $c$ vehicles and $N$ 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 $c$ small so as to minimize costs, but large enough to ensure that the blocking probability is tolerable.

Formula

Let

* $c > 0$ be the (integer) number of servers.
* $N > c$ be the (integer) number of sources of traffic;
* $\lambda > 0$ be the idle source arrival rate (i.e., the rate at which a free source initiates requests);
* $h > 0$ be the average holding time (i.e., the average time it takes for a server to handle a request);

Then, the probability of blocking is given by
:$P=\frac.$
By rearranging terms, one can rewrite the above formula as
:$P = \frac$
where $_2 F_1$ is the Gaussian Hypergeometric function.

Computation

There are several recursions that can be used to compute $P$ in a numerically stable manner.

Alternatively, any numerical package that supports the hypergeometric function can be used. Some examples are given below.

Python with SciPy

from scipy.special import hyp2f1
P = 1.0 / hyp2f1(1, -c, N - c, -1.0 / (Lambda * h))

MATLAB with th
Symbolic Math Toolbox
''

P = 1 / hypergeom( , -c N - c, -1 / (Lambda * h))

Unknown source arrival rate

In practice, it is often the case that the source arrival rate $\lambda$ is unknown (or hard to estimate) while $\alpha > 0$, the offered traffic per-source, is known.
In this case, one can substitute the relationship
:$\lambda h=\frac$
between the source arrival rate and blocking probability into the Engset formula to arrive at the fixed point equation
:$P = f(P)$
where
:$f(P) = \frac.$

Computation

While the above removes the unknown $\lambda$ 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 $f$. Alternatively, it is possible to use one of bisection or Newton's method, for which a
open source implementation
is available.

References

{{reflist

Queueing theory