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

$P={\frac {{\binom {N-1}{c}}\left(\lambda h\right)^{c}}{\sum _{i=0}^{c}{\binom {N-1}{i}}\left(\lambda h\right)^{i}}}.$

By rearranging terms, one can rewrite the above formula as

$P={\frac {1}{{}_{2}F_{1}(1,-c;N-c;-1/(\lambda h))}}$

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 the Symbolic Math Toolbox

P = 1 / hypergeom([1, -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

$\lambda h={\frac {\alpha }{1-\alpha (1-P)}}$

between the source arrival rate and blocking probability into the Engset formula to arrive at the fixed point equation

$P=f(P)$

where

$f(P)={\frac {1}{{}_{2}F_{1}(1,-c;N-c;1-P-1/\alpha )}}.$

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 an open source implementation is available.