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^[1]

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^[2]

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^[3] 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. / hyp2f1(1, -c, N - c, -1. / (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 $f$ .^[2]

Alternatively, it is possible to use one of

bisection, which converges unconditionally,^[2] or
Newton's method, which is proven to converge for $\alpha \leq 1$ ^[2] (in practice, it also works for $\alpha >1$ ).

An open source implementation is available for these methods.

References

↑ Tijms, Henk C. (2003). A first course in stochastic models. John Wiley and Sons. doi:10.1002/047001363X.
1 2 3 4 Azimzadeh, Parsiad; Carpenter, Tommy (2016). "Fast Engset computation". Operations Research Letters. 44 (3): 313–318. arXiv:1511.00291. doi:10.1016/j.orl.2016.02.011. ISSN 0167-6377.
↑ Zukerman, Moshe (2000). "An Introduction to Queueing Theory and Stochastic Teletraffic Models" (pdf). Retrieved 2012-11-27.

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