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 ${\displaystyle c}$  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.

Formula

Let

• ${\displaystyle c>0}$  be the (integer) number of servers.
• ${\displaystyle N>c}$  be the (integer) number of sources of traffic;
• ${\displaystyle \lambda >0}$  be the idle source arrival rate (i.e., the rate at which a free source initiates requests);
• ${\displaystyle h>0}$  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]

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

By rearranging terms, one can rewrite the above formula as^[2]

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

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

Computation

There are several recursions^[3] that can be used to compute ${\displaystyle P}$  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 ${\displaystyle \lambda }$  is unknown (or hard to estimate) while ${\displaystyle \alpha >0}$ , the offered traffic per-source, is known. In this case, one can substitute the relationship

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

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

${\displaystyle P=f(P)}$ 

where

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

Computation

While the above removes the unknown ${\displaystyle \lambda }$  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 ${\displaystyle f}$ .^[2] Alternatively, it is possible to use one of bisection or Newton's method, for which an open source implementation is available.

References

1. Tijms, Henk C. (2003). A first course in stochastic models. John Wiley and Sons. doi:10.1002/047001363X.
2. 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.
3. Zukerman, Moshe (2000). "An Introduction to Queueing Theory and Stochastic Teletraffic Models" (pdf). Retrieved 2012-11-27.