< List of probability distributions >* Borel Distribution*

## What is the Borel Distribution?

The Borel distribution, named after the French mathematician Émile Borel [1], is a discrete probability distribution that is similar to the Poisson distribution.

The distribution helps us to understand the likelihood of outcomes, especially in branching processes and queueing theory. Other uses include the modeling of extinction events, queues at businesses and traffic jams on highways.

For example, we can use a Borel distribution to calculate how likely it is that any given individual or population will have a certain number of descendants over time. The formula factors in the average number of offspring per organism and their respective probabilities as well as any inherent uncertainty associated with these factors (such as whether or not an organism might die before it reaches reproductive age).

## Borel Distribution Properties

The probability mass function (pmf) is [2]:

for n = 1, 2, 3,… where μ ∈ [0, 1] and ! is a factorial.

Borel showed in 1943 that the number of customers served in an M/D/1 queuing system during a busy period follows this distribution. In this notation, the M stands for Markovian; an M/D/1 queue has Poisson arrival process, a deterministic service time distribution, and a single server [3].

Other properties:

Note that sometimes lambda (λ) is used in the formula instead of mu (μ) and *m *may be seen instead of *n *(e.g., [4]). Other forms of the PMF exist, such as that given by consul and Famoye [5].

The Borel distribution is a special case of the Borel-Tanner distribution when n = 1. The Borel-Tanner distribution shows the number of customers served in one queue, before it vanishes given random customer arrival time and constant serving time. The probability that Y customers will be served before the queue in a Borel-Tanner distribution vanishes (equal to y) is [6]

**P(Y = y) = (n + y – 1) choose (y) * (1 – β) ^{n} * β^y**

where

- n = initial number of customers in the queue,
- β = constant arrival time of customers,
- λ = constant rate of customer arrival times.

## Difference Between Poisson and Borel Distribution

The Borel distribution and the Poisson distribution are both discrete probability distributions used to model the number of events occurring in a fixed interval of time. The Borel distribution is a special case of the Poisson distribution when the average rate of events is equal to one. The two distributions have slightly different assumptions and are used in different scenarios.

For example, both distributions share the following assumptions:

- Events occur independently.
- The average rate of events is constant over time.
- Negligible probability of multiple events occurring at the same time.

However, in the Poisson distribution, the number of events in disjoint intervals is independent. This is not the case with the Borel distribution.

Applications are also different. The more general Poisson distribution can be used for a wide range of scenarios, while the Borel distribution is most commonly used in queuing theory and reliability engineering.

## Applications for Businesses

The Borel distribution can help businesses make better decisions regarding resource allocation and personnel management. For example, businesses can use the Borel distribution to estimate how many customer service inquiries they should expect at any given time, which helps them to plan for staffing needs. Similarly, businesses can use this model to forecast resource demand in order to more accurately estimate procurement costs.

## References

- Borel, Émile (1942). “Sur l’emploi du théorème de Bernoulli pour faciliter le calcul d’une infinité de coefficients. Application au problème de l’attente à un guichet”. C. R. Acad. Sci. 214: 452–456. ^ Jump up to:a b
- Groen, S. (2017). Borel Distribution. Retrieved April 10, 2023 from: https://professorfrancken.nl/association/news/17-09-29-borel-borel-borel-distribution
- M/D/1 queueing system.
- Stochastic models of road traffic. Retrieved April 10, 2023 from: https://www.maths.usyd.edu.au/u/richardc/traffic.html#anchor699348
- Consul, P. & Famoye, F. (2006). Lagrangian Probability Distributions. Birkhäuser.
- Johnson, N. et al. (2005). Univariate Discrete Distributions. Wiley.