Borel distribution

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The Borel distribution is a discrete probability distribution used in contexts such as branching processes and queueing theory. Named after the French mathematician Émile Borel [1], it helps us understand the likelihood of outcomes in certain situations. One of those outcomes is extinction—if the number of offspring an organism has is Poisson-distributed, and if the average number of offspring of each organism is no bigger than 1, then its descendants will ultimately become extinct. Let’s explore further how this distribution works.

The Basics of a Borel Distribution

A Borel distribution is a type of discrete probability distribution that is similar to the Poisson distribution. In addition to extinction events, the Borel distribution can be used to predict queues at banks or supermarkets and traffic jams on highways.

As an example, the formula for calculating a Borel distribution involves factoring in both the average number of offspring per organism and their respective probabilities. The formula also takes into account any inherent uncertainty associated with these factors (such as whether or not an organism might die before it reaches reproductive age). By using this formula, we can calculate how likely it is that any given individual or population will have a certain number of descendants over time.

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

borel distribution PMF

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 the 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:

  • Mean = 1 / (1 – μ)3
  • Variance = μ / (1 – μ)3.

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]:

0 < λ < 1.

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]


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

Applications for Businesses

The applications for businesses are vast—the Borel distribution can help companies make better decisions regarding resource allocation and personnel management. For example, when dealing with customer service inquiries, companies can use this model to estimate how many inquiries they should expect at any given time so they can plan appropriately for staffing needs. Similarly, businesses can use this model to forecast resource demand in order to more accurately estimate procurement costs.

In sum, the Borel distribution is a powerful tool for predicting outcomes in various scenarios involving organisms and populations over time. It was first introduced by French mathematician Émile Borel and has since been applied across numerous industries from banking to retail sales. Businesses use this model to better allocate resources for personnel management and procurement needs—allowing them to make more informed decisions about their operations. Overall, understanding what a Borel distribution is can prove very useful in making predictions about future events.


[1]  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

[2] Groen, S. (2017). Borel Borel Borel Distribution. Retrieved April 10, 2023 from:

[3] M/D/1 queueing system.

[4] Stochastic models of road traffic. Retrieved April 10, 2023 from:

[5] Consul, P. & Famoye, F. (2006). Lagrangian Probability Distributions. Birkhäuser.

[6] Johnson, N. et al. (2005). Univariate Discrete Distributions. Wiley.

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