Birth-Death Processes explained for betting

In one sentence

A birth-death process tracks a number that goes up by one (a birth) or down by one (a death) at random moments, which is a natural model for a queue of unmatched bets at a single price.

How it works

Pick one price on the ladder, say 3.50 in the pink column. New unmatched back bets join the queue there from time to time; those are births. Bets leave the queue when they are matched or cancelled; those are deaths. The queue length wanders up and down.

If new bets arrive faster than they leave, the queue tends to grow and the price level becomes a wall. If they leave faster, the queue tends to empty, and when it hits zero the best price moves by a tick. That is how the model links the queue to price changes.

The simplest version, the M/M/1 queue, assumes a steady arrival rate and a steady departure rate. It is a special kind of Markov chain where you can only step up or down by one.

The maths

\rho = \frac{\lambda}{\mu}, \qquad P(n) = (1 - \rho)\,\rho^{n}

P(\text{reach } N \text{ before } 0 \mid \text{start at } k) = \frac{1 - r^{k}}{1 - r^{N}}, \quad r = \frac{\mu}{\lambda}

λ: the rate at which new bets join the queue (per minute).
μ: the rate at which bets leave (matched or cancelled).
ρ: the ratio of arrivals to departures; the queue is stable only if it is below 1.
P(n): the long-run chance the queue holds exactly n bets.
k: the current number of bets in the queue; N: a larger level you care about.
r: departures relative to arrivals.

In plain English: if bets leave faster than they arrive, the queue is usually short, and it is much more likely to empty than to build into a wall.

Worked betting example

At a price level, new bets arrive at 4 per minute and leave at 5 per minute.

ρ = 4 ÷ 5 = 0.8.

Average queue length: ρ ÷ (1 − ρ) = 0.8 ÷ 0.2 = 4 bets.
Chance the level is empty at a random moment: 1 − 0.8 = 20%.
Chance of 10 or more bets waiting: 0.8 to the power 10 ≈ 10.7%.

Right now 3 bets are queued. Will it build to a 10-bet wall or empty first (so the price ticks)?

r = 5 ÷ 4 = 1.25.

Chance of reaching 10 first: (1 − 1.25³) ÷ (1 − 1.25¹⁰) ≈ 11.5%.

Chance of emptying first ≈ 88.5%.

So a trader reading this level as "support" would be wrong nearly nine times in ten under these rates. The catch is that the rates must be measured, and they rarely stay steady for long.

Where it's good

Estimating how long a price level will hold before the market ticks.
Deciding whether to join a queue or move to a neighbouring price.
Building simple simulators of the ladder to test trading bots.
In-play states where a count moves one step at a time, such as a running goal lead.

Limitations and pitfalls

Counting bets rather than pounds is a simplification; one £5,000 bet is not the same as one £2 bet.
Arrival and departure rates change sharply near kick-off and after goals and red cards.
Real queues interact: what happens at 3.50 depends on 3.45 and 3.55, which a single-level model ignores.
The long-run formulas assume the queue has settled, which pre-match markets often never do.
Rates estimated from one market may not carry over to another league, time of day or price range.
It explains mechanics; on its own it rarely delivers an edge after commission.

How to build it

Estimate λ and μ by counting queue joins and exits from Betfair Stream API data (betfairlightweight).
numpy is enough to simulate queues; the queueing library "ciw" or simpy handle more complex versions.
Tip: model queues in pound-sized chunks, for example £50 units, so a big bet counts as several arrivals.

Queue position applies this to your own place in line.
Poisson processes describe the random arrivals and departures.
Markov chains are the general family this belongs to.
Brownian motion and random walks use the same one-step-up, one-step-down idea for prices.