What is the average length of a queue of cars? (2023)
20 days ago
- #queue-theory
- #probability
- #simulation
- The article explores the average length of car queues on a twisty mountain road where overtaking is impossible.
- Initially, an intuitive but incorrect approach suggests the average queue length is 2 cars, based on independent and identically distributed (i.i.d.) vehicle speeds.
- Simulation results contradict the initial theory, showing much longer average queue lengths (around 11 cars) and even extreme cases up to 22,849 cars.
- The correct solution involves conditioning on the speed of the first car in the queue, leading to a different probability distribution for queue lengths.
- The corrected probability distribution for queue lengths is derived as \( p(N=n) = \frac{1}{n(n+1)} \), which matches simulation results.
- The expected queue length under the correct distribution diverges to infinity, indicating that very long queues are inevitable under the given assumptions.