  Bus systems involve complex bus-bus and bus-passengers interactions. We study
the problem of assigning buses to bus stops to minimise the average waiting
time of passengers, W. An analytical theory for two specific cases of
interactions is formulated: normal situation where all buses board passengers
from every bus stop, versus novel express buses where disjoint subsets of
non-interacting buses serve disjoint subsets of bus stops. Our formulation
allows exact calculation of W for general loops in the two cases examined.
Compared with regular buses, we present scenarios where express buses show
improvement in W. Useful insights are obtained from our theory: 1) there is a
minimum number of buses needed, 2) splitting a crowded bus stop into two less
crowded ones always increases W for regular buses, 3) changing the destination
of passengers and location of bus stops do not influence W. In the second part,
we introduce a reinforcement-learning platform that overcomes limitations of
our analytical method to search for better allocations of buses to bus stops
that minimise W. Compared with the previous cases, any possible interaction
between buses is allowed, unlocking novel emergent strategies. We apply this
tool to a simple toy model and three empirically-motivated bus loops, based on
data collected from the Nanyang Technological University shuttle bus system. In
the simplified model, we observe an unexpected strategy emerging that could not
be analysed with our mathematical formulation and displays chaotic behaviour.
The possible configurations in the three empirically-motivated scenarios are
approximately 10^11, 10^11 and 10^20, so a brute-force approach is impossible.
Our algorithm reduces W by 12% to 32% compared with regular buses and 12% to
29% compared with express buses. This tool has practical applications because
it works independently of the specific characteristics of a bus loop.

Physics and Society