Bus bunching as a synchronisation phenomenon

Bus bunching is a perennial phenomenon that not only diminishes the efficiency of a bus system, but also prevents transit authorities from keeping buses on schedule. We present a physical theory of buses serving a loop of bus stops as a ring of coupled self-oscillators, analogous to the Kuramoto model. Sustained bunching is a repercussion of the process of phase synchronisation whereby the phases of the oscillators are locked to each other. This emerges when demand exceeds a critical threshold. Buses also bunch at low demand, albeit temporarily, due to frequency detuning arising from different human drivers’ distinct natural speeds. We calculate the critical transition when complete phase locking (full synchronisation) occurs for the bus system, and posit the critical transition to completely no phase locking (zero synchronisation). The intermediate regime is the phase where clusters of partially phase locked buses exist. Intriguingly, these theoretical results are in close correspondence to real buses in a university’s shuttle bus system.


I. ANALYTICAL DERIVATION OF THE PHASE TRANSITION TO COMPLETE PHASE LOCKING OF ALL N BUSES SERVING M STAGGERED BUS STOPS IN A LOOP
Consider N = 2 buses with natural angular frequencies ω 1 > ω 2 (ω i = 2πf i = 2π/T i ) serving M = 1 bus stop in a loop. Suppose that the coupling k := s/l is strong enough such that these two buses are phase locked. (Recall that s and l are the people arrival and loading rates, respectively.) In that case, these two buses would always bunch at the bus stop and share the loading of people. Once everybody has been picked up, the two buses leave together, with the faster one pulling away. After one revolution, the faster one returns to the bus stop and begins picking up people. But before finishing, the slower one arrives (because k is strong enough such that there are many people waiting at the bus stop) and the two buses share loading. These two buses are in such an equilibrium which repeats over and over. In τ shared is the duration when these two buses share loading. The total number of people to be picked up is s times the total time elapsed from (a) to (d), which is T 2 + τ shared . These people are picked up by: 1. Only the fast bus = l(T 2 − T 1 ).
2. Shared by the fast and slow buses = 2lτ shared .
The critical transition between no phase locking and phase locking is when τ shared = 0. In that case, = 1 − f 2 f 1 (4) So with M bus stops, each bus stop multiplies the coupling strength. Hence, only one M -th of the coupling strength with one bus stop is required when there are M bus stops.
Let us now consider N buses with angular frequencies ω 1 > · · · > ω N serving M = 1 bus stop in a loop, and we know that having M staggered bus stops would be one M -th of k c for M = 1. The total number of people to pick up is s(T N + τ shared ), since all buses have to wait for the slowest bus to reach the bus stop, and then all buses would share the load over the duration τ shared . These people are picked up by: 1. Only the first bus = l(T 2 − T 1 ).
The critical transition between complete and partial phase locking is when τ shared = 0. In that case, Thus, the critical transition between complete and partial phase locking for the general case of N buses serving M staggered bus stops in a loop is: where ω N /ω i = f N /f i = T i /T N .