A sorter belt is a conveyor with fixed parcel pitch s, so it can carry at most one parcel every s / v seconds. Two induction feeders add parcels at random (Poisson) times whenever the slot in front of them is free; pushers divert each parcel into the lane for its destination. A lane holds K parcels and is emptied by a packer with random (exponential) service. If the lane is full the controller either lets the parcel ride the return loop for another lap (and rejects it after four laps) or stops the whole belt until a place opens.
capacity C = 60 v / s parcels/min (s = 0.6 m pitch, v in m/s); tick dt = s / v
bound (stop) X <= min( lambda, C, min_i mu_i / p_i ) head-of-line: the slowest packer sets the pace
bound (loop) X <= sum_i min( mu_i, min(lambda, C) p_i ) an upper bound: laps also use belt slots
lane = M/M/1/K rho = lambda_i / mu_i, P_block = (1 - rho) rho^K / (1 - rho^(K+1))
Little's law L = lambda W (mean parcels inside = induction rate x mean time inside)
Monte Carlo mean +- t(0.975, n-1) s / sqrt(n) over n independent seeded runs of the same model
- Belt speed - below the capacity C the throughput equals the offered load and a faster belt only shortens the time inside; above it a faster belt raises the ceiling until the packers or the offered load take over (the chart shows the knee).
- Merge priority - parcels already on the belt or the loop always win; feeder A is upstream of feeder B, so when the belt is nearly full feeder B waits first and its queue grows.
- Full lane - stopping the belt keeps every parcel but one slow lane throttles all three; recirculating keeps the other lanes moving but lapping parcels take belt slots, so near saturation the loop fills, induction slows and rejects appear (try Loop saturation).
- Same model in the 2D twin - one shared physics module with a seeded random generator; the 3D page and the plan view only draw it.
- Honest simplifications - one parcel per slot whatever its size, instant pushers and scanners, no acceleration, gap control or mis-sorts, exponential packer times, independent Poisson arrivals, a fixed 52-slot ring. It is a teaching model, not a sorter design or a rating.