Parcel Sorter

What is modelled

Parcels are placed on empty carriers at the induction point. Each parcel has a destination chute; when its carrier passes that chute the parcel is dropped if there is room, otherwise it stays on and goes round again. Operators empty each chute at random moments.

Equations

• Belt capacity v / pitch carriers per second.
• A chute with Poisson arrivals, exponential emptying and room for K parcels is an M/M/1/K queue: Pblock = (1−ρ) ρK / (1 − ρK+1).
• Mean content L = ρ/(1−ρ) − (K+1)ρK+1/(1−ρK+1) and Little law L = λeff W.

Checks run in the page

• A 400000-event simulation of one chute against the blocking formula.
• Little law and the formula for the mean content.
• The belt capacity against parcels inserted in 10 s.
• Conservation of parcels: inserted, on the belt, in chutes, emptied.

Honest simplifications

Real arrivals at a chute are not exactly Poisson (a recirculating parcel returns after a fixed loop time), induction is limited only by the fill rate, and there is no merge, jam, scanner error or gap control. The queue formulas are checked on an isolated chute.

• Press One hot chute and see chute 1 turn red and the recirculation grow.
• Press Hot chute, big buffer and compare the blocking.
• Slow the operators down to see the whole sorter lose throughput.