Baggage Scanner Line

What is modelled

The scanner stage of a baggage system is a queue with c identical servers fed from a single line. Bags arrive at random, each scan takes a random time with a given mean, and the line grows sharply as the load approaches the capacity of the scanners. The scanner is a generic black box with a service time; no screening physics or procedure is modelled.

Equations

• Load ρ = λ s / c for arrival rate λ, mean scan time s and c scanners.
• Erlang C, the chance of waiting: C(c, a) = (ac/c!) (c/(c−a)) / [Σk<c ak/k! + (ac/c!) (c/(c−a))] with offered load a = λ s.
• Mean wait Wq = C(c, a) / (c/s − λ); for other variability the Allen-Cunneen factor (1 + cs²)/2 is applied.
• One server (Pollaczek-Khinchine): Wq = ρ (1 + cs²) s / (2 (1 − ρ)); Little's law Lq = λ Wq.

Checks run in the page

• Erlang C evaluated two different ways.
• A hidden simulation of 160,000 bags against Erlang C.
• Fixed and random scan times for one scanner against the Pollaczek-Khinchine formulas.
• Little's law from the queue length counted at random moments.

Honest simplifications

Only the scanner stage is modelled: no belts, sorter, tugs, flight deadlines or mishandled bags. Arrivals are Poisson and the scan times are gamma distributed. The multi-server formula for non-exponential scan times is an approximation, so only the exponential case is checked against simulation.

• Press Rush hour, near the limit, then Overloaded, and watch the line grow without limit.
• Press Fixed scan time (M/D/c) to see the wait fall to about half of the random case.
• Press Add scanners and read how much a few extra servers cut the wait.