Twinworks Queueing Check

How it works

Closed-form queueing results for a single workstation: utilization, mean waiting time, queue length and time in system for M/M/1, M/M/c (Erlang C), M/D/1, M/G/1 (Pollaczek–Khinchine) and the Kingman G/G/1 approximation. Optionally runs the Twinworks simulation of the same station and reports whether the theory lies inside the simulated confidence interval.

• M/M/1: Wq = ρ/(μ−λ). M/M/c: Wq = C(c, a)/(cμ−λ) with the Erlang C probability C computed by a stable recursion. M/G/1: Wq = λE[S²]/(2(1−ρ)) (Pollaczek–Khinchine), M/D/1 is the special case SCV = 0. Kingman: Wq ≈ ((ca²+cs²)/2)·(ρ/(1−ρ))·te.
• L = λW and Lq = λWq (Little’s law).
• Verification builds a source → unlimited queue → station → sink model, simulates long replications (≈ 20 000 arrivals each, 5 % warm-up) and checks whether the theoretical Wq lies inside the Student-t interval of the simulated mean wait.

Formulas assume steady state, infinite queue capacity and FIFO service. Kingman is an approximation.

Sources

• Gross, Shortle, Thompson & Harris: Fundamentals of Queueing Theory, 4th ed. (Wiley, 2008) (Wiley)
• Kingman, J. F. C.: The single server queue in heavy traffic (Proc. Cambridge Phil. Soc. 57, 1961)
• Hopp & Spearman: Factory Physics — VUT equation (Waveland Press)