Production line digital twin in 3D

A serial production line is simulated event by event: parts flow through stations separated by finite buffers; a station that finishes while the next buffer is full is blocked, one with nothing to do is starved, and machines break down and are repaired at random. Everything on screen comes from that one simulation.

Formulas used

Availability     A = MTBF / (MTBF + MTTR)            (failures counted while the machine works)
Station capacity mu_i = A * P / T_i                   T_i ideal cycle time, P running speed
Line capacity    min_i mu_i  (the bottleneck)         throughput can never exceed it
OEE = A x P x Q  A = (planned - down)/planned,  P = ideal cycle x count / run time,  Q = good / total
Little's law     L = lambda * W                       WIP = throughput x flow time
M/M/1            L = rho / (1 - rho),  rho = lambda / mu
M/G/1 (P-K)      L = rho + rho^2 (1 + c_s^2) / (2 (1 - rho))
Balancing        n_min = ceil(sum t / C),  efficiency E = sum t / (n C),  balance delay = 1 - E
Ranked positional weight: weight of a task = its time + times of all tasks that must follow it

Honest simplifications

• Failures are operation-dependent (the clock runs only while the machine is working), repairs are exponential, and there is no preventive maintenance, set-up time, rework loop or operator model.
• Scrap is removed at the station where it occurs; real lines usually inspect at a few points.
• The twelve lamp-assembly tasks and their times are an illustrative example, not a measured plant.
• Saturated mode assumes raw material is always available; demand mode feeds a Poisson stream into an unlimited first queue.

Numerical checks on this page

(1) A single machine at utilisation 0.7 must reproduce M/M/1 and M/E2/1 queue lengths (6 million simulated seconds each). (2) Little's law on your line must hold within 2 %. (3) The bottleneck running flat out must match A x P / T. (4) Throughput must never exceed the bottleneck capacity and must approach it with large buffers. (5) Both balancing methods must respect precedence and takt, and use at least the theoretical minimum number of stations.