Reliability and Layout: Breakdowns, Repairs and When a Backup Machine Helps

The problem: one breakdown stops the line

Picture three machines in a row on a factory floor. Each part goes through the first, then the second, then the third. If any one of them breaks, the whole line stops until it’s fixed. Machines do break, at random, and every repair takes time.

A plant manager’s question is simple: what share of the year will this line actually run? Engineers call that share availability. It sets how much the plant can ship. It also decides whether a backup machine is worth buying. I built the lab so you can watch the answer play out, breakdown by breakdown, next to the math that predicts it.

Two numbers describe each machine. MTBF, the mean time between failures, is how long it runs on average before it breaks. MTTR, the mean time to repair, is how long a fix takes on average. Say a machine runs 120 hours between breakdowns and takes 4 hours to fix. On average it’s working 120 out of every 124 hours, or about 96.8% of the time. That’s the formula A = MTBF / (MTBF + MTTR). Longer runs between breakdowns and faster repairs both push it up.

Layout changes the answer

When every machine has to work, engineers say the machines are in series. The line runs only while all of them are up at once. Say they break independently, so one breaking doesn’t change the odds for the others. Then their availabilities multiply. Three machines at 96.8% give a line that runs about 90.6% of the time. Now put a backup next to one machine. Engineers call that in parallel. That step stops only when both machines are broken at the same moment, which is far rarer. If the two break independently, that step is down (1−A₁)(1−A₂) of the time. So its availability is 1 − (1−A₁)(1−A₂).

The lab’s line puts machines in series, with the option of doubling up the middle step. It also runs that same layout as a simulation that jumps from one breakdown or repair to the next. Every machine breaks and gets fixed on its own random clock. Product moves only while every step has at least one working machine. With independent failures, the simulated share of up time settles onto what the formula predicts. That’s the check that the math and the animation describe the same line. The layout can matter as much as how good any single machine is.

The controls, and one shared cause

The MTBF and MTTR sliders set each machine’s availability. The layout decides how those values combine: three in a row, or a backup beside M2. The backup formula gives its full boost only when two things are true. The two machines must break independently, and either one must be able to do the job alone. The shared-cause slider breaks the first of those. Engineers call this common-cause failure and model it with a beta factor, β. A fraction β of the failures of M2 and its backup, M2B, are shared shocks, like a power dip. Each shock knocks out every machine in the pair that’s still running. Each machine’s own availability doesn’t change. But the doubled-up step is down more often than (1−A)² predicts. Across the lab’s range (β up to 30%), the line with the backup still runs more of the time than three in a row. It just runs less than the independent formula claims. The lab shows three values side by side: the formula, the exact beta-factor value and the simulated value. The simulation settles on the beta-factor value, not the formula.

What the model leaves out

Real life can break the simple formula in several ways. Failures can share a cause, and the switch over to a backup doesn’t always work. Planned maintenance takes machines offline too. And breakdowns may not follow the simplest random pattern, called exponential. In that pattern, a machine is just as likely to break in the next hour whether it was fixed yesterday or a month ago. Parts can be slow to arrive, and a machine can limp along at part speed instead of being simply up or down. The lab covers only the first of these, shared causes, with the simplest beta-factor shock model. It works out the exact answer with a three-state Markov chain. That’s a model that tracks whether the pair has two, one or no machines running. Then it checks that answer with the simulation. The simulation makes its own simplifications. Run times and repair times follow that simple exponential pattern. Machines can break even while the line is stopped, since breakdowns run on calendar time. And each machine has its own repair crew. Bigger systems may need fault trees (diagrams of how small failures add up to a big one), larger Markov models or a more detailed simulation.

How I check it

Availability must stay between zero and one. Raising MTBF while MTTR stays the same must never lower a machine’s availability. A perfect backup pair can’t be less available than either machine on its own. Adding another must-work machine in series can’t make the line more available. With independent failures, the simulated long-run share of up time must settle onto the formula, within its margin of error (the confidence interval). With a shared cause, it must settle onto the beta-factor value instead, which is below the independent formula. And a fixed seed, the starting number for the random draws, must replay the exact same breakdowns.