The Press That Breaks

Question

A print shop's press earns 30 of profit per job. After each job, independently with probability 0.1, the press breaks and must be repaired for 120 before the next job can run. What is the shop's long-run average profit per job?

Accepted Answer

Per-job accounting (linearity). Every job earns 30 and triggers a 120 repair with probability 0.1, so each job's expected contribution is 30 - 0.1 times 120 = 30 - 12 = 18. Approach 2, renewal cycles. A cycle is a run of jobs ending in a breakdown. The number of jobs per cycle is geometric with mean frac10.1 = 10, so a cycle earns 10 times 30 = 300 gross and pays exactly one 120 repair: frac300 - 12010 = 18 per job. The two routes agreeing is the check, and the cycle view is the one that generalizes when repairs also cost time. The trap is hand-waving the repair into "about one repair every ten jobs, so it's small": the expected charge per job is a full frac12010 = 12, which is 40% of the gross margin. Rare-but-chunky costs deserve exact amortization, not adjectives. This is the simplest renewal-reward computation there is, and it is exactly how a desk amortizes irregular lumpy costs into a per-unit number: exchange fines per million orders, a server rebuild per so-many restarts, a clearing break per so-many trades. Long-run cost per use equals (probability of the event per use) times (cost of the event), and any strategy whose per-trade edge is smaller than that amortized number i

The Press That Breaks

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