OPS.10.1:5.2 - Four hours on average, with a missed-deadline risk
Two independent jobs are ready at time zero. One server processes them in fixed order without interruption. Each service takes one or three hours with probability one half, independently of the other service. Both jobs must be complete by hour four.
| Service durations, hours | Last completion, hour | Probability | Both complete by four? |
|---|---|---|---|
| 1, 1 | 2 | 1/4 | Yes |
| 1, 3 | 4 | 1/4 | Yes |
| 3, 1 | 4 | 1/4 | Yes |
| 3, 3 | 6 | 1/4 | No |
Mean last completion is four hours; the probability of meeting the deadline is three quarters. Replacing both durations by their mean of two hours would produce a single on-time schedule and discard that risk.
An additional independent server capable of the same work would allow both to start at zero and finish by three in every listed case. That is a conditional alternative; obtaining it is another operating action. Without it, a deadline of six covers all cases in this bounded model.
Now keep both marginal service distributions but make their durations equal, perhaps because a shared job condition affects both. The only outcomes are (1,1) and (3,3), each with probability one half. Mean last completion remains four; deadline probability falls to one half. The changed dependence reopens the probability result without changing the two means.