A watch roster rotates through every slot only if the number of slots in a day and the number of people share no common factor: with 12 two-hour slots a day, 5 or 7 people each stand every slot in turn, while 4 or 6 or 12 people stand the same slots for ever. A rotation that is fair in this sense still cannot give a person 8 hours of unbroken rest on a one-post watch with 5 people, because each person’s watches come only 8 hours apart.
Part of Watch and Patrol Scheduling
Rotating schedules move each person through the different hours of the day so nobody is left with the small hours, and the number theory of the roster decides whether they do.
What Is Derived And What Is Recalled
The scheduling results below are arithmetic that anyone can check: they depend on the slot count, the number of people and nothing else. The physiological statements (how fast the body clock can shift, which direction of rotation is easier) are recalled from sleep and shift-work research and I have not been able to check them here; they are marked.
The Round-Robin Rule
Number the 12 two-hour slots of day 0 as 0 to 11 and continue the count day after day, so slot n of the whole run is 12 x day + hour-slot. Give slot n to person (n mod S), where S is the number of people. Each person then stands every S-th slot, which means:
- the time between one watch and the next is S slots of 2 hours, minus the 2 hours on watch: 2S - 2 hours off, always;
- the pattern repeats after S / gcd(12, S) days, where gcd is the greatest common divisor.
Worked Example: Four, Five And Seven People
| People S | gcd(12, S) | What happens | Hours off between watches |
|---|---|---|---|
| 4 | 4 | Never rotates: each person stands the same 3 slots daily | 6 |
| 5 | 1 | Everyone stands every slot once every 5 days | 8 |
| 7 | 1 | Everyone stands every slot once every 7 days | 12 |
Four people. gcd(12, 4) = 4, so every day person 0 stands slots 0, 4 and 8 (0000-0200, 0800-1000, 1600-1800), person 1 stands 0200-0400, 1000-1200, 1800-2000, person 2 stands 0400-0600, 1200-1400, 2000-2200 and person 3 stands 0600-0800, 1400-1600, 2200-2400. Persons 1 and 2 hold the 0200 to 0600 hours for good, and none of them ever has more than 6 hours off between watches.
Five people. gcd(12, 5) = 1. Person 0’s watches over five days are:
- day 0: 0000-0200, 1000-1200, 2000-2200
- day 1: 0600-0800, 1600-1800
- day 2: 0200-0400, 1200-1400, 2200-2400
- day 3: 0800-1000, 1800-2000
- day 4: 0400-0600, 1400-1600
Over the five days person 0 stands each of the 12 slots exactly once, 12 watches in all, and every other person does the same. Night slots (2200 to 0600) are slots 11, 0, 1 and 2; each person gets 4 of the 20 in five days (60 slots, 5 people, 12 each, of which 4 are night). This is perfectly fair. But every gap is exactly 8 hours, so the watch recurs every 10 hours, a cycle that does not match the 24-hour day.
What The Fair Rota Costs In Sleep
A gap of 8 hours between watches must hold the walk to bed, falling asleep, waking before the watch, the 5-minute early arrival at handover and everything else the person needs (a meal, a wash, other work). An assumed 1 hour of that leaves about 7 hours of sleep at the very best. And the clock phase moves around: in the five-day cycle above, person 0’s longest possible sleep windows run 0200-1000, 1200-2000, 2200-0600, 0800-1600 and 1800-0200 in turn, each starting 10 hours after the last. The sleep goes round the whole clock every five days, which the body clock follows poorly. It is fair, but not restful.
The Rolling Rota In The Parent Article
The parent’s rolling schedule moves each person 2 hours later every day, so the cycle is 12 days. Two issues:
- Speed against the body clock. Shifting 2 hours a day is faster than the body clock is usually said to adjust, about an hour a day (recalled). Delaying the day (later each day) is generally easier than advancing it (recalled). On both counts a 2-hour daily delay is hard going, and a cycle that returns to the starting slot after 12 days resets the adjustment.
- It does not follow the staffing table. If four people each shift two hours later every day, each person’s three daily watches are 8 hours apart and always leave a maximum of 6 hours off, whatever the day. The parent’s promise of 8 hours of sleep for every person is not met by four people on one post however the rota rolls.
Getting 8 Contiguous Hours: Clustering
The only way four people can each have an 8-hour unbroken block on a single post is to cluster each person’s three watches into a 16-hour window. Exhaustive search found 11,472 ways to divide the 12 slots among four people, three slots each, with no person on more than two slots in a row and every person’s three slots leaving an 8-hour free stretch. One example (start times):
| Person | Watches | Longest free stretch |
|---|---|---|
| A | 0000-0200, 0200-0400, 0600-0800 | 0800 to 2400 (16 h) |
| B | 0400-0600, 0800-1000, 2200-2400 | 1000 to 2200 (12 h) |
| C | 1400-1600, 1800-2000, 2000-2200 | 2200 to 1400 (16 h) |
| D | 1000-1200, 1200-1400, 1600-1800 | 1800 to 1000 (16 h) |
Everyone has an 8-hour block, but A and B, who cover the dark hours, can sleep only between about 0800 and 2200. Sleep in the day is generally shorter and lighter than night sleep (recalled). If A and B swap with C and D weekly, each person’s clock is dragged round the day every fortnight. The conclusion is not that the arithmetic fails but that a single continuously watched post with four people costs somebody their night’s sleep. Reduce what you man before you reduce anyone’s sleep (Watch Rotation Systems).
Method: Choosing A Roster
Step 1: Count slots per day and people. Compute the gcd.
Step 2: If the gcd is above 1, break the symmetry by adding an offset each day or by giving the odd extra slot to a different person, otherwise the roster does not rotate.
Step 3: List each person’s longest gap and compare with the sleep they need (Sleep Cycle Management).
Step 4: Choose between clustering, pairing and fewer manned hours rather than between fairness and sleep.
Step 5: Re-run the numbers when someone is ill. An absence changes S.
Common Mistakes
- Equating a fair rota with a restful one. Fairness only means equal burdens.
- A divisor in the roster. 4, 6 or 12 people on 12 slots give a fixed roster whatever you call it.
- Forgetting the odd-length watch. The dog watches in the parent’s naval scheme are what make 7 watches a day rotate with two sections: gcd(7, 2) = 1, so each section stands every watch once in two days.
What’s Next
- Learn next: Sleep Cycle Management — what rest a roster must protect
- Combine with: Two-Person Shifts — adding people at the hours that matter