Clearing the Testing Backlog
A defence aptitude testing centre runs three sessions each weekday, 25 seats per session. Ten per cent of all seats are held for retests; the remainder are offered to candidates on the backlog. Historically 20 per cent of booked candidates do not attend, and a no-show seat cannot be reused that day. The backlog currently stands at 610 candidates, and new bookings arrive at a steady 55 per weekday. Extra sessions can be added on request: each has 25 seats, carries no retest allocation, and shows the same 20 per cent no-show rate. No testing occurs on Sundays or public holidays, and there are none in the planning window. Extra weekday sessions behave the same as extra Saturday sessions.
1. Calculate how many backlog candidates actually sit a test on a normal weekday, and the resulting net change in the backlog per weekday.
Read the worked answer
Seats: 3 x 25 = 75. Retest allocation: 10% x 75 = 7.5, so about 7-8 seats; take 7.5 for the estimate. Seats offered to the backlog: 75 - 7.5 = 67.5. Actual attendance: 67.5 x 0.80 = 54 candidates per weekday.
Net change: arrivals 55 - sittings 54 = +1 per weekday.
The backlog is not falling. The centre is running at almost exactly break-even, so the headline capacity of 75 seats is misleading: two separate leakages (retest hold, no-shows) remove about 28 per cent of it.
2. Project the backlog over the next 10 weekdays with no extra sessions. What does this imply about any plan that relies on normal capacity?
Read the worked answer
Net change is +1 per weekday, so over 10 weekdays: 610 + 10 = 620.
The backlog grows, slowly but monotonically. Any plan that assumes the queue clears itself at normal capacity is wrong, and a plan that assumes it merely 'holds steady' is also slightly wrong. The deficit is small per day but compounding, which is the trap: a 1-per-day leak looks like rounding error and is not.
Implication: no extra sessions means no reduction, ever, regardless of the planning horizon.
3. A surge leaves a backlog of 200 that must be cleared within 5 weekdays, while still absorbing 55 new bookings per weekday. How many extra weekday sessions per day are required? Show that one fewer is not enough.
Read the worked answer
Required net reduction: 200 / 5 = 40 per weekday.
Current net is +1, so extra sessions must deliver 41 additional sittings per weekday.
Each extra session yields 25 x 0.80 = 20 sittings.
- 2 extra sessions: 40 sittings, net reduction = 54 + 40 - 55 = 39/day. Over 5 days: 195. Short by 5.
- 3 extra sessions: 60 sittings, net reduction = 54 + 60 - 55 = 59/day. Over 5 days: 295. Clears with 95 to spare.
Answer: 3 extra sessions per weekday.
The judgement being tested is that the marginal session delivers 20 sittings, not 25, and that the target is a rate (40/day) plus the leak (+1), not just 200/5 divided by 25.