Diagnostic queues: worked out examples

To investigate the morning queue, you ask for demand and capacity by time of day. The interviewer shares Exhibit 1: requests and usable slots by window. It shows average weekday demand across three equal scheduling windows. Slots are comparable and already reflect usable staffed capacity; do not apply another productivity deduction.

The immediate question is why the morning is overloaded and whether the service needs more total daily capacity.

Before reading further, calculate both the daily balance and the balance in each window.

Daily requests are 96 against 120 usable slots: 80% demand relative to capacity. There are 24 more daily slots than requests in aggregate.

That ratio does not, by itself, tell you actual utilisation. Actual utilisation would depend on how many appointments are delivered and which capacity denominator is used. Requests can wait, remain unserved or move between windows.

The morning contains eight more requests than slots. Its demand-to-capacity ratio is 48 ÷ 40 = 120%. Midday and afternoon each have 16 spare slots relative to their 24 requests.

The operational finding is specific: the problem is concentrated in the morning, despite spare capacity later.

First test a scheduling change, rather than treating it as a new observation: suppose eight morning requests can appropriately move to midday. The proposed schedule below is calculated from Exhibit 1, with daily demand and capacity unchanged.

That removes the whole-window capacity shortfall without adding daily slots. It does not prove that nobody waits within a window; exact arrival and appointment timing still matter.

Ask which requests need the morning, which can move, and whether patients and staff can use the later slots. Capacity is useful only when it fits the timing and type of need.

If only four requests can move, the morning still has 44 requests for 40 slots. You have reduced the gap to four, not solved it. Quantify the residual before recommending the schedule change as sufficient.

As a separate option, return to the original 48, 24 and 24 requests. Suppose their timing is fixed but eight usable slots can move from midday to morning by changing the staffing pattern, while preserving daily capacity and service conditions.

The capacity profile becomes 48 morning, 32 midday and 40 afternoon. Against 48, 24 and 24 requests, each window has enough capacity in aggregate.

This can be a better response when the demand is time-sensitive and staff schedules are flexible. The calculation does not establish that the shift is feasible. Ask about the personnel, equipment and preparation that make a slot usable in that window.

Avoid a vague “optimise staffing” recommendation. Say how much capacity needs to move, from where and to where. Here, moving eight usable slots from midday to morning closes the observed gap.

For a third scenario, return to the original demand and capacity in Exhibit 1 and allow unfinished morning work to wait. The eight excess requests then join midday’s 24, creating 32 requests against 40 slots. If all are suitable for the later window and can be served there, the backlog could clear by the end of midday.

But the table alone cannot tell you whether an individual waits ten minutes or several hours. It does not show arrival times, appointment lengths within the window, priority rules or whether equipment is shared across appointment types.

The useful next analysis depends on the decision. To choose whether to buy another scanner, the daily and window-level capacity comparison is a strong starting point. To redesign patient arrival times, you need the finer timing data.

Matching 48 average requests with exactly 48 morning slots removes the average shortfall. It leaves no headroom for a busier morning or lost capacity.

If demand varies materially, examine its distribution and the service target before treating an average match as sufficient. Include attendance and rebooking when comparing required with available capacity.

For an interview, state the implication plainly: “This fixes the observed average mismatch. I would check busy-day demand and capacity interruptions before finalising the rota.” Then ask for the information that affects the choice, rather than adding a generic uncertainty disclaimer.

I would start with a scheduling and staffing test rather than buying additional daily capacity on the evidence shown:

“The morning is eight slots short, while later windows have spare capacity. I would first test moving eight suitable requests later or shifting eight usable slots into the morning. If neither is feasible, the remaining morning gap defines the extra capacity we need.”

That recommendation follows directly from the exhibit and provides alternatives tied to the operating constraint.

Try the emergency-department congestion drill for another bottleneck problem. For the arithmetic from scheduled hours to completed appointments, read clinic capacity maths.