Cash Counter Queue During Pension Week
On a Monday, a branch with three cash counters receives customers at an average of 90 per hour between 10 a.m. and 2 p.m. Each counter can serve 35 customers per hour. At 11 a.m. one counter is closed for two hours for cash verification, leaving two counters. The branch manager may open a fourth counter only if the queue is expected to exceed 25 customers at any time or if average waiting time exceeds 15 minutes. Customers arrive uniformly, and service is first-come-first-served. The manager also knows that the 2 p.m. deadline for government pension payments will add 40 customers between 1 p.m. and 2 p.m. The branch opens at 10 a.m. with no queue. The manager must decide whether to open the fourth counter at 11 a.m. or at 1 p.m.
1. What queue builds between 11 a.m. and 1 p.m. if no fourth counter is opened? Show the rate calculation.
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From 10 a.m. to 11 a.m., three counters serve at 3 × 35 = 105 customers per hour, while arrivals are 90 per hour. Capacity exceeds arrivals, so no queue builds; the branch starts 11 a.m. with an empty queue.
From 11 a.m. to 1 p.m., only two counters remain open: capacity = 2 × 35 = 70 customers per hour. Arrivals are 90 per hour. The net build-up is 90 − 70 = 20 customers per hour. Over two hours, the queue reaches 20 × 2 = 40 customers by 1 p.m.
That already exceeds the manager's 25-customer threshold. The average wait is also a problem: with 40 people waiting and two counters serving 70 per hour, the approximate wait is 40/70 hour ≈ 34 minutes, well above the 15-minute trigger.
2. Should the fourth counter be opened at 11 a.m. or at 1 p.m.? Use the manager's thresholds.
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It should be opened at 11 a.m. If the fourth counter opens at 11 a.m., the branch has three active counters (the fourth plus the two remaining after the closure): capacity = 3 × 35 = 105 per hour against 90 arrivals, so the queue does not build. From 1 p.m. to 2 p.m., with the closed counter reopened plus the fourth counter, capacity is 4 × 35 = 140 per hour against 130 arrivals (90 normal + 40 pension), so the queue still does not build.
If the manager waits until 1 p.m., the queue is already 40 at 1 p.m. Opening the fourth counter then gives capacity 140 per hour against 130 arrivals, so the queue stops growing quickly but is not cleared; it remains around 40. The 25-customer and 15-minute thresholds are breached before 1 p.m., so the decision must be taken at 11 a.m.
3. If the fourth counter is opened at 1 p.m. only, what queue remains at 2 p.m.?
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At 1 p.m. the queue is already 40 customers from the two-hour closure period. From 1 p.m. to 2 p.m., the reopened counter plus the fourth counter give four active counters: capacity = 4 × 35 = 140 customers per hour. Arrivals are 90 normal + 40 pension = 130 customers per hour. The net change is 130 − 140 = −10 per hour, so in theory the queue could fall by 10, but the backlog is 40 and the branch has only one hour.
A simple flow calculation gives a remaining queue of about 40 − 10 = 30 customers at 2 p.m., still above the 25-customer threshold. The precise number depends on when within the hour the pension customers arrive, but the key judgement is that opening at 1 p.m. is too late to clear the backlog. Opening at 11 a.m. avoids the 40-person queue altogether.