The Cafeteria Line Balance
A school cafeteria serves 240 students in a 30-minute lunch period. There are 3 serving lines. Each student takes an average of 20 seconds at a line once they reach the front. Students arrive at a steady rate of 8 per minute across the whole period. The cafeteria manager notices that lines get long around minute 10 and then clear out by minute 25. She is considering adding a 4th line, which would cost the school $150 per day to staff. A survey shows that students who wait more than 5 minutes complain.
1. Using the arrival rate and service rate, determine whether 3 lines can keep the average wait under 5 minutes, and explain your reasoning.
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Each line serves one student every 20 seconds, so 3 students per minute per line. With 3 lines, the total service rate is 9 students per minute. Arrivals are 8 per minute, so the system has spare capacity of 1 student per minute. Since arrival rate (8) is below total service rate (9), average wait stays low and the queue does not grow without bound. The 3 lines are sufficient on average. The mid-period backup is a temporary surge, not a capacity failure.
2. The manager says the 4th line will eliminate all complaints. Is that a sound conclusion? What evidence would you want?
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No. Adding a line raises service capacity to 12 per minute, but the complaint threshold depends on wait time during the surge, not just average capacity. The 10-minute backup suggests arrivals may not be perfectly steady - they likely cluster. A 4th line helps only if the surge is the cause. Before spending $150/day, gather per-minute arrival counts to confirm the surge timing and size. If the surge is brief and under 5 minutes of wait, the 4th line may not be worth the cost. The conclusion overreaches from 'lines are long' to 'all complaints end.'