Queues at a Single-Vendor Stall: Waiting through the Morning Peak, the Point at Which Customers Leave Faster than They Are Served, and a Rule for a Second Pair of Hands

A single-vendor stall serves one customer at a time, while arrivals rise through the morning, peak, and fall away. This paper computes queue length and waiting time for a stated arrival profile and a service rate of 30 customers per hour, using steady-state queue formulas for each hour and a minute-by-minute fluid calculation across the morning. At 20 arrivals per hour the mean wait before service is 4.0 minutes. At 28 per hour it is 28.0 minutes, with a mean of 13.07 customers waiting. In the peak hour, at 35 arrivals per hour against service of 30 per hour, no steady state exists and the queue grows while that rate lasts. Adding abandonment, with mean patience of 10 minutes, waiting customers leave at 6 per hour for each customer waiting, so they leave faster than the vendor serves once the queue passes 5. In the fluid morning, abandonment also caps the peak queue at 0.83 and limits morning abandonment to 4.39 customers. A second pair of hands for three peak hours costs $45.00, retains those 4.39 customers worth $10.53, and returns −$34.47 on the stated morning. A scan over sustained rates shows help paying only at high rates: over three hours at a sustained 38 per hour it retains 22.67 customers worth $54.40, for a net $9.40. The originated rule is to hire peak help only when the sustained arrival rate is high enough that retained customers at their contribution exceed the wage — on these terms, between a sustained 35 and 38 per hour, not on the strength of one busy hour in a moderate morning.