The Market-Day Weighing Queue: Why More Weigh Stations Cannot Fix a Compressed Arrival Pulse, and the Staggered Schedule That Can

At many produce markets, every farmer must pass a weigh station before selling, and most farmers arrive in the same early-morning hours. The queue that forms is not a minor inconvenience: a perishable load loses value for every hour it waits in the heat. This paper models a market day with 240 farmers, a service time averaging 4.0 minutes per farmer at one weigh station, and arrivals compressed into a two-hour morning window, and derives how many parallel weigh stations are needed to keep the 95th-percentile wait at or below 30 minutes. Under the compressed pulse, seven stations are required; four stations — the most a stated capital budget of $15,000 can buy at $3,500 per station — leave a 95th-percentile wait of 118.62 minutes, a mean wait of 58.21 minutes, and an expected spoilage loss of $1,373.64 per market day on tomato loads worth $240 each. No station count within the stated budget fixes the tail of the waiting-time distribution under the compressed pulse. The alternative that does work costs less, not more: dividing the same 240 farmers into eight hourly collection slots of 30 farmers meets the same bound with three stations (95th-percentile wait 26.30 minutes, expected spoilage $251.67 per day, capital $10,500, within budget). Total daily cost, counting station running costs and spoilage together, falls from $1,468.96 for four stations under compressed arrivals to $323.16 for three stations under staggered arrivals. The sizing rule and its limit are both stated: parallel capacity can fix a queue whose arrival rate it can exceed for long enough, but when the whole day's demand arrives in two hours, rescheduling the arrivals is worth more than any affordable amount of added capacity.