How Many Baristas?

Here's a puzzle. Forty eight customers arrive per hour. Each drink takes about two minutes. That's ninety six minutes of work per hour. So two baristas have a hundred and twenty minutes available, and that's plenty, right? They'd be only eighty percent busy. Now watch what happens.

• 48 customers an hour
• Two baristas look like enough
• Eighty percent busy

Simulating the queue

No formula is needed. We simulate it. Random gaps between customers, and random service times. Each customer goes to whichever barista gets free first, and we record how long they waited. We run two thousand hours, for two, three, and four baristas. Run it. Look at that. With two baristas, the average wait is over seven minutes, and on the slow days, twenty five. Add one person, and it falls to half a minute. That's a huge change. Queues don't get worse gently. As you get close to full capacity, waiting times explode. That's why a road that's ninety percent full is a traffic jam, and a hospital which runs at a hundred percent can't cope with a bad day.

import numpy as np

def simulate(baristas, per_hour=48, minutes_each=2.2, hours=2000, seed=0):
    rng = np.random.default_rng(seed)
    gaps = rng.exponential(60 / per_hour, int(per_hour * hours))
    free = np.zeros(baristas)
    t, waits = 0.0, []
    for gap in gaps:
        t += gap
        i = free.argmin()
        start = max(t, free[i])
        waits.append(start - t)
        free[i] = start + rng.exponential(minutes_each)
    return np.mean(waits), np.quantile(waits, .95)

for n in [2, 3, 4]:
    mean, worst = simulate(n)
    print(f"{n} baristas: average wait {mean:4.1f} min, slow days {worst:4.1f} min")

2 baristas: average wait  7.6 min, slow days 25.3 min
3 baristas: average wait  0.6 min, slow days  3.4 min
4 baristas: average wait  0.1 min, slow days  0.9 min

The lesson of utilisation

Managers love to see staff busy all the time. But the maths says that a team at full capacity has endless queues. Some slack isn't waste. It's what absorbs randomness. Whenever you've got a server, a support team, or a machine, keep usage well below a hundred percent. Around seventy or eighty percent is a common target for good response times.

Simulate or calculate?

You can solve simple queues with formulas. But real cafes have breaks, rushes, and people who walk away. Simulation handles all of that with a few lines of Python. So whenever a system has randomness and waiting, simulate it. It's how airports, hospitals, and call centres plan their staffing.

Summary