Kind of a textbook description of queue theory if you were taking a stochastic processes class. If the author really is suggesting this for use in retail line situations, he is out of touch because... retail managers have no idea what queue theory is. If they did, they would be making much more money, and not working in retail. Staffing is more of a function of how many people you can get on the clock without getting yelled at by regional managers (who also do not know queue theory) for overstaffing.

Retailers do use queue theory (here's an old article about it[1]), it just has to come from the management down rather than individual store managers.

1: https://www.nytimes.com/2007/06/23/business/23checkout.html

That’s not true at all! A good GM understands throughput and may get bonuses based on that.

The retailer I worked at in the 90s also tracked the ring rate and time per customer for cashiers, and made scheduling and hour decisions based on performance.

Thats great and all, but also not queue theory.

What's the formula for making more money by knowing queueing theory?

(Not the application to retail management, which is straightforward, but the application to getting a better job.)

Maybe less literal than you are thinking. A background in mathematics opens you to many lucrative careers.

This is classic open-loop queuing theory with Poisson arrivals. Nothing really works like that. There's always a "rejection rate", where people leave the line before being serviced. Or, in networking, the connection times out. Or, for a shopping site or a real store, you get an "abandoned cart". In real life, this has a lot to do with how many sales you lose.

In the early days of the Internet, people were trying to analyze datagram networks as open-loop systems with Poisson arrivals. That's because everybody read Kleinrock back then. His thesis was on Western Union Plan 55-A, which was Sendmail made out of telephone relays and paper tape punches feeding paper tape readers as buffers. A mail system behaves much more like an open loop system - people don't mail less as the mail propagation time goes up. That was the wrong model for IP messages, because TCP connections add a closed loop system on top of IP.

Seems to me the big problem with any kind of retail queue is highly inconsistent service times. If you have a consistent time/ customer, it's easy to predict. As soon as you have some customers taking 20 minutes and some taking 2 minutes all the simple math goes out the window. This is why some people get so infuriated when they see the person in the front of the line break out a fist full of coupons, food stamps, or wants to pay with a check. It's also why so many businesses use a single queue with multiple tellers/ cashiers.

This happens a lot in retail, the DMV, post office, etc. The best way to model it is with a meta-process where servers randomly disappear and reappear. This also captures the infuriating phenomenon of tellers going on break or (worse) looking like they're available when they're not because they have to stop and count the till or something.

"Seems to me the big problem with any kind of retail queue is highly inconsistent service times."

The article mentions that as one of the two problems: "If customer arrivals were exactly evenly spaced and each took exactly 10 minutes to serve, there would be no problem."

"It's also why so many businesses use a single queue with multiple tellers/ cashiers."

The article implicitly assumes a single queue. But, as long as any empty queue is immediately filled by someone switching from another queue, this assumption doesn't affect the expected total # of people waiting.

Right, variance of wait time -- not just the mean -- is definitely a(n anti-)desideratum, for exactly the reason you gave[1]. I don't know how much emphasis academic queueing theory puts on this though.

[1] To put it another way, you can assume entrants plan for the line to take less than the 95th (or whatever) percentile time; the lower you get this figure, they better they can allocate their time for other tasks.

It's all fun and games until it's not a bank with one teller, but a LabCorp with one phlebotomist, and you're not there to give a blood sample, but a urine sample.

Or you get some one like me with uncommon tests I once had 75% of my local hospitals phlebotomist's working out how to do this particular test.

Did a fun project for my MBA. A bike parking facility downtown, where you can park your bike securely, showers, lockers, towel service, ironing boards, buy some coffee and a snack. Raised $80k for it...needed 100. Anyways

We modeled the showering process. Had some data on how long people took to shower. Needed something like 3 showers per gender to get good throughput

One of the MBA profs didn't like this for some reason. Didn't believe it would be enough (without any numbers to back it up). We were like, well if we add a fourth shower we can accommodate dozens of more people. He liked that even less

Why gender the showers in the first place? Seems like added complication for little to no benefit by splitting into separate queues.

I would guess the showers were inside of a gendered locker room.

If you had them as self-contained shower rooms with changing area, that would probably be better from a personal standpoint and also alleviate the issue you raised.

Not the OP though, so who knows.

we did a survey of cyclists and gendered bathrooms were much preferred. this was 9 years ago though...

Community consultation is the right thing to do! I wonder if modelling might have predicted better service times for the majority gender with non-gendered bathrooms, and that might have swung the survey result.

Yeah it really slows throughput, especially if your users aren’t 50/50 men women. There are locker room layouts that can accommodate mixed genders, like ones where each shower is in a changing stall, e.g. https://i.pinimg.com/originals/1a/ec/31/1aec3156f4f31ca42540...

An event venue near me does a genius thing with the bathrooms.

The restrooms have single line of stalls. There's a moveable wall between the stalls to change the allocation of how many men's stalls there are versus women's. That makes it so if you're running an event that tends to attract more people of one gender than the other, you can create more toilets for that gender. Quilting expo? More women's stalls. PC gaming event? More men's.

A thread from 2015: https://news.ycombinator.com/item?id=9652882

Why I will never live anywhere with roommates, or have an event, anywhere with one bathroom!

Meh, it's not that bad. I lived my 6 years of college in a dorm with 5 people lodged in one room - 15ish m2. One bathroom in each two room duplex containing two sinks, one private toilet and one private shower. Surprisingly, it worked really well. We rarely had to que and if someone was using a part of the bathroom you could still use the rest of the facilities.

The teller post and queueing theory ideas are included here: https://github.com/joelparkerhenderson/queueing_theory

What does it mean that service times are exponential?

Good question! It means that the probably of finishing in the next increment of time is always the same. Just got up to the teller? There's, say, a 20% chance that you're done within the next minute. Been at the teller's desk for 30 minutes already? Still a 20% chance that you're going to be wrapped up in the next minute.

This might be saying the same thing in a slightly different way. But does it also mean that if we plot "service times" on the X axis and frequency on the Y, then in the first quadrant we get something high on the left, getting lower as it goes right, and asymptotically approaching zero as it goes to infinity on the right?

It means that the service time is modeled as an exponential random variable. See https://en.m.wikipedia.org/wiki/Exponential_distribution .

Think about radioactive half-life. Half the atoms will decay in X minutes. Half the remainder in 2X minutes. Half the remainder in 3X etc.

Likewise you'll serve half your customers within X minutes. Half the remainder within 2X, half the remainder within 3X etc.