One game I play while out & about is applying Little’s law of queuing theory & operations management to the ‘queues’ I see around me in many forms.

Little’s law (not to be confused with the equally valuable Littlewood’s Law!) is one of those statistical laws which looks too simple to be true: ‘the number of average customers’ (length of queue, L) is ‘the new customer arrival rate’ (λ) times ‘the average wait’ (W), or formally:

That is, if it takes half an hour to deal with the DMV clerk and there’s 100 people arriving per hour, then there will be a line of 50 people waiting, because 50 = 0.5 × 100. Simple as that!

This is useful for planning when you know all 3 variables; but you can also easily use Little’s law when you only know 2⁄3 variables, because the algebra is so simple you can intuitively solve for 1⁄3 variables in whichever direction you like: arrival rate + customer count → average wait, arrival rate + average wait → customer count, or customer count + average wait → arrival rate.

It applies to them all: grocery checkout lines, gas stations, to-do lists, email inboxes, coffee shops, banks, post offices, fast food drive-throughs, movie theaters, concerts, financial floats, ecology population sizes or sustainability, COVID disease cases, number of active satellites in orbit, number of customer support employees you need, how many cars there are on your current highway…

If you have a system which takes in things over time and does something with them, then you can try applying Little’s law in some direction.

It provides a nice distraction while waiting, and I find that it’s comforting to have a better understanding of the system I am now a part of (waiting times always feel much longer when more uncertain), and sometimes when you turn the Little’s law crank, you’ll learn something you didn’t know:

1. If you walk into a restaurant and you see lots of people have recently arrived (high arrival rate) and there aren’t many tables full of waiting customers (low number of customers), then you can relax:

   the food must be served quickly (ie. a low average wait → fast turnover), because otherwise all those arriving customers would overflow the restaurant’s few tables

2. If you walk into a Disney World ride and see a sign warning you that it’s a 2-hour wait from your point (high average wait) and you can see that 50 people have arrived in the few minutes that you’ve been in the line (high arrival rate), then you know to expect the worst…

   …and you won’t be so unpleasantly surprised by the theme park designer’s clever little architecture tricks—like when you turn the corner & suddenly see a whole new labyrinth full of waiting people (high number of customers)

   <p>And there are a <em>very</em> high number of customers at Disney theme parks which must be served these days, causing lines to become extraordinarily long. The last time I was at Disney World &amp; Universal in the late 2010s, I was shocked how long the lines were and how much time we were all wasting standing around—it seemed like a truly cursèd <a href="https://en.wikipedia.org/wiki/All-pay_auction">all-pay</a> <a href="https://en.wikipedia.org/wiki/Dollar_auction">dollar auction</a>, like going to college, where we competed to waste the most time. Despite the extraordinary accomplishments of Disney World (not a single mosquito!), I noted that most people did not seem as happy as they could’ve been if they had instead either stayed home or paid more money to skip waits—suggesting Disney is struggling to make the <a href="https://en.wikipedia.org/wiki/Market_clearing">market clear</a> without taking undeserved flak.</p>
   <p>No matter how much shortages of things like GPUs or Taylor Swift concert tickets or Air Jordan sneakers is the fault of consumers, they will blame anyone but themselves—scalpers, Ticketmaster, ‘bots’, the original manufacturer or artist—<em>anyone</em> but themselves!</p>
   <p>I, for one, resolved to do my part to help the market clear by not revisiting (although descriptions of the newer <em>Star Wars</em> expansions like <a href="https://en.wikipedia.org/wiki/Star_Wars:_Galaxy%27s_Edge"><em>Star Wars</em>: Galaxy’s Edge</a> did tempt me).</p>

3. If you show up at the barber where you can see 3 other customers and you recall getting your hair cut takes half an hour, then you know to expect a few more customers to arrive during your cut.

4. If you are rushed for time on something time-sensitive, you can use it to cut your losses. At one point post-COVID I arrived somewhat late to an opera broadcast, having not bought an e-ticket in advance, and was trapped waiting in the unexpectedly combined (and long) ticket-food line.

   Because of Little’s law, I could estimate fairly well how long the wait was going to be, that I was going to miss some of the best parts, and knew to cut my losses.

What if we only know 1⁄3 variables? This is less useful because there is a tradeoff between the missing 2 variables explaining the one known variable (you can’t solve for two unknowns), but you can still sometimes use this in a Fermi estimate or to probe our prior beliefs.

For example, I see deer occasionally in my area, and they don’t seem to be the same ones; and I’ve read that deer life expectancy is 3–4 years; if I assume that I’m seeing 1% of deer (by loose analogy to the Internet’s 1% rule), then that would seem to imply that there must be a huge amount of turnover and young fawns I am never seeing (in order to have a life expectancy across all the deer of only 3 years). And that itself seems a bit unlikely, and so suggests that maybe I am seeing the same deer multiple times or I’m seeing more of the local deer than I realized.

I’ve found that the more I look, the more I find Little’s law everywhere.