Read a line

A line grows by the gap between arrivals and what a step finishes, every hour the gap lasts; a step close to full has long, lumpy lines, and only more capacity or fewer arrivals shortens one.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

You will explain a line as arrivals outpacing a step, compute how busy the step is, how fast its line grows and how long it is after a rush, plot it growing, and choose a change that actually shortens it.

2. What you already have

You can work out what a step can finish in an hour. Customers do not arrive at that rate on cue. When they arrive faster, they wait, and the waiting is a line.

3. Words for this lesson

4. A line is the gap between two rates

At lunchtime about 72 customers an hour reach Neighborhood Kitchen's register, and the register can serve 60. Each hour, 12 more arrive than are served, and they wait. After an hour, 12 are waiting; after an hour and a half, 18.

$$\text{line growth per hour} = \text{arrivals} - \text{what the step finishes}$$

How busy the register is: 72 ÷ 60 × 100 = 120 percent. Anything over 100 percent means the line grows for as long as the rush lasts, and no amount of hurrying or apologizing changes that: only a faster register, a second register, or fewer people arriving at once.

Many common fixes do nothing to the arithmetic. A bigger waiting area holds more of the line. A sign apologizing makes the wait feel better. Numbered tickets make it fairer. Each can be worth doing, and none of them changes how many are served each hour.

Term	What it means
Arrival rate	How many people or jobs reach a step in an hour.
Service rate	How many the step can finish in an hour: its capacity.
Utilization	Arrivals over capacity, as a percentage: how busy the step is.
Line	The people or jobs waiting.
Growth	Arrivals less service, each hour.
Clearing time	How long a line takes to empty once arrivals drop.

Another way: table

Neighborhood Kitchen's register at lunchtime.

	Per hour
Customers arriving	72
Customers the register serves	60
Line grows by	12
How busy the register is	120 percent

Another way: steps

Count arrivals an hour at the step, during the busy period.
Work out the step's capacity an hour.
Divide arrivals by capacity for how busy it is.
Take capacity from arrivals for the growth each hour.
Multiply the growth by the hours of the rush for the line.
Shorten it only by raising capacity or spreading arrivals.

5. Why close to full is already too full

It is tempting to aim for a step that is exactly as busy as it can be, at 100 percent, because nothing is wasted. In practice arrivals are lumpy: people come in bursts, and some take longer than others. A step running at 95 percent on average spends parts of every hour above 100 percent, and its line grows in those parts and has almost no slack to clear afterwards. The result is a line that seems to appear from nowhere and linger for ages.

So a step that customers line up for should normally have spare room: a register that can serve 60 an hour copes comfortably with 45 or 50 arriving, and struggles badly with 58.

6. The method, step by step, and how to check it

Count arrivals. At the step, during the busy period, in fifteen-minute blocks. A tally on a clipboard is enough; multiply a fifteen-minute count by four for an hourly rate.

Work out capacity. From the capacity lesson: sixty times the people over the minutes a customer takes.

Work out the growth. Arrivals less capacity, each hour. If it is negative, the line shrinks by that much.

Work out the line. Growth times the hours of the rush, starting from whatever line there was at the start.

Check against the floor. Count the line at the end of the rush and compare with the figure. If the real line is shorter, some people gave up and left, which is a cost of its own. If it is longer, either arrivals were higher or service slower than the counts showed.

7. Why each step is allowed

Taking the difference of the two rates is allowed because each hour, arrivals add people to the line and service removes them. What is left after an hour is what arrived less what was served, added to whatever was there before.

Multiplying by the hours is allowed when the two rates stay about the same through the rush. If arrivals change partway, work out each part separately and add the results, as the chart does when the rush ends.

Ignoring fixes that do not change either rate is allowed because they cannot change the difference. A bigger waiting area changes where the line stands, not how fast it grows.

8. How long a line takes to clear

When the rush ends, the line does not vanish. If arrivals drop to 40 an hour and the register still serves 60, the line shrinks by 20 an hour. Eighteen people waiting at half past one take 18 ÷ 20, just under an hour, to clear. Customers who arrive at a quarter to two still find a line, even though the rush is over.

This is why lines feel worse than the arithmetic of the rush alone suggests: they outlast it. A business that adds a second register only for the peak half-hour may find the line still there at two o'clock; keeping it open until the line clears is part of the fix.

9. Spreading arrivals

Sometimes the cheaper fix is to change when people arrive, not how fast they are served. Neighborhood Kitchen can take pre-orders for collection at set times, so a third of the lunch customers arrive at 11:45 or 1:15 instead of 12:30. A repair shop can book drop-offs in fifteen-minute slots on Monday morning instead of letting everyone come at opening. A cleaning company can stagger client calls by asking for a preferred callback time.

Each of these lowers the arrival rate at the peak without adding capacity. The arithmetic is the same: if peak arrivals fall below what the step can serve, the line stops growing.

10. Lines the customer cannot see

Not every line has people standing in it. Phone calls waiting on hold, emails waiting for a reply, bikes waiting in the rack and orders waiting to be packed are all lines, and they grow by the same arithmetic. A repair shop with eight bikes in the rack and a mechanic who finishes two an hour has a four-hour line, whether or not anyone is waiting at the counter.

Count these lines too. They decide how long a customer waits for the result, which is often what they complain about. The same two rates, arrivals and service, tell the owner whether the line will grow or shrink today.

11. Making the wait better while fixing it

While the rates are being fixed, the wait can still be made easier. Tell people how long it will be, honestly. Let them do something useful while waiting, such as filling in the drop-off form or looking at the menu. Serve in a clear order so nobody feels skipped.

These matter, because a wait that is explained and fair feels shorter than one that is not. But they are not the fix. If arrivals stay above capacity, the line keeps growing, and the most pleasant waiting area in town will not stop people giving up and leaving. Do both: make the wait decent now, and change the rates for next week.

12. People who give up

A long line loses customers. Some look at it and walk past; others join, wait a few minutes and leave. Either way the sale is gone, and often the next visit too. These lost customers make the line look shorter than the arithmetic says, which can fool an owner into thinking the problem is smaller than it is.

Count them when you count arrivals: a tally of people who looked in and left, or who left the line. If the arithmetic says 18 should be waiting and only 10 are, roughly 8 people gave up, and at an average order of 12 dollars, that is about a hundred dollars of lunch lost in one rush. A figure like that makes the case for a second register far better than the length of the line alone.

13. Reading the line every week

Lines change as the business changes. A new office building opens nearby and lunch arrivals jump; a new menu item takes longer to make and the register's capacity falls. Count arrivals and the line at the busiest hour once a week, and write both down next to the step's capacity.

When arrivals start creeping toward capacity, say past 85 percent, plan the fix before the line appears: book extra help for that hour, open pre-orders, or simplify the slowest item. Acting at 85 percent is cheap; acting at 120 percent, after regulars have started going elsewhere, is not.

14. In the world: a Tampa bagel shop splits its morning rush

A bagel shop in Tampa had a line out the door from 7:30 to 8:30 every weekday. The owner counted for a week: about 90 customers arrived in that hour, and the counter, with two people taking orders and making bagels, served about 70. The line grew by 20 in the hour and took another 40 minutes to clear, by which time some regulars had started going elsewhere.

Rather than hire a third person for one hour a day, the owner added online ordering with pickup slots every ten minutes from 7:00, capped at eight orders a slot, and prepared those orders between walk-in customers in the quieter half hour before 7:30. About a quarter of the morning customers moved to the slots, many of them earlier.

Walk-in arrivals in the peak hour fell to about 65, below the counter's 70. The line stopped growing, the morning sales held steady, and the regulars came back. The counter was no faster; the arrivals were spread.

15. In the world: queueing theory

The mathematics of waiting lines, called queueing theory, was developed for telephone exchanges in the early twentieth century and is now used by banks, hospitals and call centers. One of its main results is the one in this lesson: as a server's utilization approaches one hundred percent, average waits grow very quickly, which is why well-run services keep spare capacity at their busiest steps.

16. Where this goes wrong

Lines are caused by impatient customers. They are caused by arrivals outpacing service.

A step at 100 percent is efficient. With lumpy arrivals, it guarantees long waits.

If we hurry, the line will clear. Only if what is finished each hour rises above what arrives.

17. Bright Home Cleaning's Monday phone

18. Northside Repairs' Monday check-in

19. Neighborhood Kitchen's lunch register

20. Your turn: Maya's open studio day

21. Guided practice

At lunchtime $65$ customers an hour arrive at Neighborhood Kitchen's register, which can serve $59$ an hour. Which change would stop the line growing?

22. Guided practice

Complete the worked solution: at Neighborhood Kitchen's breakfast counter, about $61$ customers an hour arrive during the morning rush, and the counter can serve $50$ an hour. The rush lasts $3$ hours and there is no line at the start. Find how busy the counter is as a percentage, how much the line grows each hour, and the line at the end of the rush.

23. Guided practice

At Bright Home Cleaning, people arrive at the booking phone on Monday at about $20$ an hour, and it can finish $16$ an hour. How busy is the step, as a percentage of what it can finish?

24. Practice

At Maya's Ceramics, people arrive at the register at the Christmas fair at about $30$ an hour, and it can finish $24$ an hour. The rush lasts $4$ hours and there is no line at the start. How many people are waiting at the end of it?

25. Practice

Customers arrive at Monica's stall at $24$ an hour on Saturday morning, and the stall serves $20$ an hour. There is no line at eight o'clock. Plot the line after $1$, $2$ and $3$ hours.

26. Practice

A pharmacy in Orlando has a pickup counter staffed by $2$ technicians, each taking $3$ minutes a customer. From five to seven in the evening, about $51$ customers an hour arrive. There is no line at five o'clock. How many people are waiting at seven? Fill in each figure on the sheet.

27. Somewhere new

A post office counter always has a long line at lunchtime. Mark every explanation that accounts for it.

28. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

29. Test question

At Bright Home Cleaning, people arrive at the booking phone on Monday at about $20$ an hour, and it can finish $16$ an hour. The rush lasts $2.5$ hours. Fill in how busy the step is (percent), how many the line grows by each hour, and the line at the end of the rush.

30. What you can do now

You can say why a line forms and how long it will get from two rates. Tell someone why a bigger waiting area does not shorten a line. Next: planning for the peak that comes every week.

	Value
How busy, percent
Growth each hour
Line at the end