Agile — Scrum (sprints, roles, ceremonies), Kanban: Worked examples

This page does one job: take the three "engines" of Agile math — velocity forecasting (Scrum), Little's Law (Kanban), and the burndown line (Scrum's progress ruler) — and run them through every case they can throw at you. Nice clean numbers, ugly fractions, zeros, degenerate teams, real-world word problems, and one exam trap. If a scenario exists, it's below.

If any word here feels unfamiliar, that's fine — we rebuild every idea from zero as we hit it. Parent: Agile — Scrum & Kanban.

The scenario matrix

Before working anything, let's list every kind of case these formulas produce. A "case class" is a situation where the behaviour of the formula changes — a new sign, a rounding jump, a division that misbehaves, or a word-problem wrapping.

#	Case class	What's special about it	Which example
C1	Velocity, clean divide	backlog ÷ velocity is a whole number → no rounding	Ex 1
C2	Velocity, fractional divide	leftover points → ceiling jumps up	Ex 2
C3	Velocity, noisy data	one outlier sprint → why the mean smooths it	Ex 3
C4	Velocity, degenerate	a zero-point sprint (team blocked) in the average	Ex 4
C5	Little's Law, solve for $W$	given $L, λ$ → find wait time	Ex 5
C6	Little's Law, WIP cut	lower $L$ , same $λ$ → shorter $W$ (the core Kanban claim)	Ex 6
C7	Little's Law, limiting case	$L \to 0$ and $λ \to 0$ → what "empty board" means	Ex 7
C8	Word problem, mixed units	rate given per day, answer wanted in weeks → unit conversion	Ex 8
C9	Exam twist	a subtle "which lever?" question that punishes plug-and-chug	Ex 9
C10	Burndown line	reading "ahead/behind" off the ideal line, all day-values	Ex 10 (figure)

Every row is covered below. Let's go.

Velocity examples (Machine 1)

Ex 1 — Case C1: the clean divide

Ex 2 — Case C2: the fractional divide (ceiling bites)

Ex 3 — Case C3: noisy data, why the mean

Ex 4 — Case C4: degenerate — a zero-point sprint

Worked example Statement

The team was blocked by an outage one sprint: completions were $24, 0, 21, 15$ . Backlog $B = 180$ .

Forecast: does the zero sprint break the average? Should we drop it?

Step 1 — keep the zero, it's real data. $\overset{v}{ˉ} = (24 + 0 + 21 + 15) /4 = 60/4 = 15$ . Why this step? Outages happen; a forecast should include the risk that they happen again. Deleting the 0 would over-promise. The mean of 15 already "prices in" the bad sprint.

Step 2 — forecast. $180/15 = 12.0$ , $⌈ 12.0 ⌉ = 12$ . Why this step? We turn the smoothed speed into a sprint count exactly as before — "backlog ÷ velocity, then ceiling". Here $180/15$ lands on a whole number, so the ceiling leaves it at 12.

Answer: 12 sprints.

Verify: $12 \times 15 = 180$ ✓. Sanity on the degenerate value: $\overset{v}{ˉ} = 15 > 0$ , so the division is safe. (The truly broken case would be $\overset{v}{ˉ} = 0$ — every sprint scored 0 — which means the forecast is undefined / infinite: a team that completes nothing never finishes. The formula honestly refuses to divide by zero, which is the correct answer: "never".)

Little's Law examples (Machine 2)

Ex 5 — Case C5: solve for wait time

Ex 6 — Case C6: cut WIP, same throughput → faster (the core claim)

Worked example Statement

Same board, $λ = 5$ cards/week. You impose a WIP limit so only $L = 10$ cards may be in progress. New lead time?

Forecast: fewer cards on the board — does each finish sooner even though the team's speed (5/week) didn't change? Guess yes/no.

Step 1 — recompute $W$ . $W = 10/5 = 2$ weeks. Why this step? Only $L$ changed; $λ$ stayed 5. So we just re-divide.

Step 2 — interpret. Lead time dropped from 3 weeks to 2 weeks — a $1/3$ cut — with the same throughput. Why does capping WIP speed things up? A card no longer waits behind 14 others; it waits behind 9. Less queueing = shorter total stay. This is the reason Kanban enforces WIP limits.

Answer: 2 weeks (down from 3).

Verify: forward check $L = 5 \times 2 = 10$ ✓. And note throughput unchanged: still 5/week both times — we made delivery faster without working faster. ✓

Ex 7 — Case C7: the limiting cases

Worked example Statement

Two extremes on a Kanban board: (a) The board is empty: $L = 0$ , $λ = 5$ . (b) The team is fully blocked: $λ = 0$ , $L = 8$ . What does $W$ do in each?

Case (a) — empty board. $W = 0/5 = 0$ weeks. Why this makes sense: no cards means no card is waiting — average wait is 0. The formula behaves gracefully at zero WIP.

Case (b) — zero throughput. $W = 8/0 =$ undefined (→ ∞). Why this makes sense: if nothing ever leaves ( $λ = 0$ ), a card on the board waits forever. Division by zero is the maths correctly screaming "infinite lead time". This is the mirror of Ex 4's zero-velocity team: a system that never delivers has no finite answer.

Answer: (a) $W = 0$ ; (b) $W$ infinite/undefined.

Verify: (a) forward: $L = λW = 5 \times 0 = 0$ ✓. (b) there's no finite $W$ with $L = λW \Rightarrow 8 = 0 \cdot W = 0$ , which is impossible — confirming no finite lead time exists. ✓

Ex 8 — Case C8: word problem with mixed units

Ex 9 — Case C9: the exam twist ("which lever?")

Worked example Statement

A manager says: "Lead time is 4 weeks; I want it halved to 2 weeks. Just tell the team to work twice as fast." The board has $L = 20$ , $λ = 5$ /week. Is doubling throughput the only way? What's the cheaper lever?

Forecast: the parent note hinted throughput is "hard to force". So what else can move $W$ ?

Step 1 — confirm current $W$ . $W = 20/5 = 4$ weeks. ✓ matches the manager.

Step 2 — write $W$ in terms of both levers. $W = L / λ$ . To halve $W$ , you may either halve $L$ or double $λ$ — the ratio is what matters. Why this step? $W$ depends on the ratio $L / λ$ , not on either alone. Two knobs, same effect.

Step 3 — the cheap lever: halve $L$ . Set WIP limit $L = 10$ (same $λ = 5$ ): $W = 10/5 = 2$ weeks. Why cheaper? Cutting WIP is a policy (a number on the board) — free and instant. "Work twice as fast" means doubling $λ$ to 10/week — hiring or heroics, expensive and often impossible.

Answer: No, doubling throughput isn't the only way. Halving WIP ( $L : 20 \to 10$ ) halves lead time to 2 weeks for free — the Kanban lever.

Verify: cheap path forward: $L = λW = 5 \times 2 = 10$ ✓. Expensive path forward: $λ = 10$ , $L = 20 \Rightarrow W = 20/10 = 2$ ✓. Both reach 2 weeks; only one is free. This is the trap: plug-and-chug on $λ$ misses the WIP lever.

Burndown example (Machine 3 — the ideal line)

Ex 10 — Case C10: reading ahead/behind at every day

Figure — Agile — Scrum (sprints, roles, ceremonies), Kanban

Worked example Statement

Sprint: $W = 40$ points, $D = 10$ days. Compute the ideal remaining at $d = 0, 5, 10$ . On day 5 the team actually has 25 points left. Ahead or behind?

Forecast: ideal at the midpoint should be half done — so 20 left. Real is 25. Which side?

Step 1 — endpoints. $R (0) = 40 (1 - 0) = 40$ ; $R (10) = 40 (1 - 1) = 0$ . Why this step? Sanity check the line: it must start full (40, nothing done) and end empty (0, all done). In the figure, the black straight line runs from the top-left corner (day 0, 40 points) down to the bottom-right (day 10, 0 points) — that diagonal is the ideal.

Step 2 — midpoint. $R (5) = 40 (1 - \frac{5}{10}) = 40 \times 0.5 = 20$ points. Why this step? The ideal is the reference; we need its exact value where we'll compare — halfway through, so we expect half the work gone. The black line passes through $(5, 20)$ , marked on the figure.

Step 3 — compare to reality. Actual $= 25$ , ideal $= 20$ . Since $25 > 20$ , the red actual line sits above the black ideal at day 5. Why "above = behind"? Above means more work remaining than planned → you're behind schedule. (Below the line = ahead.) In the figure, look at the red actual line poking above the black ideal at day 5, and the red double-arrow marking the "gap = 5" between them — that vertical gap is exactly how many points behind you are.

Answer: behind by $25 - 20 = 5$ points at day 5.

How to read the whole figure: any day the red line is above black → behind; on black → exactly on plan; below black → ahead. The steeper the red drops, the faster the team is burning work.

Verify: slope of ideal $= - W / D = - 40/10 = - 4$ pts/day, so from 40 it should read $40 - 4 \times 5 = 20$ ✓ (matches Step 2). Gap $= 5$ points behind ✓.

Recall Self-test (reveal after guessing)

Backlog 250, velocity 20 → sprints ::: $⌈ 12.5 ⌉ = 13$ Board $L = 15$ , $λ = 5$ /wk → lead time $W$ ::: $15/5 = 3$ weeks Cut WIP to $L = 10$ (same $λ = 5$ ) → new $W$ ::: $10/5 = 2$ weeks Zero-throughput board ( $λ = 0$ ) → $W$ ::: undefined / infinite (nothing ever leaves) Ideal burndown at day 5 of a 40-pt / 10-day sprint ::: $40 (1 - 5/10) = 20$ points To halve lead time for free, which lever? ::: halve WIP $L$ (throughput is expensive to raise)

Related: Little's Law · Project Estimation & Story Points · Continuous Integration & Delivery (CI-CD) · Lean Manufacturing · parent topic