Prove Little's law on a finite horizon

ProofIntermediate⏱ 1 hourAI Systemslittle's lawsample patharea argument

Problem

Let items arrive at times $a_1 \le a_2 \le \cdots$ and depart at $d_i = a_i + W_i$ , with $N(t)$ the number present at time $t$ .

Prove the exact sample-path identity $\int_0^T N(t)\,dt \;=\; \sum_{i\ge1}\bigl|[a_i,d_i)\cap[0,T]\bigr|$ for every $T > 0$ , with no probabilistic assumptions.
Deduce the two-sided bound $\sum_{i: d_i \le T} W_i \le \int_0^T N \le \sum_{i: a_i \le T} W_i$ .
Conclude $\overline L = \lambda\overline W$ under the hypothesis that $A(T)/T \to \lambda$ and $\frac1n\sum_{i\le n}W_i \to \overline W$ , being explicit about where each hypothesis is used.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints — 0 of 2 revealed

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Problem	Field	Type	Difficulty	Time
Apply Little's law to a serving target	LLM Ops	Proof	Intermediate	30 min
Recover Little's law from an event log run	AI Systems	Coding	Intermediate	1 hour
Simulate M/M/1 and verify the utilisation wall run	AI Systems	Coding	Intermediate	1 hour