Little's law
📐FORMULA
$$L = \lambda,W$$
- L: the average number of items in the system
- λ: the average arrival rate (in a stable system, also the rate at which items leave)
- W: the average time an item spends in the system
Named after John Little, who proved it in 1961. It asks for almost nothing: no particular pattern of arrivals, no particular order of service, only a system that is stable over the long run, with as much leaving as arriving.
My backlog as a queue. Rearranged, W = L / λ. Suppose a backlog holds 120 items and the team finishes or deletes 8 a week:
- W = 120 / 8 = 15 weeks
That is the average wait for something entering today, whatever position I rank it in. Re-ranking changes which item waits fifteen weeks. It does not change the fifteen. Only two things do: a bigger outflow, or a smaller stock.
This is 💎 A backlog is a stock and only the outflow shrinks it with numbers on it, and 🧩 Stocks and flows with one equation. I worked it out on a Sunday for 📝 Issue 42 - Your backlog is a stock, not a list and cut it from the draft, because one formula halves a newsletter's readership and the bathtub was doing the work already.
If more arrives than leaves, the system is not stable, L grows without limit and the law has nothing to say except that I have a different problem.
Open questions
- Does a backlog ever count as stable? Mine has grown every quarter I have measured it.
- Deleting is an outflow. Is an item deleted after forty weeks a success of the queue or a failure of the intake?
- The same law for a dispatcher's exception list: arrivals per hour, minutes per exception. Parked in 🔭 Queueing theory for support and dispatch.