Little's law: L = λ × W
How many packets are sitting inside a router right now? Watch packets arrive in bursts, wait in a queue and go out one at a time, then count them. The average number inside always equals the arrival rate times the average time each one stays: L = λ × W. Use it on a router, a link, a café and a web server.
5 quick questions
Pick an answer to see why. Your score saves with your progress.
01Packets arrive at a router at 1,000 packets per second, and each packet spends 2 ms inside it on average. How many packets are inside on average?
L = λ × W = 1,000 per second × 0.002 seconds = 2 packets. Change milliseconds to seconds first so the units cancel.
02A café gets 30 customers an hour, and each stays 20 minutes on average. How many customers are inside at a typical moment?
30 per hour is 0.5 per minute. L = 0.5 per minute × 20 minutes = 10 customers.
03A web server handles 200 requests per second and has 50 requests open on average. How long does one request take on average?
Rearrange the law: W = L ÷ λ = 50 ÷ 200 per second = 0.25 s.
04What does Little's law need to be true?
Little's 1961 proof makes no assumption about the arrival pattern, the service times or the order of service. It only needs the long-run averages to exist, which fails if the queue grows forever.
05A link gets 1,000 packets per second and takes 0.8 ms to send each one. What fraction of the time is it busy?
Draw the box around the link alone: it holds λ × S = 1,000 × 0.0008 = 0.8 packets on average. It holds 0 or 1, so it is busy 80% of the time. This is its utilisation.
Glossary
- System
- Any box with a way in and a way out that you choose to count: a router's queue, one link, a whole network, a café.
- Arrival rate (λ)
- How many items enter the system per unit of time, averaged over a long time. λ is the Greek letter lambda.
- Time in system (W)
- How long one item stays inside on average, from arriving to leaving: waiting time plus service time.
- Number in system (L)
- How many items are inside on average over time. Little's law: L = λ × W.
- Service time (S)
- How long the server spends on one item; for a link, packet size ÷ link speed.
- Utilisation (ρ)
- The fraction of time a server is busy: ρ = λ × S. It must stay below 1, or the queue grows without limit.
- Stable system
- A system where, in the long run, items leave as fast as they arrive, so the averages settle down.
- Erlang
- The unit of traffic intensity in telephony (ITU-T E.600): arrival rate × mean holding time, the average number of busy lines.
Notes and sources
- The arrival pattern (8 packets repeating every 8 ms) is illustrative; every average shown is computed from it exactly.
- The drawing runs 1,000 times slower than the network, so one millisecond takes one second on screen.
- Little's law is exact over any period that starts and ends with the system empty, and holds for long-run averages in any stable system (Little 1961; Little 2011).
- In telephone traffic, arrival rate × mean holding time is the traffic intensity in erlangs (ITU-T E.600): the average number of busy lines.
- The café and web-server numbers are round illustrative values.
- J. D. C. Little, A Proof for the Queuing Formula: L = λW, Operations Research 9(3), 1961
- J. D. C. Little, Little's Law as Viewed on Its 50th Anniversary, Operations Research 59(3), 2011
- L. Kleinrock, Queueing Systems, Vol. 1: Theory (Wiley, 1975), §2.1
- M. Harchol-Balter, Performance Modeling and Design of Computer Systems (Cambridge, 2013), ch. 6
- D. Bertsekas and R. Gallager, Data Networks, 2nd ed., §3.2 Little's theorem
- MIT OpenCourseWare 6.262 Discrete Stochastic Processes (R. Gallager)
- ITU-T E.600: Terms and definitions of traffic engineering
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