The parts of a queue
Every queue has arrivals, a waiting line and servers. At a router's output port the customers are packets. Here they arrive at λ (lambda) = 600 packets per second: on average one every 1.67 ms.
The server does the work; here it is the outgoing link. Sending one 1,500-byte packet takes 12,000 bits ÷ 12 Mb/s = 1 ms. So the service rate μ (mu) is 1,000 packets per second.
Utilisation ρ (rho) = λ ÷ μ = 600 ÷ 1,000 = 0.6: the link is busy 60% of the time (the lamp) and idle 40%. With perfectly even arrivals each packet leaves before the next comes, so nobody waits.
Randomness and load
Real arrivals are random: sometimes several come close together and must wait, even though the link is idle 40% of the time. With random arrivals and 1 ms service, packets wait 0.75 ms on average, and 0.45 are in line.
Raise arrivals to 900 per second: ρ = 0.9. The link is still idle 10% of the time, yet the average wait jumps from 0.75 ms to 4.5 ms, 6 times longer, with 4.05 packets in line.
Now 1,500 packets arrive each second but only 1,000 can leave: ρ = 1.5. The line grows by 500 packets every second until the buffer fills and packets are dropped. With any randomness, even λ = μ grows without limit.
More servers, and names
Add a second link of the same speed: c = 2 servers share one line, so ρ = λ ÷ (c × μ) = 1,500 ÷ 2,000 = 0.75. The backlog drains, because a queue stays stable only while λ < c × μ.
Engineers name queues A/S/c: arrival pattern / service pattern / number of servers. M means random (Markov, memoryless), D fixed (deterministic), G general (any pattern). Random arrivals, fixed 1 ms packets, two links: M/D/2.
The same three numbers fit a supermarket (shoppers, tills), a web server (requests, worker threads) and a phone exchange (calls, lines), where Erlang began this maths in 1909. Measure λ and μ, count c, and check ρ < 1.
The arithmetic
;; step 1/9 · arrivals · arrive ;; arrivals → waiting line a queueing system = arrivals + a waiting line + one or more servers λ (lambda) arrival rate 600 packets/s 1 / λ mean gap 1,000 ms / 600 = 1.667 ms customers can be packets, shoppers, web requests or phone calls
The animation shows one short stretch of arrivals; queue lengths in it come and go, while the meter gives long-run averages. The tray draws at most 8 envelopes.