The model
Packets arrive at random times, wait in one line, and one server sends them in order. Here the server is a router's output link. Queueing theory calls this M/M/1: random arrivals, random service times, one server.
The link sends an average 1,250-byte packet in 1 ms at 10 Mb/s, so it can serve μ = 1,000 packets per second. Packet sizes vary, so each service time is random; 1 ms is the average.
Packets arrive at λ = 500 per second on average. Utilisation, or load, is ρ = λ/μ = 500 / 1,000 = 0.5: the link is busy half the time. Arrivals still come in random bursts.
For M/M/1 the average number of packets in the system is L = ρ/(1−ρ), and the average time each spends there is W = 1/(μ−λ). At ρ = 0.5: L = 1 packet, W = 2 ms, twice the service time.
Turning up the load
Raise arrivals to 800 per second: ρ = 0.8. Now L = 0.8/0.2 = 4 packets and W = 1/(1,000 − 800) = 5 ms, five times the service time. The link is 60% busier, but delay is 2.5 times longer.
At 900 per second, ρ = 0.9: L = 9 packets and W = 10 ms, ten times the service time. Going from 80% to 90% load adds 12.5% more traffic but doubles the delay.
At 950 per second, ρ = 0.95: L = 19 packets and W = 20 ms. Another 5.6% of traffic doubles the delay again. The tray shows at most 8 waiting; the count gives the real average.
At 990 per second, ρ = 0.99: L = 99 packets and W = 100 ms, a hundred times the 1 ms service time. The link carries only 10% more than at 90%, yet each packet takes ten times as long.
What it means
Time in the system, counted in service times, is 1/(1−ρ). The spare capacity, 1−ρ, is what clears random bursts. At 50% load half the time is spare; at 99% only 1% is, so each burst takes far longer to drain.
If packets arrive as fast as the link sends them or faster (ρ ≥ 1), there is no steady state: the line keeps growing. At 1,050 per second it grows by about 50 a second until the buffer is full and the router drops packets.
That is why networks are run well below full load. At 50% a packet spends 2 ms in the system; at 99%, 100 ms. Adding capacity, or keeping queues short with active queue management, keeps delay low when traffic bursts.
The arithmetic
;; step 1/11 · one-server · model ;; arrivals → internet Kendall notation A/S/c: arrival process / service times / number of servers M = Markovian (memoryless): Poisson arrivals, exponential service times 1 = one server; the line is first in, first out, with unlimited room arrivals ─► queue ─► server ─► out
The tray draws at most 8 waiting envelopes plus the one being sent; the count gives the real average number in the system. L and W are long-run averages. At any moment the line may be empty or much longer: at ρ = 0.9 the chance of finding 10 or more packets already there is about 35%.