Random arrivals
Many users send requests to one server. Nobody plans when; each person acts on their own. On average λ (lambda) = 4 requests arrive per second. A pin on the timeline marks each arrival. In this first second, 4 came.
Gaps, counts and memory
The gap between arrivals averages 1/λ = 0.25 s, but varies wildly: here 0.05 s, then 0.50 s. Short gaps are the most common: about a third are under 0.1 s. This spread is the exponential distribution.
Count the arrivals in any one second and you get a Poisson number: average 4, often 2 to 6. Here: 4. Exactly 4 happens 20% of the time; none at all, 1.8%.
At 3.76 s, nothing has arrived for 0.4 s. Is one now "due"? No. Users don't coordinate, so the wait from here is still exponential, mean 0.25 s. The past gap tells you nothing. This is memorylessness.
Bursts and queues
Now 8 arrive in one second, twice the average. Nobody caused it: random times bunch up. At least 8 in a second has a 5.1% chance, so expect it about once every 20 seconds.
The server finishes one request every 0.2 s, no faster. The clump arrived faster than that, so a line forms and waits grow. The request that came at 6.02 s waits 0.80 s, four times its own service time.
Utilisation ρ (rho) = λ ÷ μ = 4 ÷ 5 = 0.8: the server is busy 80% of the time. Spaced evenly, every 0.25 s, nobody would wait. Arriving at random, the average wait is 0.4 s, twice the service time.
Where Poisson fits
Say 400 people each send about once every 100 s, independently. Together that is 4 per second, and their merged times behave like a Poisson process: no one person's habits show.
Poisson bursts smooth out when you count over longer windows. Real packet traffic does not: Leland and colleagues (1994) found Ethernet traffic equally bursty at every time scale (self-similar). Real queues grow longer than Poisson predicts; new user sessions still look Poisson.
The arithmetic
;; step 1/9 · average-rate · t = 0–1 s ;; users → server λ = 4 arrivals per second (the average rate) arrivals in 0–1 s: 0.28, 0.32, 0.38, 0.61 s server: 0.2 s per request, so at most μ = 5 per second the times come from one random draw (seeded generator), not hand-placed
The arrival times are one random draw (seeded generator, λ = 4 per second, 37 arrivals in 10 s); another draw gives different times with the same statistics. The tray shows at most 7 waiting requests; this sample never has more than 5.