Phone trunks, support agents, EV chargers, ICU beds — wherever "all busy" means turned away rather than queued, you're running a loss system. Agner Erlang worked out the arithmetic for Copenhagen's telephone exchange in 1917, and it still sizes anything that says no instead of please hold: given the traffic and the number of lines, his B formula gives the share of arrivals that hit a busy signal. Sizing is a tradeoff — every line you add idles most of the day, every line you don't turns callers away. Set the lines and read off the blocking, or set the blocking you'll tolerate and read off the minimum lines — and notice how big pools get away with running far hotter than small ones.
| Lines | Blocking | Lost / day | Busy / line |
|---|---|---|---|
| 3 | 53% | 1,271 | 78% |
| 4 | 40% | 956 | 75% |
| 5 | 28% | 684 | 72% |
| 6you | 19% | 460 | 67% |
| 7 | 12% | 289 | 63% |
| 8 | 7% | 168 | 58% |
| 9 | 3.7% | 90 | 53% |
| 10 | 1.8% | 44 | 49% |
| 11 | 0.83% | 20 | 45% |
| 12 | 0.34% | 8.3 | 42% |
| 13 | 0.13% | 3.2 | 38% |
Read down the blocking column: each extra line saves a shrinking absolute number of calls, yet the percentage falls off a cliff — a steady ratio per line, which is why the chart is log-scaled. And the bigger the pool, the hotter it runs: at 1% blocking, 2 erlangs of traffic needs 7 lines sitting 71% idle, while 50 erlangs gets by with 64 lines at 78% busy each. Pool two trunk groups into one and the combined group needs fewer lines at the same target — that's why consolidation wins. All of this assumes a blocked call vanishes; the moment callers can hold and wait instead, you're in M/M/c territory — the queue cliff.