Operations
Erlang B Trunk Blocking & Erlang C Multi-Server Call Center Delay (M/M/c)
Calculate Agner Krarup Erlang's exact c=3 server Blocking Probability B(3, A), queueing Delay Probability C(3, A) = 3*B / (3 - A*(1 - B)), and Average Speed of Answer (ASA in seconds) from offered traffic Erlangs A = lambda / mu.
Staff call-center pods, ICU beds, and telecom trunks with exact M/M/c queueing!
Offered traffic intensity in Erlangs A = lambda / mu measures the average number of simultaneously busy servers (e.g., lambda = 40 calls/hr with mean handle time 3 min -> mu = 20 calls/hr -> A = 2.0 Erlangs). In a pure-loss system with NO waiting queue (M/M/c/c trunk lines or hospital beds), Agner Krarup Erlang's B Formula gives exact blocking probability: for c = 3 servers, B(3, A) = (A^3 / FACT(3)) / (1 + A + (A^2 / 2) + (A^3 / FACT(3)))!
When callers wait in an infinite FIFO queue instead of getting a busy signal (the M/M/c Call Center model, stable whenever A < c), Erlang's C Probability of Delay relates directly to Erlang B via the exact algebraic bridge: C(c, A) = (c * B(c, A)) / (c - A * (1 - B(c, A)))! Dividing C(c, A) by spare server capacity (c * mu - lambda) and multiplying by 3,600 yields Average Speed of Answer (ASA) in seconds!
| 3-Agent Pod Load (c = 3 Agents, Service mu = 20/hr) | Offered Erlangs A = lambda / mu | Erlang B Blocking B(3, A) | Erlang C Wait Prob & ASA (Sec) |
|---|---|---|---|
| Moderate Load (lambda = 24/hr) | 1.20 Erlangs (40% util) | 8.98% | 14.12% wait | ASA = 14.1 sec |
| Heavy Peak Load (lambda = 40/hr) | 2.00 Erlangs (66.7% util) | 21.05% (= 4/19) | 44.44% (= 4/9) | ASA = 80.0 sec |
| Overloaded Pod (lambda = 60/hr) | 3.00 Erlangs (>= c!) | 34.62% (if loss trunk) | Unstable Queue (A >= c)! |
Exact c = 3 Server Erlang C Probability of Delay C(3, A)
Computes A = B2 / B3, evaluates B(3, A) using FACT(3), and transforms to Erlang C delay probability after checking queue stability B2 < 3 * B3.
=IF(OR(B2 < 0, B3 <= 0), "Invalid Arrival or Service Rate", IF(B2 >= (3 * B3), "Unstable Queue (Offered Erlangs A >= 3)", ROUND((3 * (((((B2 / B3) ^ 3) / FACT(3)) / (1 + (B2 / B3) + (((B2 / B3) ^ 2) / 2) + (((B2 / B3) ^ 3) / FACT(3)))))) / (3 - ((B2 / B3) * (1 - (((((B2 / B3) ^ 3) / FACT(3)) / (1 + (B2 / B3) + (((B2 / B3) ^ 2) / 2) + (((B2 / B3) ^ 3) / FACT(3)))))))) , 4)))Erlang B Loss Probability B(3, A) & Average Speed of Answer ASA (Seconds)
Evaluates exact 3-channel Erlang B blocking probability B(3, A) = (A^3 / 6) / (1 + A + A^2/2 + A^3/6).
=ROUND((((B2 / B3) ^ 3) / FACT(3)) / (1 + (B2 / B3) + (((B2 / B3) ^ 2) / 2) + (((B2 / B3) ^ 3) / FACT(3))), 4)Try the example
Use the original values or change them to test your understanding.