Erlang B

353. Erlang B

The Erlang B formula: probability that all servers in an M/M/c/c queue (Poisson arrivals, exponential service, $𝑐$ servers, $𝑐$ capacity = no queue) are busy. Arriving customers facing all-busy are blocked / lost.

353.1. Formula

𝐵 (𝑐, 𝑎) = \frac{\frac{𝑎^{𝑐}}{𝑐!}}{\sum_{𝑘 = 0}^{𝑐} \frac{𝑎^{𝑘}}{𝑘!}}

where $𝑎 = \frac{𝜆}{𝜇}$ is the offered load (in Erlangs).

$𝐵 (𝑐, 𝑎)$ is the blocking probability — proportion of arrivals turned away.

353.2. Telephony origin

Developed by A. K. Erlang for telephone networks (1917). $𝑐$ servers = trunk lines; blocked calls are dropped (no queue — caller hears busy signal).

Used for over a century to size phone networks: how many trunk lines do I need to keep blocking probability below some threshold (typically 1% or 0.1%)?

353.3. Worked example

$𝜆 = 50$ calls/hour, average call lasts $3$ minutes = $0.05$ hr. So $𝑎 = 50 \cdot 0.05 = 2.5$ Erlangs.

How many trunks to keep blocking below 1%?

$𝑐$	$𝐵 (𝑐, 𝑎) \approx$
3	$35 %$
5	$9 %$
7	$1.3 %$
8	$0.4 %$

Need $𝑐 = 8$ trunks for $< 1 %$ blocking with offered load 2.5 Erlangs.

353.4. Recursive formula (numerically stable)

𝐵 (𝑐, 𝑎) = \frac{𝑎 𝐵 (𝑐 - 1, 𝑎)}{𝑐 + 𝑎 𝐵 (𝑐 - 1, 𝑎)}, 𝐵 (0, 𝑎) = 1

This recursion avoids the factorials that overflow in direct computation. Standard implementation.

353.5. Erlang B vs Erlang C

Model	System	Blocked customer
Erlang B (M/M/c/c)	c servers, no queue	lost (dropped)
Erlang C (M/M/c)	c servers, infinite queue	waits

Erlang B is the blocking probability (proportion lost). Erlang C is the waiting probability (proportion who must wait > 0).

For the same parameters, Erlang B < Erlang C.

353.6. Where it shows up

Telephone trunk sizing (historical, but still relevant for VoIP gateways)
Connection limits on servers (max concurrent connections)
Hospital bed capacity (refused-admission probability)
Parking lot sizing (turn-away rate)
Toll booth design

353.7. Insensitivity to service distribution

Surprisingly: Erlang B is insensitive to the service-time distribution — only the mean matters. So even if call durations are highly variable, Erlang B’s blocking probability formula stays exact. Holds because of PASTA (Poisson Arrivals See Time Averages) + reversibility.

353.8. See also

Erlang C — with queue
M/M/1 — single-server with queue
M/M/c — multi-server with queue
Erlang A — with abandonment