Erlang B Finite-Server Loss

Worked Example

Consider a system with offered load A = 2 and c = 2 servers.

1. First, we initialize the blocking probability recursion with B_0 = 1.
2. Next, we calculate B_1 = (2 · 1)/(1 + 2 · 1) = 2/3.
3. Then, we calculate the blocking probability for 2 servers: B_2 = (2 · 2/3)/(2 + 2 · 2/3) = (4/3)/(10/3) = 0.4.

Thus, the blocking probability is 0.4. The carried load is A(1 - B_2) = 2(1 - 0.4) = 1.2, and the mean number of occupied servers is also 1.2. The server utilization is 1.2 / 2 = 0.6.

Experiment

To explore the Erlang B model, use the interactive module at https://riano.app/module/erlang-b-finite-server-loss.

1. Change the Offered load select to observe how increasing the traffic increases the blocking probability.
2. Change the Servers select to see how adding capacity reduces the blocking probability and increases the carried load.

Limitations

The Erlang B formula assumes Poisson arrivals and independent exponential service times. The system has no waiting room (zero queue capacity), meaning any arrival finding all c servers busy is immediately lost. The offered load must be non-negative (A ≥ 0), and the number of servers must be a positive integer (c ≥ 1). This model does not apply to systems where arrivals queue after loss, such as the M/M/1 model.

References

The stable recursive calculation of the Erlang B formula is discussed in https://arxiv.org/abs/2308.07344.