Loss Networks

Analysis of networks with blocking.

The lossn module implements algorithms for loss networks where arrivals can be blocked when resources are unavailable.

Key function categories:

Loss Network Analysis Algorithms.

Native Python implementations for analyzing loss networks using Erlang formulas and related methods.

Key algorithms:

lossn_erlangfp: Erlang fixed-point algorithm for loss networks erlang_b: Erlang B blocking probability erlang_c: Erlang C delay probability

lossn_erlangfp(nu, A, c, tol=1e-8, max_iter=1000)[source]

Erlang fixed point approximation for loss networks.

Calls (jobs) on route (class) r arrive according to Poisson rate nu_r. Call service times on route r have unit mean.

The link capacity requirements are:

sum_r A[j,r] * n[j,r] < c[j]

for all links j, where n[j,r] counts calls on route r on link j.

Parameters:

  • nu (ndarray) – Arrival rates vector (R,) for each route.

  • A (ndarray) – Capacity requirement matrix (J, R) - A[j,r] is capacity required on link j by route r.

  • c (ndarray) – Capacity vector (J,) - c[j] is capacity of link j.

  • tol (float) – Convergence tolerance.

  • max_iter (int) – Maximum iterations.

Returns:

  • qlen: Mean queue-length for each route (R,)

  • loss: Loss probability for each route (R,)

  • eblock: Blocking probability for each link (J,)

  • niter: Number of iterations

Return type:

Tuple of (qlen, loss, eblock, niter) where

Example

>>> nu = np.array([0.3, 0.1])
>>> A = np.array([[1, 1], [1, 4]])  # 2 links, 2 routes
>>> c = np.array([1, 3])
>>> qlen, loss, eblock, niter = lossn_erlangfp(nu, A, c)
erlang_b(offered_load, servers)[source]

Compute Erlang B blocking probability.

The Erlang B formula gives the probability that an arriving call is blocked in an M/M/c/c loss system.

Parameters:

  • offered_load (float) – Traffic intensity (arrival rate * service time).

  • servers (int) – Number of servers.

Returns:

Blocking probability.

Return type:

float

Example

>>> erlang_b(10.0, 12)  # Offered load 10 Erlang, 12 channels
0.1054...
erlang_c(offered_load, servers)[source]

Compute Erlang C delay probability.

The Erlang C formula gives the probability that an arriving call must wait in an M/M/c queue.

Parameters:

  • offered_load (float) – Traffic intensity (arrival rate * service time).

  • servers (int) – Number of servers.

Returns:

Delay probability (probability of waiting).

Return type:

float

Example

>>> erlang_c(10.0, 12)  # Offered load 10 Erlang, 12 agents