How to Define Routing with a Transition Matrix¶
There are a number of methods of defining how customers are routed from one node to another after service. This is done with the routing keyword when creating a network.
One common method is by defining a transition matrix, common when defining Jackson networks [JJ57]. A transition matrix is an \(n \times n\) matrix (where \(n\) is the number of nodes in the network) such that the \((i,j)\text{th}\) element corresponds to the probability of transitioning to node \(j\) after service at node \(i\).
For example, in a three node network, a transition matrix might look like:
- This represents that:
customers finishing service at node 1 enter node 2 with probability 0.3, and envet node 3 with probability 0.7;
customers finishing service at node 2 are certain to enter node 3;
customers finishing service at node 3 re-enter node 3 with probability 0.2, and with probability 0.8 leave the system.
In Ciw there are two ways to define this, with a list of lists, or a routing object.
List of Lists¶
To define this with a list of lists, we can use:
>>> import ciw
>>> N = ciw.create_network(
... arrival_distributions=[
... ciw.dists.Exponential(1),
... ciw.dists.Exponential(1),
... ciw.dists.Exponential(1)
... ],
... service_distributions=[
... ciw.dists.Exponential(2),
... ciw.dists.Exponential(2),
... ciw.dists.Exponential(2)
... ],
... number_of_servers=[3, 3, 3],
... routing=[
... [0.0, 0.3, 0.7],
... [0.0, 0.0, 0.1],
... [0.0, 0.0, 0.2]
... ]
... )
Routing Object¶
To define with with a routing object we can make use of the built-in TransitionMatrix routing object, and give it the list of lists above:
>>> N = ciw.create_network(
... arrival_distributions=[
... ciw.dists.Exponential(1),
... ciw.dists.Exponential(1),
... ciw.dists.Exponential(1)
... ],
... service_distributions=[
... ciw.dists.Exponential(2),
... ciw.dists.Exponential(2),
... ciw.dists.Exponential(2)
... ],
... number_of_servers=[3, 3, 3],
... routing=ciw.routing.TransitionMatrix(
... transition_matrix=[
... [0.0, 0.3, 0.7],
... [0.0, 0.0, 0.1],
... [0.0, 0.0, 0.2]
... ]
... )
... )
Routing objects can be much more flexible that this, allowing logic based routing in addition to probabilistic based routing. More information can be found here.
