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13.7.3 Important Concepts

This section presents some basic concepts that will prove useful to analyse the
dynamics of computer models. The notation used here follows the excellent book
on stochastic processes written by Kulkarni (1995).

Definition 1 Accessibility A state j is said to be accessible from state i if starting at
state i there is a chance that the system may visit state j at some point in the future.
By convention, every state is accessible from itself. Formally, a state j is said to be
accessible from state i if for some n 0, p (n) i,j >0.
Note that j is accessible from i ¤ j if and only if there is a directed path from i
to j in the transition diagram. In that case, we write i ! j.If i ! j we also say that
i leads to j. As an example, in the THMC represented in Fig. 13.10, s 2 is accessible
from s 12 but not from s 5 . Note that the definition of accessibility does not depend on
the actual magnitude of p (n) i,j , only on whether it is exactly zero or strictly positive.
Definition 2 Communication A state i is said to communicate with state j if i ! j
and j ! i.
If i communicates with j, we also say that i and j communicate and write i $ j.
As an example, note that in the simple random walk presented in Sect. 13.7.1,
every state communicates with every other state. It is worth noting that the relation
‘communication’ is transitive, i.e.

i $ j; j $ k ) i $ k:

Definition 3 Communicating Class A set of states C S is said to be a
communicating class if:
• Any two states in the communicating class communicate with each other.
Formally,

i 2 C; j 2 C ) i $ j

•The set C is maximal, i.e. no strict superset of a communicating class can be a
communicating class. Formally,

i 2 C; i $ j ) j 2 C

As an example, note that in the simple random walk presented in Sect. 13.7.1,
there is one single communicating class that contains all the states. In the THMC
represented in Fig. 13.10, there are four communicating classes: fs 2 g, fs 5 g, fs 10 g,
fs 1 , s 3 , s 4 , s 6 , s 7 , s 8 , s 9 , s 11 , s 12 g.