Stochastic process fundamentals - Modelling with Generalised Stochastic Petri Nets - page 281

for all A J ∈ S 0 , and also

X

η 0 J = 1 (A.62)

A J ∈S 0

where

X

η 0 J = lim n→∞ P { X n ∈ A J } =

j∈A J η j

(A.63)

X X

p 0 IJ =

p ij ν i|I (A.64)

i∈A I j∈A J

η i

ν i|I = (A.65)

P

k∈A I η k

Note that equations ( A.60) to (A.65) hold for all DTMCs, regardless of
the fulfillment of the lumpability condition, so that it is possible to evalu-
ate the steady-state distribution over macrostates from ( A.61) and ( A.62) ,
provided that we can evaluate the quantities p 0 IJ from (A.64) and ( A.65) .
Obviously, we do not want to evaluate the steady-state distribution of the
process { X n ,n ≥ 0 } and then plug the result into (A.64) and ( A.65) , since
the reason to aggregate states into macrostates is exactly to avoid the so-
lution of this problem due to the large number of states. In some cases,
however, it is possible to evaluate the p 0 IJ without solving for the η i .
The lumpability condition (A.59) is su cient to very simply obtain the p 0 IJ
without solving for the η i . In this case ( A.61) becomes

X X

η 0 J η 0 I i ∈ A I

= p ij (A.66)

A I ∈S 0

j∈A J

A.5.2 Aggregation in Continuous Time Markov Chains
Consider a finite, ergodic CTMC { X(t),t ≥ 0 } with state space S = { 1, 2, ··· ,N } ,
infinitesimal generator Q, transition probability matrix P(t), state distri-
bution η(t), and equilibrium distribution η.
Define also in this case a partition of S by aggregating states into macrostates
A I such that ( A.55) holds.
A new stochastic process { Y (t),t ≥ 0 } can be defined on the set of macrostates,
with state space S 0 = { A 1 ,A 2 , ··· ,A M } . It can be shown that the condition
for this new process to be an ergodic CTMC is
X

q ij = q 0 IJ ∀ i ∈ A I (A.67)

j∈A J

The q 0 IJ are the entries of the infinitesimal generator Q 0 of the process
{ Y (t),t ≥ 0 } . This equation represents the condition for the lumpability
of the CTMC { X(t),t ≥ 0 } with respect to the partition S 0 . This condition
also implies that

X

p ij (t) = p 0 IJ (t) ∀ i ∈ A I (A.68)

j∈A J