Invariant Measures for Random Walks
in the Quarter-Plane
Yanting Chen, Richard J. Boucherie and Jasper Goseling
Previous Results
Introduction
We consider homogeneous random walks in the quarterplane. Classes of walks are known for which the invariant measure is a geometric product-form or a countable
linear combination of such terms. We study invariant
measures that can be expressed as arbitrary linear combinations of geometric terms.
↑j
q(1,1)
q(−1,1) q(0,1) q(1,1)
q(1,0)
q(−1,0)
v(1)
q(1,1)
• For i > 0, j > 0, the invariant measure m(i, j) can
always be represented by (*).
• Suppose q(i, j) are given. For any Γ, |Γ| = 1, h and
v can be found such that (*) is an invariant measure.
• There are random
↑ρ
2
walks for which
pairwise coupled terms
|Γ| = ∞, Γ consisting of pairwise
coupled terms, is an
→ρ1
invariant measure.
v(−1)
q(1,−1)
q(−1,−1) q(0,−1) q(1,−1)
q(1,1)
Results
q(−1,1) q(0,1) q(1,1)
v(1)
(0,0)
h(1)
h(−1)
h(1)
→i
Theorem: If 1 < |Γ| < ∞, then (*) can not
be an invariant measure.
Problem Statement
↑ρ2
↑ρ2
(0,1)
(0,1)
The balance equation for states (i, j), i > 0, j > 0
m(i, j) =
X
m(i − k, j − l)q(k, l)
k,l
induces the set
C = {(ρ1, ρ2) |
j
i
ρ1 ρ2
=
X
k,l
(0,0)
j−l
i−k
ρ1 ρ2 q(k, l)}
of pairs (ρ1, ρ2) of potential candidates that can be
considered in measures of the form
m(i, j) =
X
(ρ1,ρ2)∈Γ
↑ρ2
j
i
α(ρ1, ρ2)ρ1ρ2,
↑ρ2
Γ
(0,1)
Γ⊂C
C
→ρ1
(0,0)
(1,0)
→ρ1
Theorem: If |Γ| = ∞, then Γ must consist of
pairwise-coupled terms.
(*)
↑ρ2
↑ρ2
(0,1)
(0,1)
Γ
(0,1)
(1,0)
C
(0,0)
(1,0)
→ρ1
(0,0)
(1,0)
→ρ1
Problem Statement:
• For what values of |Γ| can (*) be an invariant
measure?
• What is the structure of Γ that is required?
(0,0)
(1,0)
→ρ1
(0,0)
(1,0)
→ρ1
Future Work
• Γ of uncountable cardinality should be explored.
• Random walks in higher dimensions will be of interest.
Stochastic Operations Research Group
{Y.Chen, R.J.Boucherie, J.Goseling}@utwente.nl