Dependence in Lag for Markov Chains on Partially Ordered State Spaces with Applications to Degradable Networks

Abstract

We study the property of dependence in lag for Markov chains on countable partially ordered state spaces and give conditions which ensure that a process is monotone in lag. In case of linearly ordered state spaces, proofs are based on the Lorentz inequality. However, we show that on partially ordered spaces Lorentz inequality is only true under additional assumptions. By using supermodular-type stochastic orders we derive comparison inequalities that compare the internal dependence structure of processes with that of their speeding-down versions.

Applications of the results are presented for degradable exponential networks in which the nodes are subject to random breakdowns and repairs. We obtain comparison results for the breakdown processes as well as for the queue length processes that are not even Markovian on their own.

Keywords:

• Dependence in lag
• Dependence ordering
• Jackson networks
• Lorentz inequality
• Networks with breakdowns and repairs
• Supermodular functions

AMS 2000 Subject Classification:

• 60K25

ACKNOWLEDGMENT

We are grateful to Hans Daduna and Ryszard Szekli for many helpful discussions on the paper's subject and to the referees for their useful comments. Work supported by DAAD/KBN grant number D/04/25554.