 Abstract | Long-range dependence in discrete and continuous time Markov
chains over a countable state space is defined via embedded
renewal processes brought about by visits to a fixed state. In the
discrete time chain, solidarity properties are obtained and
long-range dependence of functionals are examined. On the other
hand, the study of LRD of continuous time chains is defined via
the number of visits in a given time interval. Long-range
dependence of Markov chains over a non-countable state space is
also carried out through positive Harris chains. Embedded renewal
processes in these chains exist via visits to sets of states
called proper atoms.
Examples of these chains are presented, with particular attention
given to long-range dependent Markov chains in single-server
queues, namely, the waiting times of GI/G/1 queues and queue
lengths at departure epochs in M/G/1 queues. The presence of
long-range dependence in these processes is dependent on the
moment index of the lifetime distribution of the service times.
The Hurst indexes are obtained under certain conditions on the
distribution function of the service times and the structure of
the correlations. These processes of waiting times and queue sizes
are also examined in a range of M/P/2 queues via simulation (here, P denotes a Pareto distribution). |
