48 research outputs found
Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process
We consider a discrete-time two-dimensional process
on with a supplemental process on a finite set,
where individual processes and are both skip free.
We assume that the joint process is
Markovian and that the transition probabilities of the two-dimensional process
are modulated depending on the state of the background
process . This modulation is space homogeneous except for the
boundaries of . We call this process a discrete-time
two-dimensional quasi-birth-and-death (2D-QBD) process and, under several
conditions, obtain the exact asymptotic formulae of the stationary distribution
in the coordinate directions.Comment: 54 page
Asymptotic approximations for stationary distributions of many-server queues with abandonment
A many-server queueing system is considered in which customers arrive
according to a renewal process and have service and patience times that are
drawn from two independent sequences of independent, identically distributed
random variables. Customers enter service in the order of arrival and are
assumed to abandon the queue if the waiting time in queue exceeds the patience
time. The state of the system with servers is represented by a
four-component process that consists of the forward recurrence time of the
arrival process, a pair of measure-valued processes, one that keeps track of
the waiting times of customers in queue and the other that keeps track of the
amounts of time customers present in the system have been in service and a
real-valued process that represents the total number of customers in the
system. Under general assumptions, it is shown that the state process is a
Feller process, admits a stationary distribution and is ergodic. It is also
shown that the associated sequence of scaled stationary distributions is tight,
and that any subsequence converges to an invariant state for the fluid limit.
In particular, this implies that when the associated fluid limit has a unique
invariant state, then the sequence of stationary distributions converges, as
, to the invariant state. In addition, a simple example is
given to illustrate that, both in the presence and absence of abandonments, the
and limits cannot always be
interchanged.Comment: Published in at http://dx.doi.org/10.1214/10-AAP738 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org