178 research outputs found
A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications
The main contribution of this paper is to present a new sufficient condition
for the subexponential asymptotics of the stationary distribution of a
GI/GI/1-type Markov chain without jumps from level "infinity" to level zero.
For simplicity, we call such Markov chains {\it GI/GI/1-type Markov chains
without disasters} because they are often used to analyze semi-Markovian queues
without "disasters", which are negative customers who remove all the customers
in the system (including themselves) on their arrivals. In this paper, we
demonstrate the application of our main result to the stationary queue length
distribution in the standard BMAP/GI/1 queue. Thus we obtain new asymptotic
formulas and prove the existing formulas under weaker conditions than those in
the literature. In addition, applying our main result to a single-server queue
with Markovian arrivals and the -bulk-service rule (i.e., MAP//1 queue), we obatin a subexponential asymptotic formula for the
stationary queue length distribution.Comment: Submitted for revie
Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments
This paper considers the tail asymptotics for a cumulative process sampled at a heavy-tailed random time . The main contribution of
this paper is to establish several sufficient conditions for the asymptotic
equality as , where and is a certain
positive constant. The main results of this paper can be used to obtain the
subexponential asymptotics for various queueing models in Markovian
environments. As an example, using the main results, we derive subexponential
asymptotic formulas for the loss probability of a single-server finite-buffer
queue with an on/off arrival process in a Markovian environment
Phantom distribution functions for some stationary sequences
The notion of a phantom distribution function (phdf) was introduced by
O'Brien (1987). We show that the existence of a phdf is a quite common
phenomenon for stationary weakly dependent sequences. It is proved that any
-mixing stationary sequence with continuous marginals admits a
continuous phdf. Sufficient conditions are given for stationary sequences
exhibiting weak dependence, what allows the use of attractive models beyond
mixing. The case of discontinuous marginals is also discussed for
-mixing.
Special attention is paid to examples of processes which admit a continuous
phantom distribution function while their extremal index is zero. We show that
Asmussen (1998) and Roberts et al. (2006) provide natural examples of such
processes. We also construct a non-ergodic stationary process of this type
At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift
In Chapter 2 we introduce a classification of Markov chains with
asymptotically zero drift, which relies on relations between first and second
moments of jumps. We construct an abstract Lyapunov functions which looks
similar to functions which characterise the behaviour of diffusions with
similar drift and diffusion coefficient.
Chapter 3 is devoted to the limiting behaviour of transient chains. Here we
prove converges to and normal distribution which generalises papers by
Lamperti, Kersting and Klebaner. We also determine the asymptotic behaviour of
the cumulative renewal function.
In Chapter 4 we introduce a general strategy of change of measure for Markov
chains with asymptotically zero drift. This is the most important ingredient in
our approach to recurrent chains.
Chapter 5 is devoted to the study of the limiting behaviour of recurrent
chains with the drift proportional to . We derive asymptotics for a
stationary measure and determine the tail behaviour of recurrence times. All
these asymptotics are of power type.
In Chapter 6 we show that if the drift is of order then moments
of all orders are important for the behaviour of stationary
distributions and pre-limiting tails. Here we obtain Weibull-like asymptotics.
In Chapter 7 we apply our results to different processes, e.g. critical and
near-critical branching processes, risk processes with reserve-dependent
premium rate, random walks conditioned to stay positive and reflected random
walks.
In Chapter 8 we consider asymptotically homogeneous in space Markov chains
for which we derive exponential tail asymptotics
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