14,697 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
Zero-automatic queues and product form
We introduce and study a new model: 0-automatic queues. Roughly, 0-automatic
queues are characterized by a special buffering mechanism evolving like a
random walk on some infinite group or monoid. The salient result is that all
stable 0-automatic queues have a product form stationary distribution and a
Poisson output process. When considering the two simplest and extremal cases of
0-automatic queues, we recover the simple M/M/1 queue, and Gelenbe's G-queue
with positive and negative customers
Corrected phase-type approximations of heavy-tailed queueing models in a Markovian environment
Significant correlations between arrivals of load-generating events make the
numerical evaluation of the workload of a system a challenging problem. In this
paper, we construct highly accurate approximations of the workload distribution
of the MAP/G/1 queue that capture the tail behavior of the exact workload
distribution and provide a bounded relative error. Motivated by statistical
analysis, we consider the service times as a mixture of a phase-type and a
heavy-tailed distribution. With the aid of perturbation analysis, we derive our
approximations as a sum of the workload distribution of the MAP/PH/1 queue and
a heavy-tailed component that depends on the perturbation parameter. We refer
to our approximations as corrected phase-type approximations, and we exhibit
their performance with a numerical study.Comment: Received the Marcel Neuts Student Paper Award at the 8th
International Conference on Matrix Analytic Methods in Stochastic Models 201
Commuting matrices in the sojourn time analysis of MAP/MAP/1 queues
Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their sojourn time distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have an order matrix exponential representation for their sojourn time distribution, where is the size of the background continuous time Markov chain, the sojourn time distribution of the latter class allows for a more compact representation of order .
In this paper we unify these two results and show that the key step exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue. We prove, using two different approaches, that the required matrices do commute and identify several other sets of commuting matrices. Finally, we generalize some of the results to queueing systems with batch arrivals and services
Sample path large deviations for multiclass feedforward queueing networks in critical loading
We consider multiclass feedforward queueing networks with first in first out
and priority service disciplines at the nodes, and class dependent
deterministic routing between nodes. The random behavior of the network is
constructed from cumulative arrival and service time processes which are
assumed to satisfy an appropriate sample path large deviation principle. We
establish logarithmic asymptotics of large deviations for waiting time, idle
time, queue length, departure and sojourn-time processes in critical loading.
This transfers similar results from Puhalskii about single class queueing
networks with feedback to multiclass feedforward queueing networks, and
complements diffusion approximation results from Peterson. An example with
renewal inter arrival and service time processes yields the rate function of a
reflected Brownian motion. The model directly captures stationary situations.Comment: Published at http://dx.doi.org/10.1214/105051606000000439 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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