3,526 research outputs found
Fluid Approximation of a Call Center Model with Redials and Reconnects
In many call centers, callers may call multiple times. Some of the calls are
re-attempts after abandonments (redials), and some are re-attempts after
connected calls (reconnects). The combination of redials and reconnects has not
been considered when making staffing decisions, while ignoring them will
inevitably lead to under- or overestimation of call volumes, which results in
improper and hence costly staffing decisions. Motivated by this, in this paper
we study call centers where customers can abandon, and abandoned customers may
redial, and when a customer finishes his conversation with an agent, he may
reconnect. We use a fluid model to derive first order approximations for the
number of customers in the redial and reconnect orbits in the heavy traffic. We
show that the fluid limit of such a model is the unique solution to a system of
three differential equations. Furthermore, we use the fluid limit to calculate
the expected total arrival rate, which is then given as an input to the Erlang
A model for the purpose of calculating service levels and abandonment rates.
The performance of such a procedure is validated in the case of single
intervals as well as multiple intervals with changing parameters
Markov-modulated Brownian motion with two reflecting barriers
We consider a Markov-modulated Brownian motion reflected to stay in a strip
[0,B]. The stationary distribution of this process is known to have a simple
form under some assumptions. We provide a short probabilistic argument leading
to this result and explaining its simplicity. Moreover, this argument allows
for generalizations including the distribution of the reflected process at an
independent exponentially distributed epoch. Our second contribution concerns
transient behavior of the reflected system. We identify the joint law of the
processes t,X(t),J(t) at inverse local times.Comment: 13 pages, 1 figur
Dynamic Service Rate Control for a Single Server Queue with Markov Modulated Arrivals
We consider the problem of service rate control of a single server queueing
system with a finite-state Markov-modulated Poisson arrival process. We show
that the optimal service rate is non-decreasing in the number of customers in
the system; higher congestion rates warrant higher service rates. On the
contrary, however, we show that the optimal service rate is not necessarily
monotone in the current arrival rate. If the modulating process satisfies a
stochastic monotonicity property the monotonicity is recovered. We examine
several heuristics and show where heuristics are reasonable substitutes for the
optimal control. None of the heuristics perform well in all the regimes.
Secondly, we discuss when the Markov-modulated Poisson process with service
rate control can act as a heuristic itself to approximate the control of a
system with a periodic non-homogeneous Poisson arrival process. Not only is the
current model of interest in the control of Internet or mobile networks with
bursty traffic, but it is also useful in providing a tractable alternative for
the control of service centers with non-stationary arrival rates.Comment: 32 Pages, 7 Figure
Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic
A result of Ward and Glynn (2005) asserts that the sequence of scaled offered
waiting time processes of the queue converges weakly to a
reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the
traffic intensity approaches one. As a consequence, the stationary distribution
of a ROU process, which is a truncated normal, should approximate the scaled
stationary distribution of the offered waiting time in a queue;
however, no such result has been proved. We prove the aforementioned
convergence, and the convergence of the moments, in heavy traffic, thus
resolving a question left open in Ward and Glynn (2005). In comparison to
Kingman's classical result in Kingman (1961) showing that an exponential
distribution approximates the scaled stationary offered waiting time
distribution in a queue in heavy traffic, our result confirms that
the addition of customer abandonment has a non-trivial effect on the queue
stationary behavior.Comment: 29 page
Real convergence and regime-switching among EU accession countries
Real convergence among the ten EU 2004 accession economies is investigated with respect to long-run real interest parity. We employ a novel approach where unit-root tests for real interest differentials are embedded within a Markov regime-switching framework. Whereas standard univariate unit-root tests provide mixed support for parity, we find parity is present in all cases where differentials either switch between regimes of stationary and non-stationarity behaviour, or between alternative regimes of stationarity characterized by differing degrees of persistence. Further insights are obtained from the inferred probabilities of being in each regime, and the regime-switching nature of the differential variances
The ODE method for stability of skip-free Markov chains with applications to MCMC
Fluid limit techniques have become a central tool to analyze queueing
networks over the last decade, with applications to performance analysis,
simulation and optimization. In this paper, some of these techniques are
extended to a general class of skip-free Markov chains. As in the case of
queueing models, a fluid approximation is obtained by scaling time, space and
the initial condition by a large constant. The resulting fluid limit is the
solution of an ordinary differential equation (ODE) in ``most'' of the state
space. Stability and finer ergodic properties for the stochastic model then
follow from stability of the set of fluid limits. Moreover, similarly to the
queueing context where fluid models are routinely used to design control
policies, the structure of the limiting ODE in this general setting provides an
understanding of the dynamics of the Markov chain. These results are
illustrated through application to Markov chain Monte Carlo methods.Comment: Published in at http://dx.doi.org/10.1214/07-AAP471 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Scaling Analysis and Evolution Equation of the North Atlantic Oscillation Index Fluctuations
The North Atlantic Oscillation (NAO) monthly index is studied from 1825 till
2002 in order to identify the scaling ranges of its fluctuations upon different
delay times and to find out whether or not it can be regarded as a Markov
process. A Hurst rescaled range analysis and a detrended fluctuation analysis
both indicate the existence of weakly persistent long range time correlations
for the whole scaling range and time span hereby studied. Such correlations are
similar to Brownian fluctuations. The Fokker-Planck equation is derived and
Kramers-Moyal coefficients estimated from the data. They are interpreted in
terms of a drift and a diffusion coefficient as in fluid mechanics. All partial
distribution functions of the NAO monthly index fluctuations have a form close
to a Gaussian, for all time lags, in agreement with the findings of the scaling
analyses. This indicates the lack of predictive power of the present NAO
monthly index. Yet there are some deviations for large (and thus rare) events.
Whence suggestions for other measurements are made if some improved
predictability of the weather/climate in the North Atlantic is of interest. The
subsequent Langevin equation of the NAO signal fluctuations is explicitly
written in terms of the diffusion and drift parameters, and a characteristic
time scale for these is given in appendix.Comment: 6 figures, 54 refs., 16 pages; submitted to Int. J. Mod. Phys. C:
Comput. Phy
Stability of a Markov-modulated Markov Chain, with application to a wireless network governed by two protocols
We consider a discrete-time Markov chain , , where
the -component forms a Markov chain itself. Assume that is
Harris-ergodic and consider an auxiliary Markov chain whose
transition probabilities are the averages of transition probabilities of the
-component of the -chain, where the averaging is weighted by the
stationary distribution of the -component.
We first provide natural conditions in terms of test functions ensuring that
the -chain is positive recurrent and then prove that these conditions
are also sufficient for positive recurrence of the original chain .
The we prove a "multi-dimensional" extension of the result obtained. In the
second part of the paper, we apply our results to two versions of a
multi-access wireless model governed by two randomised protocols.Comment: 23 page
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