74 research outputs found
Two-sided reflected Markov-modulated Brownian motion with applications to fluid queues and dividend payouts
In this paper we study a reflected Markov-modulated Brownian motion with a
two sided reflection in which the drift, diffusion coefficient and the two
boundaries are (jointly) modulated by a finite state space irreducible
continuous time Markov chain. The goal is to compute the stationary
distribution of this Markov process, which in addition to the complication of
having a stochastic boundary can also include jumps at state change epochs of
the underlying Markov chain because of the boundary changes. We give the
general theory and then specialize to the case where the underlying Markov
chain has two states. Moreover, motivated by an application of optimal dividend
strategies, we consider the case where the lower barrier is zero and the upper
barrier is subject to control. In this case we generalized earlier results from
the case of a reflected Brownian motion to the Markov modulated case.Comment: 22 pages, 1 figur
Sojourn time in a single server queue with threshold service rate control
We study the sojourn time in a queueing system with a single exponential
server, serving a Poisson stream of customers in order of arrival. Service is
provided at low or high rate, which can be adapted at exponential inspection
times. When the number of customers in the system is above a given threshold,
the service rate is upgraded to the high rate, and otherwise, it is downgraded
to the low rate. The state dependent changes in the service rate make the
analysis of the sojourn time a challenging problem, since the sojourn time now
also depends on future arrivals. We determine the Laplace transform of the
stationary sojourn time and describe a procedure to compute all moments as
well. First we analyze the special case of continuous inspection, where the
service rate immediately changes once the threshold is crossed. Then we extend
the analysis to random inspection times. This extension requires the
development of a new methodological tool, that is "matrix generating
functions". The power of this tool is that it can also be used to analyze
generalizations to phase-type services and inspection times.Comment: 16 pages, 13 figure
Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence
We analyze the behavior of closed product-form queueing networks when the
number of customers grows to infinity and remains proportionate on each route
(or class). First, we focus on the stationary behavior and prove the conjecture
that the stationary distribution at non-bottleneck queues converges weakly to
the stationary distribution of an ergodic, open product-form queueing network.
This open network is obtained by replacing bottleneck queues with per-route
Poissonian sources whose rates are determined by the solution of a strictly
concave optimization problem. Then, we focus on the transient behavior of the
network and use fluid limits to prove that the amount of fluid, or customers,
on each route eventually concentrates on the bottleneck queues only, and that
the long-term proportions of fluid in each route and in each queue solve the
dual of the concave optimization problem that determines the throughputs of the
previous open network.Comment: 22 page
A short note on the monotonicity of the Erlang C formula in the Halfin-Whitt regime
We prove a monotonicity condition satisfied by the Erlang C formula when computed in
the Halfin-Whitt regime. This property was recently conjectured in Janssen et al. [2011
Discounted optimal stopping of a Brownian bridge, with application to American options under pinning
Mathematically, the execution of an American-style financial derivative is
commonly reduced to solving an optimal stopping problem. Breaking the general
assumption that the knowledge of the holder is restricted to the price history
of the underlying asset, we allow for the disclosure of future information
about the terminal price of the asset by modeling it as a Brownian bridge. This
model may be used under special market conditions, in particular we focus on
what in the literature is known as the "pinning effect", that is, when the
price of the asset approaches the strike price of a highly-traded option close
to its expiration date. Our main mathematical contribution is in characterizing
the solution to the optimal stopping problem when the gain function includes
the discount factor. We show how to numerically compute the solution and we
analyze the effect of the volatility estimation on the strategy by computing
the confidence curves around the optimal stopping boundary. Finally, we compare
our method with the optimal exercise time based on a geometric Brownian motion
by using real data exhibiting pinning.Comment: 29 pages, 9 figures. Supplementary material: 5 R scripts, 4 RData
file
First passage process of a Markov additive process, with applications to reflection problems
In this paper we consider the first passage process of a spectrally negative
Markov additive process (MAP). The law of this process is uniquely
characterized by a certain matrix function, which plays a crucial role in
fluctuation theory. We show how to identify this matrix using the theory of
Jordan chains associated with analytic matrix functions. Importantly, our
result also provides us with a technique, which can be used to derive various
further identities. We then proceed to show how to compute the stationary
distribution associated with a one-sided reflected (at zero) MAP for both the
spectrally positive and spectrally negative cases as well as for the two sided
reflected Markov-modulated Brownian motion; these results can be interpreted in
terms of queues with MAP input.Comment: 16 page
Equilibrium strategies in a tandem queue under various levels of information
We analyze from an economical point of view a tandem network with two nodes. We look at different situations, that is, when customers upon their arrival are no informed, partially informed or totally informed about the state of the system. For each case, we look for the strategy that optimizes the individual net benefit. In addition, for the totally unobservable case, we also study the strategy what would be socially optimal, i.e. maximizing the overall welfare
Sojourn time in a single-server queue with threshold service rate control
We study the sojourn time in a queueing system with a single exponential server,serving a Poisson stream of customers in order of arrival. Service is provided at a low or high rate,which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace transform of the stationary sojourn time and describe a procedure to compute all moments as well. First we analyze the special case of continuous inspection, where the service rate immediately changes once the threshold is crossed. Then we extend the analysis to random inspection times.This extension requires the development of a new methodological tool, that is, matrix generating functions. The power of this tool is that it can also be used to analyze generalizations to phase-type services and inspection times.This author’s research was partially supported by the
Spanish Ministry of Education and Science Grants MTM2010-16519, SEJ2007-64500, MTM2013-42104-P (FEDER funds); and the Dutch Star grant of October 201
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