9,968 research outputs found
The compensation approach for walks with small steps in the quarter plane
This paper is the first application of the compensation approach to counting
problems. We discuss how this method can be applied to a general class of walks
in the quarter plane with a step set that is a subset of
in the interior of . We
derive an explicit expression for the counting generating function, which turns
out to be meromorphic and nonholonomic, can be easily inverted, and can be used
to obtain asymptotic expressions for the counting coefficients.Comment: 22 pages, 5 figure
New steps in walks with small steps in the quarter plane
In this article we obtain new expressions for the generating functions
counting (non-singular) walks with small steps in the quarter plane. Those are
given in terms of infinite series, while in the literature, the standard
expressions use solutions to boundary value problems. We illustrate our results
with three examples (an algebraic case, a transcendental D-finite case, and an
infinite group model).Comment: 47 pages, 8 figures, to appear in Annals of Combinatoric
Counting walks in a quadrant: a unified approach via boundary value problems
The aim of this article is to introduce a unified method to obtain explicit
integral representations of the trivariate generating function counting the
walks with small steps which are confined to a quarter plane. For many models,
this yields for the first time an explicit expression of the counting
generating function. Moreover, the nature of the integrand of the integral
formulations is shown to be directly dependent on the finiteness of a naturally
attached group of birational transformations as well as on the sign of the
covariance of the walkComment: 28 pages; 6 figure
Power series approximations for two-class generalized processor sharing systems
We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability beta, and a customer of queue 2 is served with probability 1-beta. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z (1),z (2)) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z (1),z (2)) in beta. The first coefficient of this power series corresponds to the priority case beta=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Pad, approximation
Steady-state analysis of shortest expected delay routing
We consider a queueing system consisting of two non-identical exponential
servers, where each server has its own dedicated queue and serves the customers
in that queue FCFS. Customers arrive according to a Poisson process and join
the queue promising the shortest expected delay, which is a natural and
near-optimal policy for systems with non-identical servers. This system can be
modeled as an inhomogeneous random walk in the quadrant. By stretching the
boundaries of the compensation approach we prove that the equilibrium
distribution of this random walk can be expressed as a series of product-forms
that can be determined recursively. The resulting series expression is directly
amenable for numerical calculations and it also provides insight in the
asymptotic behavior of the equilibrium probabilities as one of the state
coordinates tends to infinity.Comment: 41 pages, 13 figure
Erlang arrivals joining the shorter queue
We consider a system in which customers join upon arrival the shortest of two single-server queues. The interarrival times between customers are Erlang distributed and the service times of both servers are exponentially distributed. Under these assumptions, this system gives rise to a Markov chain on a multi-layered quarter plane. For this Markov chain we derive the equilibrium distribution using the compensation approach. The obtained expression for the equilibrium distribution matches and re??nes heavy-traffic approximations and tail asymptotics obtained earlier in the literature. Keywords: random walks in the quarter plane, compensation approach, join the shorter queue, tail asymptotic
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