50 research outputs found
The invariant measure of homogeneous Markov processes in the quarter-plane: Representation in geometric terms
We consider the invariant measure of a homogeneous continuous-time Markov process in the quarter-plane. The basic solutions of the global balance equation are the geometric distributions. We first show that the invariant measure can not be a finite linear combination of basic geometric distributions, unless it consists of a single basic geometric distribution. Second, we show that a countable linear combination of geometric terms can be an invariant measure only if it consists of pairwise-coupled terms. As a consequence, we obtain a complete characterization of all countable linear combinations of geometric distributions that may yield an invariant measure for a homogeneous continuous-time Markov process in the quarter-plane
The Invariant Measure of Random Walks in the Quarter-plane: Representation in Geometric Terms
We consider the invariant measure of homogeneous random walks in the
quarter-plane. In particular, we consider measures that can be expressed as a
finite linear combination of geometric terms and present conditions on the
structure of these linear combinations such that the resulting measure may
yield an invariant measure of a random walk. We demonstrate that each geometric
term must individually satisfy the balance equations in the interior of the
state space and further show that the geometric terms in an invariant measure
must have a pairwise-coupled structure. Finally, we show that at least one of
the coefficients in the linear combination must be negative
Linear programming error bounds for random walks in the quarter-plane
We consider approximation of the performance of random walks in the quarter-plane. The approximation is in terms of a random walk with a product-form stationary distribution, which is obtained by perturbing the transition probabilities along the boundaries of the state space. A Markov reward approach is used to bound the approximation error. The main contribution of the work is the formulation of a linear program that provides the approximation error
Energy-delay tradeoff in wireless network coding
A queueing model for wireless communication network in which network coding is employed is introduced. It is shown that networks with coding are closely related to queueing networks with positive and negative customers. Analytical upper and lower bounds on the energy consumption and the delay are obtained using a Markov reward approach. The tradeoff between minimizing energy consumption and minimizing delay is investigated. Exact expressions are given for the minimum energy consumption and the minimum delay attainable in a network
A Linear Programming Approach to Error Bounds for Random Walks in the Quarter-plane
We consider the approximation of the performance of random walks in the
quarter-plane. The approximation is in terms of a random walk with a
product-form stationary distribution, which is obtained by perturbing the
transition probabilities along the boundaries of the state space. A Markov
reward approach is used to bound the approximation error. The main contribution
of the work is the formulation of a linear program that provides the
approximation error
Energy consumption in coded queues for wireless information exchange
We show the close relation between network coding and queuing networks with negative and positive customers. Moreover, we develop Markov reward error bounding techniques for networks with negative and positive customers. We obtain bounds on the energy consumption in a wireless information exchange setting using network coding
Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms
We consider the invariant measure of homogeneous random walks in the
quarter-plane. In particular, we consider measures that can be expressed as an
infinite sum of geometric terms. We present necessary conditions for the
invariant measure to be a sum of geometric terms. We demonstrate that, under a
mild regularity condition, each geometric term must individually satisfy the
balance equations in the interior of the state space. We show that the
geometric terms in an invariant measure must be the union of finitely many
pairwise-coupled sets of infinite cardinality. We further show that for the
invariant measure to be a sum of geometric terms, the random walk cannot have
transitions to the north, northeast or east. Finally, we show that for an
infinite weighted sum of geometric terms to be an invariant measure at least
one of the weights must be negative