arXiv subjects

Xinwei Bai

Publications and source records attributed to Xinwei Bai.

5 recordsLinked to original sources

arXiv2019

Bounds on the bias terms for the Markov reward approach

An important step in the Markov reward approach to error bounds on stationary performance measures of Markov chains is to bound the bias terms. Affine functions have been successfully used for these bounds for various models, but there are also models for which it has not been possible to establish such bounds. So far, no theoretical results have been available that guarantee bounds on the bias terms. We consider random walks in the positive orthant and provide sufficient conditions under which quadratic and/or geometric functions can be used to bound the bias terms. In addition, we provide a linear programming framework that establishes the quadratic bounds as well as the resulting bound on the stationary performance.

math.PR↗

arXiv2018

A linear programming approach to Markov reward error bounds for queueing networks

In this paper, we present a numerical framework for constructing bounds on stationary performance measures of random walks in the positive orthant using the Markov reward approach. These bounds are established in terms of stationary performance measures of a perturbed random walk whose stationary distribution is known explicitly. We consider random walks in an arbitrary number of dimensions and with a transition probability structure that is defined on an arbitrary partition of the positive orthant. Within each component of this partition the transition probabilities are homogeneous. This enables us to model queueing networks with, for instance, break-downs and finite buffers. The main contribution of this paper is that we generalize the linear programming approach of [1] to this class of models.

math.PR↗

arXiv2017

Performance measures for the two-node queue with finite buffers

We consider a two-node queue modeled as a two-dimensional random walk. In particular, we consider the case that one or both queues have finite buffers. We develop an approximation scheme based on the Markov reward approach to error bounds in order to bound performance measures of such random walks in terms of a perturbed random walk in which the transitions along the boundaries are different from those in the original model and the invariant measure of the perturbed random walk is of product-form. We then apply this approximation scheme to a tandem queue and some variants of this model, for the case that both buffers are finite. We also apply our approximation scheme to a coupled-queue in which only one of the buffers has finite capacity.

math.PR↗

arXiv2017

Inhomogeneous perturbation and error bounds for the stationary performance of random walks in the quarter plane

A continuous-time random walk in the quarter plane with homogeneous transition rates is considered. Given a non-negative reward function on the state space, we are interested in the expected stationary performance. Since a direct derivation of the stationary probability distribution is not available in general, the performance is approximated by a perturbed random walk, whose transition rates on the boundaries are changed such that its stationary probability distribution is known in closed form. A perturbed random walk for which the stationary distribution is a sum of geometric terms is considered and the perturbed transition rates are allowed to be inhomogeneous. It is demonstrated that such rates can be constructed for any sum of geometric terms that satisfies the balance equations in the interior of the state space. The inhomogeneous transitions relax the pairwise-coupled structure on these geometric terms that would be imposed if only homogeneous transitions are used. An explicit expression for the approximation error bound is obtained using the Markov reward approach, which does not depend on the values of the inhomogeneous rates but only on the parameters of the geometric terms. Numerical experiments indicate that inhomogeneous perturbation can give smaller error bounds than homogeneous perturbation.

math.PR↗

arXiv2017

Optimization of Caching Devices with Geometric Constraints

It has been recently advocated that in large communication systems it is beneficial both for the users and for the network as a whole to store content closer to users. One particular implementation of such an approach is to co-locate caches with wireless base stations. In this paper we study geographically distributed caching of a fixed collection of files. We model cache placement with the help of stochastic geometry and optimize the allocation of storage capacity among files in order to minimize the cache miss probability. We consider both per cache capacity constraints as well as an average capacity constraint over all caches. The case of per cache capacity constraints can be efficiently solved using dynamic programming, whereas the case of the average constraint leads to a convex optimization problem. We demonstrate that the average constraint leads to significantly smaller cache miss probability. Finally, we suggest a simple LRU-based policy for geographically distributed caching and show that its performance is close to the optimal.

cs.NI↗