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

A linear programming approach to error bounds for random walks in the quarter-plane

summary:We consider the steady-state behavior of random walks in the quarter-plane, in particular, the expected value of performance measures that are component-wise linear over the state space. Since the stationary distribution of a random walk is in general not readily available we establish upper and lower bounds on performance in terms of another random walk with perturbed transition probabilities, for which the stationary distribution is a geometric product-form. The Markov reward approach as developed by van Dijk is used to bound the perturbation error. The main contribution of the work is the formulation of finite linear programs that provide upper and lower bounds to the performance of the original random walk. Most importantly, these linear programs establish bounds on the bias terms. This leverages an important drawback in the application of the Markov reward approach, which in existing literature is based on meticulously crafted bounds on the bias terms

Erlang loss bounds for OT-ICU systems

In hospitals, patients can be rejected at both the operating theater (OT) and the intensive care unit (ICU) due to limited ICU capacity. The corresponding ICU rejection probability is an important service factor for hospitals. Rejection of an ICU request may lead to health deterioration for patients, and for hospitals to costly actions and a loss of precious capacity when an operation is canceled.\ud There is no simple expression available for this ICU rejection probability that takes the interaction with the OT into account. With c the ICU capacity (number of ICU beds), this paper proves and numerically illustrates a lower bound by an $M|G|c|c$ system and an upper bound by an $M|G|c-1|c-1$ system, hence by simple Erlang loss expressions.\ud The result is based on a product form modification for a special OT–ICU tandem formulation and proved by a technically complicated Markov reward comparison approach. The upper bound result is of particular practical interest for dimensioning an ICU to secure a prespecified service quality. The numerical results include a case study.\u

Proceedings / 6th International Symposium of Industrial Engineering - SIE 2015, 24th-25th September, 2015, Belgrade

editors Vesna Spasojević-Brkić, Mirjana Misita, Dragan D. Milanovi

Proceedings / 6th International Symposium of Industrial Engineering - SIE 2015, 24th-25th September, 2015, Belgrade

editors Vesna Spasojević-Brkić, Mirjana Misita, Dragan D. Milanovi