Applications of robust optimization to queueing and inventory systems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 105-111).This thesis investigates the application of robust optimization in the performance analysis of queueing and inventory systems. In the first part of the thesis, we propose a new approach for performance analysis of queueing systems based on robust optimization. We first derive explicit upper bounds on performance for tandem single class, multiclass single server, and single class multi-server queueing systems by solving appropriate robust optimization problems. We then show that these bounds derived by solving deterministic optimization problems translate to upper bounds on the expected steady-state performance for a variety of widely used performance measures such as waiting times and queue lengths. Additionally, these explicit bounds agree qualitatively with known results. In the second part of the thesis, we propose methods to compute (s,S) policies in supply chain networks using robust and stochastic optimization and compare their performance. Our algorithms handle general uncertainty sets, arbitrary network topologies, and flexible cost functions including the presence of fixed costs. The algorithms exhibit empirically practical running times. We contrast the performance of robust and stochastic (s,S) policies in a numerical study, and we find that the robust policy is comparable to the average performance of the stochastic policy, but has a considerably lower standard deviation across a variety of networks and realized demand distributions. Additionally, we identify regimes when the robust policy exhibits particular strengths even in average performance and tail behavior as compared with the stochastic policy.by Alexander Anatolyevich Rikun.Ph.D

Performance analysis of queueing networks via robust optimization

Performance analysis of queueing networks is one of the most challenging areas of queueing theory. Barring very specialized models such as product-form type queueing networks, there exist very few results that provide provable nonasymptotic upper and lower bounds on key performance measures. In this paper we propose a new performance analysis method, which is based on the robust optimization. The basic premise of our approach is as follows: rather than assuming that the stochastic primitives of a queueing model satisfy certain probability laws—such as i.i.d. interarrival and service times distributions—we assume that the underlying primitives are deterministic and satisfy the implications of such probability laws. These implications take the form of simple linear constraints, namely, those motivated by the law of the iterated logarithm (LIL). Using this approach we are able to obtain performance bounds on some key performance measures. Furthermore, these performance bounds imply similar bounds in the underlying stochastic queueing models. We demonstrate our approach on two types of queueing networks: (a) tandem single-class queueing network and (b) multiclass single-server queueing network. In both cases, using the proposed robust optimization approach, we are able to obtain explicit upper bounds on some steady-state performance measures. For example, for the case of TSC system we obtain a bound of the form C(1 – {rho})–1 ln ln((1 – {rho})–1) [C(1-p) superscript -1 ln ln ((1 - p) superscript -1)]on the expected steady-state sojourn time, where C is an explicit constant and {rho} is the bottleneck traffic intensity. This qualitatively agrees with the correct heavy traffic scaling of this performance measure up to the ln ln((1 – {rho})–1) [ln ln((1 - p) superscript -1)] correction factor.National Science Foundation (U.S.) (Grant DMI-0556106)National Science Foundation (U.S.) (Grant CMMI-0726733