A sequential update algorithm for computing the stationary distribution vector in upper block-Hessenberg Markov chains

This paper proposes a new algorithm for computing the stationary distribution vector in continuous-time upper block-Hessenberg Markov chains. To this end, we consider the last-block-column-linearly-augmented (LBCL-augmented) truncation of the (infinitesimal) generator of the upper block-Hessenberg Markov chain. The LBCL-augmented truncation is a linearly-augmented truncation such that the augmentation distribution has its probability mass only on the last block column. We first derive an upper bound for the total variation distance between the respective stationary distribution vectors of the original generator and its LBCL-augmented truncation. Based on the upper bound, we then establish a series of linear fractional programming (LFP) problems to obtain augmentation distribution vectors such that the bound converges to zero. Using the optimal solutions of the LFP problems, we construct a matrix-infinite-product (MIP) form of the original (i.e., not approximate) stationary distribution vector and develop a sequential update algorithm for computing the MIP form. Finally, we demonstrate the applicability of our algorithm to BMAP/M/$\infty$ queues and M/M/$s$ retrial queues.Comment: The typo in Abstract has been correcte

Error bounds for last-column-block-augmented truncations of block-structured Markov chains

This paper discusses the error estimation of the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of block-structured Markov chains (BSMCs) in continuous time. We first derive upper bounds for the absolute difference between the time-averaged functionals of a BSMC and its LC-block-augmented truncation, under the assumption that the BSMC satisfies the general $f$-modulated drift condition. We then establish computable bounds for a special case where the BSMC is exponentially ergodic. To derive such computable bounds for the general case, we propose a method that reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the properties of the bounds through the numerical results on an M/M/$s$ retrial queue, which is a representative example of LD-QBDs. Finally, we present computable perturbation bounds for the stationary distribution vectors of BSMCs.Comment: This version has fixed the bugs for the positions of Figures 1 through

On the numerical solution of Kronecker-based infinite level-dependent QBD processes

Cataloged from PDF version of article.Infinite level-dependent quasi-birth-and-death (LDQBD) processes can be used to model Markovian systems with countably infinite multidimensional state spaces. Recently it has been shown that sums of Kronecker products can be used to represent the nonzero blocks of the transition rate matrix underlying an LDQBD process for models from stochastic chemical kinetics. This paper extends the form of the transition rates used recently so that a larger class of models including those of call centers can be analyzed for their steady-state. The challenge in the matrix analytic solution then is to compute conditional expected sojourn time matrices of the LDQBD model under low memory and time requirements after truncating its countably infinite state space judiciously. Results of numerical experiments are presented using a Kronecker-based matrix-analytic solution on models with two or more countably infinite dimensions and rules of thumb regarding better implementations are derived. In doing this, a more recent approach that reduces memory requirements further by enabling the computation of steady-state expectations without having to obtain the steady-state distribution is also considered. (C) 2013 Elsevier B.V. All rights reserved

On the numerical solution of Kronecker-based infinite level-dependent QBD processes

Infinite level-dependent quasi-birth-and-death (LDQBD) processes can be used to model Markovian systems with countably infinite multidimensional state spaces. Recently it has been shown that sums of Kronecker products can be used to represent the nonzero blocks of the transition rate matrix underlying an LDQBD process for models from stochastic chemical kinetics. This paper extends the form of the transition rates used recently so that a larger class of models including those of call centers can be analyzed for their steady-state. The challenge in the matrix analytic solution then is to compute conditional expected sojourn time matrices of the LDQBD model under low memory and time requirements after truncating its countably infinite state space judiciously. Results of numerical experiments are presented using a Kronecker-based matrix-analytic solution on models with two or more countably infinite dimensions and rules of thumb regarding better implementations are derived. In doing this, a more recent approach that reduces memory requirements further by enabling the computation of steady-state expectations without having to obtain the steady-state distribution is also considered. © 2013 Elsevier B.V. All rights reserved

Infrastructure Scaling and Pricing

Infrastructure systems play a crucial role in our daily lives. They include, but are not limited to, the highways we take while we commute to work, the stadiums we go to watch games, and the power plants that provide the electricity we consume in our homes. In this thesis we study infrastructure systems from several different perspectives with a focus on pricing and scalability. The pricing aspect of our research focuses on two industries: toll roads and sports events. Afterwards, we analyze the potential impact of small modular infrastructure on a wide variety of industries. We start by analyzing the problem of determining the tolls that maximize revenue for a managed lane operator -- that is, an operator who can charge a toll for the use of some lanes on a highway while a number of parallel lanes remain free to use. Managing toll lanes for profit is becoming increasingly common as private contractors agree to build additional lane capacity in return for the opportunity to retain toll revenue. We start by modeling the lanes as queues and show that the dynamic revenue-maximizing toll is always greater than or equal to the myopic toll that maximizes expected revenue from each arriving vehicle. Numerical examples show that a dynamic revenue-maximizing toll scheme can generate significantly more expected revenue than either a myopic or a static toll scheme. An important implication is that the revenue-maximizing fee does not only depend on the current state, but also on anticipated future arrivals. We discuss the managerial implications and present several numerical examples. Next, we relax the queueing assumption and model traffic propagation on a highway realistically by using simulation. We devise a framework that can be used to obtain revenue maximizing tolls in such a context. We calibrate our framework by using data from the SR-91 Highway in Orange County, CA and explore different tolling schemes. Our numerical experiments suggest that simple dynamic tolling mechanisms can lead to substantial revenue improvements over myopic and time-of-use tolling policies. In the third part, we analyze the revenue management of consumer options for tournaments. Sporting event managers typically only offer advance tickets which guarantee a seat at a future sporting event in return for an upfront payment. Some event managers and ticket resellers have started to offer call options under which a customer can pay a small amount now for the guaranteed option to attend a future sporting event by paying an additional amount later. We consider the case of tournament options where the event manager sells team-specific options for a tournament final, such as the Super Bowl, before the finalists are determined. These options guarantee a final game ticket to the bearer if his team advances to the finals. We develop an approach by which an event manager can determine the revenue maximizing prices and amounts of advance tickets and options to sell for a tournament final. Afterwards, for a specific tournament structure we show that offering options is guaranteed to increase expected revenue for the event. We also establish bounds for the revenue improvement and show that introducing options can increase social welfare. We conclude by presenting a numerical application of our approach. Finally, we argue that advances made in automation, communication and manufacturing portend a dramatic reversal of the ``bigger is better'' approach to cost reductions prevalent in many basic infrastructure industries, e.g. transportation, electric power generation and raw material processing. We show that the traditional reductions in capital costs achieved by scaling up in size are generally matched by learning effects in the mass-production process when scaling up in numbers instead. In addition, using the U.S. electricity generation sector as a case study, we argue that the primary operating cost advantage of large unit scale is reduced labor, which can be eliminated by employing low-cost automation technologies. Finally, we argue that locational, operational and financial flexibilities that accompany smaller unit scale can reduce investment and operating costs even further. All these factors combined argue that with current technology, economies of numbers may well dominate economies of unit scale