Block-Structured Supermarket Models

Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging. This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinite-dimensional system of differential vector equations satisfied by the expected fraction vector, where "the invariance of environment factors" is given as an important result. (2) Introducing the phase-type structure to the operator semigroup and to the mean-field limit, and a Lipschitz condition can be obtained by means of a unified matrix-differential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times. Thus the results of this paper give new highlight on understanding influence of non-Poisson inputs and of non-exponential service times on performance measures of more general supermarket models.Comment: 65 pages; 7 figure

Super-Exponential Solution in Markovian Supermarket Models: Framework and Challenge

Marcel F. Neuts opened a key door in numerical computation of stochastic models by means of phase-type (PH) distributions and Markovian arrival processes (MAPs). To celebrate his 75th birthday, this paper reports a more general framework of Markovian supermarket models, including a system of differential equations for the fraction measure and a system of nonlinear equations for the fixed point. To understand this framework heuristically, this paper gives a detailed analysis for three important supermarket examples: M/G/1 type, GI/M/1 type and multiple choices, explains how to derive the system of differential equations by means of density-dependent jump Markov processes, and shows that the fixed point may be simply super-exponential through solving the system of nonlinear equations. Note that supermarket models are a class of complicated queueing systems and their analysis can not apply popular queueing theory, it is necessary in the study of supermarket models to summarize such a more general framework which enables us to focus on important research issues. On this line, this paper develops matrix-analytical methods of Markovian supermarket models. We hope this will be able to open a new avenue in performance evaluation of supermarket models by means of matrix-analytical methods.Comment: Randomized load balancing, supermarket model, matrix-analytic method, super-exponential solution, density-dependent jump Markov process, Batch Markovian Arrival Process (BMAP), phase-type (PH) distribution, fixed poin

Nonlinear Markov Processes in Big Networks

Big networks express various large-scale networks in many practical areas such as computer networks, internet of things, cloud computation, manufacturing systems, transportation networks, and healthcare systems. This paper analyzes such big networks, and applies the mean-field theory and the nonlinear Markov processes to set up a broad class of nonlinear continuous-time block-structured Markov processes, which can be applied to deal with many practical stochastic systems. Firstly, a nonlinear Markov process is derived from a large number of interacting big networks with symmetric interactions, each of which is described as a continuous-time block-structured Markov process. Secondly, some effective algorithms are given for computing the fixed points of the nonlinear Markov process by means of the UL-type RG-factorization. Finally, the Birkhoff center, the Lyapunov functions and the relative entropy are used to analyze stability or metastability of the big network, and several interesting open problems are proposed with detailed interpretation. We believe that the results given in this paper can be useful and effective in the study of big networks.Comment: 28 pages in Special Matrices; 201

In-plant logistics systems modeling with SysML

Up till now Systems Modeling Language (SysML) has mostly been used to model physical systems of interest. This paper shows how SysML can also be used to represent an abstract model. In this application a mathematical cost model is represented using the SysML plugin for the software MagicDraw. ParaMagic, a plugin in MagicDraw supplementary to SysML, links to Mathematica to solve the model. SysML is a formal language and offers a very intuitive graphical representation. It is therefore a useful medium to create a domain specific language for a field of knowledge. The comprehensiveness of the language, which makes it possible to incorporate specification, analysis, design, verification, and validation of systems, makes it a very valuable tool for collaboration on large projects

Food Access in Petersburg, Virginia: Final Report and Recommendations

The City of Petersburg has long suffered with issues of limited access to food and food insecurity. Food deserts, or areas underserved by retail food options, are prevalent throughout the City. As a result, the Robert Wood Johnson Foundation has ranked the city last of Virginia\u27s 133 counties in their annual health rankings. For the Fall 2019 semester, students from Virginia Commonwealth University\u27s L. Douglas Wilder School of Government and Public Affairs, through Dr. John Accordino\u27s Urban Commercial Revitalization course, focused on planning solutions to address food deserts in commercial areas, with the City of Petersburg being one of their clients. The class assessed the potential for commercial revitalization and made five recommendations

The law of one price and the role of market structure

This paper examines the role of market structure on the persistence of price deviations from the LOP using monthly actual product prices of 47 items collected from three different types of markets in Istanbul over 1993:01-2008:12. After showing the importance of market structure on the distribution of relative prices, we implement threshold autoregressive models. We find significant differences in average threshold estimates across markets which we explain referring to differing menu costs in each market. Yet, we find no differences in average half-life estimates across markets. We argue that this is due to low search costs in Istanbul. Robustness checks verify our findings