Winfried K. Grassmann

Regenerative Analysis and Steady State Distributions for Markov Chains

We apply regenerative theory to derive certain relations between steady state probabilities of a Markov chain. These relations are then used to develop a numerical algorithm to find these probabilities. The algorithm is a modification of the Gauss-Jordan method, in which all elements used in numerical computations are nonnegative; as a consequence, the algorithm is numerically stable.

Transient solutions in markovian queueing systems

Abstract This paper discusses the methods available to find transient solutions for huge, but sparse Markov processes, as they arise in connection with queueing systems. The methods discussed include Runge-Kutta, Lious method and randomization. It is shown that all these methods are closely related, but that the method of randomization is superior to the other two methods. Our own experience and experience of others clearly indicate that all the methods mentioned above are viable for finding transient solutions in problems having 600 states or more.

Equilibrium distribution of block-structured Markov chains with repeating rows

In this paper we consider two-dimensional Markov chains with the property that, except for some boundary conditions, when the transition matrix is written in block form, the rows are identical except for a shift to the right. We provide a general theory for finding the equilibrium distribution for this class of chains. We illustrate theory by showing how our results unify the analysis of the M/G/ 1 and GI/M/ 1 paradigms introduced by M. F. Neuts.

Transient solutions in Markovian queues: An algorithm for finding them and determining their waiting-time distributions

Abstract An algorithm for finding transient solutions in Markovian queues is given and it is applied to find transient solutions for the M/M/1 queue. This algorithm can also be applied to solve waiting-time problems as is shown by an example.

Means and variances of time averages in Markovian environments

Abstract Means and variances of time averages are important for cost minimization, hypothesis testing, estimation, initial bias problems in simulation and run length determination. This paper demonstrates how these measures can be calculated efficiently in transient Markovian systems. The method proposed here can be used to analyse systems of significant size, that is, systems with 1000 states or more. Numerical results are also given.

An analytical solution for a tandem queue with blocking

The model considered in this paper involves a tandem queue with two waiting lines, and as soon as the second waiting line reaches a certain upper limit, the first line is blocked. Both lines have exponential servers, and arrivals are Poisson. The objective is to determine the joint distribution of both lines in equilibrium. This joint distribution is found by using generalized eigenvalues. Specifically, a simple formula involving the cotangent is derived. The periodicity of the cotangent is then used to determine the location of the majority of the eigenvalues. Once all eigenvalues are found, the eigenvectors can be obtained recursively. The method proposed has a lower computational complexity than all other known methods.

Queueing Analysis of a Jockeying Model

In this paper, we solve a type of shortest queue problem, which is related to multibeam satellite systems. We assume that the packet interarrival times are independently distributed according to an arbitrary distribution function, that the service times are Markovian with possibly different service rates, that each server has its own buffer for packet waiting, and that jockeying among buffers is permitted. Packets always join the shortest buffers. Jockeying takes place as soon as the difference between the longest and shortest buffers exceeds a preset number not necessarily 1. In this case, the last packet in a longest buffer jockeys instantaneously to the shortest buffers. We prove that the equilibrium distribution of packets in the system is modified vector geometric. Expressions of main performance measures, including the average number of packets in the system, the average packet waiting time in the system, and the average number of jockeying, are given. Based on these solutions, numerical results are computed. By comparing the results for jockeying and nonjockeying models, we show that a significant improvement of the system performance is achieved for the jockeying model.

Computation of Steady-State Probabilities for Infinite-State Markov Chains with Repeating Rows

In this paper we consider Markov chains with these properties. The transition matrix is banded, and except for some boundary conditions, when the transition matrix is written in block form, the rows are identical except for a shift to the right. Some authors have used variants of the state reduction method to solve special cases. We present a general algorithm and some of the theoretical underpinnings of these methods. In particular, we give a rigorous proof of convergence. We also provide a simple method to norm the probabilities such that their sum is unity. We describe the connection between this new technique and the matrix-iterative methods of M. F. Neuts. The paper concludes with some numerical examples. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

A System Dynamics Approach to Study Virtual Communities

In recent years, extensive studies of many interesting aspects of virtual community dynamics promoted a better understanding of this area. One of the most challenging problems facing builders of virtual communities is the design of incentive mechanisms that can ensure user participation. However, running virtual community experiments in the real world is expensive, and requires a great deal of motivation from users. In this paper we advocate a system dynamics approach to simulate the overall behaviors of participants in the communities, which can provide insights into the user motivation process, incentive mechanism evaluation and community development. A simulation model for a virtual community called Comtella is presented, and the results are very promising

Matrix Analytic Methods

This chapter shows how to find the equilibrium probabilities in processes of GI/M/1 type, and M/G/1 type, and GI/G/1 type by matrix analytic methods. GI/M/1-type processes are Markov chains with transition matrices having the same structure as the imbedded Markov chain of a GI/M/1 queue, except that the entries are matrices rather than scalars. Similarly, M/G/1 type processes have transition matrices of the same form as the imbedded Markov chain of the M/G/1 queue, except that the entries are matrices. In the imbedded Markov chain of the GI/M/1 queue, all columns but the first have the same entries, except that they are displaced so that the diagonal block entry is common to all. Similarly, in the M/G/1 queue, all rows except the first one are equal after proper centering.