Finding the stationary distribution of a finite-state, discrete time, irreducible Markov chain is equivalent to seeking the left eigenvector corresponding to the eigenvalue 1 of a transition matrix P of order n. Grassmann, Taksar and Heyman [1] introduced a direct algorithm, the GTH algorithm, to find the steady-state vector π. Later O’Cinneide [2] showed that the computed vector π has low componentwise relative error. In order to reach high performance over a large class of computers, O’Leary and Wu [3] developed a block form algorithm, the block-GTH algorithm, and successfully demonstrated the efficiency of the algorithm on vector pipeline machines and on workstations with cache memory.

/pdf/a-parallel-implementation-of-the-block-gth-algorithm-56lv9x4cbr.pdf

A Parallel Implementation of the Block-GTH algorithm

Congestion in Wireless Multimedia Sensor Networks (WMSN) may cause packet loss, buffer overflow and increases queueing delay, which has a significant negative impact on the network performance and reliability. This problem imposes more resource constraints that involve energy consumption and buffer size to meet additional requirements such as desired buffer size and limited amount of packet loss. One of the fundamental issues of congestion problem in such networks is the queue management. In fact, handling congestion without considering flexible buffer management and queueing methods, stay insignificant against dropping rate and packet loss. Consequently, a novel congestion control approach is needed to overcome these problems. In this paper, a new queue management mechanism called: Priority Queue Management based Explicit Congestion Control (PQMECC) is proposed and modeled using Stochastic Automata Networks, which are appropriate for modeling and analyzing complex systems with interacting components and complex synchronization. The congestion control technique used in PQMECC is based on priority, load repartition mechanism and explicit congestion notification which is generated only if the congestion reaches the threshold value.

On the applicability of SAN in modeling and analyzing a Priority Queue Management based congestion control for WMSNs

Dependability is an all-encompassing definition for reliability, availability, safety and security, and is required in computer applications such as safetycritical control systems for road vehicles, airplanes and medical devices, and business-critical systems for e-commerce and financial transactions. To assure high dependability of systems, redundancy has been widely applied and plays an important role in enhancing system reliability. In general, there are two commonly-used types of system designs; component (or subsystem) redundancy and environmental redundancy. The component redundancy is the use of additional components or subsystems beyond the number actually required for the system to operate reliably, such as k-out-of-n redundant systems with spares. In the environmental redundancy, the system re-executes some of its initialization procedures to obtain a fresh environment that might in turn make the system less prone to failures, such as rejuvenation techniques that reboot the system before failures occur. In fact, redundancy increases not only the complexity of a system, but also the complexity of associated problems such as common-mode error. Thus, in order to detect the optimal design of systems, model-based analysis is important in the system design. Fault trees (FTs), reliability block diagrams (RBDs), Markov chains (e.g., discretetime Markov chain (DTMC) and continuous-time Markov chain (CTMC))

/pdf/a-study-on-sensitivity-approaches-for-dependable-systems-2g87scxelf.pdf

A Study on Sensitivity Approaches for Dependable Systems Design

We consider queueing systems with multiple classes of customers and heteroge- neous servers where customers have the flexibility of being processed by more than one server and servers possess the capability of processing more than one customer class. We provide a unified framework for the modeling and analysis of these systems under arbitrary customer and server flexibility and for a rich set of control policies that includes customer/server-specific priority schemes for server and customer selection. We use our models to generate several insights into the effect of system configuration and control policies. In particular, we examine the relationship between flexibility, control policies and throughput under varying assumptions for system parameters. © 2004 Wiley Periodicals, Inc. Naval Research Logistics 51: 755-782, 2004. In this paper, we consider the modeling and analysis of flexible queueing systems. We use the term flexible queueing systems to refer to systems with heterogeneous servers and multiple customer classes where customers have the flexibility of being processed by more than one server and servers possess the capability of processing more than one customer class. Customer classes can vary in demand rate and routing flexibility. Servers can vary in service rates and service flexibility. The assignment of customers to servers is determined by a server selection rule, that can be customer-specific, and the selection of the next customer to serve is determined by a queue selection rule, that can be server-specific. An example of a flexible queueing system is shown in Figure 1. Flexible queueing systems arise in a variety of contexts, including manufacturing (Buzacott and Shanthikumar (8)), telecommunication networks (Ross (23)), computer systems (Kleinrock

Modeling and analysis of flexible queueing systems

We describe how an aggregation/disaggregation method for finding quasi-stationary distributions of continuous-time Markov chains can be implemented on a massively parallel computer. The method is similar to an algebraic multigrid, using restriction operators that depend on the current iteration of the solution, and Jacobi smoothers at each level of the multigrid. The method is illustrated using a simple epidemic model, and the performance compared to a sequential implementation as the size of the population increases.

Parallel implementation of an aggregation/disaggregation method for evaluating quasi-stationary behavior in continuous-time Markov chains

We are concerned with computing the steady-state distribution it of an finite irreducible Markov chain with transition matrix P. We will use the GTH [1]algorithm which has excellent numerical properties which have been demonstrated empirically[2] and mathematically [3] We have at our disposal workstations and a massively parallel computer; we want to see how execution times on the latter compare to execution times on the former. Embedded in this endeavor is an exploration of how to harness the massively parallel computer to work on the GTH algorithm. Our main conclusions are: 


Our massively parallel computer can solve a problem with one thousand states one hundred times as fast as a serial computer. Extrapolation of our experience using one-eighth of the available memory indicates that for a problem with eleven thousand states,the serial machine would require 104 as much time as the parallel machine


Having enough memory to store the transition matrix is the limiting factor for our parallel computer.


When the transition matrix has the block tridiagonal form, our parallel computer can store many thousands of states (depending on the block size; 16 × 106 states can be stored when the blocks are 2 × 2) and compute the steady-state distribution in a few hours. The 16 × 106 state example can be done in 24 hours.

Parallel Implementation of the GTH Algorithm for Markov Chains

The Grassmann–Taksar–Heyman algorithm is a direct algorithm for computing the steady-state distribution of an finite irreducible Markov chain. We describe our experience in implementing this algorithm on a single-instruction multiple-data parallel processor computer. Our main conclusions are that a lower-level language has a performance advantage compared to Fortran, and that data storage is the limiting factor that determines the largest problem that can be solved. As a consequence, we devote considerable attention to storing a block tridiagonal transition matrix.

A Parallel Implementation of the GTH Algorithm

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Parallel Implementation of the GTH Algorithm for Markov Chains

A Parallel Implementation of the GTH Algorithm