On the benefits of using functional transitions and Kronecker algebra

Abstract

Much attention has been paid recently to the use of Kronecker or tensor product modelling techniques for evaluating the performance of parallel and distributed systems. While this approach facilitates the description of such systems and mimimizes memory requirements, it has suffered in the past from the fact that computation times have been excessively long. In this paper we propose a suite of modelling strategems and numerical procedures that go a long way to alleviating this drawback. Of particular note are the benefits obtained by using functional transitions that are implemented via a generalized tensor algebra. Examples are presented which illustrate the reduction in computation time as each suggested improvement is deployed.

Memory-efficient Kronecker algorithms with applications to the modelling of parallel systems

“…We present results obtained from two classical parallel systems chosen from the literature ( [7,20,33]). The first model, called mutex1, performs resource sharing with the use of functions; the second, mutex2, represents the same model, but with functions replaced by synchronizing transitions.…”

“…In the SANs formalism, the use of functions allows a decrease in the size of the product state space. The use of a generalized tensor algebra ( [7,33]) permits tensor operations on matrices to have functional characteristics. However, the cost of matrix evaluation is high and so we try to limit their number.…”

Efficient vector-descriptor product exploiting time-memory trade-offs

“…These runs were statistically handled to obtain a 95% confidence level. The five classes of examples presented here are the following SAN models in the descriptor format using only classical tensor algebra, i.e., considering only models without functional rates: (i) Resource Sharing (RS) model [4] -a classical example of resource sharing with different network configurations since P is the number of processes (matrices with two states: idle and occupied) and R is the number of occupied resources (a matrix with R + 1 states). The model descriptor presents (2P ) synchronizing events, totaling (4P ) tensor products with P + 1 matrices.…”

Memory-efficient kronecker algorithms with applications to the modelling of parallel systems

“…We present results obtained from two classical parallel systems chosen from the literature ( [6,19,31]). The first model, called mutex1, performs resource sharing with the use of functions; the second, mutex2, represents the same model, but with functions replaced by synchronizing transitions.…”

“…In the SANs formalism, the use of functions allows a decrease in the size of the product state space. The use of a generalized tensor algebra ( [6,31]) permits tensor operations on matrices to have functional characteristics. However, the cost of matrix evaluation is high and so we try to limit their number.…”