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

Abstract

The description of large state spaces through stochastic structured modeling formalisms like stochastic Petri nets, stochastic automata networks and performance evaluation process algebra usually represent the infinitesimal generator of the underlying Markov chain as a Kronecker descriptor instead of a single large sparse matrix. The best known algorithms used to compute iterative solutions of such structured models are: the pure sparse solution approach, an algorithm that can be very time efficient, and almost always memory prohibitive; the Shuffle algorithm which performs the product of a descriptor by a probability vector with a very impressive memory efficiency; and a newer option that offers a trade-off between time and memory savings, the Split algorithm. This paper presents a comparison of these algorithms solving some examples of structured Kronecker represented models in order to numerically illustrate the gains achieved considering each model's characteristics.

Symbolic Solution of Kronecker-Based Structured Markovian Models

“…Such solution may extend structured model's application scope to problems that require a much faster solution than the ones available with traditional solutions [1], [2]. These solutions deliver predictions for large models and complex models within hours and even days, which is too long to many applications such as autonomic computing predictors [13].…”

“…Returning to our example, this operation is exemplified by adding a scaled version of the first row to the third one 2 . It starts by computing the value of the element below the pivot, which is α, as well as the current elements of the first row, which are:…”

“…I shows the results for classic SAN models available in the literature [8], [10]- [12]. In this table, it is indicated the number, name and source of each model, its product state space size (M ), the time to solve it performing the numerical solution through Split algorithm and power iterations [2] (num. sol.…”

“…Structured formalisms cope quite efficiently with memory needs to store very large stochastic models, but the time needed to achieve stationary solution is frequently prohibitive [1], [2]. The performance evaluation community usually tackle this limitation with approximations either based on numerical solutions [3], or discrete event simulations [4], [5].…”

SANGE – Stochastic Automata Networks Generator. A tool to efficiently predict events through structured Markovian models

“…The output is in the format of a .san file, that can be directly fed into a SAN solver, such as PEPS [5] that performs state of art Kronecker solutions [6].…”