On compact solution vectors in Kronecker-based Markovian analysis

Abstract

State based analysis of stochastic models for performance and dependability often requires the computation of the stationary distribution of a multidimensional continuous-time Markov chain (CTMC). The infinitesimal generator underlying a multidimensional CTMC with a large reachable state space can be represented compactly in the form of a block matrix in which each nonzero block is expressed as a sum of Kronecker products of smaller matrices. However, solution vectors used in the analysis of such Kronecker-based Markovian representations require memory proportional to the size of the reachable state space. This implies that memory allocated to solution vectors becomes a bottleneck as the size of the reachable state space increases. Here, it is shown that the hierarchical Tucker decomposition (HTD) can be used with adaptive truncation strategies to store the solution vectors during Kronecker-based Markovian analysis compactly and still carry out the basic operations including vector-matrix multiplication in Kronecker form within Power, Jacobi, and Generalized Minimal Residual methods. Numerical experiments on multidimensional problems of varying sizes indicate that larger memory savings are obtained with the HTD approach as the number of dimensions increases.

Efficient Transient Analysis of a Class of Compositional Fluid Stochastic Petri Nets

“…In these areas compact representations to approximate higher dimensional tensors in a more efficient way have been developed [17], [23]. These representations have recently also been applied to the stationary analysis of Markov models [5], [19]. We extend the approaches here to the transient analysis of SFSPNs.…”

“…However, for the HTD representation a well defined approximation algorithm exists. This algorithm computes for each matrix in the tree a singular value decomposition which allows one to truncate the ranks according to a well defined error bound (for details see [5], [21]). The local truncation operation in each node results in an optimal local approximation of the exact representation in terms of the 2-norm [14].…”

“…For this purpose, recently developed data structures for higher dimensional tensors can be used [17]. We use a Hierarchical Tucker Decomposition (HTD) which has already been applied for the analysis of PDEs [20] and for the stationary analysis of Markov chains [5]. Here we extend the approaches to the transient analysis of FSPNs.…”

On compact vector formats in the solution of the chemical master equation with backward differentiation

“…HTD may be considered as a generalization of the TT decomposition in which basis and core matrices are related through a tree structure with logarithmic depth in the number of dimensions that reduces the complexity of the Tucker decomposition by hierarchically splitting the core tensor into core matrices. An advantage of the HTD representation for our purposes is that the multiplication of a compact solution vector in HTD format with a sum of Kronecker products is a well‐defined operation and has recently been used in the steady‐state analysis of Kronecker‐based CTMCs . In the CME setting, such a Kronecker‐based Markovian analysis with vectors in HTD format is possible under the separability assumption of transition rates .…”

“…An advantage of the HTD representation for our purposes is that the multiplication of a compact solution vector in HTD format with a sum of Kronecker products is a well-defined operation 49 and has recently been used in the steady-state analysis of Kronecker-based CTMCs. 50 In the CME setting, such a Kronecker-based Markovian analysis with vectors in HTD format is possible under the separability assumption of transition rates. 7,25,32,43,51,52 To the best of our knowledge, the HTD format has not been employed in the transient analysis of the CME so far.…”