Kronecker-Based Infinite Level-Dependent QBD Processes

Abstract

Markovian systems with multiple interacting subsystems under the influence of a control unit are considered. The state spaces of the subsystems are countably infinite, whereas that of the control unit is finite. A recent infinite level-dependent quasi-birth-and-death model for such systems is extended by facilitating the automatic representation and generation of the nonzero blocks in its underlying infinitesimal generator matrix with sums of Kronecker products. Experiments are performed on systems of stochastic chemical kinetics having two or more countably infinite state space subsystems. Results indicate that, even though more memory is consumed, there are many cases where a matrix-analytic solution coupled with Lyapunov theory yields a faster and more accurate steady-state measure compared to that obtained with simulation. R l = Q l,l+1 (−Q l+1,l+1 − R l+1 Q l+2,l+1 ) −1 for l ≥ 0,

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

“…Model A biological process associated with metabolite synthesis involving two metabolites and two enzymes is considered . In this model, molecules interact through the nine transition classes given in Table .…”

“…In this paper, we propose a BDF k solver for the CME, which uses a prediction vector to truncate the countably infinite state space at each time step. Then, the truncated generator matrix associated with the CME at each time step is represented compactly, and exactly, using a sum of Kronecker products of matrices associated with molecules . This exact representation, which has been used in the Kronecker‐based modeling and analysis of finite multidimensional CTMCs for many years, is already compact and does not require a low‐rank approximation for compactness in the HTD format.…”

“…Let be the stochastic process associated with the d ‐dimensional CTMC, where is the set of states that molecules of type h can be in, and is the product state space of the system, which is potentially infinite. The underlying CTMC can be described by J transition classes each corresponding to a reaction channel as in the next definition …”

“…It is possible to represent the infinitesimal generator matrix underlying the multidimensional CTMC as a sum of Kronecker products of matrices associated with molecules. The following relates molecules and transition classes by defining transition matrices …”