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.

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

“…A biological process associated with metabolite synthesis involving two metabolites and two enzymes is considered. 55,68 In this model, molecules interact through the nine transition classes given in Table 1. Metabolites S 1 and S 2 are synthesized respectively by enzymes S 3 and S 4 through the first two transition classes.…”

“…Now, observe that the matrices S ( ) h ( n,h ,  n,h ) and D ( ) h ( n,h ,  n,h ) that contribute to Q n are not only finite but also sparse, and those associated with molecules that do not change their states in transition class j are identity. 55,56 The elements of the vectors obtained at time step n − 1 (i.e., backward difference vectors ∇ 0 p n−1 , … , ∇ k p n−1 and prediction vector p (0) n ) correspond to states in the truncated state space  n−1 . These vectors need to be updated so that they become incident to  n (see Line 5 of Algorithm 2).…”

“…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. 55 This exact representation, which has been used in the Kronecker-based modeling and analysis of finite multidimensional CTMCs 56 for many years, is already compact and does not require a low-rank approximation for compactness in the HTD format. Following recent results from the steady-state analyses of Kronecker-based CTMCs with compact vectors, 50 however, solution vectors in the HTD format are employed in the proposed BDFk solver with the exact, compact, and truncated generator matrix in Kronecker form during transient analysis.…”

On the numerical solution of Kronecker-based infinite level-dependent QBD processes

“…In many cases, the ergodicity of an LDQBD process can be established by relatively easy to check conditions on the 1-dimensional CTMC defined over its levels [12]. For computational purposes however, it is preferable to consider Lyapunov function methods as discussed in [13,14], so that lower and higher level numbers (called Low and High, respectively) can be computed as in [15][16][17] between which a specified percentage of the steady-state probability mass lies when the LDQBD is ergodic. It is the latter approach we follow here.…”

“…However, it has been shown in [17] that systems of stochastic chemical kinetics can be modeled as LDQBD processes with the level number determined by the maximum value among the countably infinite variables. Inspired by hierarchical Markovian models (HMMs) introduced in [22], the result in [17] has been taken one step further in [16] by providing a Kronecker-based representation for the nonzero blocks Q l,l−1 , Q l,l , Q l,l+1 at each level to cope with the multidimensionality of the product state space of variables and its reachability. As suggested in [27] and as observed to be the best overall choice in [17], again R High is set to 0 to initiate the computational procedure associated with the matrices of conditional expected sojourn times.…”