Optimal Aggregation of Components for the Solution of Markov Regenerative Processes

Abstract

The solution of non-ergodic Markov Renewal Processes may be reduced to the solution of multiple smaller sub-processes (components), as proposed in [4]. This technique exhibits a good saving in time in many practical cases, since components solution may reduce to the transient solution of a Markov chain. Indeed the choice of the components might significantly influence the solution time, and this choice is demanded in [4] to a greedy algorithm. This paper presents a computation of an optimal set of components through a translation into an integer linear programming problem (ILP). A comparison of the optimal method with the greedy one is then presented.

Efficient model checking of the stochastic logic CSLTA

“…Although any acyclic set of components ensures that P is in RNF as per Equation 1, the work in [17] provides evidence that the choice of the components can heavily influence the Component Method performance. Following [30] we consider three component classes, based on the cost of computing the component outgoing probability (component solution for short). A components is of class C E if no state of the component enables a general event: the component is a CTMC and the computation of the outgoing probabilities amounts to computing the steady-state solution of the frontier states.…”

“…The experiments in [17] suggest that working with the smallest components or with very large components may lead to inefficiency, and therefore the paper defines an optimality criteria for component aggregation, recalled in the following definitions. The paper identifies an heuristic for the construction of an optimal partition, while an optimal solution based on linear integer programming is given in [30].…”

“…aggregation does not change the complexity of the required solution. The original definition of the three component classes given in [30](Sec.2), can be simplified when the MRgP is a M×A or M×Z, based on the observation that clock resets always lead to states of S g1 , and that the firing of g k without clock reset enables g k+1 .…”

“…A component Z j of the region graph Z is classified in exactly one of the following m + 2 classes: The definition of D g1 is slightly complex (as for the C g1 case), since it is necessary to exclude the presence in Z j of a reset edge in the component, either by an inner edge or by a time-elapse edge followed by a path of boundary edges that goes back to the same Z j component. The above problem is solved, using the technique presented in [30], as in the case of the MRgP component optimization problem of Definition 22.…”

“…Algorithms 3 and 4 can then be modified to work with an optimal set of components of Z, according to the definition above. Note that the size of Z is typically much smaller than that of M×Z and therefore the computation of an optimal set of components is usually feasible (for example by reducing the optimization to the solution of the ILP problem presented in [30] for MRgP), while for M×A in most cases only a sub-optimal partition can be found, based on the heuristic for MRgP defined in [17]. A natural question then arises: the components {M×Z j } 1≤j≤J , generated from an optimal set of components {Z j } 1≤j≤J are an optimal set of component for the MRgP M×Z?…”