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.
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Notes
- 1.
This logic implication is not in standard ILP form. It can be transformed [10] in ILP form as follows. Let U be a constant greater than |V|. Add a new variable \(k_{v,w}\) subject to these constraints: \(0 \le k_{v,w} \le 1\), \(U k_{v,w} - U < x_v -x_w \le U k_{v,w}\) and \(\forall D' \in \varvec{\mathcal {D}}\setminus \{D\}\) add \(y^{D'}_v \le y^{D'}_w + U k_{v,w}\).
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Amparore, E.G., Donatelli, S. (2016). Optimal Aggregation of Components for the Solution of Markov Regenerative Processes. In: Agha, G., Van Houdt, B. (eds) Quantitative Evaluation of Systems. QEST 2016. Lecture Notes in Computer Science(), vol 9826. Springer, Cham. https://doi.org/10.1007/978-3-319-43425-4_2
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