A Kronecker Algebra Formulation for Markov Activity Networks with Phase-Type Distributions

Abstract

The application of theoretical scheduling approaches to the real world quite often crashes into the need to cope with uncertain events and incomplete information. Stochastic scheduling approaches exploiting Markov models have been proposed for this class of problems with the limitation to exponential durations. Phase-type approximations provide a tool to overcome this limitation. This paper proposes a general approach for using phase-type distributions to model the execution of a network of activities with generally distributed durations through a Markov chain. An analytical representation of the infinitesimal generator of the Markov chain in terms of Kronecker algebra is proposed, providing a general formulation for this class of problems and supporting more efficient computation methods. This entails the capability to address stochastic scheduling in terms of the estimation of the distribution of common objective functions (i.e., makespan, lateness), enabling the use of risk measures to address robustness.

A branch-and-bound approach to minimise the value-at-risk of the makespan in a stochastic two-machine flow shop

“…Creemers (2015Creemers ( , 2018 addressed the P o s t -p r i n t resource-constrained project scheduling problem (RCPSP) using specific classes of phase-type distributions to cope with non-exponential processing times and, grounding on this, optimal scheduling policies were derived based on continuoustime Markov chain models. Angius et al (2021) proposed a general approach for modelling the execution of a network of activities with generally distributed processing times through a Markov chain and general phase-type distributions. All these works leverage the capability to estimate the distribution of an objective function (e.g., the makespan) to enable risk measures to address robustness.…”

“…Based on the states in Figure 3, the infinitesimal generator of the CTMC representing the execution of the network of activities with phase-type distributions of the processing times can be obtained using a Kronecker algebra approach (Angius et al 2021). Thus, the makespan of the network of activities is the time to absorption of the described CTMC, whose distribution can be calculated according to:…”

“…1. taking advantage of a modular definition of the infinitesimal generator matrix (Angius et al 2021) to reuse the estimation operated on partial schedules within the exploration of the same branching tree.…”

“…However, exploiting risk measures as the optimisation criterion ( e.g., minimising the value-at-risk of the makespan) entails estimating the distribution of the considered objective function, which could be difficult even for relatively small scheduling problems (Dodin 1985(Dodin , 1996. Markovian Activity Network (MAN), which exploits a Markovian model to represent the execution of the activities in the network as a Continuous Time Markov Chain (CTMC), is a powerful tool to address this estimation problem (Kulkarni and Adlakha 1986) and has been extended to model generally distributed processing times by using phase-type distributions (Angius et al 2021).…”

A Robust Scheduling Framework for Re-Manufacturing Activities of Turbine Blades

“…To overcome this difficulty, a Markovian Activity Network(MAN) approach can be used to support the analytical estimation of this distribution. Basic MANs require the processing times of the activities to follow exponential distributions [39] but can be extended to cope with general distributions, approximated by phase-type distributions [40]. Grounding on this, the distribution of objective function based on the completion times of the activities (e.g., the makespan), enables the use of risk measures to address robustness of scheduling.…”