Stochastic stability in Max-Product and Max-Plus Systems with Markovian Jumps

Abstract

We study Max-Product and Max-Plus Systems with Markovian Jumps and focus on stochastic stability problems. At first, a Lyapunov function is derived for the asymptotically stable deterministic Max-Product Systems. This Lyapunov function is then adjusted to derive sufficient conditions for the stochastic stability of Max-Product systems with Markovian Jumps. Many step Lyapunov functions are then used to derive necessary and sufficient conditions for stochastic stability. The results for the Max-Product systems are then applied to Max-Plus systems with Markovian Jumps, using an isomorphism and almost sure bounds for the asymptotic behavior of the state are obtained. A numerical example illustrating the application of the stability results on a production system is also given.proposed in various applications involving timing, such as communication and traffic management, queueing systems, production planning, multi-generation energy systems, et.). Recently, the use of the closely related class of Max-Product systems (systems which satisfy the superposition principle in the Max-Product algebra) was proposed as a tool for the modelling of cognitive processes, such as detecting audio and visual salient events in multimodal video streams (Maragos and Koutras (2015)). Max-Plus and Max-Product algebras have also computational uses involving Optimal Control problems (McEneaney (2006)) and estimation problems in probabilistic models such as the max-sum algorithm in Probabilistic Graphical models and the Viterbi algorithm in Hidden Markov Models (eg. Bishop (2006)).In this work, we study stochastic Max-Plus and Max-Product systems, where the system matrices depend on a finite state Markov chain. For the Max-Plus systems we focus on the asymptotic growth rate, whereas for the Max-Product systems on stochastic stability. A motivation to study Max-Plus systems with Markovian jumps is to model production systems, where the processing or holding times are random variables (not necessarily independent) or there are random failures and repairs, modeled as a Markov chain. The results on max-product stochastic systems will be used as an intermediate step. An independent motivation to study Max-Product systems is the modeling of cognitive processes interrupted by random events. Similar problems with Markovian delays for linear systems were studied in Beidas and Papavassilopoulos (1993), for random failures in Papavassilopoulos (1994) and for nonlinear time varying systems in Beidas and Papavassilopoulos (1995), in the context of distributed parallel optimization and routing applications. In the current work, we try to exploit the special (Max-Product or Max-Plus) structure of the system.At first, deterministic Max-Product systems are considered and their asymptotic stability is characterized using Lyapunov functions.The Lyapunov function derived can be also used to study systems which are not linear in the Max-Product algebra. We then study Max-Product systems with Markovian Jumps and derive sufficient conditions for their stochastic s...

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