This work examines almost sure stability of a pure random delay system whose delay time is modeled by a finite state continuous-time Markov chain with two-time scales. The Markov chain contains a fast-varying part and a slowly-changing part. Using the properties of the weighted occupation measure of the Markov chain, it is shown that the overall system s almost sure-asymptotic stability can be obtained by using the averaged delay. This feature implies that even if some longer delay times may destabilize the system individually, the system may still be stable if their impact is balanced. In other words, the Markov chain becomes a stabilizing factor. Numerical results are provided to demonstrate our results.
Wu, F., Yin, G. G., & Wang, L. Y. (2012). Stability of a pure random delay system with two-time-scale Markovian switching. Journal of Differential Equations, 253(3), 878 - 905. https://doi.org/10.1016/j.jde.2012.04.017