We investigate queueing networks in a random environment. The impact of the evolving environment on the network is by changing service capacities (upgrading and/or degrading, breakdown, repair) when the environment changes its state. On the other side, customers departing from the network may enforce the environment to jump immediately. This means that the environment is nonautonomous and therefore results in a rather complex two-way interaction, especially if the environment is not itself Markov. To react to the changes of the capacities we implement randomised versions of the well-known deterministic rerouteing schemes 'skipping' (jump-over protocol) and `reflection' (repeated service, random direction). Our main result is an explicit expression for the joint stationary distribution of the queue-lengths vector and the environment which is of product form.

This paper investigates the efficiency and properties of limit cycle walking, running, and skipping of a planar, active, telescopic-legged rimless wheel. First, we develop the robot equations of motion and design an output following control for the telescopic-legs' action. We then numerically show that a stable walking gait can be generated by asymmetrizing the impact posture. Second, we numerically show that a stable running gait can be generated by employing a simple feedback control of the control period, and compare the properties of the generated running gait with those of the walking gait. Furthermore, we find out another underlying gait called skipping that emerges as an extension of the walking gait. Through numerical analysis, we show that the generated skipping gaits are inherently stable and are less efficient than the other two gaits.