Per-queue stability analysis of a random access system

up response of the pendulum starting at nearly the vertically downward position, with the remaining initial conditions zero. Notice that the response is very fast without any initial swinging of the pendulum. The following remarks are in order. • It is clear from the simulations that the stabilization mechanism of our controller consists of spinning-up the disk inertia to lift the pendulum, which might impose some unrealistic values to the disk speed. This should be contrasted with the alternative method of [9]—also studied in [1], [3]—where the energy is first pumped-up through a balancing motion before lifting the pendulum. Two drawbacks of the latter approach are the slow convergence and the need to switch the controller close to the upward position. From the theoretical viewpoint both methods also differ, our controller (as well as the one reported in [8]) stabilizes the equilibrium point, while the energy-pumping methods stabilizes the homoclinic orbit, hence the need for the switching. • Although we have solved the stabilization problem of the system (10) with any prescribed saturation of the control, when we come back to the original disk inertia pendulum (9), we have to add sin(x 1) to the above control. So the above procedure does not give an answer to the problem where the maximal torque that the motor can deliver is smaller than the maximal gravity torque. Simulations and experiments have shown that stability cannot be guaranteed if we impose this saturation limit. REFERENCES [1] K. Astrom and K. Furuta, " Swinging up a pendulum by energy control, " in Proc. Adding integrations, saturated controls and global asymptotic stabilization for feedforward systems, " IEEE Trans. Abstract—In this note, we have extended previous studies of the system stability of buffered ALOHA systems to study an individual queue's stability , i.e., per-queue stability. The main result obtained in this work is a necessary and sufficient per-queue stability condition, which can be computed analytically only for several cases. For other noncomputable cases, we have evaluated several inner and outer bounds. They are generally quite tight for not-so-asymmetric systems.