10 problems

Consider the queueing system with on-demand servers and its fluid limits, whose trajectories have a local stability property when the corresponding dynamic system away from the bou…

Let $z$ be a fluid limit with initial condition $z(0)=x$, where $x=\lim_n x^n/|x^n|$, and let $V_i$ denote the neighborhood associated with coordinate $i$. The right derivative at…

Let $\widetilde{\bm{X}}^{(n)}(\infty)$ be the stationary random vector of the scaled finite system, and let $\bm{y}(\infty)$ denote the equilibrium point of the fluid model. Interc…

Intermediate time-scale conjecture. The non-trivial behavior expected at a time scale intermediate between $N$ and $N^2$ should occur in any network topology.

Let $x(t)$) be a fluid solution with fixed point $x^\star$, and let $\lambda$ be the arrival rate. Theorem Global Stability proves exponential convergence to $x^\star$ when…

Consider the queueing system with on-demand servers and its fluid limits, and let $gamma$ be the parameter governing the invitation scheme. Large-$\gamma$ global-stability conjectu…

Consider the fluid-limit dynamics of the on-demand agent service system with system matrix $A^{-}$. Non-Hurwitz parameter-dependence conjecture. If $A^{-}$ is not Hurwitz, the syst…

Consider the fluid-limit dynamics of the on-demand agent service system, represented by switched linear systems with system matrices $A^{-}$ and $A^{+}$. A matrix is Hurwitz when a…

Let $x$ be a fluid solution with initial condition $x(0)$, let ${\cal O}$ be the set of initial conditions whose trajectories oscillate indefinitely, and let $u^*$ denote a periodi…

Rationality-removal conjecture. Assumption that $r_{1,2}$ and $r_{2,1}$ are rational positive numbers can be removed.