11 problems

Let $\xi=(\xi_s)_{s\geq 0}$ be the Lévy process in the factorisation … and let $\overline{F}_\Psi(x)=\mathbb{P}(I_\Psi>x)$ denote the tail of the exponential functional $I_\Psi$. R…

FCFS strong tail optimality conjecture. FCFS is strongly tail optimal for class-I job sizes: for every scheduling algorithm $\pi$,

Let $Q_1$ and $Q_2$ be the equilibrium queue lengths in the two-queue random walk, and let $Q$ be the equilibrium queue length in the corresponding $Geo/Geo/2$ model with one queue…

Assume $E[\log\rho_0]<0$, and assume the stronger definition of $\kappa$ together with $E[\rho_0^\kappa\log\rho_0]<\infty$ and a non-lattice distribution for $\log\rho_0$, as descr…

Tail-equivalence conjecture. There exists a constant $c\in(0,\infty)$ such that

Boundary-case tail conjecture. The same conclusion should hold when

Modulus conjecture for $b$. The coefficient always satisfies

Gaussian-tail conjecture. (a) $N_{n}(\infty)$ exists for each $n$; (b) the sequence $\{\tilde{N}_{n}(\infty):n\in\mathbb{N}\}$ weakly converges to $\check{N}(\infty)$; and (c)

Tail-dependence conjecture. The limiting tail behavior of $\tilde{N}_{n}(\infty)$ depends on the service time distribution only through its first two moments, and on the patience t…

Gaussian-tail conjecture. The limit queue length process has a Gaussian tail, and the tail depends on the service time distribution only through its first two moments.

Let $\xi=(\xi_s)_{s\geq 0}$ be a Lévy process and let … For some constant $c\in(0,\infty)$, tail-equivalence conjecture. … The conjecture is motivated by the heuristic that large v…