13 problems

Consider a fork-join network of four LLM agents indexed by $j\in\{0,1,2,3\}$. Requests arrive at agent $0$ with rate $\lambda$, fork into tasks processed by agents $1$ and $2$, and…

Consider Markovian bandwidth-sharing networks operating under $b1$-fair rate allocation policies, with linear topologies and $b1 = 1$ (Proportional Fairness), under the local traff…

A bandwidth sharing network has routes indexed by $i\in\mathbb{I}$ and resources indexed by $j\in\mathbb{J}$, with capacity-resource matrix $A$, service-rate vectors $\mu^r$ and…

Large-system response-time conjecture. Provided that $\rho(\tau)<1$,

Let $n$ be the number of nodes, and let $k$ denote the pre-selected number of nodes participating in selective multicast. Write $\Delta^S_{(k)}$ for the original average age functi…

Let $B=(B_1,B_2)$ be the two-dimensional Brownian motion on a filtered probability space $(\mathnormal{\Omega},\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P})$, and consider the…

Let $G(n,r)$ be the random graph model under consideration, with edge-probability parameter $r^2$, and let $\operatorname{LoP}$ denote the graph property that the local pooling fac…

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…

Let $J$ be the number of stations, let $\rho_j$ denote the traffic intensity at station $j$, and let denote the network stability condition … for $i\in\{0,\ldots,I-1\}$. Stability…

Consider a closed multi-class product-form queueing network in which the number of customers grows to infinity while remaining proportionate on each route or class. A queue is non-…

Quadratic-workload optimality conjecture. The schedule minimizing this long-run average quadratic workload will be the cone schedule with matrix $\bB$. This conjecture proposes a p…

A cone schedule is a policy for selecting service vectors in the processing system according to a cone determined by a matrix $\bB$, generalizing MaxWeight schedules, which corresp…

Threshold-interval conjecture. In such networks, there should again be a threshold interval that distinguishes two regimes for the optimal sensing range.