71 problems

Let $u$ be the transition rate from the lounge to the queue, let $\beta$ satisfy … where $\eta>0$ is constant, and let $A=\lfloor\eta\mu/\alpha\rfloor$. For each $u$, write…

Let $\alpha,\beta\in(0,1]$, and let $X_{\alpha}$ and $X_{\beta}$ be independent Mittag-Leffler random variables with power indices $\alpha$ and $\beta$, respectively, and equal tim…

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…

Mean-field busy-probability comparison. For $<$,

Consider a multiple-access channel in discrete time with Poisson arrivals of mean birth rate $\lambda>0$. A backoff protocol is specified by a send sequence…

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…

Let $j$ be the number of stations, and let $O^{(i)}_{n,n+k}$ denote the overlap time at station $i$ for the $n^{th}$ and $(n+k)^{th}$ customers in an $(M/M/\infty)^j$ tandem queue…

M/M/1-PS AAoI conjecture. The average age of information is

Structured-vector optimality conjecture. The vector $\boldsymbol{v}^*$ is an optimal solution of the paper's continuous minimization problem.

Non-integer convex-hull conjecture. Every vector satisfying that characterization should belong to

Non-integer characterization conjecture. A vector $\boldsymbol{v}\in\Lambda(v)$ is optimal if and only if it has this decomposition and satisfies, for the relevant indices $l$,

Non-integer optimality conjecture. The set

Integer-case convex-hull conjecture. Every such vector should satisfy

Let $Q$ be the queue length and $S$ the server speed, and let $u$ be the measurement-rate parameter tending to zero. Define … the generating function of the scaled pair…

Consider the static-routing optimization problem … where $\widetilde{\boldsymbol{\Pi}}^{\textup{SR}}$ is the class of stationary randomized policies that route every arrival to som…

Consider an input-queued switch operating under a scheduling algorithm that achieves state space collapse. Let $L(\boldsymbol\theta)$ and $\mathbf M(\boldsymbol\theta)$ denote the…

Aldous's conjecture. In the queue-free setting, no backoff protocol is stable for any positive value of $\lambda$.

Let the standing assumptions of the paper hold. Let ${\cal S}_{\text{LP}}$ denote the set of solutions to the linear program, and suppose that this set contains multiple solutions.…

Let $N_i^*$ be the truncation levels, let $\bar{t}$ be the time parameter, and let $\relax\epsilon_1(N_1^*,N_2^*,\bar{t})$ and $\relax\epsilon_2(N_1^*,N_2^*,\bar{t})$ denote the tw…

Fix a network and a tail-exponent vector ${\boldsymbol{\gamma}}$. Let ${\boldsymbol{\lambda}}^*$ range over nonnegative nominal arrival vectors, and let 0-RJF and G-RDS denote the…

Fix a network with queue $m$, tail-exponent set $\Gamma$, and the notions of RDS and special $\epsilon$-Lyapunov function as in the preceding corollary. Converse Lyapunov conjectur…

Uniform minimization conjecture. For every $q\geq 0$, the coefficient under the uniform policy equals the minimum over all edge-selection probability vectors:

Profit-driven regime conjecture. The scaled steady-state variable should satisfy

Large reshuffling-system conjecture. As $N\to\infty$: (i) the number of useful packets on a link at a communication epoch converges in distribution to…

Delay-profile convergence conjecture. For $D=\mathrm{OU}$ or $D=\mathrm{RU}$, and for a given load $\lambda<1$, as $N\to\infty$,