292 problems

We consider a single-queue system with one fast server and two identical slow servers. Jobs arrive according to a Poisson process with rate $\lambda$; the fast server has exponenti…

Consider the stochastic bipartite matching model with arrival probabilities … where $\rho\in(0,1)$, and let the limiting regime be $\rho\downarrow 0$, in which the mean waiting tim…

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…

Consider the proposed dispatching policy for a system with renewal arrivals and generally distributed service times. Stability conjecture. The policy is always stable, even with re…

Let $\pi^N$ denote the stationary measure of the occupancy process in the $N$-th system, let $\qq^*$ be the unique fixed point of the limiting deterministic dynamical system…

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…

Consider a stable first-come-first-served $GI/GI/s$ queue with $s$ identical servers, i.i.d. interarrival times of finite mean $a$, i.i.d. service times of finite mean $b$, mutuall…

Consider a single-server system on the unit circle $\mathbb{R}/\mathbb{Z}$: customers arrive according to a Poisson process of rate $\lambda$, each at an independent uniformly dist…

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 $m(s,t)$ denote the number of calls in progress past point $s\in\mathbb{R}$ at time $t\geq 0$ in a one-dimensional loss network with call-attempt rate $\lambda$, call-length di…

Coffman–Denning conjecture. If $C(B)>1$, then the expected sojourn time in the FB queue with service time $B$ is smaller than that in an FB queue with generic service time $B'$ hav…

Poisson Hypothesis. As $M,N\to\infty$, every fixed $k$-tuple of servers becomes asymptotically mutually independent and identically distributed; the flow $\mathcal{F}_{M,N}$ to a g…

Let $(x_t^n)_{t\geq 0}$ be a sequence of semimartingales whose diffusion approximation, when it exists, is a solution of an Itô equation … Here $w_t$ is a vector-valued Wiener proc…

Consider constrained homogeneous random walks in ${\cal Z}_+^d$ and the associated problems of computing stationary probability distributions and large-deviation rates. The source…

A multiclass queueing network is a queueing system with multiple customer classes, and a priority or First-In-First-Out policy is a scheduling rule that selects which waiting custo…

Consider a closed network of $M$ servers and $N$ clients, with the ratio tending to a fixed value … For large $M$ and $N$, the network defines a complicated Markov process. Weak Po…

Uniqueness conjecture. Equation (11) has a unique solution.

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

Tail-optimality conjecture. In this setting, $\gamma$-CounterBoost is tail optimal among all policies, not only among Contextual CounterBoost policies.

MJSQ–JSQ efficiency conjecture. Based on preliminary calculations, MJSQ and JSQ should deliver comparable efficiency at the level of the $k$ shortest queues for any fixed $k$.

JSQ steady-state scaling conjecture. It has been conjectured that both the minimum and the average steady-state queuelengths scale like

Density-free heavy-traffic limit conjecture. The density condition is not needed for the heavy-traffic limit theorem: under the remaining hypotheses, $\{\nu^{(n)}\}$ converges weak…

Let $\mathcal{N}$ denote the target state set of the two-queue system, and consider anti-cooperative social network structures. Persistence conjecture. Anti-cooperative social netw…

Let $M_{Q,t}/D/\infty$ denote the potentially coupled point-process sampler with deterministic service times, let $M_{Q,t}/M_t/\infty$ denote the corresponding potentially coupled…

An acknowledgement-based contention-resolution protocol is a protocol in which a message's waiting time may be chosen from a non-geometric distribution depending on its collision c…