22 problems

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 dynamic averaging process on a cycle, in which load is continually introduced through weighted arrivals and neighboring vertices locally average their loads. Let the g…

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 $k$.

Let $m$ balls be allocated across $n$ bins and repeatedly reallocated by the repeated balls-into-bins process: in each round, one ball is removed from each non-empty bin and each r…

Critical-exponent conjecture. The results established for every $\alpha<0.5$ should continue to hold for $\alpha=0.5$.

Consider a cluster with $N$ servers, indexed so that server $1$ is fastest and server $N$ is slowest. Let $\boldsymbol{ll}=(ll_1,ll_2,ots,ll_N)$ denote the optimal buffer lengths a…

Consider a graphical allocation process in which balls are allocated greedily to one of the two vertices of a requested edge, and let the load fluctuations denote the fluctuations…

Let the $N$-server systems use the join-the-shortest-of-$d$-queues routing algorithm, and let their stationary distributions be denoted by $\pi_N$. Suppose that the hydrodynamic eq…

LL($d,K$) waiting-time limit conjecture. The limiting value

Waiting-time conjecture. The average waiting time of load balancing algorithms in $\Pi$ is

Universal scaling conjecture. The following results hold for any load balancing algorithm in $\Pi$:

Let the combinatorial model be the service-system model with server selections of maximum cardinality $d$, and let the corresponding supermarket model use a JSQ($d$) policy with id…

Message-rate conjecture. Access to incoming job sizes should permit a policy with vanishing queueing delay even when its message rate is strictly less than the arrival rate…

Let $m=\Omega(n)$ balls be allocated to $n$ bins in the two-thinning setting, and let the overseer two-thin the allocations. More-balls conjecture. The asymptotically optimal maxim…

Let $n$ bins receive balls in the two-thinning setting, where the overseer may iteratively reject up to $k$ suggested allocations for each ball. More-choice conjecture. The asympto…

Consider the SQ$(d,N)$ system analyzed with finite buffer size $I$, and let $I=\infty$ denote the corresponding system with unbounded buffers. Infinite-buffer conjecture. The resul…

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…

The proposed proximity-aware two choices scheme redirects requests using nearby servers' cache contents and the queue lengths of two randomly chosen servers within a neighborhood o…

The paper considers a large-scale heterogeneous system with multiple independent routers and the PULL-2 load-distribution algorithm, which uses pull-messages and occasional pull-re…

Consider the finite-server heterogeneous service system with vector packing constraints, and an algorithm in which each arriving customer samples $d$ servers uniformly at random an…

Let $A_u$ and $A_c$ denote the linearization matrices governing the underloaded and critically loaded fluid models, respectively, for a system with general service-rate parameters…

Storage-efficiency conjecture. If $l=2$ and $q-1=c\ln n$, then the load-balancing modulation code has storage efficiency $\gamma=k$ with probability $1-o(1)$ as…