37 problems

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…

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

Alternative diffusion-limit conjecture. The process $L^{(r)}_{\rm dif}(\cdot)$ can nevertheless weakly converge to a process other than $Z(\cdot)$.

Vanishing-threshold-mass conjecture.

Limiting-distribution conjecture. The limiting distribution is truncated exponential or uniform below the threshold level and exponential above it.

The heavy-traffic limit involves the scalar-valued nonlinear term $R(\cdot)$ in the coupled stochastic integral equation. Local time decomposition conjecture. There may be underlyi…

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…

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.…

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

In an M/G/1 queue with unknown job sizes, let M-SERPT denote the scheduling policy that prioritizes jobs according to their expected remaining processing time, and let its approxim…

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 $\Pi$:

Explicit heavy-traffic limit conjecture. As $\rho\uparrow1$,

Empty-system large-deviation conjecture. As $\lambda^{\mathrm{f}}_{\mathrm{tot}}\to\infty$,

Let $\theta>0$, let $\boldsymbol{X}=(X_1,X_2)$ denote the stationary queue-length vector, and let $\rho$ be the traffic intensity. The notation $\boldsymbol{X}/[-log(1-\rho)]$ deno…

Consider a stable $GI/GI/n$ queue with traffic intensity $\rho$, and seek a universal upper bound on its relevant performance measure that scales as $1/(1-\rho)$. Such a bound is t…

Let $GI/GI/n$ queues be indexed by a sequence approaching heavy traffic, with the number of servers $n$ held fixed. For such a queue, let the traffic intensity be … where $\mu_A$ a…

Let $W_{i_1,i_2,\dots,i_M}$ be the path time along the customer route $i_1,i_2,\dots,i_M$, and let $\mathcal{W}^{\textit{fluid}}_{i_1,i_2,\dots,i_M}$ be the corresponding standardi…

Let $W_i$ be the waiting time of a type-$i$ customer, and let $\mathcal{W}^{\textit{fluid}}_i$ be the standardised waiting time of a type-$i$ particle in the fluid model. Let…

Let $L_i$ be the queue length at queue $i$, let $\mathcal{L}^{\textit{fluid}}_{i,k}$ be the standardised number of type-$i$ particles in the fluid model during visit period $V_k$,…

Let $C_i$ denote the cycle time seen by an arbitrary customer or arrival at queue $i$, and let $\bm{C_i}$ denote its scaled length-biased version. Let $\alpha$ and $\mu$ be as defi…

Let $V$ be the total amount of work in the system at the beginning of a cycle, and let $r$, $\rho$, $\tilde{b}_i$, $\tilde{B}_i$, $\tilde{A}_i$, $\tau$, and $\beta$ have the meanin…

Let $\eta_n$ be the system-size scaling parameter and let $d$ be the number of queues sampled by each arriving customer. Consider queues whose lengths are of order … The process tr…

Let $\eta_n$ denote the system-size scaling parameter, let $d$ be the number of queues sampled by each arriving customer, and let $\omega_n$ be the auxiliary sequence used to descr…