11 problems

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

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

Let $N$ denote the number of customers in a Hawkes/M/$\infty$ queue, with ${\mathscr J}(u)=e^{-\mu u}$ and $h(u)=e^{-u}$ for $u\geqslant 0$. Assume that the service-time distributi…

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 $p$ be fixed in $(0,1)$, let $w(\lambda)$ denote the size of the lookahead window, and let $C^*_{\Pi_{w(\lambda)}}(p,\lambda)$ be the best achievable delay under policies…