15 problems

Let $\mathcal{M}$ be the polling-system Markov chain, let $\ell$ be the boundary level, and let $\pi(x,y,s)$ denote its stationary probability at state $(x,y,s)$. In the spiral-spi…

Let ${\pi} \in {\cal T}_s$ be a switching policy, and let $\Theta_1^\mu(t)$ denote the number of customers in queue $Q_1$ at time $t$. Let $\Phi_k^\mu$ be the server-visit epoch to…

Let $\lambda_k=2$, $\mu_k=8$, and $s_k=2$ for $k=1,2,3$, let $\psi$ be linear with $c_1=c_2=1$, and consider $\mathcal N=\{1\}$. For a system with three queues and basic table…

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…

Consider a polling system with two queues, exhaustive service, no switch-over times, and exponential service times with rate $\mu$ in each queue. Customers arrive according to a Po…

Let ${\cal P}$ be a regular polling policy, and let $\lambda_1$ and $\lambda_2$ be the intensities of the arrival processes in lanes $1$ and $2$, respectively. Let $s$ be the servi…

Consider polling systems with Poisson arrivals and dynamic visit orders, meaning that there is no predetermined order in which the queues are served. A switch-over has a regulating…

Let a cyclic polling system have general renewal arrival processes and deterministic switch-over times. The system is considered in the regime where the deterministic switch-over t…

Let ${\cal L}$ be the random walk associated with routing matrices $P$ and $\wt P$, and let conditions … denote the necessary rate and flux conditions introduced in the paper. Ergo…

Fluid-model distributional dependence conjecture. If $l_2>1$, then $\bar{U}'_4(t)$ is not determined solely by $J(t)$ and the model parameters, but may also depend on the distribut…

Distributional stability conjecture. A necessary and sufficient condition for stability may depend on the whole distributions of the primitive sequences, rather than only on their…