12 problems

Consider two systems on the same lattice dimension $d$ with the same number of types $n$, having parameter vectors $(p_1,p_2,\ldots,p_n)$ and…

Let $S$ be the limiting distribution of the search cost associated to a sequence $\omega$ of independent and identically distributed positive random variables. Let $S_1$ be a rando…

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

Let $a$ be odd, and let the initial state $z$ be either $\frac{a-1}{2}$ or $\frac{a+1}{2}$, each with probability $\frac{1}{2}$. Consider the gambler's ruin game in which the gambl…

Consider the state space $S=\{0,1,2\}$ with its natural total order, and the probability kernel … with equilibrium distribution $\pi=[1/3,1/3,1/3]$. The kernel is sm-regular when i…

Let a Markov chain have a transition kernel or transition matrix, and let sm-regular mean that its finite-dimensional distributions become smaller in supermodular order when the sa…

Let $\lambda(t)$ be a stationary and ergodic baseline intensity, and for each modulation rate $c\in(0,\infty)$ define $\lambda_c(t)=\lambda(ct)$. Consider the queue driven by the c…

Let $Q^b_1(\infty)$, $Q_1^h(\infty)$ and $Q_1^J(\infty)$ denote the stationary number of needy patients in the Erlang-R model with blocking, holding and the closed ward, respective…

Consider a sequence of coupon collections indexed by $N$, with $U_j^N$ denoting the number of unfilled coupons in a brother's album after the main collector has a complete set. A s…

Let $n\geq 1$, let $c\in\{1,\ldots,n\}$, and let $p=(p_1,\ldots,p_n)\in(0,1)^n$ satisfy $p_1+\cdots+p_n\leq 1$. Write … Here $T_{c,n}(p)$ denotes the time needed in the generalized…

Let $\beta$ denote the random path-loss exponent in the heterogeneous cellular network model, and let $\widetilde{S}$ denote the propagation variable introduced earlier. Stochastic…

Let $T(\mathcal{Y}_3)$ be the relevant triangular domain, let $\mathcal{X}_m$ and $\mathcal{X}_n$ be two random samples from the uniform distribution…