We discuss the asymptotic behavior of time-inhomogeneous Metropolis chains for solving constrained sampling and optimization problems. In addition to the usual inverse temperature schedule (βn)n∈[hollow N]*, the type of Markov processes under consideration is controlled by a divergent sequence (θn)n∈[hollow N]* of parameters acting as Lagrange multipliers. The associated transition probability matrices (Pβn,θn)n∈[hollow N]* are defined by Pβ,θ = q(x, y)exp(−β(Wθ(y) − Wθ(x))+) for all pairs (x, y) of distinct elements of a finite set Ω, where q is an irreducible and reversible Markov kernel and the energy function Wθ is of the form Wθ = U + θV for some functions U,V : Ω → [hollow R]. Our approach, which is based on a comparison of the distribution of the chain at time n with the invariant measure of Pβn,θn, requires the computation of an upper bound for the second largest eigenvalue in absolute value of Pβn,θn. We extend the geometric bounds derived by Ingrassia and we give new sufficient conditions on the control sequences for the algorithm to simulate a Gibbs distribution with energy U on the constrained set [Ω with tilde above] = {x ∈ Ω : V(x) = minz∈Ω V(z)} and to minimize U over [Ω with tilde above].

We propose a particular class of transition probability matrices for discrete-time Markov chains with a closed form to compute the stationary distribution. The stochastic monotonicity properties of this class are established. We give algorithms to construct monotone, bounding matrices belonging to the proposed class for the variability orders. The accuracy of bounds with respect to the underlying matrix structure is discussed through an example.

We consider the optimal control of two parallel servers in a two-stage tandem queuing system with two flexible servers. New jobs arrive at station 1, after which a series of two operations must be performed before they leave the system. Holding costs are incurred at rate h1 per unit time for each job at station 1 and at rate h2 per unit time for each job at station 2.

The system is considered under two scenarios; the collaborative case and the noncollaborative case. In the prior, the servers can collaborate to work on the same job, whereas in the latter, each server can work on a unique job although they can work on separate jobs at the same station. We provide simple conditions under which it is optimal to allocate both servers to station 1 or 2 in the collaborative case. In the noncollaborative case, we show that the same condition as in the collaborative case guarantees the existence of an optimal policy that is exhaustive at station 1. However, the condition for exhaustive service at station 2 to be optimal does not carry over. This case is examined via a numerical study.

We consider a modulated fluid system with a finite state-space Markov chain Jt as modulating process and general state-dependent net input rates. We derive differential equations for the transient and the stationary distribution of (Wt, Jt), where Wt is the content process, and the corresponding Laplace transforms with respect to time. Moreover, we study the level hitting times of Wt. Our results lead to explicit formulas in the case of two modulating states.

In this article, we establish stochastic comparisons between normalized spacings of generalized order statistics. These comparisons allow us to extend and unify some results obtained by other authors for ordinary order statistics and record values. Furthermore, we can compare spacings of different models (i.e., between ordinary order statistics and sequential order statistics, record values and Pfeifer's record values, and so forth).

Maymin proved that for binary systems, the path-cut lower bound is sharper than the minimax lower bound for highly reliable components, which was originally conjectured by Barlow and Proschan. In this note, an example is constructed to illustrate that this, in general, is not the case for multistate systems. However, an affirmative result is obtained under mild conditions we imposed. Examples are given to illustrate the applications of our results.

Consider a finite-capacity telecommunications link to which connection requests arrive in a Poisson process. Each connection carried on the link earns a certain amount of revenue for the link's manager. Now, assume that the manager is offered the opportunity to buy or sell a unit of the link's allocated capacity. Assuming that the manager has a knowledge of the current number of connections on the link, we demonstrate a method of calculating the buying and selling prices.

We study an M/M/∞ queuing system in which each arrival has a random urgency and is admitted if and only if it is more urgent than all individuals currently receiving service. The system represents, for example, a less-than-magnanimous emergency facility. For this system, and also for a closely related “Parody,” we study the busy period distribution and, to a lesser extent, occupancy. Both exact and heavy traffic results are given.

Given a graph G on n vertices with average degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that, if the edge connectivity k(G) satisfies k(G) [Gt] d/log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

We study a model motivated by the minesweeper game. In this model one starts with the percolation of mines on the lattice ℤd, and then tries to find an infinite path of mine-free sites. At every recovery of a free site, the player is given some information on the sites adjacent to the current site. We compare the parameter values for which there exists a strategy such that the process survives to the critical parameter of ordinary percolation. We then derive improved bounds for these values for the same process, when the player has some complexity restrictions in computing his moves. Finally, we discuss some monotonicity issues which arise naturally for this model.

This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [12, 11] and another studied by Tsilevich [30, 31] and Mayer-Wolf, Zeitouni and Zerner [21]. The invariance under this split-and-merge transformation of the interval partition generated by a suitable Poisson process yields a simple proof of the recent result of [21] that a Poisson–Dirichlet distribution is invariant for a closely related fragmentation–coagulation process. Uniqueness and convergence to the invariant measure are established for the split-and-merge transformation of interval partitions, but the corresponding problems for the fragmentation–coagulation process remain open.

We present improved lower and upper bounds for the time constant of first-passage percolation on the square lattice. For the case of lower bounds, a new method, using the idea of a transition matrix, has been used. Numerical results for the exponential and uniform distributions are presented. A simulation study is included, which results in new estimates and improved upper confidence limits for the time constants.

For two stochastically dependent random variables X and Y taking values in {0,…, m−1}, we study the distribution of the random residue U = XY mod m. Our main result is an upper bound for the distance Δm = supx∈[0,1] [mid ] P(U/m [les ] x)−x[mid ]. For independent and uniformly distributed X and Y, the exact distribution of U is derived and shown to be stochastically smaller than the uniform distribution on {0,…, m−1}. Moreover, in this case Δm is given explicitly.

Let q be an integer with q [ges ] 2. We give a new proof of a result of Erdös and Turán determining the proportion of elements of the finite symmetric group Sn having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group An. In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.

We apply these results to estimate the proportion of elements of order 2f in Sn, and of order 3f in An and Sn, where gcd(2, f) = 1, and gcd(3, f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in An or Sn, given as a black-box group with an order oracle, is discussed.

We consider k-uniform set systems over a universe of size n such that the size of each pairwise intersection of sets lies in one of s residue classes mod q, but k does not lie in any of these s classes. A celebrated theorem of Frankl and Wilson [8] states that any such set system has size at most (ns) when q is prime. In a remarkable recent paper, Grolmusz [9] constructed set systems of superpolynomial size Ω(exp(c log2n/log log n)) when q = 6. We give a new, simpler construction achieving a slightly improved bound. Our construction combines a technique of Frankl [6] of ‘applying polynomials to set systems’ with Grolmusz's idea of employing polynomials introduced by Barrington, Beigel and Rudich [5]. We also extend Frankl's original argument to arbitrary prime-power moduli: for any ε > 0, we construct systems of size ns+g(s), where g(s) = Ω(s1−ε). Our work overlaps with a very recent technical report by Grolmusz [10].