The move-to-root (MTR) heuristic is a self-organizing rule that attempts to keep a binary search tree in near-optimal form. It is a tree analogue of the move-to-front (MTF) scheme for self-organizing lists. Both heuristics can be modeled as Markov chains. We show that the MTR chain can be derived by lumping the MTF chain and give exact formulas for the transition probabilities and stationary distribution for MTR. We also derive the eigenvalues and their multiplicities for MTR.

The move-to-root heuristic is a self-organizing rule that attempts to keep a binary search tree in near-optimal form. It is a tree analogue of the move-to-front scheme (also known as the weighted random-to-top card shuffle or Tsetlin library) for self-organizing lists. We study convergence of the move-to-root Markov chain to its stationary distribution and show that move-to-root converges two to four times faster than move-to-front for many examples. We also discuss asymptotics for expected search cost. For equal weights, $\operatorname{cn}/\ln n$ steps are necessary and sufficient to drive the maximum relative error to 0.

A bound is given for a reversible Markov chain on the probability that the occupation measure of a set exceeds the stationary probability of the set by a positive quantity.

We give a unified simple proof of recent results of Alexander concerning the rate of convergence of the mean length of the longest common subsequence of two random sequences and the rate of convergence of the expected value of certain passage times in percolation theory.

It is now known that the usual traffic condition (the nominal load being less than 1 at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and first-buffer-first-served preemptive resume discipline in a reentrant line are positive Harris recurrent under the usual traffic condition.

In this paper we treat a discrete-time generalized semi-Markov process in which simultaneous deaths of more than one lifetime and simultaneous arrivals are permitted and service scheduling is probabilistic. Necessary and sufficient conditions for insensitivity are derived and a simple algorithmic procedure provided whereby the equilibrium probabilities can be furnished when insensitivity holds. Some particular cases of special interest are examined in detail.

We consider a state-dependent generalization of the exponential Benes model of single-source buffer system in which the source process consists of alternating transmission and idle periods. Martingale methods are applied for analyzing limit nonstationary behavior of the buffer content process, when the buffer is loaded and depleted, with rates proportional to a large parameter $N$. Depending on traffic conditions, defined by parameters of the model, different types of approximations are established for the buffer content. We show that in heavy traffic the buffer content grows linearly in $N$, whereas the deviations of the order $\sqrt{N}$ from the deterministic limit are approximated by the Gaussian diffusion process. In moderate traffic the buffer content grows as $\sqrt{N}$, and the normalized buffer content is approximated by a diffusion process with reflection at zero. In the case of normal traffic, we show that the buffer utilization tends to the ratio of "the input-to-output rate." Moreover, we show that the main contribution to the utilization comes from arbitrary small buffer content.

In this note we observe that the recent argument of Ekhaus and Gray combined with the approach of Liggett and Shiga shows that the limit from passing a stationary ergodic arrival process of rate $\alpha < 1$ through a sequence of independent, rate one, exponential server queues is a Poisson process of rate $\alpha$. This builds on work of Liggett and Shiga and Anantharam.

We consider a branching random walk on the real line, with mean family size greater than 1. Let $B_n$ denote the minimal position of a member of the $n$th generation. It is known that (under a weak condition) there is a finite constant $\gamma$, defined in terms of the distributions specifying the process, such that as $n \rightarrow \infty$, we have $B_n = \gamma n + o(n)$ a.s. on the event $S$ of ultimate survival. Our results here show that (under appropriate conditions), on $S$ the random variable $B_n$ is strongly concentrated and the $o(n)$ error term may be replaced by $O(\log n)$.

We prove a diffusion limit theorem in the sense of weak convergence of measure-valued processes for a population age model first studied by Kendall. We show that in the diffusion limit scaling, the population structured in age groups behaves in the same way as the total population size, but with an exponential weight. A particular feature of the limiting process is that in general it is discontinuous at time zero.

Consider predicting an integral of a stochastic process based on $n$ observations of the stochastic process. Among all linear predictors, an optimal quadrature rule picks the $n$ observation locations and the weights assigned to them to minimize the mean squared error of the prediction. While optimal quadrature rules are usually unattainable, it is possible to find rules that have good asymptotic properties as $n \rightarrow \infty$. Previous work has considered processes whose local behavior is like $m$-fold integrated Brownian motion for $m$ a nonnegative integer. This paper obtains some asymptotic properties for quadrature rules based on median sampling for processes whose local behavior is not like $m$-fold integrated Brownian motion for any $m$.

For a wide sense stationary random field $\Phi = \{\phi(x): x \in R^2\}$, we investigate the asymptotic errors made in the numerical integration of line integrals of the form $\int_\Gamma f(x)\phi(x)d\sigma(x)$. It is shown, for example, that if $f$ and $\Gamma$ are smooth, and if the spectral density $\rho(\lambda)$ satisfies $\rho(\lambda)\approx k|\lambda|^{-4}$ as $\lambda \rightarrow \infty$, then there is a constant $c'$ with $N^3E|\int_\Gamma f(x)\phi(x)d\sigma(x) - \sum \beta_{j\varphi}(x_j)|^2 \geq c'N^{-3}$ for all finite sets $\{x_j: 1 \leq j\leq N\}$ and all choices of coefficients $\{\beta_j\}$. And, if any fixed parameterization $x(t)$ of $\Gamma$ is given and the integral $\int^1_0 f(x(t))\phi(x(t))|x'(t)|dt$ is numerically integrated using the midpoint method, the exact asymptotics of the mean squared error is derived. This leads to asymptotically optimal designs, and generalizes to other power laws and to nonstationary and nonisotropic fields.

Option replication in a discrete-time framework with transaction costs and its limit is discussed. First the notion of an efficient strategy is introduced, and then it is shown that an efficient strategy is the best strategy if it exists. It is also shown that the limit of the cost for option replication is given by a certain formula when the unit of time $\Delta t$ tends to zero and the transaction costs tends to zero on the order of $\sqrt{\Delta t}$.

We consider a one-dimensional Brownian motion with a constant drift, moving among Poissonian obstacles. In the case where the drift is below some critical value we characterize the limiting distribution of the process under the conditional probability measure that the particle has survived up to time $t$. Unlike the situation where the drift equals zero, we show in particular that in the presence of a constant drift, the process in natural scale feels the "boundary."

The object of this paper is to show that under certain auxiliary assumptions a stationary autoregressive sequence has a best predictor in mean square that is linear if and only if the sequence is minimum phase or is Gaussian when all moments are finite.

Imposing a total order on a regular two-dimensional discrete random field induces an orthogonal decomposition of the random field into two components: a purely indeterministic field and a deterministic field. The deterministic component is further orthogonally decomposed into a half-plane deterministic field and a countable number of mutually orthogonal evanescent fields. Each of the evanescent fields is generated by the column-to-column innovations of the deterministic field with respect to a different nonsymmetrical-half-plane total-ordering definition. The half-plane deterministic field has no innovations, nor column-to-column innovations, with respect to any nonsymmetrical-half-plane total-ordering definition. This decomposition results in a corresponding decomposition of the spectral measure of the regular random field into a countable sum of mutually singular spectral measures.

We consider finite state nonhomogeneous Markov chains with one-step transition probabilities roughly proportional to powers of a small parameter, converging to zero. We examine asymptotic properties of trajectories. The analysis is based on the so-called orders of recurrence. Transient states, recurrent classes and periodic subclasses can be identified in terms of the matrix of powers. This leads to a complete description of the tail sigma field. Our theorems generalize the classical results for homogeneous chains and can also be applied to chains generated by stochastic algorithms of the "simulated annealing" type.

Suppose a box contains $m$ balls, numbered from 1 to $m$. A random number of balls are drawn from the box, their numbers are noted and the balls are then returned to the box. This is done repeatedly, with the sample sizes being iid. Let $X$ be the number of samples needed to see all the balls. This paper uses Markov-chain coupling to derive a simple but typically very accurate approximation for $EX$ in terms of the sample size distribution. The approximation formula generalizes the formula found by Polya for the special case of fixed sample sizes.

We study the following two-person zero-sum game. $n$ random numbers are drawn independently from a continuous distribution known to both players. Player 2 observes all the numbers and selects an order to present them to the opponent. Player 1 learns the numbers sequentially as they are presented and may stop learning whenever he/she pleases. If the stop occurred at the number that is the $k$th largest among all $n$ numbers, Player 1 pays the amount $q(k)$ to Player 2, where $q(1) \leq \cdots \leq q(n)$ is a given payoff function. Player 1 aims to minimize the expected payoff; Player 2 aims to maximize it. We find an explicit solution of the game for a wide class of payoff functions including those $q$'s typically considered in the context of best choice problems.

For binary random variables $\sigma_1, \sigma_2, \ldots, \sigma_n$ that satisfy the well-known FKG condition, we show that the variances and covariances satisfy $\operatorname{Var}(\sigma_j) \operatorname{Cov}(\sigma_i, \sigma_k) \geq \operatorname{Cov}(\sigma_i, \sigma_j)\operatorname{Cov}(\sigma_j, \sigma_k),\quad 1 \leq i, j, k \leq n.$ This generalizes and improves a result by Graham for ferromagnetic Ising models with nonnegative external fields.