We consider a very general type of $d$-station open queueing network, with multiple customer classes and a more or less arbitrary service discipline at each station, but restricted by the requirement that customers always flow from lower numbered stations to higher numbered ones. To approximate the behavior of such a queueing network under heavy traffic conditions, a corresponding Brownian network model is proposed and it is shown that the approximating Brownian model reduces to a $d$-dimensional reflected Brownian motion $W$ whose state space is the nonnegative orthant. A necessary and sufficient condition for $W$ to have a product form stationary distribution (that is, a stationary distribution with independent components) and a probabilistic interpretation for that condition are given. Our interpretation involves a notion of quasireversibility analogous to that introduced by Kelly and elaborated by Walrand in their brilliant analysis of product form solutions for conventional queueing network models. Three illustrative queueing network models are discussed in detail and the analysis of these examples shows how a Brownian network approximation may have a product form stationary distribution even when the original or exact model is intractable. Particularly intriguing in that regard are two examples involving non-Poisson inputs, deterministic routing, deterministic service times and processor-sharing service disciplines.

We analyze the most commonly used method for shuffling cards. The main result is a simple expression for the chance of any arrangement after any number of shuffles. This is used to give sharp bounds on the approach to randomness: $\frac{3}{2} \log_2 n + \theta$ shuffles are necessary and sufficient to mix up $n$ cards. Key ingredients are the analysis of a card trick and the determination of the idempotents of a natural commutative subalgebra in the symmetric group algebra.

A continuous-time, consumption/investment problem with constant market coefficients is considered on a finite horizon. A dual problem is defined along the lines of Part 1. The value functions for both problems are proved to be solutions to the corresponding Hamilton-Jacobi-Bellman equations and are provided in terms of solutions to linear, second-order, partial differential equations. As a consequence, a mutual fund theorem is obtained in this market, despite the prohibition of short-selling. If the utility functions are of power form, all these results take particularly simple forms.

Let $X_i$ be positive i.i.d. random variables (or more generally a uniformly mixing positive-valued ergodic stationary process). The $r$-scan process induced by $\{X_i\}$ is $R_i = \sum^{i+r-1}_{k=i} X_k, i = 1, 2, \ldots, n - r + 1$. Limiting distributions for the extremal order statistics among $\{R_i\}$ suitably normalized (and appropriate threshold values $a = a_n$ and $b = b_n$) are derived as a consequence of Poisson approximations to the Bernoulli sums $N^-(a) = \sum^{n+r-1}_{i=1} W^-_i(a)$ and $N^+(b) = \sum^{n-r+1}_{i=1}W^+_i(b)$, where $W^-_i(a) \lbrack W^+_i(b) \rbrack = 1$ or 0 according as $R_i \leq a (R_i > b)$ occurs or not. Applications include limit theorems for $r$-spacings based on i.i.d. uniform $\lbrack 0, 1 \rbrack$ r.v.'s, for extremal $r$-spacings based on i.i.d. samples from a general density and for the $r$-scan process with a variable time horizon.

An ($n$ men-$n$ women) stable marriage problem is studied under the assumption that the individual preferences for a marriage partner are uniformly random and mutually independent. We show that the total number of stable matchings (marriages) is at least $(n/\log n)^{1/2}$ with high probability (whp) as $n \rightarrow \infty$ and also that the total number of stable marriage partners of each woman (man) is asymptotically normal with mean and variance close to $\log n$. It is proved that in the male (female) optimal stable marriage the largest rank of a wife (husband) is whp of order $\log^2 n$, while the largest rank of a husband (wife) is asymptotic to $n$. Earlier, we proved that for either of these extreme matchings the total rank is whp close to $n^2/\log n$. Now, we are able to establish a whp existence of an egalitarian marriage for which the total rank is close to $2n^{3/2}$ and the largest rank of a partner is of order $n^{1/2} \log n$. Quite unexpectedly, the stable matchings obey, statistically, a "law of hyperbola": namely, whp the product of the sum of husbands' ranks and the sum of wives' ranks in a stable matching turns out to be asymptotic to $n^3$, uniformly over all stable marriages. The key elements of the proofs are extensions of the McVitie-Wilson proposal algorithm and of Knuth's integral formula for the probability that a given matching is stable, and also a notion of rotations due to Irving. Methods developed in this paper may, in our opinion, be found useful in probabilistic analysis of other combinatorial algorithms.

We consider random tries constructed from sequences of i.i.d. random variables with a common density $f$ on $\lbrack 0, 1 \rbrack$ (i.e., paths down the tree are carved out by the bits in the binary expansions of the random variables). The depth of insertion of a node and the height of a node are studied with respect to their limit laws and their weak and strong convergence properties. In addition, laws of the iterated logarithm are obtained for the height of a random trie when $\int f^2 < \infty$. Finally, we study two popular improvements of the trie, the $\mathrm{PATRICIA}$ tree and the digital search tree, and show to what extent they improve over the trie.

This paper is concerned with two stochastic processes, namely, a Bernoulli excursion and a tied-down random walk. Three random variables are defined for these processes, each variable representing the area of a random set determined by one of the processes. The aim is to find the distributions and the moments of these random variables and to determine their asymptotic behavior. The results derived for random walks are applied to the theory of order statistics to determine the asymptotic behavior of the moments and the distributions of two statistics which measure the deviation between two empirical distribution functions.

The goal of this paper is to assess the improvement in performance that might be achieved by optimally scheduling a multiclass open queueing network. A stochastic process is defined whose steady-state mean value is less than or equal to the mean number of customers in a queueing network under any arbitrary scheduling policy. Thus, this process offers a lower bound on performance when the objective of the queueing network scheduling problem is to minimize the mean number of customers in the network. Since this bound is easily obtained from a computer simulation model of a queueing network, its main use is to aid job-shop schedulers in determining how much further improvement (relative to their proposed policies) might be achievable from scheduling. Through computational examples, we identify some factors that affect the tightness of the bound.

Let $p(x)$ and $q(x)$ be density functions and let $(p \ast q)(x)$ be their convolution. Define $w(x) = -(d/dx)\log q(x) \text{and} v(x) = -(d/dx)\log p(x).$ Under the hypothesis of the regular oscillation of the functions $w$ and $v$, the asymptotic form of $(p \ast q)(x)$, for $x \rightarrow \infty$, is obtained. The results are applied to a model previously introduced by the author for the estimation of the distribution of HIV latency time.

A universal bound for the maximal expected reward is obtained for stopping a sequence of independent random variables where the reward is a nonincreasing function of the rank of the variable selected. This bound is shown to be sharp in three classical cases: (i) when maximizing the probability of choosing one of the $k$ best; (ii) when minimizing the expected rank; and (iii) for an exponential function of the rank.