Ora E. Percus

Probability of Self-Nonself Discrimination

The binding of antibody or T-cell receptors to antigen occurs by a generalized lock and key fit of portions of the two structures. The question that we address here is: how large should the complementary regions on the two structures be? In order to estimate the size of an optimal receptor combining region, we assume that the mammalian immune system over evolutionary time has been presented with a large random set of foreign antigens that occur on common pathogens, which it must recognize, and a smaller random set of self-antigens which a mature organism must not recognize. Evolution is imagined to have coevolved the receptors in a fashion such as to maximize the probability that this task is performed. The probability of a random receptor-antigen match is estimated from this condition. For protein antigens, the genesis of the probability is traced to the complementarity of sufficiently long sequences of amino acids on the two molecules involved, and computed accordingly. The result is quite insensitive to the population sizes inserted, and results in an estimated 13-site binding region, in agreement with experimental information.

Theory and application of Marsaglia's monkey test for pseudorandom number generators

A theoretical analysis is given for a new test, the “Monkey” test, for pseudorandom number sequences, which was proposed by Marsaglia. Selected results, using the test on several pseudorandom number generators in the literature, are also presented.

PROBABILITY BOUNDS ON THE SUM OF INDEPENDENT NONIDENTICALLY DISTRIBUTED BINOMIAL RANDOM VARIABLES

The cumulative distribution of the sum of independent binomial random variables is investigated. After writing down exact expressions for these quantities, we develop a sequence of increasingly tight upper and lower bounds, given various characteristics of the underlying set of probabilities. The major tool in each case is a transformation of the probability set for which the cumulative distributions act as Lyapounov functions. Our most sophisticated bounds, in which the first two cumulatives are given, are computed for a number of sets of probabilities and compared with familiar results in the literature. They are uniformly superior.

Elementary properties of clock-regulated queues

A single clock-regulated queue is fed by two Bernoulli streams of messages. The distributions of queue length, waiting time, and output for such a system are obtained. This is generalized as well to the case of a queue of finite capacity.

Performance analysis of clock-regulated queues with output multiplexing in three different 2×2 crossbar switch architectures

Abstract Switches in interconnection networks for highly parallel shared memory computer systems may be implemented with different internal buffer structures. For a 2 × 2 synchronous switch, previous studies have often assumed a switch composed of two queues, one at each output, each of which has unbounded size and may accept two inputs every clock cycle. Hardware implementations may actually use simpler queue designs and will have bounded size. Two additional models for a 2 × 2 switch using queues that may accept only one input at a time are analyzed. The first uses four queues, one for each input/output pair. The second uses two queues, one at each input. In each case, a multiplexer blocks one queue if two queues desire the same output, making these models more difficult to analyze than the previous model. Maximum bandwidth, expected queue length, expected waiting time, and queue length distribution are presented for both models, with unbounded queue size and with queue size equal to 1.

The amplitudes of viral blips in HIV-1 infected patients treated with antiretroviral therapy are power-law distributed

We previously reported that in patients treated with highly active antiretroviral therapy (HAART) who achieve viral load (VL) suppression, low fluctuations of viral load over the threshold of detection (viral blips) more than 4 weeks apart occur at random, with a frequency that does not change with longer times of observation. The etiology of viral blips is currently unknown, but viral blip frequency inversely correlates with the decay of the latent reservoir, whose stability has been proposed as the major hurdle to HIV eradication. We show here that the distribution of viral blip amplitudes observed in a group of 272 patients successfully treated with highly active antiretroviral therapy appears to be power-law distributed. Such a distribution can be theoretically generated by randomly sampling the arrival of asynchronous and overlapping elementary pulses of viremia, with asymptotic exponential decay of kinetics, thus suggesting that the low fluctuations of viremia observed in patients during HAART treatment is, in part, a discrete phenomenon consistent with random activation of latently infected cells or release of virus and infected cells into the blood compartment from unknown sites of active viral replication.

On the Bechhofer–Kulkarni Stopping Rule for Sequential Clinical Trials

A stopping rule recently introduced by Bechhofer and Kulkarni for sequential clinical trials is considered in the special case of relative evaluation of two procedures. Figures of merit are introduced for such trials and evaluated explicitly in terms of Legendre polynomial series by means of a sequence of generating functions modified to automatically include the required protocol. The expected trial duration is also evaluated asymptotically to reinforce the point that asymptotic methods can be effective in such situations for extremely short sequences.

Piecewise Homogeneous random walk with a moving boundary

We study a random walk with nearest neighbor transitions on a one-dimensional lattice. The walk starts at the origin, as does a dividing line which moves with constant speed

One-dimensional random walk in alternating homogeneous domains

\gamma

Generating functions for a class of one-dimensional random walks

, but the outward transition probabilities