Krishnamurthi Ravishankar

Broadcasting on [0, L ]

Abstract We study the problem of broadcasting in a system where n nodes are placed on a line of length L, independently uniformly distributed. We assume that every node is equipped with a transmitter whose radius of transmission is 1. We further assume that simultaneous broadcasts by neighbouring nodes results in garbled messages. The system is synchronous and time is slotted with nodes transmitting only during a slot. We present an algorithm for broadcasting and compute the expected number of time steps required for it to complete. The algorithm is shown to be optimal. We also present and analyze an algorithm, to be executed by every node, that identifies the subset of nodes to serve as transmitters for a broadcast originating at the origin. We assume that the nodes are initially unaware of the topology of the system.

Asymptotically optimal gossiping in radio networks

Abstract We study the problem of gossiping in a system where n nodes are placed on a line of length Ln independently uniformly distributed. We assume that every node is equipped with a transmitter whose radius of transmission is 1. We further assume that simultaneous transmissions by neighboring nodes results in garbled messages. We present an algorithm for gossiping and show that it works in asymptotically optimal time. We assume that the system is synchronous and time is slotted and that nodes transmit only during a slot.

Gossiping on a ring with radios

We study the problem of gossiping where n nodes equipped with radios are placed on a ring of circumference L. Each radio has a transmission range of 1 and we assume that simultaneous transmissions by neighboring nodes results in garbled messages. We present an algorithm for gossiping and show that it works in asymptotically optimal time.

Supercriticality conditions for asymmetric zero-range process with sitewise disorder

We discuss necessary and sufficient conditions for the convergence of disordered asymmetric zero-range process to the critical invariant measures.

Interface fluctuations in the two-dimensional weakly asymmetric simple exclusion process

We consider the two-dimensional weakly asymmetric simple exclusion process, where the asymmetry is along the X-axis. The generator for such a process can be written as [var epsilon]--2L0+[var epsilon]--1L[alpha], [var epsilon]>0, where L0 and L[alpha] are the generators for the nearest neighbor symmetric simple exclusion and totally asymmetric simple exclusion, respectively. We prove propagation of chaos and convergence to Burgers equation with viscosity in the limit as [var epsilon] goes to zero. The density fluctuation field converges to a generalized Ornstein-Uhlenbeck process. The covariance kernel for a class of travelling wave solutions is consistent with a phase boundary which fluctuates according to a linear stochastic partial differential equation.

Central Limit Theorem for Time to Broadcast in Radio Networks

We study the problem of broadcasting in a system where nodes are equipped with radio transmitters with constant radius of transmission. A message originating at a node has to be transmitted to all the other nodes in the system. We prove the central limit theorem and the law of large numbers for the number of time steps required to complete a broadcast for the case when the nodes are placed on a line independently uniformly distributed. We show that the number of time steps required to broadcast is 3 n /4 in probability.

A Random Walk with Collapsing Bonds and Its Scaling Limit

Abstract We introduce a new self-interacting random walk on the integers in a dynamic random environment and show that it converges to a pure diffusion in the scaling limit. We also find a lower bound on the diffusion coefficient in some special cases. With minor changes the same argument can be used to prove the scaling limit of the corresponding walk in ℤ d .