Connectivity in one-dimensional geometric random graphs: Poisson approximations, zero-one laws and phase transitions

Consider n points (or nodes) distributed uniformly and independently on the unit interval [0,1]. Two nodes are said to be adjacent if their distance is less than some given threshold value.For the underlying random graph we derive zero-one laws for the property of graph connectivity and give the asymptotics of the transition widths for the associated phase transition. These results all flow from a single convergence statement for the probability of graph connectivity under a particular class of scalings. Given the importance of this result, we give two separate proofs; one approach relies on results concerning maximal spacings, while the other one exploits a Poisson convergence result for the number of breakpoint users.This work was prepared through collaborative participation in the Communications and Networks Consortium sponsored by the U. S. Army Research Laboratory under the Collaborative Technology Alliance Program, Cooperative Agreement DAAD19-01-2-0011

Martingale relations for the M⧸GI⧸1 queue with Markov modulated Poisson input

AbstractThis paper is concerned with single server queueing systems with renewal service process and Poisson arrivals modulated by a finite-state Markov chain. Exponential martingales are associated with a chain embedded at service completion epochs in the stochastic process describing the joint evolution of the number of customers in the queue and the state of the environment. The analysis of these martingales leads to a new and unified treatment of various known results concerning the stability condition and the steady state statistics, as well as to several new properties. Noteworthy among them are a conservation law that relates the duration of the busy period to the state of the environment at the end of the busy period, and some absolute continuity properties with respect to other queues of the same type