An urn contains m balls of value $-1$ and p balls of value $+1$. The balls are drawn randomly without replacement until the urn is emptied. Before each ball is drawn, the player is asked whether or not she would like to accept the next ball drawn, with the payout being the value of the ball. We present results related to the random version of this $(m,p)$ acceptance urn, where the total number of balls n is known and the number of balls with value $+1$ is unknown and random. We also consider the ruin problem for the $(m,p)$ urn: The player has b dollars available to play the $(m,p)$ urn and is ruined if at some point she has lost b dollars. We find a necessary condition for ruin, and use this condition to give upper bounds for the ruin probability. We also examine the ruin problem for the random acceptance urn, finding the ruin probability provided the initial distribution satisfies some additional conditions.

The rapidly increasing energy consumption by computing and communications equipment is a significant economic and environmental problem that needs to be addressed. Ethernet network interface controllers (NICs) in the US alone consume hundreds of millions of US dollars in electricity per year. Most Ethernet links are underutilized and link energy consumption can be reduced by operating at a lower data rate. In this paper, we investigate adaptive link rate (ALR) as a means of reducing the energy consumption of a typical Ethernet link by adaptively varying the link data rate in response to utilization. Policies to determine when to change the link data rate are studied. Simple policies that use output buffer queue length thresholds and fine-grain utilization monitoring are shown to be effective. A Markov model of a state-dependent service rate queue with rate transitions only at service completion is used to evaluate the performance of ALR with respect to the mean packet delay, the time spent in an energy-saving low link data rate, and the oscillation of link data rates. Simulation experiments using actual and synthetic traffic traces show that an Ethernet link with ALR can operate at a lower data rate for over 80 percent of the time, yielding significant energy savings with only a very small increase in packet delay.

Rapidly increasing energy use by computing and communications equipment is a significant problem that needs to be addressed. Ethernet network interface controllers (NICs) consume hundreds of millions of US$ in electricity per year. Most Ethernet links are underutilized and link power consumption can be reduced by operating at lower data rates. An output buffer threshold policy to change link data rate in response to utilization is investigated. Analytical and simulation models are developed to evaluate the performance of Adaptive Link Rate (ALR) with respect to mean packet delay and time spent in low data rate with Poisson traffic and 100 Mb/s network traces as inputs. A Markov model of a state-dependent service rate queue with rate transitions only at service completion is developed. For the traffic traces, it is found that a link can operate at 10 Mb/s for over 99% of the time yielding energy savings with no user-perceivable increase in packet delay.

Given a graph G=(V,E) with n vertices, m edges, and a family of $\kappa$ pairs of vertices in V, we are interested in finding for each pair (a i , b i ) a path connecting a i to b i such that the set of $\kappa$ paths so found is edge disjoint. (For arbitrary graphs the problem is ${\cal NP}$-complete, although it is in ${\cal P}$ if $\kappa$ is fixed.) We present a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths (up to constant factors) in the random graph G n,m for all edge densities above the connectivity threshold. (The graph is chosen first; then an adversary chooses the pairs of endpoints.) Our results give the first tight bounds for the edge-disjoint paths problem for any nontrivial class of graphs.