The optimal receiver operating characteristic (ROC) curve, giving the maximum probability of detection as a function of the probability of false alarm, is a key information-theoretic indicator of the difficulty of a binary hypothesis testing problem (BHT). It is well known that the optimal ROC curve for a given BHT, corresponding to the likelihood ratio test, is determined by the probability distribution of the observed data under each of the two hypotheses. In some cases, these two distributions may be unknown or computationally intractable, but independent samples of the likelihood ratio can be observed. This raises the problem of estimating the optimal ROC for a BHT from such samples. The maximum likelihood estimator of the optimal ROC curve is derived, and it is shown to converge almost surely to the true optimal ROC curve in the Lévy metric, as the number of observations tends to infinity. Finite sample size bounds are obtained for three other estimators: the classical empirical estimator, based on estimating the two types of error probabilities from two separate sets of samples, and two variations of the maximum likelihood estimator called the split estimator and fused estimator, respectively. The maximum likelihood estimator is observed in simulation experiments to be considerably more accurate than the empirical estimator, especially when the number of samples obtained under one of the two hypotheses is small. The area under the maximum likelihood estimator is derived; it is a consistent estimator of the area under the true optimal ROC curve.

The optimal receiver operating characteristic (ROC) curve, giving the maximum probability of detection as a function of the probability of false alarm, is a key information-theoretic indicator of the difficulty of a binary hypothesis testing problem (BHT). It is well known that the optimal ROC curve for a given BHT, corresponding to the likelihood ratio test, is theoretically determined by the probability distribution of the observed data under each of the two hypotheses. In some cases, these two distributions may be unknown or computationally intractable, but independent samples of the likelihood ratio can be observed. This raises the problem of estimating the optimal ROC for a BHT from such samples. The maximum likelihood estimator of the optimal ROC curve is derived, and it is shown to converge to the true optimal ROC curve in the Lévy metric, as the number of observations tends to infinity. A classical empirical estimator, based on estimating the two types of error probabilities from two separate sets of samples, is also considered. The maximum likelihood estimator is observed in simulation experiments to be considerably more accurate than the empirical estimator, especially when the number of samples obtained under one of the two hypotheses is small. The area under the maximum likelihood estimator is derived; it is a consistent estimator of the true area under the optimal ROC curve.

Recovery of the causal structure of dynamic networks from noisy measurements has long been a problem of intense interest across many areas of science and engineering. Many algorithms have been proposed, but there is no work that compares the performance of the algorithms to converse bounds in a non-asymptotic setting. As a step to address this problem, this paper gives lower bounds on the error probability for causal network support recovery in a linear Gaussian setting. The bounds are based on the use of the Bhattacharyya coefficient for binary hypothesis testing problems with mixture probability distributions. Comparison of the bounds and the performance achieved by two representative recovery algorithms are given for sparse random networks based on the Erdős-Rényi model.

This paper considers the problem of scheduling real-time traffic in wireless networks. We consider ad hoc wireless networks with general conflict graph-based interference model and single-hop traffic. Each packet is associated with a deadline and will be dropped if it is not transmitted before the deadline. The number of packet arrivals in each time-slot and the maximum delay before the deadline are independent and identically distributed across time. We require a minimum fraction of packets to be delivered. At each link, we assume the link keeps track of the difference between the minimum number of packets that need to be delivered so far and the number of packets that are actually delivered, which we call the deficit. The largest-deficit-first (LDF) policy schedules links in descending order according to their deficit values, which is a variation of the longest-queue-first (LQF) policy for non-real-time traffic. We prove that the efficiency ratio of LDF, which is the fraction of the throughput region that LDF can achieve for given traffic distributions, can be lower-bounded by a quantity that we call the real-time local-pooling factor (R-LPF). We further prove that a lower bound on the R-LPF can be related to the weighted sum of the service rates, with a special case of 1/(β+1) by considering the uniform weight, where β is the interference degree of the conflict graph. We also propose a heuristic consensus algorithm that can be used to obtain a good weight vector for such lower bounds for given network topology.

In many computing and networking applications, arriving tasks have to be routed to one of many servers, with the goal of minimizing queueing delays. When the number of processors is very large, a popular routing algorithm works as follows: select two servers at random and route an arriving task to the least loaded of the two. It is well-known that this algorithm dramatically reduces queueing delays compared to an algorithm which routes to a single randomly selected server. In recent cloud computing applications, it has been observed that even sampling two queues per arriving task can be expensive and can even increase delays due to messaging overhead. So there is an interest in reducing the number of sampled queues per arriving task. In this paper, we show that the number of sampled queues can be dramatically reduced by using the fact that tasks arrive in batches (called jobs). In particular, we sample a subset of the queues such that the size of the subset is slightly larger than the batch size (thus, on average, we only sample slightly more than one queue per task). Once a random subset of the queues is sampled, we propose a new load balancing method called batch-filling to attempt to equalize the load among the sampled servers. We show that our algorithm dramatically reduces the sample complexity compared to previously proposed algorithms.

We consider ad hoc wireless networks with real-time traffic, and study the capacity requirement of a low-complexity scheduling policy, called largest-deficit-first (LDF), for achieving the same quality of service (QoS) as the optimal policy in networks with unit capacity. We derive theoretical upper and lower bounds for general traffic. The bounds depend on the interference degree of the network and the max/min delay bound ratio. The performance of LDF is further evaluated using simulations and compared with two other algorithms. The simulation results show that LDF significantly outperforms the other two algorithms.

This paper considers the problem of scheduling real-time traffic in wireless networks. We consider an ad hoc wireless network with general interference and general one-hop traffic. Each packet is associated with a deadline and will be dropped if it is not transmitted before the deadline expires. The number of packet arrivals in each time slot and the length of a deadline are both stochastic and follow certain distributions. We only allow a fraction of packets to be dropped. At each link, we assume the link keeps track of the difference between the minimum number of packets that need to be delivered and the number of packets that are actually delivered, which we call deficit. The largest-deficit-first (LDF) policy schedules links in descending order according to their deficit values, which is a variation of the largest-queue-first (LQF) policy for non-real-time traffic. We prove that the efficiency ratio of LDF can be lower bounded by a quantity that we call the real-time local-pooling factor (R-LPF). We further prove that given a network with interference degree β, the R-LPF is at least 1/(β+1), which in the case of the one-hop interference model translates into an R-LPF of at least 1/3.