Abstract-We determine the asymptotic scaling for the per user throughput in a large hybrid ad hoc network, i.e., a network with both ad hoc nodes, which communicate with each other via shared wireless links of capacity bits/s, and infrastructure nodes which in addition are interconnected with each other via high capacity links. Specifically, we consider a network model where ad hoc nodes are randomly spatially distributed and choose to communicate with a random destination. We identify three scaling regimes, depending on the growth of the number of infrastructure nodes, relative to the number of ad hoc nodes , and show the asymptotic scaling for the per user throughput as becomes large. We show that when log the per user throughput is of order log and could be realized by allowing only ad hoc communications, i.e., not deploying the infrastructure nodes at all. Whenever log log , the order for the per user throughput is and, thus, the total additional bandwidth provided by infrastructure nodes is effectively shared among ad hoc nodes. Finally, whenever log , the order of the per user throughput is only log , suggesting that further investments in infrastructure nodes will not lead to improvement in throughput. The results are shown through an upper bound which is independent of the routing strategy, and by constructing scenarios showing that the upper bound is asymptotically tight.Index Terms-Ad hoc wireless networks, capacity scaling, hybrid wireless networks, throughput.
We propose an algorithm to calculate confidence intervals for the values of hedging parameters of discretely exercisable options using Monte-Carlo simulation. The algorithm is based on a combination of the duality formulation of the optimal stopping problem for pricing discretely exercisable options and Monte-Carlo estimation of hedging parameters for European options. We show that the width of the confidence interval for a hedging parameter decreases, with an increase in the computer budget, asymptotically at the same rate as the width of the confidence interval for the price of the option. The method can handle arbitrary payoff functions, general diffusion processes, and a large number of random factors. We also present a fast, heuristic, alternative method and use our method to evaluate its accuracy.
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