Fractional Brownian motion (fBm) emerged as a useful model for self-similar and long-range dependent Internet traffic. Approximate performance measures are known from large deviations theory for single queuing systems with fBm through traffic. In this paper we derive end-to-end performance bounds for a through flow in a network of tandem queues under fBm cross traffic. To this end, we prove a rigorous sample path envelope for fBm that complements previous approximate results. We find that both approaches agree in their outcome that overflow probabilities for fBm traffic have a Weibullian tail. We employ the sample path envelope and the concept of leftover service curves to model the remaining service after scheduling fBm cross traffic at a system. Using composition results for tandem systems from the stochastic network calculus we derive end-to-end statistical performance bounds for individual flows in networks under fBm cross traffic. We discover that these bounds grow in O(n (log n)^(1 (2-2H))) for n systems in series where H is the Hurst parameter of the fBm cross traffic. We show numerical results on the impact of the variability and the correlation of fBm traffic on network performance.
In contrast, the derivation of performance bounds using the stochastic network calculus builds on sample path arguments, such as the backlog bound ), and requires a bound for @math for all @math , i.e. a sample path envelope of the form Such sample path envelopes are constructed in @cite_21 using Boole's inequality under the assumption of a time scale limit @math , i.e. by summing the constant point-wise overflow probabilities @math over all @math . The time scale in this context can be regarded as a constraint on the duration of busy periods. In case of fBm the duration of busy periods has, however, been found to grow extremely fast with @math @cite_8 .
@cite_21 @cite_8
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The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the network as a whole in terms of a probabilistic bound. The presented network service curve permits the calculation of statistical end-to-end delay and backlog bounds for broad classes of arrival and service distributions. The benefits of the derived service curve are illustrated for the exponentially bounded burstiness (EBB) traffic model. It is shown that end-to-end performance measures computed with a network service curve are bounded by O(Hlog H), where H is the number of nodes traversed by a flow. Using currently available techniques that compute end-to-end bounds by adding single node results, the corresponding performance measures are bounded by O(H3). This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP. We present a technique for computing upper bounds on the distribution of individual per-session performance measures such as delay and buffer occupancy for networks in which sessions may be routed over several “hops.” Our approach is based on first stochastically bounding the distribution of the number of packets (or cells) which can be generated by each traffic source over various lengths of time and then “pushing” these bounds (which are then shown to hold over new time interval lengths at various network queues) through the network on a per-session basis. Session performance bounds can then be computed once the stochastic bounds on the arrival process have been characterized for each session at all network nodes. A numerical example is presented and the resulting distributional bounds compared with simulation as well as with a point-valued worst-case performance bound. The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the network as a whole in terms of a probabilistic bound. The presented network service curve permits the calculation of statistical end-to-end delay and backlog bounds for broad classes of arrival and service distributions. The benefits of the derived service curve are illustrated for the exponentially bounded burstiness (EBB) traffic model. It is shown that end-to-end performance measures computed with a network service curve are bounded by spl Oscr (H log H), where H is the number of nodes traversed by a flow. Using currently available techniques, which compute end-to-end bounds by adding single node results, the corresponding performance measures are bounded by spl Oscr (H sup 3 ).