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 @cite_21 a connection between effective bandwidths and effective envelopes is established. In contrast to asymptotic results for large buffers from large deviations theory, effective envelopes in conjunction with the stochastic network calculus @cite_5 @cite_33 @cite_21 @cite_14 @cite_30 @cite_1 can provide non-asymptotic performance bounds. Moreover, recent stochastic network calculus provides methods for derivation of stochastic leftover service curves as well as for composition of tandem systems. Effective envelopes @math are statistical upper bounds of the cumulative arrivals @math of the form An envelope for fBm traffic is derived in @cite_8 @cite_35 @cite_21 as
@cite_30 @cite_35 @cite_14 @cite_33 @cite_8 @cite_21 @cite_1 @cite_5
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The class of Gupta-Kumar results, which predict the throughput capacity in wireless networks, is restricted to asymptotic regimes. This tutorial presents a methodology to address a corresponding non-asymptotic analysis based on the framework of the stochastic network calculus, in a rigorous mathematical manner. In particular, we derive explicit closed-form results on the distribution of the end-to-end capacity and delay, for a fixed source-destination pair, in a network with broad assumptions on its topology and degree of spatial correlations. The results are non-asymptotic in that they hold for finite time scales and network sizes, as well as bursty arrivals. The generality of the results enables the research of several interesting problems, concerning for instance the effects of time scales or randomness in topology on the network capacity. We study a noiseless broadcast link serving @math users whose requests arise from a library of @math files. Every user is equipped with a cache of size @math files each. It has been shown that by splitting all the files into packets and placing individual packets in a random independent manner across all the caches prior to any transmission, at most @math file transmissions are required for any set of demands from the library. The achievable delivery scheme involves linearly combining packets of different files following a greedy clique cover solution to the underlying index coding problem. This remarkable multiplicative gain of random placement and coded delivery has been established in the asymptotic regime when the number of packets per file @math scales to infinity. The asymptotic coding gain obtained is roughly @math . In this paper, we initiate the finite-length analysis of random caching schemes when the number of packets @math is a function of the system parameters @math , and @math . Specifically, we show that the existing random placement and clique cover delivery schemes that achieve optimality in the asymptotic regime can have at most a multiplicative gain of 2 even if the number of packets is exponential in the asymptotic gain @math . Furthermore, for any clique cover-based coded delivery and a large class of random placement schemes that include the existing ones, we show that the number of packets required to get a multiplicative gain of @math is at least @math . We design a new random placement and an efficient clique cover-based delivery scheme that achieves this lower bound approximately. We also provide tight concentration results that show that the average (over the random placement involved) number of transmissions concentrates very well requiring only a polynomial number of packets in the rest of the system parameters. .We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks. 1. Introduction. This paper is a continuation of Dai and Lin (2005), in which maximum pressure policies are shown to be throughput optimal for a class of stochastic processing networks. Throughput optimality is an important, first-order objective for many networks, but it ignores some key secondary performance measures like queueing delays experienced by jobs in these networks. In this paper we show that maximum pressure policies enjoy additional optimality properties; they are asymptotically optimal in minimizing a certain workload or holding cost of a stochastic processing network. Stochastic processing networks have been introduced in a series of three papers by Harrison (2000, 2002, 2003). In Dai and Lin (2005) and this paper we consider a special class of Harrison’s model. This class of stochastic processing networks is much more general than multiclass queueing networks that have been a subject of intensive study in the last 20 years; see, for example, Harrison (1988), Williams In this paper, we undertake the first study of statistical multiplexing from the perspective of approximation algorithms. The basic issue underlying statistical multiplexing is the following: in high-speed networks, individual connections (i.e., communication sessions) are very bursty, with transmission rates that vary greatly over time. As such, the problem of packing multiple connections together on a link becomes more subtle than in the case when each connection is assumed to have a fixed demand. We consider one of the most commonly studied models in this domain: that of two communicating nodes connected by a set of parallel edges, where the rate of each connection between them is a random variable. We consider three related problems: (1) stochastic load balancing, (2) stochastic bin-packing, and (3) stochastic knapsack. In the first problem the number of links is given and we want to minimize the expected value of the maximum load. In the other two problems the link capacity and an allowed overflow probability p are given, and the objective is to assign connections to links, so that the probability that the load of a link exceeds the link capacity is at most @math . In bin-packing we need to assign each connection to a link using as few links as possible. In the knapsack problem each connection has a value, and we have only one link. The problem is to accept as many connections as possible. For the stochastic load balancing problem we give an O(1)-approximation algorithm for arbitrary random variables. For the other two problems we have algorithms restricted to on-off sources (the most common special case studied in the statistical multiplexing literature), with a somewhat weaker range of performance guarantees. A standard approach that has emerged for dealing with probabilistic resource requirements is the notion of effective bandwidth---this is a means of associating a fixed demand with a bursty connection that "represents" its distribution as closely as possible. Our approximation algorithms make use of the standard definition of effective bandwidth and also a new one that we introduce; the performance guarantees are based on new results showing that a combination of these measures can be used to provide bounds on the optimal solution.