STATE SPACE COLLAPSE AND DIFFUSION APPROXIMATION FOR A NETWORK OPERATING UNDER A FAIR BANDWIDTH SHARING POLICY
STATE SPACE COLLAPSE AND DIFFUSION APPROXIMATION FOR A NETWORK OPERATING UNDER A FAIR BANDWIDTH SHARING POLICY
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公平带宽共享策略下运行的网络的状态空间崩溃和扩散近似
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
L. N.H.
中科院分区：
文献类型：
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作者：
K. F.P.;L. N.H.
我们考虑由Massoulié和Roberts[36]提出的Internet拥塞控制的连接级模型，该模型表示网络中存在的随机变化的流数量。这里，根据Mo和Walrand[37](α∈(0，∞))提出的加权α公平带宽共享策略，在弹性文档传输之间公平地共享带宽。假设泊松到达和指数分布的文档大小，我们关注的是高流量模式，在该模式中，每个资源上的平均负载大约等于其容量。这一随机模型的流体模型(或泛函大数近似)是由两位作者在先前的工作[29]中推导和分析的。本文利用文献[29]中所建立的流体模型解的长时间性态，导出了一个称为乘性状态空间塌陷的性质，它粗略地表明，在扩散尺度下，随机模型的流计数过程可以近似地恢复为工作量过程的连续提升。在带宽的加权比例公平共享(α=1)和温和的局部业务条件下，我们展示了如何将乘性状态空间崩溃与不变性原理[23]相结合来建立工作负载过程的扩散近似，从而得到流计数过程的近似。在这种情况下，负载扩散在多面体锥体内部表现为布朗运动，并通过在边界处的反射被限制在锥体内，其中反射方向在任何给定的边界面上是恒定的。当所有权重相等(比例公平分享)时，这种扩散具有乘积形式的不变分布。如果后者是可积的，则它产生扩散的唯一平稳分布，该分布具有关于独立的对偶随机变量的显著简单的解释，每个对偶随机变量对应于网络的每个资源。我们能够将这个乘积形式的结果扩展到文档大小作为指数的有限混合分布的情况，以及包括多路径路由的模型。我们指出了将扩散近似结果推广到α6=1的值的一些困难。
We consider a connection-level model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted α-fair bandwidth sharing policy introduced by Mo and Walrand [37] (α ∈ (0,∞)). Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [29] by two of the authors. Here we use the long time behavior of the solutions of this fluid model established in [29] to derive a property called multiplicative state space collapse, which loosely speaking shows that in diffusion scale the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. Under weighted proportional fair sharing of bandwidth (α = 1) and a mild local traffic condition, we show how multiplicative state space collapse can be combined with an invariance principle [23] to establish a diffusion approximation for the workload process and hence to yield an approximation for the flow count process. In this case, the workload diffusion behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary, where the direction of reflection is constant on any given boundary face. When all of the weights are equal (proportional fair sharing), this diffusion has a product form invariant distribution. If the latter is integrable, it yields the unique stationary distribution for the diffusion which has a strikingly simple interpretation in terms of independent dual random variables, one for each of the resources of the network. We are able to extend this product form result to the case where document sizes are distributed as finite mixtures of exponentials, and to models that include multi-path routing. We indicate some difficulties related to extending the diffusion approximation result to values of α 6= 1.