Real and Complex Analysis

Cloud data centers host diverse applications, mixing workloads that require small predictable latency with others requiring large sustained throughput. In this environment, today's state-of-the-art TCP protocol falls short. We present measurements of a 6000 server production cluster and reveal impairments that lead to high application latencies, rooted in TCP's demands on the limited buffer space available in data center switches. For example, bandwidth hungry "background" flows build up queues at the switches, and thus impact the performance of latency sensitive "foreground" traffic.To address these problems, we propose DCTCP, a TCP-like protocol for data center networks. DCTCP leverages Explicit Congestion Notification (ECN) in the network to provide multi-bit feedback to the end hosts. We evaluate DCTCP at 1 and 10Gbps speeds using commodity, shallow buffered switches. We find DCTCP delivers the same or better throughput than TCP, while using 90% less buffer space. Unlike TCP, DCTCP also provides high burst tolerance and low latency for short flows. In handling workloads derived from operational measurements, we found DCTCP enables the applications to handle 10X the current background traffic, without impacting foreground traffic. Further, a 10X increase in foreground traffic does not cause any timeouts, thus largely eliminating incast problems.

/pdf/data-center-tcp-dctcp-4d3pj8xdal.pdf

Data center TCP (DCTCP)

Network protocols in layered architectures have historically been obtained on an ad hoc basis, and many of the recent cross-layer designs are also conducted through piecemeal approaches. Network protocol stacks may instead be holistically analyzed and systematically designed as distributed solutions to some global optimization problems. This paper presents a survey of the recent efforts towards a systematic understanding of layering as optimization decomposition, where the overall communication network is modeled by a generalized network utility maximization problem, each layer corresponds to a decomposed subproblem, and the interfaces among layers are quantified as functions of the optimization variables coordinating the subproblems. There can be many alternative decompositions, leading to a choice of different layering architectures. This paper surveys the current status of horizontal decomposition into distributed computation, and vertical decomposition into functional modules such as congestion control, routing, scheduling, random access, power control, and channel coding. Key messages and methods arising from many recent works are summarized, and open issues discussed. Through case studies, it is illustrated how layering as Optimization Decomposition provides a common language to think about modularization in the face of complex, networked interactions, a unifying, top-down approach to design protocol stacks, and a mathematical theory of network architectures

/pdf/layering-as-optimization-decomposition-a-mathematical-theory-3t9xhz4cfy.pdf

Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures

| Network protocols in layered architectures have historically been obtained on an ad hoc basis, and many of the recent cross-layer designs are also conducted through piecemeal approaches. Network protocol stacks may instead be holistically analyzed and systematically designed as distributed solutions to some global optimization problems. This paper presents a survey of the recent efforts towards a systematic understanding of "layering" as "optimization decomposition," where the overall communication network is modeled by a generalized network utility maximization problem, each layer corresponds to a decomposed subproblem, and the interfaces among layers are quantified as functions of the optimization variables coordinating the subproblems. There can be many alternative decompositions, leading to a choice of different layering architectures. This paper surveys the current status of horizontal decomposition into distributed computation, and vertical decomposition into functional modules such as congestion control, routing, scheduling, random access, power control, and channel coding. Key messages and methods arising from many recent works are summarized, and open issues discussed. Through case studies, it is illustrated how "Layering as Optimization Decomposition" provides a common language to think about modularization in the face of complex, networked interactions, a unifying, top-down approach to design protocol stacks, and a mathematical theory of network architectures.

Layering as optimization decomposition: A mathematical theory of network architectures : There are various ways that network functionalities can be allocated to different layers and to different network elements, some being more desirable than others. The intellectual goal of the research surveyed by this article is to provide a theoretical foundation for these architectural decisions in networking

Different continuous-time models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuous-time model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then mean-reverts strongly when far away from the mean. The volatility is higher when away from the mean.

Testing Continuous-Time Models of the Spot Interest Rate

In large multiplexers with many TCP flows, the aggregate traffic flow behaves predictably; this is a basis for the fluid model of Misra, Gong and Towsley V. Misra et al., (2000) and for a growing literature on fluid models of congestion control. In this paper we argue that different fluid models arise from different buffer-sizing regimes. We consider the large buffer regime (buffer size is bandwidth-delay product), an intermediate regime (divide the large buffer size by the square root of the number of flows), and the small buffer regime (buffer size does not depend on number of flows). Our arguments use various techniques from queueing theory. We study the behaviour of these fluid models (on a single bottleneck Kink, for a collection of identical long-lived flows). For what parameter regimes is the fluid model stable, and when it is unstable what is the size of oscillations and the impact on goodput? Our analysis uses an extension of the Poincare-Linstedt method to delay-differential equations. We find that large buffers with drop-tail have much the same performance as intermediate buffers with either drop-tail or AQM; that large buffers with RED are better at least for window sizes less than 20 packets; and that small buffers with either drop-tail or AQM are best over a wide range of window sizes, though the buffer size must be chosen carefully. This suggests that buffer sizes should be much much smaller than is currently recommended.

Buffer sizes for large multiplexers: TCP queueing theory and instability analysis

This article describes how control theory has been used to address the question of how to size the buffers in core Internet routers. Control theory aims to predict whether the is stable, i.e. whether TCP flows are desynchronized. If flows are desynchronized then small buffers are sufficient [14 ]; the theory here shows that small buffers actually promote desynchronization--a virtuous circle.

Part II: control theory for buffer sizing

We perform the necessary calculations to determine the stability and asymptotic forms of solutions bifurcating from steady state in a nonlinear delay differential equation with a single discrete delay. The results are used to examine the loss of local stability in a selection of congestion control algorithms employed over a single link. In particular, we analyze the fair and the delay-based dual algorithms. Explicit conditions are derived to ensure the onset of stable limit cycles as these algorithms just lose local stability. Further, we are able to quantify the effect parameters of the system have on the amplitude of the bifurcating periodic solutions.

Local bifurcation analysis of some dual congestion control algorithms

Rate control protocols that utilise explicit feedback from routers are able to achieve fast convergence to an equilibrium which approximates processor-sharing on a single bottleneck link, and hence such protocols allow short flows to complete quickly. For a network, however, processor-sharing is not uniquely defined but corresponds with a choice of fairness criteria, and proportional fairness has a reasonable claim to be the network generalization of processor-sharing.In this paper, we develop a variant of RCP (rate control protocol) that achieves α-fairness when buffers are small, including proportional fairness as the case α = 1. At the level of theoretical abstraction treated, our model incorporates a general network topology, and heterogeneous propagation delays. For our variant of the RCP algorithm, we establish a simple decentralized sufficient condition for local stability.An outstanding question for explicit congestion control is whether the presence of feedback based on queue size is helpful or not, given the presence of feedback based on rate mismatch. We show that, for the variant of RCP considered here, feedback based on queue size may cause the queue to be less accurately controlled.A further outstanding question for explicit congestion control is the scale of the step-change in rate that is necessary at a resource to accommodate a new flow. We show that, for the variant of RCP considered here, this can be estimated from the aggregate flow through the resource, without knowledge of individual flow rates.

http://ccr.sigcomm.org/online/files/p53-v38n3k-rainaA.pdf

Stability and fairness of explicit congestion control with small buffers

In this paper, we use time-series modeling to forecast taxi travel demand, in the context of a mobile application-based taxi hailing service. In particular, we model the passenger demand density at various locations in the city of Bengaluru, India. Using the data, we first shortlist time-series models that suit our application. We then analyse the performance of these models by using Mean Absolute Percentage Error (MAPE) as the performance metric. In order to improve the model performance, we employ a multi-level clustering technique where we aggregate demand over neighboring cells/geohashes. We observe that the improved model based on clustering leads to a forecast accuracy of 80% per km2. In addition, our technique obtains an accuracy of 89% per km2 for the most frequently occurring use case.

Gaurav Raina

Papers

Buffer sizes for large multiplexers: TCP queueing theory and instability analysis

Part II: control theory for buffer sizing

Local bifurcation analysis of some dual congestion control algorithms

Stability and fairness of explicit congestion control with small buffers

A multi-level clustering approach for forecasting taxi travel demand