Gaurav Raina

We study Compound TCP (C-TCP), the transport protocol in the Windows operating system, in different buffer sizing regimes along with Drop-Tail and Random Exponential Marking (REM). The buffer sizing regimes we focus on are the widely deployed bandwidth-delay rule and a small buffer regime. The performance metrics we consider are stability of the queue size, queuing delay, link utilisation and packet loss. © 2013 IFIP.

We study the performance of Compound TCP with Random Early Detection (RED) in three different limiting regimes. In the first regime, averaging over the queue size helps to decides the probability of dropping packets. Then, we consider a model where averaging over the queue size is not performed, but the queue is modelled as an integrator. Finally, we consider a model where the threshold for dropping packets is so small that it is not possible to model the queue as an integrator. In these three regimes, we derive sufficient, as well as necessary and sufficient conditions for local stability. These conditions help to capture the dependence of protocol and network parameters on system stability. We also show that in the event of loss of local stability, the Compound TCP-RED system undergoes a Hopf bifurcation which would lead to limit cycles. Some of the analytical results are corroborated using packet-level simulations.

The increase in queueing delays in the Internet motivates the study of transmission control protocol (TCP) and queue management policies. This paper studies Compound TCP with a Proportional–Integral (PI) policy for queue management at the Internet routers. A nonlinear fluid model for the Compound TCP-PI system is considered. For this model, a sufficient condition for local stability, which yields some design guidelines, is derived. Fragility analysis of this model highlights that even marginal variations in the PI parameters may induce instability. Next, a regime where the integral component, of the PI policy, operates over a small time-scale is considered. For a fluid model befitting this regime, the necessary and sufficient condition for local stability is derived. It is explicitly shown that, when this condition is violated, the system undergoes a Hopf bifurcation, which would lead to limit cycles. Furthermore, a detailed local bifurcation analysis is conducted to characterize the type of the Hopf bifurcation and determine the orbital stability of the limit cycles. Packet-level simulations corroborate the analytical insight. The analysis and simulations demonstrate some drawbacks of the PI policy. Following this, a simple threshold-based queue policy, which has desirable stability properties, is proposed for queue management at routers. Furthermore, a simulation-based performance evaluation reveals that this policy outperforms PI. Through a combination of theory and simulations, it is shown that the threshold-based policy can ensure system stability, while ensuring reduced queueing delays.

We analyze Compound TCP, the default protocol in the Windows operating system, along with an Exponential-RED (E-RED) queue policy. The E-RED queue policy specially aims for high link utilization. Our setup considers two sets of TCP flows, each having a different round-trip time, operating over a single bottleneck link. For this system, we first derive a sufficient condition for local stability. The stability condition reveals that the link gain needs to scale with the maximum round-trip time in the system. Additionally, the Compound parameter α needs to be chosen judiciously if stability is to be ensured. We then computationally show that, as parameters vary, the system can undergo a Hopf bifurcation. This bifurcation alerts us about the emergence of limit cycles, in the system dynamics, as stability is just lost. Finally, we exhibit the emergence of such limit cycles, in the queue size, via packet-level simulations. These limit cycles can result in the loss of link utilization and should be avoided.

The analysis of transport protocols, along with queue management policies, forms an important aspect of performance evaluation for the Internet. In this article, we study Compound TCP (C-TCP), the default TCP in the Windows operating system, along with the Exponential-RED (E-RED) queue policy and the widely used Drop-Tail queue policy. We consider queuing delay, link utilization and the stability of the queue size as the performance metrics. We first analyse the stability properties of a nonlinear model for C-TCP coupled with the E-RED queue policy. We observe that this system, in its current form, may be difficult to stabilize as the feedback delay gets large. Further, using an exogenous and non-dimensional parameter, we show that the system loses local stability via a Hopf bifurcation, which gives rise to limit cycles. Employing Poincaré normal forms and the center manifold theory, we outline an analytical framework to characterize the type of the Hopf bifurcation and to determine the orbital stability of the emerging limit cycles. Numerical examples, stability charts and bifurcation diagrams complement our analysis. We also conduct packet-level simulations, with E-RED and Drop-Tail, in small and large buffer-sizing regimes. With large buffers, E-RED can achieve small queue sizes compared with Drop-Tail. However, it is difficult to maintain the stability of the E-RED policy as the feedback delay gets large. On the other hand, with small buffers, E-RED offers no clear advantage over the simple Drop-Tail queue policy. Our work can offer insights for the design of queue policies that can ensure low latency and stability.

We study Compound TCP (C-TCP), the default TCP in the Windows operating system, with Random Exponential Marking (REM) and the widely used Drop-Tail queue policy. The performance metrics we consider are stability of the queue size, queuing delay, link utilization, and packet loss. We analyze the following models: 1) a nonlinear model for C-TCP with Drop-Tail and small buffers; 2) a stochastic variant of REM along with C-TCP; and 3) the original REM proposal as a continuous-time nonlinear model with delayed feedback. We derive conditions to ensure local stability and show that variations in system parameters can induce a Hopf bifurcation, which would lead to the emergence of limit cycles. With Drop-Tail and small buffers, the Compound parameters and the buffer size both play a key role in ensuring stability. In the stochastic variant of REM, larger thresholds for marking/dropping packets can destabilize the system. With the original REM proposal, using Poincaré normal forms and the center manifold analysis, we also characterize the type of the Hopf bifurcation. This enables us to analytically verify the stability of the bifurcating limit cycles. Packet-level simulations corroborate some of the analysis. Some design guidelines to ensure stability and low latency are outlined.

In this paper, we study the performance of Compound TCP with the classical Proportional Integral (PI) control policy implemented at routers. We first conduct a local stability analysis and derive the necessary and sufficient condition for local stability of the non-linear model of Compound with PI. We explicitly show that the system undergoes a Hopf bifurcation as it transits into instability, which would lead to the emergence of limit cycles in the queue size. We then use Poincaré normal forms and center manifold theory to provide an analytical basis to characterise the Hopf bifurcation and determine the orbital stability of the limit cycles. The analysis is complemented with numerical examples.

Compound TCP (C-TCP) is currently the default transport layer protocol in the Windows operating system. We study a non-linear fluid model of Compound along with Virtual Queue (VQ) management schemes in network routers. One can implement virtual queue policies either non-adaptively or adaptively. The Adaptive Virtual Queue (AVQ), in particular, has the objective of driving the link utilisation to a desired level. The stability analysis of non-adaptive virtual queues reveals that smaller virtual buffer sizing rules help stability. Small virtual buffers would quite naturally reduce queueing delays in routers. Some guidelines for Compound and network parameters to ensure local stability are outlined. Analysis of the AVQ policy shows that the system is prone to losing local stability with large feedback delays, high link capacities, and with variations in the AVQ damping factor. We further show that the loss of local stability would occur via a Hopf bifurcation. Based on the analysis, our recommendation is that a virtual queue, with small buffer sizes, could be an attractive queue management scheme. © 2013 IEEE.

Large and unmanaged router buffers could lead to an increase in queuing delays in the Internet, which is a serious concern for network performance and quality of service. Our focus is to conduct a performance evaluation of Compound TCP (C-TCP), in a regime where the router buffer sizes are small (i.e., independent of the bandwidth-delay product), and the queue policy is Drop-Tail. In particular, we provide buffer sizing recommendations for high speed core routers fed by well multiplexed TCP controlled flows. For this, we consider two topologies: a single bottleneck and a multi-bottleneck topology, under different traffic scenarios. The first topology consists of a single bottleneck router, and the second consists of two distinct sets of TCP flows, regulated by two edge routers, feeding into a common core router. We focus on some key dynamical and statistical properties of the underlying system. From a dynamical perspective, we first develop fluid models. A local stability analysis for these models yields a key insight: buffer sizes need to be dimensioned carefully, and smaller buffers favour stability. We also highlight that larger Drop-Tail buffers, in addition to increasing latency, are prone to inducing limit cycles in the system dynamics. These limit cycles in turn induce synchronisation among the TCP flows, which then results in a loss of link utilisation. We then empirically analyse some statistical properties of the bottleneck queues. These statistical analyses serve to validate an important modelling assumption: that in the regime considered, each bottleneck queue may be reasonably well approximated as either an M∕M∕1∕B or an M∕D∕1∕B queue. We also highlight that smaller buffers, in addition to ensuring stability and low latency, would also yield reasonable system-wide performance, in terms of throughput and flow completion times.

The analysis of TCP, along with queue management policies, forms an important aspect of performance evaluation for the Internet. In this paper, we analyse a non-linear fluid model for Compound TCP (C-TCP) coupled with an Exponential-RED (E-RED) queue management policy. Compound is an important flavor of TCP, as it is the default transport protocol in the Windows operating system. For the bifurcation analysis, we motivate an exogenous and non-dimensional bifurcation parameter. Using this parameter, we first derive the Hopf bifurcation condition for the underlying model. Then, employing Poincaré normal forms and the center manifold theory, we outline the analysis to characterise the type of the Hopf bifurcation, and determine the stability of the bifurcating periodic solutions. Some numerical analysis and stability charts complement our theoretical analysis. © 2014 IEEE.