On Modeling Feedback Congestion Control Mechanism of TCP using Fluid Flow Approximation and Queuing Theory　 Hisamatu Hiroyuki Department of Infomatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, Japan hisamatu@ics.es.osaka-u.ac.jp Thank you for your Introduction, Mr. Chairman. And now, I’m going to speak out my research. The title is “On Modeling Feedback Congestion Control Mechanism of TCP using Fluid Flow Approximation and Queuing Theory”

Background TCP (Transmission Control Protocol) Transport layer protocol Congestion control mechanism Analysis of the TCP until today Assuming a constant packet loss probability Statistical behavior Real Network Packet loss probability has changed according to packet　transmission rate At first, I’m going to introduce you Research Background. TCP, the objective of this research, is transport layer protocol in internet and has congestion control mechanism. In the literature, there have been a great number of analytical studies on TCP. Most of those studies have focused on the statistical behavior of TCP by assuming a constant packet loss probability in the network. However, the packet loss probability, in reality, has changes according to packet transmission rate from TCP connections

Objective Model the interaction between two systems as a feedback system Network seen by TCP M/M/1/m queuing system Congestion control mechanism of TCP Fluid flow approximation Background Traffic are also taken account of For modeling the changes of packet loss probability in network according to packet transmission rate from TCP connections, we model the interaction between Network and the congestion control mechanism of TCP as a feedback system; that is, both the congestion Control mechanism of TCP running on a source host and the network seen by TCP are modeled by dynamic systems In addition, our analytic model, both TCP traffic and background traffic are taken account of

Analytic Model TCP connections Take account of background traffic source hosts destination hosts router TCP traffic background traffic This figure displays our analytic model. That is, TCP connections share one bottleneck link and Background traffic except TCP Exists on the bottleneck link

Modeling total TCP and Network Modeling Network as a feedback system Network seen by TCP Congestion control mechanism of TCP seen by network Congestion control Mechanism of TCP window size Network seen by TCP Next, we model the total network as feedback system That is, we model the interaction between the congestion control mechanism of TCP The congestion control mechanism of TCP is a window based flow control mechanism, And it dynamically changes the window size according to occurrence of packet losses in the network. So we model congestion control mechanism of TCP, where it input is the packet loss probability in the network and its output is the window size of TCP On the other hand, the network seen by the TCP behaves, When the number of packets entering the network increases, some packets are waited at the buffer of the router. This sometimes causes buffer overflow, resulting in a packet loss. So the packet loss probability becomes large when the number of packet entering the network increases. Thus, we model the network seen by TCP, Where its input is window size and its output is packet loss probability packet loss probability

Modeling Network using Queuing Theory Assume bottleneck router is a Drop-Tail router Model by M/M/1/m queue Incoming traffic TCP traffic background traffic Arrival rate of the background traffic lB p Packet loss probability ・ w1 wN TCP Window size M/M/1/m m We assume there exists only a single bottleneck link in the network. And we also assume the bottleneck router adopts a Drop tail router, If the network is stationary, bottleneck router can be modeled by a single queue Let N be the number of TCP connections, w sub i is window size and r sub i is the round trip time of the ith TCP connections, the transmission rate from ithe TCP connections can be approximated by w sub i divided r sub i, taking account of the background traffic, the average packet arrival rate at the bottleneck router, lambda is given like this. And then we can obtain the packet loss probability from the queuing theory l

Changes of TCP Window Size Congestion avoidance phase Increase window size at every receipt of ACK packet Decrease window size at every detection of packet loss Detect from receipt of duplicate ACKs Detect from time-out mechanism The congestion control mechanism of TCP is quite complicated, In this research,we model only the main part of the congestion control mechanism of TCP, We model the essential behavior of TCP in its Congestion avoidance phase, the window based flow control mechanism and the loss recovery mechanism Including the fast retrancemit

Modeling TCP using Different Approaches 4 analytic models Model A1： Assume a constant packet loss probability Derive window size of TCP connection in steady state Model A2 ： Approximate A1 when packet loss probability is very small Model B： Window size change at every receipt of ACK packet and detection of packet loss Model C： Evolution of window size between two succeeding packet loss Next, we introduce 4 analytic models called A1, A2, B, C. They derived from different modeling approaches. Model A1: by assuming a constant packet loss probability in the network, describing the window size of a TCP connections in steady state. steady state is the state after enough time passed Model A2: when the packet loss probability is very small, approximated the expression of the window size of TCP connection in Model A1 Model B: Window size changes at every ACK receiving and every detecting packet loss Model C: this analytic approach uses a discrete-time model, where a time slot Corresponds to the duration between two succeeding packet losses

Simulation Model :arrival rate Bottleneck Router UDP m:buffer size Destination ：link capacity :propagation delay :5+0.5i [packet/ms] :5 [packet/ms] :5+i [ms] :5 [ms] :2 [packet/ms] N :10 m :50 [packet]

Network Model Relation between offered traffic load and packet loss probability M/M/1/m queuing system Simulation result 0.5 1 1.5 2 2.5 3 0.01 0.1 Offered Traffic Load Packet Loss Probability Simulation M/M/1 M/M/1/m This figure shows the relation between the offered traffic load and the packet loss probability. These values are measured at the bottleneck router for every 10[ms] In the figure, the packet loss probabilities obtained from well-known results of M/M/1/m And M/M/1 are also plotted. This figure shows the dynamics of the network at a relatively small time scale can be well modeled by the M/M/1/m model M/M/1/m models dynamics of network correctly

TCP Model Relation between packet loss probability and window size Congestion control mechanism of TCP Simulation result Simulation Model A1 Model A2 Model B Model C 5 10 15 20 0.001 0.01 0.1 1 Packet Loss Probability Average Window Size [packet] By comparing with simulation results, we discuss how accurate four analytic models of TCP capture the relation between the window size and packet loss probability. This figure shows the relation between the packet loss probability and the window size obtained using model A1, A2,B,C,respectively. We also plot the simulation results which are measured at the bottleneck router for every 1[s] This figure shows when the packet loss probability is less than 0.02, analytic models A1, A2, and B show good agreement with simulation results. On the other hand, when the packet loss probability is more then 0.03, analytic models B and C show Good agreement A1, A2, B show good agreement B and C show good agreement

Transient Behavior Transient Behavior Dynamics of the window size form its initial value to its equilibrium value Use Model B for Congestion control model of TCP Using the analytic model presented before, we analyze the transient behavior of TCP in the congestion avoidance phase. By the word transient behavior, we mean the dynamics of the window size from its initial values to its equilibrium value. This time we adopt the model B from our proposed model as the congestion control model of TCP time Window size

Transient Behavior Analysis Modeling the network as a discrete-time model Time slot:duration between succeeding ACK packets received Network state w(k): window size of TCP connections P(k): packet loss probability For given initial values, the evolution of the network state can be obtained we model both the congestion control mechanism of TCP and the network as Interconnected discrete-time system, where the states of these models change at every time slot. The state of the network at slot k is then fully described by the window size and the packet loss probability For given Initial values of the window size and the packet loss probability, the evolution of the window size and the packet loss probability can be numerically obtained. From model B and well known results of the M/M/1/m queuing system, The state transition equations are obtained. Rho of k is offered traffic load at the bottleneck router at slot k

Numerical Example:Case of Different Propagation Delays 0.5 1 1.5 2 5 10 15 20 25 30 35 Average Window Size [packet] Time [s] t = 10 [ms] t = 30 [ms] t = 50 [ms] We next present several numerical examples. In following numerical examples, unless explicitly noted, the initial window size is 1[packet], the initial packet loss probability is 0, the number of TCP connections are 10, the capacity of the bottleneck link is 5[packet/ms], and its propagation delay is 15[ms] and the buffer size of the bottleneck router m is 50 [packet] the evolution of the window size in the congestion avoidance phase for the propagation delay tau of 10, 30, 50 [ms], one can find that the window size becomes large as the propagation delay increases. This can be intuitively understood from the increased bandwidth delay product. In addition, as the propagation delay becomes small, Ramp-up time of the window size becomes short and convergence speed of the Window size becomes slow. This is because, from a control theoretical viewpoint, the feedback gain in the congestion avoidance phase of TCP is changed according to the round trip time. thus increasing the propagation delay implies decreasing the feedback gain When propagation delay is small Ramp-up time of the window size becomes short The window size oscillates for long

Numerical Example:Case of Different Amount of Background Traffic lB = 0 lB = 2 lB = 4.5 5 10 15 20 25 30 35 Average Window Size [packet] 0.5 1 1.5 2 Time [s] This figure shows the evolution of the window size in the congestion avoidance phase for The amount of background traffic of 0, 2 and 4.5[packet/ms] From this figure, one can find that the window size in steady state becomes small as the amount of background traffic increases, this indicating that TCP suffers less throughput when the amount of background traffic increases One can also find the increase rate of the window size is independent of the amount of background traffic. This is because, in the congestion avoidance phase, TCP increases the window size by one packet per a round trip time. When the amount of the background traffic is large The window size of steady state is small The increase rate of the window size is independent of the amount of the background traffic

Conclusion and Future Work Model the dynamics of TCP Feedback system consisting of two systems Transient behavior Analysis of TCP Propagation delay The amount of the background traffic Future work Rigorous analyses of stability and transient behavior of TCP In this paper we have modeled both the congestion control mechanism of TCP and the network as a feedback system, and have analyzed the transient behavior. Heavily dependent on the propagation delay of the bottleneck link, but is almost Independent of the amount of background traffic. We have also shown that the operation of TCP in the congestion avoidance phase is less stable when the amount of background traffic is small or the propagation delay of the bottleneck link is short. In this paper, we have analyzed the transient behavior of TCP by iteratively calculating state transitions. However, rigorous analyses of the stability and the transient behavior of TCP are possible . we are currently working on such rigorous analyse