# A Very Simple Model for TCP Throughput

Posted by on July 24th, 2013
March 18th, 2015

The Transmission Control Protocol (TCP) is probably the most popular communication protocol on the Internet today. TCP is a complex protocol, and understanding throughput behavior is hard, especially when dealing with lossy environments such as wireless networks. In this article, we look at one such model and show that it does accurately capture TCP throughput behavior even under lossy situations, typical of wireless networks. This demonstrates that it is possible to better understand and infer network characteristics such as throughputs that are directly influenced by TCP behavior.

### A simple model for TCP throughput

TCP throughput, which is the rate that data is successfully delivered over a TCP connection, is an important metric to measure the quality of a network connection. Generally speaking, TCP throughput is measured as:

$Throughput=\frac{WindowSize}{RTT}$

In 1997, Matthew Mathis et al., came up with a model for the TCP throughput, which takes into account some link characteristics, such as the Maximum Segment Size (MSS), Round-Trip Time (RTT) and packet loss. Their derivation is taken from the analysis of how TCP behaves under loss during congestion avoidance phase. A TCP sender usually sends a batch of packets before receiving any acknowledgement from the receiver. Of course if the sender sends too many packets too fast they are going to be dropped because of congestion in intermediate queues.

To minimize this, TCP has a flow control mechanism that slows down the sender’s transmission rate if the devices in the path are not able to cope with the offered rate. A congestion window (cwnd) allows the sender to have at most cwnd unacknowledged bytes at any given time, effectively controlling the rate at which data is sent. There are different algorithms to control the behavior of congestion window, the most popular of which are New Reno and CUBIC. In steady-state, cwnd grows linearly one packet per RTT and when there’s a loss event (e.g., three duplicate ACKs) cwnd is set to half its value, as indicated in the figure below, extracted from the original Mathis paper.

Assuming the path has a packet loss of probability p, the sender will be able to send an average of 1/p packets before a packet loss. Under this model cwnd never exceeds a maximum W because at approximately 1/p packets a new packet loss makes cwnd drop to half of its value again. Hence, the total data delivered at every cycle of 1/p packets, is given by $1/p=(\frac{w}{2})^2+\frac{1}{2}(\frac{w}{2})^2=\frac{3}{8}W^2$ (i.e., the area under sawtooth). Therefore, $MSS\times\frac{3}{8}W^2$ bytes are sent at every cycle of $RTT\times \frac{W}{2}$, which solving $W=\sqrt{\frac{8}{3p}}$ we arrive at the following throughput:

$T=\frac{MSS\times \frac{3}{8}W^2}{RTT\times \frac{W}{2}} = \frac{MSS/p}{RTT\sqrt{\frac{2}{3p}}}$

Collecting all the constant terms in $C=\sqrt{\frac{3}{2}}$, we get:

$T=\frac{MSS\times C}{RTT \times\sqrt{p}}$

This constant C lumps together several terms that are typically constant for a given combination of TCP implementation, ACK strategy (delayed vs non-delayed), and loss mechanism.

### A simple validation

The following is an experiment between two hosts to validate the Mathis TCP throughput model. Host A sends traffic for 30 seconds to host B on a link configured to emulate different packet loss probabilities.

We used netem utility to set up the network interface with a constant latency of 100 ms (50 ms in each host) and a loss probability from 1% to 10% (on all packets leaving host A).

Iperf was used to measure the throughput from host A to host B by transmitting data during 30 seconds (-t) and using different congestion window algorithms: CUBIC (default on linux kernel since version 2.6.19) and New Reno (the more traditional scheme).

Figure 2: Comparison of estimated throughput with loss in different models.

The graph above compares the estimated throughput (from the Mathis model) with the actual measured values under different packet loss scenarios. To filter out any outliers each data point is the median of five repeated experiments.

The graph shows that the model greatly approximates to the actual experimental measurements and that TCP throughput is very sensitive to loss. Loss is determinant because loss events regularly cap cwnd, the sender will never send more than W bytes at every RTT, effectively limiting its maximum throughput. So, for an average of 1% of packet loss, the TCP throughput will be restricted to 1,400 kbps (because at approximately every 100 packets a packet is lost and cwnd drops to half).

The shape of the curves also reveal that loss dramatically affects TCP throughput, especially for low values. For instance, the throughput drops to half from 0.1% to 0.4% of packet loss.

The Mathis model is particularly appropriate to estimate the TCP throughput in network paths with regular packet loss, which happens in steady state when there’s a constant rate of cross traffic in the path. If no congestion is observed, cwnd steadily increases until a corrupted packet is detected (e.g., wireless environment) or the receiver advertises an insufficient receive buffer. The model also works in lossy wireless links with a uniform loss probability.

Loss is not a constant characteristic of a network path. Instead, it varies according to different network conditions, such as the physical capacity of the links (and in particular of the bottleneck link) or network congestion, which delimits the available bandwidth. We talk more about network capacity and available bandwidth in a follow up post.

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