Equation-based end-to-end single-rate multicast congestion control

Abstract

Among the recently proposed single-rate multicast congestion control protocols is transmission control protocol-friendly multicast congestion control (TFMCC; Widmer and Handley 2001; Floyd et al. 2000; Widmer et al. IEEE Netw 15:28–37, 2001), which is an equation-based single-rate protocol that extends the mechanisms of the unicast TCP-friendly rate control (TFRC) protocol into the multicast domain. In TFMCC, each receiver estimates its throughput using an equation that estimates the steady-state throughput of a TCP source. The source then adjusts its sending rate according to the slowest receiver within the session (a.k.a., current-limiting receiver, CLR). TFMCC is a relatively simple, scalable, and TCP-friendly multicast congestion control protocol. However, TFMCC is hindering its throughput performance by adopting an equation derived from the unicast TFRC protocol. Further, TFMCC is slow to react to congestion conditions that usually result in a change of the CLR. This paper is motivated by these two observations and proposes an improved version of TFMCC, which we refer to as hybrid-TFMCC (or H-TFMCC for short). First, each receiver estimates its throughput using an equation that models the steady-state throughput of a multicast source controlled according to the additive increase multiplicative decrease (AIMD) approach. The second modification consists of adopting a hybrid sender/receiver-based rate control strategy, where the sending rate can be adjusted by the source or initiated by the current or a new CLR. The source monitors RTT variations on the CLR path, in order to rapidly adjust the sending rate to network conditions. Simulation results show that these modifications result in remarkable performance improvement with respect to throughput, time to react, and magnitude of oscillations. We also show that H-TFMCC remains TCP-friendly and achieves a higher fairness index than that achieved by TFMCC.

Appendix: Theoretical Analysis

1.1 Proof of Property 1

A source is said to be TCP-friendly if, on average, its sending rate is not larger than that achieved by TCP, under the same circumstances. We want to prove that H-TFMCC multicast flow is TCP-friendly to a TCP flow on the CLR path, by showing that it gets on average less than or an equal share of the bottleneck bandwidth, with the assumption that their RTT estimation, PLR and packet size S are the same under the same conditions.

Let $X_t^{\rm{TCP}}$ and $X_t^{\rm{AIMD}}$ denote, respectively, the sending rates of the TCP flow and H-TFMCC at time t, and that $X_t^{\rm{AIMD}}/X_t^{\rm{TCP}} \leqslant 1$. We assume that only these two flows are competing at the bottleneck link, on the congestion representative path. Like other TCP throughput analyses [1, 6, 12] have done, our analysis focuses only on TCP congestion avoidance behavior.

During the congestion avoidance period, in the absence of packet losses, a TCP source increases its congestion window by 1/W packet upon reception of an Ack, where W is the current congestion window size. A TCP source transmits all the packets in its congestion window once per RTT; therefore, the window grows by one packet per RTT, which corresponds to the fact that the sending rate at the source is increased by S/RTT per RTT, where S is the packet size. The H-TFMCC source increases its sending rate in the same manner. Hence, at some time t + k × RTT, and in the absence of any losses, $X_{t + k{\rm{RTT}}}^{\rm{TCP}} = X_t^{\rm{TCP}} + kS/{\hbox{RTT}}$ and $X_{t + k{\rm{RTT}}}^{\rm{AIMD}} = X_t^{\rm{AIMD}} + kS/{\hbox{RTT}}$. Therefore, $X_{t + k{\rm{RTT}}}^{\rm{AIMD}}/X_{t + k{\rm{RTT}}}^{\rm{TCP}} \leqslant 1,\forall {\hbox{k}} \geqslant 0.$

We assume that congestion is the only reason for packet losses. In the most likely case, it is a burst loss. At the reception of a CI (congestion indication), we have a combined TCP and H-TFMCC rate reduction at about the same instant. Because the TCP and H-TFMCC flows share the same path (CLR), we assume that they detect packet losses and reduce the transmission rates approximately at the same time.

Let t ₁ be the instant just before a loss occurrence, indicating the termination of a rate increase interval, and let $X_{{t_1}}^{\rm{TCP}}$ and $X_{{t_1}}^{\rm{AIMD}}$ be the sending rates of the TCP flow and H- TFMCC at that instant, such that $X_{{t_1}}^{\rm{AIMD}}/X_{{t_1}}^{\rm{TCP}} \leqslant {1}{.}$ Thus, at the instant t₁ + δ, the H-TFMCC and TCP rates will be $\beta *X_{{t_1}}^{\rm{AIMD}}$ and $X_{{t_1}}^{\rm{TCP}}/2$. Therefore, $\beta \times X_{{t_1}}^{\rm{AIMD}}/\left( {X_{{t_1}}^{\rm{TCP}}/2} \right) \leqslant 1$ for all β ≤ 0.5.

In the current implementation of H-TFMCC, we set β = 0.5. Larger values of β resulted in RTT and rate oscillations. Consequently, H-TFMCC is TCP-friendly, given a reduction factor β ≤ 0.5.

Further, if we consider the fact that TCP has lower RTT and PLR estimations than that of H-TFMCC, so that RTT_tcp < RTT_H-TFMCC and p _TCP < p _H-TFMCC, then TCP will be increasing its sending rate faster than H-TFMCC. This will further guarantee the TCP-friendliness of H-TFMCC, i.e., ${X^{\rm{AIMD}}}/{X^{\rm{TCP}}} < 1.$

1.2 Proof of property 2

Recall that the Jain’s fairness Index is defined as follows for n competing flows:

$$ {\hbox{fairness}} = \frac{{{{\left( {\sum\limits_i^n {{X_i}} } \right)}^2}}}{{n\sum\limits_{i = 1}^n {{{\left( {{X_i}} \right)}^2}} }} $$

where X _i denotes the rate of i ^th flow at the bottleneck link.

Assume that flows X ₁ to X _n − 1 are TCP flows. Thus, if the nth flow is either H-TFMCC or TFMCC:

$$ \begin{array}{*{20}{c}} {{\text{fairness}}\left( {{\text{H\_TFMCC}}} \right) = \frac{{{{\left( {\sum\limits_{i = 1}^{n - 1} {{X_i} + {X_H}} } \right)}^2}}}{{n.\left( {\sum\limits_{i = 1}^{n - 1} {X_i^2 + X_H^2} } \right)}}} \\ {{\text{fairness}}\left( {{\text{TFMCC}}} \right) = \frac{{{{\left( {\sum\limits_{i = 1}^{n - 1} {{X_i} + {X_T}} } \right)}^2}}}{{n.\left( {\sum\limits_{i = 1}^{n - 1} {X_i^2 + X_T^2} } \right)}}} \\ \end{array} $$

Let $A = \sum\limits_{i = 1}^{n - 1} {{X_i}} {\hbox{ and }}B = \sum\limits_{i = 1}^n {{{\left( {{X_i}} \right)}^2}}$ Indeed, we consider a simplified version of the TFMCC formula as done in [25], which we refer to respectively, as the simplified form of the steady-state throughput formula of TFMCC and the proposed formula of H-TFMCC:

$$ {X_T} = \frac{S}{{\sqrt {{\frac{2}{3}p}} \times {\hbox{RTT}}}}\quad {X_H} = \frac{S}{{p \times {\hbox{RTT}}}} $$

The ratio: $\frac{{{X_T}}}{{{X_H}}} = \sqrt {{\frac{3}{2}}} \times \sqrt {p} = \alpha$ so we have X _T = α.X _H where 0 < α < 1. Figure 17 shows the source rate as time progresses, as computed by Eq. 5 for H-TFMCC and Eq. 1 for TFMCC, under the same conditions.

Fig. 17

H-TFMCC compared with TFMCC

We need to show that fairness(H-TFMCC)>fairness(TFMCC), i.e.,

$$ \frac{{{{\left( {A + {X_H}} \right)}^2}}}{{n\left( {B + X_H^2} \right)}} > \frac{{{{\left( {A + {X_T}} \right)}^2}}}{{n\left( {B + X_T^2} \right)}} $$

So we need to prove that:

$$ \begin{array}{*{20}{c}} {\frac{{{\text{fairness}}\left( {{\text{TFMCC}}} \right)}}{{{\text{fairness}}\left( {{\text{H\_TFMCC}}} \right)}} = \frac{{{F_T}}}{{{F_H}}} < 1} \\ {\frac{{{F_T}}}{{{F_H}}} = \frac{{\frac{{{{\left( {A + {X_T}} \right)}^2}}}{{n\left( {B + X_T^2} \right)}}}}{{\frac{{{{\left( {A + {X_H}} \right)}^2}}}{{n\left( {B + X_H^2} \right)}}}} = \frac{{{{\left( {A + {X_T}} \right)}^2}\left( {B + {X^2}_H} \right)}}{{{{\left( {A + {X_H}} \right)}^2}\left( {B + X_T^2} \right)}}} \\ \end{array} $$

Recall that: X _T  = α × X _H where 0 < α < 1

This leads to:

$$ \frac{{{{\left( {1 + \alpha .\frac{{{X_H}}}{A}} \right)}^2}}}{{{{\left( {1 + \frac{{{X_H}}}{A}} \right)}^2}}}*\frac{{\left( {1 + \frac{{{X_H}^2}}{B}} \right)}}{{\left( {1 + {\alpha^2}\frac{{{X_H}^2}}{B}} \right)}} < 1 $$

We have, X _H  << A and X _H  << B, which is realized due to the TCP-friendliness property. Let E = X _H /A, so we obtain: ${\left( {1 + \alpha .E} \right)^2} < {\left( {1 + E} \right)^2},$Therefore,

$$ \frac{{{{\left( {1 + \alpha E} \right)}^2}}}{{{{\left( {1 + E} \right)}^2}}} < 1 $$

Which is always true for α in ]0,1[.Whereas the following expression:

$$ \frac{{\left( {1 + \frac{{{X_H}^2}}{B}} \right)}}{{\left( {1 + {\alpha^2}\frac{{{X_H}^2}}{B}} \right)}} $$

is positive and at maximum very near 1. This is due to the TCP-friendliness property: ${X_T} = \alpha \times {X_H} < < B$ Hence, the condition fairness_TFMCC < fairness_H-TFMCC is always verified