cnp3-ebook/principles/sharing — Czech @ Weblate: Evolution of the transmission rates

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	To avoid `congestion collapse`, the hosts must regulate their transmission rate [#fcredit]_ by using a `congestion control` mechanism. Such a mechanism can be implemented in the transport layer or in the network layer. In TCP/IP networks, it is implemented in the transport layer, but other technologies such as `Asynchronous Transfer Mode (ATM)` or `Frame Relay` include congestion control mechanisms in lower layers.
	Let us first consider the simple problem of a set of :math:`i` hosts that share a single bottleneck link as shown in the example above. In this network, the congestion control scheme must achieve the following objectives [CJ1989]_ :
	The congestion control scheme must `avoid congestion`. In practice, this means that the bottleneck link cannot be overloaded. If :math:`r_i(t)` is the transmission rate allocated to host :math:`i` at time :math:`t` and :math:`R` the bandwidth of the bottleneck link, then the congestion control scheme should ensure that, on average, :math:`\forall{t} \sum{r_i(t)} \le R`.
	The congestion control scheme must be `efficient`. The bottleneck link is usually both a shared and an expensive resource. Usually, bottleneck links are wide area links that are much more expensive to upgrade than the local area networks. The congestion control scheme should ensure that such links are efficiently used. Mathematically, the control scheme should ensure that :math:`\forall{t} \sum{r_i(t)} \approx R`.
	The congestion control scheme should be `fair`. Most congestion schemes aim at achieving `max-min fairness`. An allocation of transmission rates to sources is said to be `max-min fair` if :
	no link in the network is congested
	the rate allocated to source :math:`j` cannot be increased without decreasing the rate allocated to a source :math:`i` whose allocation is smaller than the rate allocated to source :math:`j` [Leboudec2008]_ .
	Depending on the network, a `max-min fair allocation` may not always exist. In practice, `max-min fairness` is an ideal objective that cannot necessarily be achieved. When there is a single bottleneck link as in the example above, `max-min fairness` implies that each source should be allocated the same transmission rate.
	To visualize the different rate allocations, it is useful to consider the graph shown below. In this graph, we plot on the `x-axis` (resp. `y-axis`) the rate allocated to host `B` (resp. `A`). A point in the graph :math:`(r_B,r_A)` corresponds to a possible allocation of the transmission rates. Since there is a `2 Mbps` bottleneck link in this network, the graph can be divided into two regions. The lower left part of the graph contains all allocations :math:`(r_B,r_A)` such that the bottleneck link is not congested (:math:`r_A+r_B<2`). The right border of this region is the `efficiency line`, i.e. the set of allocations that completely utilize the bottleneck link (:math:`r_A+r_B=2`). Finally, the `fairness line` is the set of fair allocations.
	Possible allocated transmission rates
	As shown in the graph above, a rate allocation may be fair but not efficient (e.g. :math:`r_A=0.7,r_B=0.7`), fair and efficient ( e.g. :math:`r_A=1,r_B=1`) or efficient but not fair (e.g. :math:`r_A=1.5,r_B=0.5`). Ideally, the allocation should be both fair and efficient. Unfortunately, maintaining such an allocation with fluctuations in the number of flows that use the network is a challenging problem. Furthermore, there might be several thousands flows that pass through the same link [#fflowslink]_.
	To deal with these fluctuations in demand, which result in fluctuations in the available bandwidth, computer networks use a congestion control scheme. This congestion control scheme should achieve the three objectives listed above. Some congestion control schemes rely on a close cooperation between the end hosts and the routers, while others are mainly implemented on the end hosts with limited support from the routers.
	A congestion control scheme can be modeled as an algorithm that adapts the transmission rate (:math:`r_i(t)`) of host :math:`i` based on the feedback received from the network. Different types of feedback are possible. The simplest scheme is a binary feedback [CJ1989]_ [Jacobson1988]_ where the hosts simply learn whether the network is congested or not. Some congestion control schemes allow the network to regularly send an allocated transmission rate in Mbps to each host [BF1995]_.
	Let us focus on the binary feedback scheme which is the most widely used today. Intuitively, the congestion control scheme should decrease the transmission rate of a host when congestion has been detected in the network, in order to avoid congestion collapse. Furthermore, the hosts should increase their transmission rate when the network is not congested. Otherwise, the hosts would not be able to efficiently utilize the network. The rate allocated to each host fluctuates with time, depending on the feedback received from the network. The figure below illustrates the evolution of the transmission rates allocated to two hosts in our simple network. Initially, two hosts have a low allocation, but this is not efficient. The allocations increase until the network becomes congested. At this point, the hosts decrease their transmission rate to avoid congestion collapse. If the congestion control scheme works well, after some time the allocations should become both fair and efficient.
	Evolution of the transmission rates
	Various types of rate adaption algorithms are possible. `Dah Ming Chiu`_ and `Raj Jain`_ have analyzed, in [CJ1989]_, different types of algorithms that can be used by a source to adapt its transmission rate to the feedback received from the network. Intuitively, such a rate adaptation algorithm increases the transmission rate when the network is not congested (ensure that the network is efficiently used) and decrease the transmission rate when the network is congested (to avoid congestion collapse).
	The simplest form of feedback that the network can send to a source is a binary feedback (the network is congested or not congested). In this case, a `linear` rate adaptation algorithm can be expressed as :
	:math:`rate(t+1)=\alpha_C + \beta_C rate(t)` when the network is congested
	:math:`rate(t+1)=\alpha_N + \beta_N rate(t)` when the network is not congested
	With a linear adaption algorithm, :math:`\alpha_C,\alpha_N, \beta_C` and :math:`\beta_N` are constants. The analysis of [CJ1989]_ shows that to be fair and efficient, such a binary rate adaption mechanism must rely on `Additive Increase and Multiplicative Decrease`. When the network is not congested, the hosts should slowly increase their transmission rate (:math:`\beta_N=1~and~\alpha_N>0`). When the network is congested, the hosts must multiplicatively decrease their transmission rate (:math:`\beta_C < 1~and~\alpha_C = 0`). Such an AIMD rate adaptation algorithm can be implemented by the pseudo-code below.
	Which binary feedback ?
	Two types of binary feedback are possible in computer networks. A first solution is to rely on implicit feedback. This is the solution chosen for TCP. TCP's congestion control scheme [Jacobson1988]_ does not require any cooperation from the router. It only assumes that they use buffers and that they discard packets when there is congestion. TCP uses the segment losses as an indication of congestion. When there are no losses, the network is assumed to be not congested. This implies that congestion is the main cause of packet losses. This is true in wired networks, but unfortunately not always true in wireless networks. Another solution is to rely on explicit feedback. This is the solution proposed in the DECBit congestion control scheme [RJ1995]_ and used in Frame Relay and ATM networks. This explicit feedback can be implemented in two ways. A first solution would be to define a special message that could be sent by routers to hosts when they are congested. Unfortunately, generating such messages may increase the amount of congestion in the network. Such a congestion indication packet is thus discouraged :rfc:`1812`. A better approach is to allow the intermediate routers to indicate, in the packets that they forward, their current congestion status. Binary feedback can be encoded by using one bit in the packet header. With such a scheme, congested routers set a special bit in the packets that they forward while non-congested routers leave this bit unmodified. The destination host returns the congestion status of the network in the acknowledgments that it sends. Details about such a solution in IP networks may be found in :rfc:`3168`. Unfortunately, as of this writing, this solution is still not deployed despite its potential benefits.
	Congestion control with a window-based transport protocol
	AIMD controls congestion by adjusting the transmission rate of the sources in reaction to the current congestion level. If the network is not congested, the transmission rate increases. If congestion is detected, the transmission rate is multiplicatively decreased. In practice, directly adjusting the transmission rate can be difficult since it requires the utilization of fine grained timers. In reliable transport protocols, an alternative is to dynamically adjust the sending window. This is the solution chosen for protocols like TCP and SCTP that will be described in more details later. To understand how window-based protocols can adjust their transmission rate, let us consider the very simple scenario of a reliable transport protocol that uses `go-back-n`. Consider the very simple scenario shown in the figure below.
	Consider first a window of one segment. This segment takes 4 msec to reach host `D`. The destination replies with an acknowledgment and the next segment can be transmitted. With such a sending window, the transmission rate is roughly 250 segments per second or 250 Kbps. This is illustrated in the figure below where each square of the grid corresponds to one millisecond.
	Consider now a window of two segments. Host `A` can send two segments within 2 msec on its 1 Mbps link. If the first segment is sent at time :math:`t_{0}`, it reaches host `D` at :math:`t_{0}+4`. Host `D` replies with an acknowledgment that opens the sending window on host `A` and enables it to transmit a new segment. In the meantime, the second segment was buffered by router `R1`. It reaches host `D` at :math:`t_{0}+6` and an acknowledgment is returned. With a window of two segments, host `A` transmits at roughly 500 Kbps, i.e. the transmission rate of the bottleneck link.
	Our last example is a window of four segments. These segments are sent at :math:`t_{0}`, :math:`t_{0}+1`, :math:`t_{0}+2` and :math:`t_{0}+3`. The first segment reaches host `D` at :math:`t_{0}+4`. Host `D` replies to this segment by sending an acknowledgment that enables host `A` to transmit its fifth segment. This segment reaches router `R1` at :math:`t_{0}+5`. At that time, router `R1` is transmitting the third segment to router `R2` and the fourth segment is still in its buffers. At time :math:`t_{0}+6`, host `D` receives the second segment and returns the corresponding acknowledgment. This acknowledgment enables host `A` to send its sixth segment. This segment reaches router `R1` at roughly :math:`t_{0}+7`. At that time, the router starts to transmit the fourth segment to router `R2`. Since link `R1-R2` can only sustain 500 Kbps, packets will accumulate in the buffers of `R1`. On average, there will be two packets waiting in the buffers of `R1`. The presence of these two packets will induce an increase of the round-trip-time as measured by the transport protocol. While the first segment was acknowledged within 4 msec, the fifth segment (`data(4)`) that was transmitted at time :math:`t_{0}+4` is only acknowledged at time :math:`t_{0}+11`. On average, the sender transmits at 500 Kbps, but the utilization of a large window induces a longer delay through the network.

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