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144 ../../principles/sharing.rst:905
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — The `congestion collapse` is a problem that all heterogeneous networks face. Different mechanisms have been proposed in the scientific literature to avoid or control network congestion. Some of them have been implemented and deployed in real networks. To understand this problem in more detail, let us first consider a simple network with two hosts attached to a high bandwidth link that are sending segments to destination `C` attached to a low bandwidth link as depicted below. 479/10000 · 479
145 ../../principles/sharing.rst:926
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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. 418/10000 · 418
146 ../../principles/sharing.rst:930
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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]_ : 232/10000 · 232
147 ../../principles/sharing.rst:932
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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`. 372/10000 · 372
148 ../../principles/sharing.rst:933
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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`. 417/10000 · 417
149 ../../principles/sharing.rst:934
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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 : 189/10000 · 189
150 ../../principles/sharing.rst:936
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — no link in the network is congested 35/10000 · 35
151 ../../principles/sharing.rst:937
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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]_ . 208/10000 · 208
152 ../../principles/sharing.rst:939
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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. 322/10000 · 322
153 ../../principles/sharing.rst:941
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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. 753/10000 · 753
154 ../../principles/sharing.rst:947
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Possible allocated transmission rates 37/10000 · 37
155 ../../principles/sharing.rst:950
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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]_. 506/10000 · 506
156 ../../principles/sharing.rst:952
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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. 422/10000 · 422
157 ../../principles/sharing.rst:954
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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]_. 487/10000 · 487
158 ../../principles/sharing.rst:959
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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. 991/10000 · 991
159 ../../principles/sharing.rst:965
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — Evolution of the transmission rates 35/10000 · 35
160 ../../principles/sharing.rst:968
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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). 505/10000 · 505
161 ../../principles/sharing.rst:970
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — 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 : 206/10000 · 206
162 ../../principles/sharing.rst:972
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — :math:`rate(t+1)=\alpha_C + \beta_C rate(t)` when the network is congested 74/10000 · 74
163 ../../principles/sharing.rst:973
	English ↹ ↵ NBS … “ ” ‘ ’ ﹙-﹚ ‐ – — :math:`rate(t+1)=\alpha_N + \beta_N rate(t)` when the network is *not* congested 80/10000 · 80