Regarding Scenario 7, the bottleneck in the system is the 64kbps full-duplex link 2 3. We will assume that all acknowledgment rates are limited by their respective data rates by

where denotes a data flow from source *x* to destination*y* and
denotes the corresponding acknowledgment flow in the reverse
direction.

At time *t=0*, the flows and commence. The optimum
rates for both are:

Time | Optimum Rate (bps) | |

0<=t | 64,000 | 1706.66 |

At time *t=12*, the flows and commence. As
the link 2 3 is the bottleneck, we can write

Obviously (2) is the limiting equation, and by sharing bandwidth equally to the flows, we obtain the rates:

Time | Optimum Rate (bps) | |||

0<=t<12 | 64,000 | 1706.66 | ||

12<=t | 32,000 | 853.33 | 32,000 | 853.33 |

At time *t=18*, the flows and commence. Here, the
flows are in the opposite directions to the previous four flows. We
can write:

Distributing bandwidth fairly, we have and . Assume both links are fully utilised and substituting for the acknowledgment rates using (1), we have:

Substituting (8) into (6) gives , and the table:

Time | Optimum Rate (bps) | |||||

0<=t<12 | 64,000 | 1706.66 | ||||

12<=t<18 | 32,000 | 853.33 | 32,000 | 853.33 | ||

18<=t | 31,168.83 | 831.2 | 31,168.83 | 831.2 | 62,337.66 | 1,662.34 |

The six flows continue to transmit until time *t=12* when the flows
and commence. Again, distributing bandwidth fairly,
we have and .
Assume both links are fully utilised and substituting for the acknowledgment
rates using (1), we have:

Substituting (11) into (9) gives , and the table:

Time | Optimum Rate (bps) | |||||||

0<=t<12 | 64,000 | 1706.66 | ||||||

12<=t<18 | 32,000 | 853.33 | 32,000 | 853.33 | ||||

31,168.83 | 831.2 | 31,168.83 | 831.2 | 62,337.66 | 1,662.34 | |||

20,779.2 | 554.1 | 20,779.2 | 554.1 | 62,337.66 | 1,662.34 | 20,779.2 | 554.1 |

As flow only has 100 packets to send, it finishes quickly, and the rates return to the values before flow started:

Time | Optimum Rate (bps) | |||||||

0<=t<12 | 64,000 | 1706.66 | ||||||

12<=t<18 | 32,000 | 853.33 | 32,000 | 853.33 | ||||

31,168.83 | 831.2 | 31,168.83 | 831.2 | 62,337.66 | 1,662.34 | |||

20,779.2 | 554.1 | 20,779.2 | 554.1 | 62,337.66 | 1,662.34 | 20,779.2 | 554.1 | |

31,168.83 | 831.2 | 31,168.83 | 831.2 | 62,337.66 | 1,662.34 |

Similarly, flow has most of the bandwidth across 3 2, so it is the next to finish, and the rate return to the values before flow started:

Time | Optimum Rate (bps) | |||||||

0<=t<12 | 64,000 | 1706.66 | ||||||

12<=t<18 | 32,000 | 853.33 | 32,000 | 853.33 | ||||

31,168.83 | 831.2 | 31,168.83 | 831.2 | 62,337.66 | 1,662.34 | |||

20,779.2 | 554.1 | 20,779.2 | 554.1 | 62,337.66 | 1,662.34 | 20,779.2 | 554.1 | |

31,168.83 | 831.2 | 31,168.83 | 831.2 | 62,337.66 | 1,662.34 | |||

32,000 | 853.33 | 32,000 | 853.33 |

The flows and continue at this rate until one or both finish.

Fri Mar 15 10:43:33 EST 1996