Theoretical Rate Calculation for Scenario 7

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 destinationy 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.

Warren Toomey
Fri Mar 15 10:43:33 EST 1996