Understanding TCP Behavior

Let T = τ + 1/C, where τ is the fixed propagation delay. Divide the time into intervals of T , starting from time t = 0. We number the intervals, starting from interval 0. Let the interval n be [nT, (n + 1)T ), for n = 0, 1, 2, .... Let us further divide each interval into identical time slots, each of which is equal to the transmission time of a fixed-size packet, 1/C. We assume CT is an integer. Hence, each time interval contains CT time slots, which are numbered as time slot 0, 1, 2, ..., CT −1. Let us first suppose the buffer size B is infinitely large, and define wn to be the maximum window size, qn to be the maximum queue length, and rn to be the total number of packets released in interval n. Let Q(t) be the queue size and W (t) be the window size at time t. The units for the above quantities are number of packets. Let the system starts at time t = 0 with an empty buffer and a window size 1. A packet is released at t = 0. The acknowledgement for the first packet arrives back at the source at t = T . Upon receiving an acknowledgement packet, the window size is increased by 1, and two new packets are immediately released. As long as the buffer becomes empty at the end of each interval upto interval n, by simple induction, we have wn = 2 n (1)