A request is successful on the first attempt with probability
1 − p. Recall that the contention window is initially set to
W, the average number of elapsed backoff slots before such
a request is (m/2) + (W − 1)/2. The first term is due to the
fact that an SS cannot start a new backoff period for its next
request immediately after the previous one in the same frame
but has to wait until the next frame; because requests slots are
uniformly chosen among the m request opportunities in each
frame, the average number of backoff slots wasted until the
next frame is m/2. The second term represents the average
number of backoff slots an SS has to wait before attempting to
send a request based on the contention-resolution mechanism
described in Section II.
If its first attempt fails, a request is successful on the second
attempt with probability p(1 − p). The average number of
elapsed backoff slots in this case is (m/2) + (αW − 1)/2.
Continuing this argument until the Rth attempt yields the
average number of elapsed backoff slots (Bavg) before the
request is successful. We have