In scenario 1, we examine the probability of underflow for a 30 second clip, given that
the buffer is filled to a target level 
before playout begins,
and the rate of the media stream is the maximum rate of the channel. We also investigate
the average playout time of the clip for various adaption functions, f.
In this scenario we assume a frame rate of 15 fps and a total buffer size of 450 frames. Let
be the state (fullness) of the buffer while
frame n is played out. Let 
be the frame-to-frame transition matrix populated as in equation 6 for
a given channel C and a given adaption function f(i). For this scenario, we set
the transition from the zero state as:
Thus, any probability that flows into state zero remains there. To find the probability of
underflow for a given target , we begin by setting
We then propagate the state forward by 
frames and find the probability that
the buffer underflows during the course of the 30 second clip by examining
.
To find the expected playout duration of the clip, we find the expected playout duration
of each frame according to f(i) and d(n) and then average this value over all the frames
in the clip. We exclude from the expectation the probability accumulated in the 
or
underflow state. The average playout rate 
is given by:
In section 4 the values calculated in this manner are compared with results found during simulation experiments.
