In Scenario 3 we examine the underflow probabilities and average latency for longer programs that are streamed at a rate that is below the maximum available bandwidth of the channel. The extra bandwidth leaves room for retransmitted packets to arrive alongside regularly scheduled ones. Because there is extra bandwidth, it is possible for the client to refill the buffer after losses occur. This is in contrast to the previous two scenarios where each lost packet represents an irreversible emptying of the client buffer that leads inevitably to an underflow.
In order to handle this scenario, we assume as Steinbach does that the good periods in the channel last long enough so that the buffer is able to regain its target level by the time a bad period begins [1]. Our departure from the previous work, however, is that while Steinbach's analysis assumes that the buffer is at the target level when burst begin, we assume that since the channel has been in the good state for a long time, the buffer is in the steady state distribution for the good state.
To find the probability of underflow, we first assume that the
the channel has been in the good state for a long time. We find the steady
state distribution of the client buffer for the transition matrix 
.
where is determined according to equation 6
for a channel 
, 
. The
transition from state 0 in is to 
as in equation 14,
because we assume that
if the buffer underflows because of jitter in the arrival times in the good state,
playout will halt and re-buffering will occur.
Next, we find the find 
, the transition matrix
for the channel 
, 
.
In ,
the transition from state zero is to zero as in
equation 11, so that probability of underflow accumulates
in state zero as propagates the buffer state forward in time.
Next, we assume the bad channel length is a geometrically distributed
random variable with mean 
. Let the random variable,
B be the burst length in frames. We find the probability of underflow
during a bad channel period as:
where
We calculate average latency as Steinbach does in equation 9.
This is an approximation. In Steinbach's analysis, the buffer is deterministically in the target
state at the beginning of the burst and empties (also deterministically) at the
rate of playout. We adjust the equation so that emptying in the bad state
occurs at rate given by the difference in the arrival rate, 
and the departure rate, 
.
