In (1) the estimate of loss probability 
and 
is based on recorded past
delays of the two streams using order statistics extended from [6]. The network delay of past w packets in each stream l has been recorded
and is denoted as 
, 
, ... , 
. The order statistics, or the sorted version of , , ... , are denoted
as 
, 
, ... 
, where
The rth order statistic is defined as
which is the probability the future delay 
is no greater than
, or the probability that packet i can be received by time .
In [6], it is shown that
which is the expected probability that packet i can be received by .
In our application, we extend (2) by defining
such that we have the extended order statistics
The definitions of 
and 
are empirically
based on the standard deviation of past delays. This solves the problem that
the expected playout probability in [6] cannot go beyond 
or go below 
. (3) is hence revised as
The expected probability corresponding to any 
equal to any 
k=0, 1, ... , w+1, can be determined directly by (5); while the expected probability associated with any in between these discrete values of 
s is found
by interpolation.
For stream l,
which is the index of the greatest 
that is no greater than . The expected probability that packet i can be received by
the deadline is
The expected of loss probability of packet i in stream l is then
which is used in the cost function in (1).
The idea of using past history to predict the loss probability of a future packet is based on the assumption that the past w delays have similar probability density function (p.d.f.), and this similarity will last for at least w samples, although the delay distribution varies in the long term. The effectiveness of this estimation depends on how close the accumulated history represents the present delay statistics.
