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5.2. BPR + : an optimized BPR scheduler
Optimized Backlog-Proportional Rate (BPR + ) is a new scheduler that considers
not only the lengths of throughput-sensitive and delay-sensitive queues, but also the
arrival times of individual packets when it assigns service rates to the queues [38]. We
were motivated by the need to know the form of an optimal scheduler for proportional
delay differentiation, even if it might be to complex to be practical. In our simulation
study, we use BPR + as a benchmark for evaluating BPR performance.
BPR + is based on a fluid traffic model. We provide here a set of definitions that
are used to characterize the fluid traffic model.
We say that a server is busy if there are packets in queues waiting to be
transmitted. The input curve R in (t) is defined as a cumulative amount of traffic that has
entered a queue since the beginning of the current busy period. The output curve R out (t) is
the cumulative amount of traffic that has been transmitted from the queue since the
beginning of the current busy period [39], [40]. An example of input and output curves is
shown in Fig. 8. At every moment, service rate of a queue is equal to the slope of its
output curve. Vertical and horizontal distance between input and output curve are current
backlog and delay, respectively.
Input and output curves for throughput-sensitive class ( R (t)
in
TS and R (t)
out
TS ) and
delay-sensitive class ( R (t)
in
DS and R (t)
out
DS ) are shown in Fig. 9. If TS queue is served with
out
rate R (τ ) at time τ, the average delay of TS backlog d TS can be calculated as a sum of
TS
experienced delay
E
d TS and expected residual delay
31
R
d TS :
E
R
d ( τ ) = d ( τ ) + d ( τ ) . (12)
TS
TS
TS