
Task Scheduling with Processing Rate Constraints 241
decisions under this stationary and randomized policy. Thus:
E [ˆe(c
∗
[k],m
∗
[k])]
E
'
I
∗
[k]+
ˆ
D(c
∗
[k],m
∗
[k])
(
= power
opt
(9.15)
E [1
∗
n
[k]]
E
'
I
∗
[k]+
ˆ
D(c
∗
[k],m
∗
[k])
(
≥ λ
n
∀n ∈{1,...,N} (9.16)
where 1
∗
n
[k] is an indicator function that is 1 if c
∗
[k]=n, and 0 else. The
numerator and denominator of (9.15) correspond to those of (9.10). Likewise,
the constraint (9.16) corresponds to (9.11).
9.1.6 Virtual Queues
Figure 9.2: An illustration of the virtual queue Q
n
[k] from equation (9.18).
To solve the problem (9.3)-(9.6), we first consider the constraints (9.4),
which are equivalent to the constraints:
λ
n
(D + I) ≤ 1
n