Opportunistic Scheduling in Cellular Systems in the Presence of ...

1772 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 3, MARCH 2012
ODEs has a unique solution for any finite time. Furthermore,
the solution is bounded for any finite time. Indeed, one has
Define
. One can rewrite the objective
function in the above optimization problem as
(43)
with being the upper bound on function , and hence
From (42), we have
This concludes the proof of the lemma.
APPENDIX D
PROOF OF LEMMA 2
Under the given hypothesis, the BS’s policy from (30) and
(31) would be . From (27), the payoff for mobile 1
under policy is
Hence, the maximum of the objective function in (43) is
achieved under the required constraints by first maximizing
the term (the constraints on this alone will be less strict)
subject to
Define the probability measure
for all possible resulting from and . Also
define
to obtain
, and then maximizing the term (while
ensuring that these variables and the optimal variables from the
previous step jointly satisfy the required constraints)
subject to
By independence, we are interested in the constrained optimization
problem:
If with ,
then clearly by definition of the best strategies ,wehave
(42)
for all , and so on. Assuming condition ,it
is easily seen that (see the first equation at the top of the page).
Further (see the second equation at the top of the page). The
above defines the joint distribution with prescribed
marginals. It is now straightforward to see that the conditional
distribution is indeed . This
completes the proof.
REFERENCES
[1] 1xEV:1x Evolution IS-856 TIA/EIA Standard Airlink Overview, Qualcomm,
Inc., 2001.
[2] High Speed Downlink Packet Access (HSDPA); Overall Description;
Stage 2 Technical Specification 3rd Generation Partnership Project;
Technical Specification Group Radio Access Network, 2008, 3GPP TS
25.308, Release 8, V8.3.0.