Page 22 - ITUJournal Future and evolving technologies Volume 2 (2021), Issue 1

ITU Journal on Future and Evolving Technologies, Volume 2 (2021), Issue 1

1 ∑ 2 2
where B < ∞ and B ≥ E{p (t) + γ (t)|Θ(t)} +
2 i i Algorithm 1: DPC
i∈N
2
2
1 ∑ 2 2 1 ∑ E{α (t) + µ (t)|Θ(t)}. 1 Input constant V , Initialization:
r
r
2 E{δ + µ (t)|Θ(t)} + 2 r r
r∈R r∈R X i (0) = 0, γ i , ∀i ∈ N, Z u (0) = 0, ∀u ∈ U.
The complete derivation of the above bound can be found 2 for t = 1 : . . . do
in Appendix A.
3 MinObj ← ∞
4 for i = 1 : (|N| + 1) do
4.1 Min‑Drift‑Plus‑Penalty Algorithm 5 p i (t) ∈ P(t), Calculate f r (t), ∀r ∈ R
∑ ∑
6 Obj ← V f r (t) + X r (t)(p i (t) − γ r )
We observe that the power allocation decision at each ∑ r∈R r∈R
time slot does not affect the value of B. The minimum DPC + Z u (t)(δ u − µ i (t))
algorithm observes the virtual queue backlogs of the vir‑ u∈U
7 if MinObj>Obj then
tual queues, the actual queue, and the channel states and
′
8 p (t) ← p(t)
makes a control action to solve the following optimization
9 MinObj ← Obj
problem
′
10 p(t) ← p (t)
∑ ∑
min X i (t)(p i (t) − γ i ) + Z u (t)(δ u − µ u (t)) 11 X j (t+1) ← max [X j (t) − γ j , 0]+p j (t), ∀j ∈ N
p(t) 12 Z u (t + 1) = max [Z u (t) − µ u (t), 0] + δ u , ∀u ∈ U
i∈N r∈R
∑
+ V f r (t) (19)a In step 1, we initialize the importance factor V and the
lengthofvirtualqueuesX i (0), ∀i ∈ N, and Z u (0), ∀u ∈ U.
r∈R
s. t. p(t) ∈ P(t). (19)b We try all the possible power allocations from the set P in
step 4, and we ind the corresponding value of the objec‑
Lemma 2. The optimal solution to problem (19) minimizes tive function in step 6. In step 7, we check if the candidate
the upper bound of the drift‑plus‑penalty expression given power allocation gives a smaller value of the objective so
in the right‑hand‑side of (18). far. In steps 11 − 12, we updated the virtual queues. Af‑
ter the search, we obtain the solution, p (t), for which the
′
objective function takes its minimum value, MinObj.
Proof. See Appendix C.
5. SIMULATION RESULTS
Theorem 2 (Optimality of DPC algorithm and queue sta‑ In this section, we provide results that show the perfor‑
bility). The DPC algorithm guarantees that the virtual and mance of the DPC algorithm regarding the packet drop
the actual queues are strongly stable and therefore, accord‑ rate and the convergence time for the time average con‑
ing to Lemma 1, the time average constraints in (12)b and straints, i.e, throughput, and average power consumption
(12)c are satis ied. In particular, the time average expected constraints. We use a MATLAB environment to perform
value of the queues is bounded as
our simulations. For the time average performance, we
6
run each algorithm for 10 slots.
( )
t−1
1 ∑ ∑ ∑ We irst show the results of a system with two users. Our
lim E {X i (t)} + E {Z u (t)} ≤
t→∞ t goal is to show the performance of DPC for different val‑
τ=0 i∈N u∈U ues of the importance factor V . Second, we compare our
∗
B + V (f (ϵ) − f opt )
. (20) proposed algorithm with a baseline algorithm called LDF
ϵ algorithm proposed in [3].
In addition, the expected time average of function f(t) is
bounded as 5.1 Performance and convergence of DPC al‑
gorithm for different values of V
t−1
1 ∑ B
lim sup E{f(τ)} ≤ f opt + . (21) In Fig. 2, we provide results for a system with two users;
t→∞ t V
τ=0 user 1: deadline‑constrained user, user 2: user with
minimum‑throughput requirements. The average power
thresholds are γ 1 = 0.7 and γ 2 = 0.65 for user 1 and user
2, respectively. The minimum‑throughput requirements
Proof. See Appendix D.
for user 2 is δ 2 = 0.4 packets/slot, and the deadlines of
the packet of user 1 is m = 10 slots.

We summarize the steps of the DPC algorithm that solves The probability of the channel being in “Good” state and
the power control problem in (19) in Algorithm 1. in “Bad” state is 0.4 and 0.6, respectively. The high level
power and the low level power is P High = 2 power units

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