Page 25 - ITUJournal Future and evolving technologies Volume 2 (2021), Issue 1

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

converges in terms of throughput requirements and the takes its maximum value that is bounded. We consider
drop rate increases. the following set‑up:

6. CONCLUSIONS AND FUTURE WORK • The scheduler allocates power at every time slot with
In this work, we considered heterogeneous traf ic with the maximum power level,
two sets of users. The irst set contains users with pack‑
ets with deadlines, and the second contains users with • γ i = P High , ∀i ∈ N,
minimum‑throughput requirements, all with a limited
power budget. We considered the packet drop rate min‑
imization with minimum‑throughput guarantees. A dy‑ • δ u = 1, ∀u ∈ U,
namic algorithm was provided that solves the schedul‑
ing problem in real time. We proved that our scheduling
• λ i = 1, ∀i ∈ N.
scheme provides a solution arbitrarily close to the opti‑
mal. Simulation results show that the proposed algorithm
outperforms the baseline algorithm, LDF, when the dead‑ Then, for the above scheduling scheme, we set B =
∑
∑
2
2
2
2
lines are short, and it is faster in terms of convergence. 1 2 E{p (t)+γ (t)|Θ(t)}+ 1 2 E{δ +µ (t)|Θ(t)}+
i
r
i
r
An interesting direction, for future work, would be the as‑ i∈N r∈R
∑
1 2 2 1 High 2 1
r
sumption that packets of the same user can have different 2 E{α (t)+µ (t)|Θ(t)} = |N +1|(P ) + |R+
r
2
2
r∈R
deadlines, and there are multiple power levels. 1| + |R + 1| < ∞. We observe that even in the scenario
1
2
in which we take the maximum values of γ i , δ u , and λ i , B
7. ACKNOWLEDGMENTS is bounded.
This work has been partially supported by the Swedish
Research Council (VR) and by CENIIT. B. PROOF OF LEMMA 1
Proof. Using the basic sample property [34, Lemma 2.1,
Appendices Chapter 2], we have
t−1 t−1
X i (t) X i (0) 1 ∑ 1 ∑
A. UPPER BOUND ON THE LYAPUNOV t − t ≥ t p i (τ) − t γ i , (27)
DRIFT OF DPC τ=0 τ=0
t−1
t−1
Z u (t) Z u (0) 1 ∑ 1 ∑
2
2
2
2
Using the fact that (max[Q − b, 0] + A) ≤ Q + A + b + t − t ≥ t δ u − t µ u (t). (28)
2Q(A − b), we rewrite (13), (14), as τ=0 τ=0
2
2
2
2
X (t + 1) ≤ X (t) + p + γ + 2X i (t)(p i (t) − γ i ), Therefore, if X i (t) and Z u (t) are rate stable 2 , so that
i
i
i
i
(23) X i (t) → 0, ∀i ∈ N, and Z u (t) → 0, ∀u ∈ U, with probabil‑
t t
2
2
2
2
Z (t + 1) ≤ Z (t) + δ + µ (t) + 2Z u (t)(δ u − µ u (t)), ity 1, then constraints (12)b and (12)c are satis ied with
u
u
u
u
(24) probability 1 [38].
respectively. Rearranging the terms in (23) and (24), di‑
viding them by two, and taking the summations, we ob‑ C. PROOF OF LEMMA 2
tain Proof. Let p(t) represent any, possibly randomized,
2
2
2
∑ X (t + 1) − X (t) ∑ p (t) + γ 2 power allocation decision made at slot t. Suppose that
i i i i
≤ p (t) is the optimal solution to problem (19), and un‑
∗
2 2
∗
∗
i∈N i∈R der action p (t) the value of f i (t) yields f (t) and that of
∑ i
∗
+ X i (t)(p i (t) − γ i ), (25) µ u (t), µ (t). Then, we have
u
i∈N
Z (t + 1) − Z (t) δ + µ (t)
∑ 2 2 ∑ 2 2
u
u
u
u
≤ ∑ ∑
2 2 V f (t) + X i(t)(p (t) − γ i) + Z u(t)(δ u − µ u (t))
∗
∗
∗
u∈U u∈U
∑ i∈N u∈U
+ Z u (t)(δ u − µ u (t)). (26) ∑ ∑
≤ V f(t) + X i(t)(p(t) − γ i) + Z u(t)(δ u − µ u(t)).
u∈U
i∈N u∈U
Taking the conditional expectations in (25) and (26), and (29)
adding them together, we obtain the bound for the Lya‑
punov drift in (18). To prove that B is bounded, we have 2 A discrete time process Q(t) is strongly stable if
to ind an example and a scheduling scheme for which B lim sup t→∞ t 1 ∑ t−1 E{|Q(τ)|} < ∞, [34].
τ=0
© International Telecommunication Union, 2021 9