Page 20 - ITUJournal Future and evolving technologies Volume 2 (2021), Issue 1

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

Table 1 – Notation Table.

N Set of the total users in the system P i (t) Set of selectable power levels of
user i
R Set of the deadline‑constrained users Z u (t) Length of throughput‑dept queue
of user u
U Set of the minimum‑throughput requirements users f r (t) Cost function for user r
th
t t slot µ r (t) Power allocation indicator of user
i
Q r (t) Number of packets in queue r P(t) Set of power constraints for p(t)
Packet arrival probability of user i D r (t) Packet drop indicator of user r
π r
α r (t) Packet arrival indicator of user r D r Packet drop rate of user r
Deadline of packet of user r p Average power consumption of
m r
r
user r
d r (t) Number of slots left before the deadline of user r X r (t) Length of virtual queue of user i
S(t) Channel states L(·) Quadratic Lyapunov function
p(t) Power allocation vector ∆(L(·)) Lyapunov drift
Allowed average power consumption for user i α(t) Packet arrival indicator vector
γ i

rate represents the average number of dropped packets not have prior knowledge about the future states of the
per time slot. The average power consumption repre‑ channel and packet arrivals in the system. Therefore, we
sents the average of transmit power over all time slots. are not able to predict the values of the objective function
The throughput represents the average served packets in the future slots in order to decide on the power allo‑
per time slot for each user u ∈ U. cation that minimizes the cost. We aim to design a func‑
These metrics are connected and we will show in the fol‑ tion whose future values are affected by the current deci‑
lowing sections how the average power consumption af‑ sion and the remaining expiration time of the packets. To
fects the packet drop rate and the throughput. this end, we introduce a function incorporating the rela‑
tive difference between the packet deadline m r and the
3. PROBLEM FORMULATION number of remaining future slots (d r (t)−1) before its ex‑
piration as described below
Ourgoalistoachievetheminimumdropratefordeadline‑
constrained users while providing a minimum through‑ m r − (d r (t) − 1)
put for each user u ∈ U and keeping the average power f r (t) , 1 {µ r (t)=0} , ∀r ∈ R. (9)
m r
consumption for every user below a threshold. To this
end, we provide the following stochastic optimization The function in (9) takes its extreme value, f r (t) = 0,
problem
when a packet of user r ∈ R is served, or f r (t) = 1 when
∑ a packet of user r ∈ R is dropped. Therefore, that func‑
min D r (8)a
p(t) tion takes the same values with those of (5) in the extreme
r∈R
cases. In addition, the function in (9) assigns the cost ac‑
s. t. p ≤ γ i , ∀i ∈ N, (8)b cording to the remaining time of a packet to expire in the
i
¯ µ u ≥ δ u , u ∈ U, (8)c intermediate states, i.e., when a packet is waiting in the
p(t) ∈ P(t), (8)d queue. The cost increases when there is less time left for
serving the packet with respect to the de ined deadline.
[ (High) ] The time average of f r (t) is
where γ i ∈ 0, P indicates the allowed average
power consumption for each user i. Also, δ u denotes the
minimum throughput requirement for each user u ∈ U. f r , lim f (t), ∀r ∈ R, (10)
r
t→∞
The constraint in (8)b ensures that average power con‑
sumption of each user i remains below γ i power units. t−1
The formulation above represents our intended goal where f (t) , 1 ∑ f r (τ) and
r t
which is the minimization of the packet drop rate. How‑ τ=0
ever, the objective function in (8)a makes the solution
∑
approach non‑trivial. The decision variable, p(t) (power f = f r (t). (11)
allocation), is optimized slot‑by‑slot for minimization of r∈R
the objective function that is de ined over an in inite time
horizon. We have to cope with one critical point: we do

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