Page 26 - ITUJournal Future and evolving technologies Volume 2 (2021), Issue 1

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

Taking the conditional expectations of (29), we have the neglecting the negative term and taking t → ∞, we have
result as t−1 ( )
1 ∑ ∑ ∑
∑
∗
∗
V E{f (t)|Θ(t)} + E{X i |Θ(t)}(t)(p (t) − γ i ) lim E {X i (t)} + E {Z u (t)} ≤
i t→∞ t
τ=0
i∈N i∈N u∈U
∗
∑ B + V (f (ϵ) − f(t))
+ E{Z u (t)|Θ(t)}(δ u − µ (t)) ≤ . (37)
∗
u
ϵ
u∈U
∑ opt , we obtain the inal result
V E{f(t)|Θ(t)} + E{X i |Θ(t)}(t)(p i (t) − γ i ) Consider that E{f(t)} ≥ f
as
i∈N
∑ t−1 ( )
+ E{Z u (t)|Θ(t)}(δ u − µ u (t)). 1 ∑ ∑ ∑
lim E {X i (t)} + E {Z u (t)} ≤
u∈U t→∞ t
τ=0 i∈N u∈U
∗
B + V (f (ϵ) − f opt )
. (38)
ϵ
D. PROOF OF THEOREM 2
This shows that the queues are strongly stable for ϵ > 0.
Proof. Suppose that a feasible policy ω exists, i.e., con‑
straints (12)b and (12)c are satis ied. Furthermore, as‑
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∗
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