32 Numerical Results
5.2 AoI Evaluations of Grant-Free Access 33
Fig. 5.4 Illustration of the allocation arrangements in the TDMA with overhead whereNSNs communicate with one FC.
according to the channel state, the amount of power harvested by EH, and the time since the system started.
34 Numerical Results
Overhead = 5, 3, 0
(a) Number of SNs:N=4.
Overhead = 5, 3, 0
(b) Number of SNs: N=7.
Fig. 5.5 Average AoI as a fuction of transmission probability in slotted ALOHA and TDMA with overhead.
Chapter 6 Conclusion
In this thesis, we considered the network where multiple SNs equipped with EH power supplies transmit their data to a common FC, aiming to create a maintenance-free wireless communications system.
Taking into account TDMA and FDMA as possible multiple-access schemes, we have reformulated the AoI minimization problems for both scenarios, analyzing their optimality conditions. Furthermore, we proposed novel resource allocation algorithms for these two schemes to solve the formulated optimization problems by leveraging the aforementioned analyses. We also discussed one of the basic random access scheme, the slotted ALOHA, and presented the formulation of the AoI and the process for data transmission. From the simulation results and previous study in [29], it is necessary to choose TDMA, FDMA or slotted ALOHA separately depending on available resources, size of the data packet, cost of overhead, and the time of packet observation in the system.
Appendix A
Proof of Proposition 1
In this appendix, we prove that equation (3.7) is convex with respect to the number of time slotsnassigned to a given SN by demonstrating that its second derivative is strictly positive under feasible parameter setups.
Indeed, differentiating equation f(n)with respect tonyields the first-order derivative as follows.
d
dnf(n) =g1(n)⇥g2(n), (A.1a) with
g1(n), bn
|h|2
⇣2gn 1⌘
+n, (A.1b)
g2(n), b
|h|2
⇣2gn 1⌘ b
|h|22gnlog2gn+1, (A.1c) where the auxiliary parametersb,BN0/Eandg,D/Bwere introduced in order to simplify the expressions.
Next, taking the second derivative of f(n)yields d2f(n)
dn2 = d
dn(g1(n)⇥g2(n)) =
z }| {0
g2(n) 2 (A.2)
⇥
✓ b
|h|2 2| {z }gn 1
0
+1
◆ b
|h|22gn(log2gn)2,
where, since b >0,g >0,|h|2>0,n>0, it follows that d2dnf(n)2 >0, completing the proof.
⌅
Appendix B
Proof of Proposition 2
Since f(n)is a convex function with respect ton, it possesses a unique minimizer inn2[0,•) located at the point where its first-order derivative is equal to 0. In turn, as per equation (A.1), the minimizer of f(n)must be a root of eitherg1(n)org2(n), with the other bounded at that point.
Suffice it therefore to show that only the functiong2(n)has a root inn2[0,•). To this end, let us first examine the following limits ofg1(n)
nlim!+0g1(n) = lim
n!+0
bn
|h|2 2gn 1
| {z }
!+•
+|{z}n
!+0
= +•, (B.1a)
n!lim+•g1(n) = lim
n!+•
bn
|h|2 2gn 1
| {z }
!+•
+|{z}n
!+•
= +•, (B.1b)
which combined with the fact that the term(2g/n 1) 0, implies that g1(n) is strictly positive inn2R, and bounded within the interval limits.
Next, consider the limits ofg2(n), namely
nlim!+0g2(n) = lim
n!+0
b
|h|2
⇣2gn 1⌘ b
|h|22gnlog2gn+1
= lim
n!+0
b
|h|22gn
| {z }
!+•
⇣1 log2gn⌘
| {z }
! •
b
|h|2+1= •, (B.2a)
38 Proof of Proposition 2 and
n!lim+•g2(n) = lim
n!+•
b
|h|2
h 2gn 1 2gnlog2gni
| {z }
!+0
+1=1, (B.2b)
which indicate thatg2(n)possesses at least one root in the intervaln2[0,•).
Finally, differentiating (A.1c) with respect tonyields d
dng2(n;h) =bg2log2(2)
|h|2n3 2gn >0, (B.3) where the last inequality follows from the fact that for E >0,D>0,B>0, N0>0 and
|h|2>0, which in turn implies thatb ,BN0/E>0 andg ,D/B>0.
Altogether, equations (B.2a), (B.2b) and (B.3) imply that the function g2(n;h) is monotonically increasing function from •to 1 and therefore has a single root inn2R,
completing the proof. ⌅
Appendix C
Proof of Proposition 3
Our objective is to obtain upper and lower bounds to the solution of the equation
g2(n;h),
z }| {
1 b
|h|2+ b
|h|2
⇣2 log2gn⌘
2gn b
|h|22gn =0, (C.1) where we have rewritten the functiong2(n;h)defined in equation (A.1c), in a manner that will prove convenient in the sequel.
Next, consider the following bound
(2 log2gn)2gn e, (C.2)
which is easily proved by definingg3(x),(2 logx)xwithx,2gn and observing that d
dxg3(x)
x=e=0, (C.3a)
d2
dx2g3(x) = 1
x <0,8x 0, (C.3b)
from which it follows immediately that the maximum value ofg3(x)is achieved at the point x=2gn =e, and given by
g3(e) = (2 loge)e=e. (C.4)
Using inequality (C.2) in equation (C.1) readily yields the following functional upper bound tog2(n;h),
g2U(n;h),1+ b
|h|2(e 1) b
|h|22gn. (C.5)
40 Proof of Proposition 3 It is obvious that the upper-bounding functiong2U(n;h)is strictly ascending monotonic on n, such that due to the strictly ascending monotonicity ofg2(n;h)itself, proved in Appendix B, it follows immediately that the root theg2U(n;h)is a lower bound on the root ofg2(n;h), that is
g2U(nL;h) =0 =) n⇤L = g log2⇣
|h|2
b +e 1⌘n⇤. (C.6)
Finally, we note that at the point 2gn =e, where the equalityg2U(n;h) =g2(n;h)holds, we have
g2U(n;h) =1 b
|h|2 >0, 8|h|2>b, (C.7) such that, again due to the monotonically ascending behaviors of both functions, it follows that as long as the condition|h|2>b is satisfied, we have
g2U(nU;h) =g2(nU;h) =) nU⇤ =glog2 n⇤, (C.8) which concludes the proof.
⌅
Appendix D
Proof of Proposition 4
Recall that the lower-boundn⇤L on the optimal transmission time that minimizes the AoI of a given SN is the root of the upper-bounding functiong2U(n;h), defined in equation (C.5), of the functiong2(n;h), defined in equation (C.1).
Directly introducing the condition |h1|2 >|h2|2, and the equivalent relation |h2|2 =
|h1|2 h, withh2R+, the difference betweeng2U(n;h1)andg2U(n;h2)yields
d(n;h1,h2),g2U(n;h1) g2U(n;h2) (D.1)
= hb
|h1|2|h2|2
⇣2gn+1 e⌘ , which is obviously monotonically decreasing onn.
The latter fact, together with the trivial limits
nlim!+0d(n;h1,h2) = +•, (D.2a)
n!lim+•d(n;h1,h2) = +0, (D.2b) implicates thatd(n;h1,h2)>0 inn2R+, and consequently thatg2U(n;h1)>g2U(n;h2).
However, since g2U(n;h1) and g2U(n;h2) are themselves monotonically ascending functions, the above also implicates that the root n⇤1L of g2U(n;h1) is smaller than the rootn⇤2L ofg2U(n;h2), which proves inequality (3.17a).
Finally, we can observe from inequality (C.8) that the upper-bounding transmission time n⇤U is independent of|h|2, such that equation (3.17b) follows trivially, concluding the proof.
⌅
Appendix E
Proof of Proposition 5
Substituting the expression for the lower boundn⇤L given in inequality (C.6) into equation (3.6) we have
k(n⇤L;h) = g
|h|2
|h|2+b(e 2) log2⇣
|h|2
b +e 1⌘. (E.1)
In order to prove implication (3.18a), it is sufficing to verify that for|h1|2>|h2|2>b k(n⇤2L;h2) k(n⇤1L;h1)
= g
|h2|2
|h2|2+b(e 2) log2⇣
|h2|2
b +e 1⌘ g
|h1|2
|h1|2+b(e 2) log2⇣
|h1|2
b +e 1⌘
=
glog2⇣
|h1|2+b(e 1)
|h2|2+b(e 1)
⌘
log2⇣
|h1|2
b +e 1⌘ log2⇣
|h2|2
b +e 1⌘
| {z }
0
+bg(e 2)
|h1|2|h2|2⇥
|h1|2log⇣
|h1|2
b +e 1⌘
|h2|2log⇣
|h2|2
b +e 1⌘ log2⇣
|h1|2
b +e 1⌘ log2⇣
|h2|2
b +e 1⌘
| {z }
0
0.
(E.2)
In turn, substituting the expression for the upper boundn⇤U given in inequality (C.8) into equation (3.6) yields
k(nU⇤;h) = bg
|h|2(e 1), (E.3)
43
from what it follows trivially thatk(n⇤1U;h1)<k(n⇤2L;h2)for|h1|2>|h2|2, which completes the proof.
⌅
Appendix F
Proof of Proposition 6
For convenience, let us first reproduce equations (3.15b) and (3.15d), namely f¯B(n⇤1,n⇤2) =1
2(k(n⇤1) +n⇤1)2+1
2(k(n⇤1) +n⇤1+n⇤2)2, (F.1a) f¯D(n⇤1,n⇤2) =1
2(k(n⇤2) +n⇤2)2+1
2(k(n⇤2) +n⇤2+n⇤1)2, (F.1b) First, with respect to inequality (3.19a), it is readily found from these equations that
f¯D(n⇤1L,n⇤2L) f¯B(n⇤1L,n⇤2L) = (F.2) 1
2{k(n⇤2L) +n⇤2L}2+1
2{k(n⇤2L) +n⇤2L+n⇤1L}2 1
2{k(n⇤1L) +n⇤1L}2 1
2{k(n⇤1L) +n⇤1L+n⇤2L}2 0,
where the last inequality follows directly from inequality (3.17a) in Proposition 4 and inequality (3.18a) in Proposition 5.
As for inequality (3.19b), we again observe that
f¯D(n⇤1U,n⇤2U) f¯B(n⇤1U,n⇤2U) = (F.3) 1
2{k(n⇤2U) +n⇤2U}2+1
2{k(n⇤2U) +n⇤2U+n⇤1U}2 1
2{k(n⇤1U) +n⇤1U}2 1
2{k(n⇤1U) +n⇤1U+n⇤2U}2 0,
where the last inequality follows from equality (3.17b) in Proposition 4 and inequality (3.18b) in Proposition 5, concluding the proof.
⌅
Appendix G
Proof of Proposition 7
Differentiating f(Bi|ni), we obtain d
dBif(Bi|ni) =
✓ aiBini
✓
2BiniDi 1
◆ +ni
◆
(G.1)
⇥
✓ aini
✓
2BiniDi 1
◆
ai2BiniDi log2DiBi
◆ , where we have implicitly definedai,N0/|hi|2Ei.
In turn, the second derivative of f(Bi|ni)is d2
dB2i f(Bi|ni) =
✓ ai2BniDi
✓
ni log2BniDi
◆
+aini+1
◆2
⇥ai2BiniDi log2DiBi
✓2ni
Bi +log2DiBi
◆
(G.2) From the latter equation, it is evident that for ai>0, Di>0,ni>0,ni>0, we have d2f(Bi|ni)/dB2i >0, which indicates that f(Bi|ni) is strictly convex with respect to Bi concluding the proof.
⌅
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