Summary

Conclusion

In order to meet a tremendous increase in mobile data traffic and reduce en-ergy consumption, power saving schemes in mobile networks are essential. In this thesis, we focused on power saving schemes for C-RAN, which is the most promising architecture to realize the future mobile networks. The main compo-nents of C-RAN are RRHs and BBUs, which are separately deployed but are deeply connected because BBUs do baseband processing and allocate radio re-sources to RRHs. However, there are only a few studies that consider power saving schemes of both RRHs and BBUs simultaneously. To realize power sav-ing in the whole C-RAN, a new power savsav-ing scheme considersav-ing both RRHs and BBUs is needed. Therefore, we provide a novel power saving scheme in C-RAN and show the evaluation results with an intensive and appropriate analysis.

Precisely, our contributions are summarized as follows.

In Chapter 2, the overview of C-RAN was introduced. Additionally, the ex-isting power saving schemes in C-RAN were described. Moreover, the problems of existing power saving schemes in C-RAN were presented. In this chapter, we made it clear that considering both RRHs and BBUs power saving schemes is needed to reduce the total power consumption of C-RAN.

model of the system to propose a scheme that reduces the total power consump-tion of C-RAN. The theoretical model based on a stochastic method with traffic threshold was used to calculate the expected number of active RRHs and BBUs.

Especially, the expected number of BBUs was formulated from the expected num-ber of active RRHs and the amount of traffic load carried by the RRHs, which connects the relationship between RRHs and BBUs. The power consumption of RRHs and BBUs was calculated through the formulated equations to propose an optimal traffic threshold that reduces the total power consumption of C-RAN.

We found the condition to minimize the expected power consumption of C-RAN and proposed a procedure for finding the optimal traffic threshold.

In Chapter 4, we evaluated our theoretical model and the proposed scheme.

In a numerical analysis, we evaluated the expected number of active RRHs and BBUs and confirmed that the relationship of power consumption between RRHs and BBUs is a tradeoff. Additionally, we showed that there is a traffic threshold that minimizes the expected power consumption of C-RAN. In a simulation anal-ysis, we compared the proposed optimal traffic threshold and the average value of traffic threshold that minimizes the power consumption of C-RAN in the con-sidered environments. We observed that there is a small difference between the proposed and the simulation average values of optimal traffic threshold. More-over, we showed that our proposed scheme can reduce the power consumption of C-RAN at various traffic load distributions. In addition, our proposed scheme can be applied to time-varying traffics and we introduced a hysteresis margin of traffic threshold to reduce power consumption even further.

This thesis proposed a scheme to reduce the total power consumption of C-RAN. It is considered that the results of this thesis will contribute to the devel-opment of future mobile networks.

In the appendix, we formulate the expected number of mode switches in a subarea as mentioned in Section 4.4.

Let us define events as follows:

• L_t: A subarea is in LC mode at timet.

• S_t: A subarea is in SC mode at timet.

• K^k_t: The mode switch occursk times in a subarea from time 0 to t.

For example,

• If L₀ →L₁ →L₂ or S₀ →S₁ →S₂, then K⁰₂.

• If L₀ →S₁ →S₂ or L₀ →L₁ →S₂

or S₀ →L₁ →L₂ or S₀ →S₁ →L₂, then K¹₂.

• If L₀ →S₁ →L₂ or S₀ →L₁ →S₂, then K²₂. Then, the probabilities are defined as follows:

• P(L_t): The probability that a subarea is in LC mode at time t.

• P(S_t): The probability that a subarea is in SC mode at time t.

• P(K^k_t): The probability that the mode switch occurs k times in a subarea from time 0 to t.

Here, we want to calculate the expected number of mode switches that occur in a subarea from time 0 tot, defined as E_t(k), and it can be written as follows:

t _k

It is simple to calculate P(K_t) if t is small. For example, in case of t = 1, there are two cases for k= 0:

• L₀ →L₁,

• S₀ →S₁,

and two cases for k= 1:

• L₀ →S₁,

• S₀ →L₁.

Thus, we can calculate the probabilities as follows:

P(K⁰₁) = P(L₁|L₀)P(L₀) + P(S₁|S₀)P(S₀), (A.2) P(K¹₁) = P(S₁|L₀)P(L₀) + P(L₁|S₀)P(S₀), (A.3) where P(X|Y) is the conditional probability of X given Y. We explain this in detail as follows:

• P(L_t|L_t−1): The probability that a subarea is in LC mode at time t when the subarea was in LC mode at time t−1. (The blue area to the left in Fig. 4.8 (a))

• P(S_t|L_t−1): The probability that a subarea is in SC mode at time t when the subarea was in LC mode at time t−1. (The purple area to the right in Fig. 4.8 (a))

• P(L_t|S_t−1): The probability that a subarea is in LC mode at time t when the subarea was in SC mode at time t−1. (The blue area to the left in Fig. 4.8 (b))

• P(S_t|S_t−1): The probability that a subarea is in SC mode at time t when the subarea was in SC mode at time t−1. (The purple area to the right in Fig. 4.8 (b))

We can present these four probabilities as integrals of the PDF as follows:

P(L_t|L_t−1) = _T^t

opt+ΔT/2 0

f_t(x)dx, (A.4)

P(S_t|L_t−1) =

_Tmax

T_opt^t +ΔT/2

f_t(x)dx, (A.5)

P(L_t|S_t−1) =

_Topt^t ₋ΔT/2 0

f_t(x)dx, (A.6)

P(S_t|S_t−1) =

_Tmax

T_opt^t −ΔT/2

f_t(x)dx. (A.7)

Therefore, we can rewrite (A.2) and (A.3) using (A.4)–(A.7) as follows:

P(K⁰₁) =

_Topt¹ +ΔT/2 0

f₁(x)dx _Topt⁰

0

f₀(x)dx+ _Tmax

T_opt¹ −ΔT/2

f₁(x)dx _Tmax

T_opt⁰

f₀(x)dx, (A.8)

P(K¹₁) = _Tmax

T_opt¹ +ΔT/2

f₁(x)dx _T⁰

opt

0

f₀(x)dx+ _T¹

opt−ΔT/2 0

f₁(x)dx _Tmax

T_opt⁰

f₀(x)dx.

(A.9) Because P(L₀) and P(S₀) are the probabilities of the initial states of a subarea, there is no ΔT (hysteresis margin) in the integration range of the term.

In the same way, we can calculate P(K^k_t) in case of t= 2, but it is more complicated because of the higher amount of cases. There are two cases ofk = 0:

• L₀ →L₁ →L₂,

• S₀ →S₁ →S₂, four cases ofk = 1:

• L₀ →L₁ →S₂,

• L₀ →S₁ →S₂,

• S₀ →S₁ →L₂, and two cases ofk = 2:

• L₀ →S₁ →L₂,

• S₀ →L₁ →S₂.

Thus, we can calculate the probabilities as follows:

P(K⁰₂) = P(L₂|L₁)P(L₁|L₀)P(L₀) + P(S₂|S₁)P(S₁|S₀)P(S₀), (A.10)

P(K¹₂) = P(S₂|L₁)P(L₁|L₀)P(L₀) + P(S₂|S₁)P(S₁|L₀)P(L₀) + P(L₂|L₁)P(L₁|S₀)P(S₀) + P(L₂|S₁)P(S₁|S₀)P(S₀),

(A.11)

P(K²₂) = P(L₂|S₁)P(S₁|L₀)P(L₀) + P(S₂|L₁)P(L₁|S₀)P(S₀). (A.12) Additionally, we can calculate P(K^k_t) in case of t= 3,4,· · · in the same way, but the number of terms that we should calculate is increasing exponentially in the order of 2^t+1, and it is not easy to divide the cases for each k.

To deal with this problem, we introduce a recurrence relation. Let us define probabilities as follows:

• P(L_t∩K^k_t): The probability that a subarea is in LC mode at time t and the mode switch occurs k times from time 0 to t.

• P(S_t∩K^k_t): The probability that a subarea is in SC mode at time t and the mode switch occurs k times from time 0 to t.

If we know the mode of a subarea att−1 and how many times the mode change occurred, we can calculate the probability mentioned above. For example,

· · · →S_t−1

k−1 times mode change

→L_t. (A.13)

(A.13) means that the mode change from time 0 to t−1 occurs k−1 times and

one time mode change occurs from time t−1 to t. On the other hand,

· · · →L_t−1

ktimes mode change

→L_t. (A.14)

(A.14) means that the mode change from time 0 to t −1 occurs k times but no mode change occurs from time t−1 to t. From (A.13) and (A.14), we can calculate P(L_t∩K^k_t) as follows:

P(L_t∩K^k_t) = P(L_t|S_t−1)P(S_t−1∩K^k−_t−1¹) + P(L_t|L_t−1)P(L_t−1∩K^k_t−1). (A.15) In the same way, P(S_t∩K^k_t) can be calculated as follows:

P(S_t∩K^k_t) = P(S_t|L_t−1)P(L_t−1∩K^k−_t−1¹) + P(S_t|S_t−1)P(S_t−1∩K^k_t−1). (A.16) Note that (A.15) and (A.16) are in case of t≥1 and k≥1. Moreover, if k > t, then K^k_t =∅ because the maximum amount of mode changes is equal to t.

We can rewrite (A.15) and (A.16) including the probability of the initial state and in case ofk = 0 as follows:

P(L_t∩K^k_t)

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

P(L₀) (t= 0 and k= 0),

P(L_t|L_t−1)P(L_t−1|L_t−2)· · ·P(L₁|L₀)P(L₀) (t≥1 and k = 0), P(L_t|S_t−1)P(S_t−1∩K^k−_t−1¹) + P(L_t|L_t−1)P(L_t−1∩K^k_t−1) (t≥1 and k ≥1), (A.17)

P(S_t∩K^k_t)

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

P(S₀) (t= 0 andk = 0),

P(S_t|S_t−1)P(S_t−1|L_t−2)· · ·P(S₁|S₀)P(S₀) (t≥1 and k= 0), P(S_t|L_t−1)P(L_t−1∩K^k−_t−1¹) + P(S_t|S_t−1)P(S_t−1∩K^k_t−1) (t≥1 and k≥1), (A.18)

Because the events L_tand S_tare exclusive, i.e. P(L_t∩S_t) = 0 and P(L_t∪S_t) = 1, we can calculate P(K^k_t) as follows:

P(K^k_t) = P(L_t∩K^k_t) + P(S_t∩K^k_t). (A.19) We evaluate (A.1) and (A.19) with numerical analysis and Monte Carlo sim-ulation. The evaluation parameters are the same as the ones mentioned in Sec-tion 4.4.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0 40 80 120 160 200 240

Probability

Number of Mode Change in a Subarea

ΔT=0%

ΔT=10%

ΔT=20%

Figure A.1: The probability of the number of mode changes in a subarea calcu-lated from (A.19).

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0 40 80 120 160 200 240

Probability

Number of Mode Change in a Subarea

ΔT=0%

ΔT=10%

ΔT=20%

Figure A.2: The probability of the number of mode changes in a subarea simulated through Monte Carlo simulation.

First, we evaluate (A.19). Fig. A.1 and Fig. A.2 show the probability of the number of mode changes in a subarea calculated from (A.19) and simulated through a Monte Carlo simulation, respectively. As shown in the two figures, the overall shape of the graph is similar. Moreover, the probability of having few mode changes is increased when ΔT increases, which means the number of mode changes decreases as ΔT increases.

0 50 100 150 200

0 5 10 15 20 25 30 35 40

Number of Mode Change in a Subarea

ΔT [%]

Simulation Result Expected Value

Figure A.3: The number of mode changes in a subarea according simulation results and numerical expected values.

80 100 120 140 160 180 200

0 2 4 6 8 10

Number of Mode Change in a Subarea

Δ T [%]

Simulation Result Expected Value

Figure A.4: Zoomed in ΔT range from 0 to 10 of Fig. A.3.

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

30 32 34 36 38 40

Number of Mode Change in a Subarea

ΔT [%]

Simulation Result Expected Value

Figure A.5: Zoomed in ΔT range from 30 to 40 of Fig. A.3.

Next, we evaluate (A.1). Fig. A.3 shows the number of mode changes in a subarea for our Monte Carlo simulation and numerical expected value, and Fig. A.4 and Fig. A.5 show zoomed in ΔT ranges from 0 to 10 and from 30 to 40 of Fig. A.3, respectively. As shown in the figures, our formulated equation, (A.19), is fits well the simulation result. Although it seems that the result are not perfectly the same and expected value is always a little higher as shown in Fig. A.5, the difference to the average value is smaller than 0.3, which means that it can be ignored when considering a long total time (t_end= 480). Therefore, our stochastic-based formulation is estimates well the number of mode changes in a subarea.

In addition to the above, we are interested in the shape of the graph as shown in Fig. A.1, which has a jagged shape. To know why this happens, we analyze the equations.

To analyze (A.19), we need a general form of the recurrence relation. However, transforming (A.19) to general form directly is difficult because the values of (A.4)–(A.7) are different (f_t(x) is different for each t). To deal with this problem, we assume thatf₀(x) = f₁(x) = f₂(x) = · · ·=f(x). Although the PDF does not change, the traffic load of a subarea can vary with time, thereby the mode change can occur. Then, (A.4)–(A.7) will be as follows:

P(L_t|L_t−1) =· · ·= P(L₁|L₀) =

_Topt+ΔT/2 0

f(x)dx, (A.20)

P(S_t|L_t−1) =· · ·= P(S₁|L₀) =

_Tmax

Topt+ΔT/2

f(x)dx, (A.21)

P(L_t|S_t−1) =· · ·= P(L₁|S₀) =

_Topt−ΔT/2 0

f(x)dx, (A.22)

P(S_t|S_t−1) =· · ·= P(S₁|S₀) =

_Tmax

Topt−ΔT/2

f(x)dx. (A.23) LetA, B, C, D, E be as follows for simplicity:

A =

_Topt−ΔT/2 0

f(x)dx, (A.24)

B =

_Topt+ΔT/2 Topt−ΔT/2

f(x)dx, (A.25)

C =

_Tmax

Topt+ΔT/2

f(x)dx, (A.26)

D = A+B (A.27)

E = B+C. (A.28)

Then, (A.17) and (A.18) can be written as follows:

P(L_t∩K^k_t) =

⎧⎨

⎩

D^tP(L₀) (k= 0),

AP(S_t−1∩K^k−_t−1¹) +DP(L_t−1∩K^k_t−1) (t≥1 and k≥1), (A.29)

P(S_t∩K^k_t) =

⎧⎨

⎩

E^tP(S₀) (k = 0),

CP(L_t−1∩K^k−_t−1¹) +EP(S_t−1∩K^k_t−1) (t≥1 and k ≥1), (A.30) Now, we will write the equations with incrementalk and then find the general form of the recurrence relation from the characteristic of the equations.

k= 0.

P(K⁰_t) = D^tP(L₀) +E^tP(S₀). (A.31) k= 1.

P(K¹_t) = P(L_t∩K¹_t) + P(S_t∩K¹_t)

=AP(S_t−1∩K⁰_t−1)

E^t−1P(S₀)

+DP(L_t−1∩K¹_t−1) +CP(L_t−1∩K⁰_t−1)

D^t−1P(L₀)

+EP(S_t−1∩K¹_t−1)

=AE^t−1P(S₀) +CD^t−1P(L₀) +DP(L_t−1 ∩K¹_t−1) +EP(S_t−1∩K¹_t−1)

=A(E^t−1+DE^t−2)P(S₀) +C(D^t−1+D^t−2E)P(L₀) +D²P(L_t−2 ∩K¹_t−2) +E²P(S_t−2∩K¹_t−2)

=A(E^t−1+DE^t−2+D²E^t−3)P(S₀) +C(D^t−1+D^t−2E+D^t−3E²)P(L₀) +D³P(L_t−3 ∩K¹_t−3) +E³P(S_t−3∩K¹_t−3)

=· · ·.

(A.32)

Ift= 1, then P(K₁) = AP(S₀) +CP(L₀).

Ift= 2, then P(K¹₂) = AP(S₀)(E+D) +CP(L₀)(D+E).

Ift= 3, then P(K¹₃) = AP(S₀)(E²+DE+D²) +CP(L₀)(D²+DE+E²).

...

Ift=t, then P(K¹_t) =AP(S₀)(E^t−1+DE^t−2+· · ·+D^t−2E+D^t−1) +CP(L₀)(D^t−1+ED^t−2+· · ·+E^t−2D+E^t−1).

Therefore,

P(K¹_t) =AP(S₀)

t−1

δ=0

D^δE^t−1−δ+CP(L₀)

t−1

δ=0

D^t−1−δE^δ, (A.33)

and

P(L_t∩K¹_t) =AP(S₀)

t−1

δ=0

D^δE^t−1−δ, (A.34)

P(S_t∩K¹_t) =CP(L₀)

t−1

δ=0

D^t−1−δE^δ. (A.35) Note that

t−1

δ=0

D^δE^t−1−δ =

t−1

δ=0

D^t−1−δE^δ, (A.36) because of the symmetry.

k= 2.

P(K²_t) = P(L_t∩K²_t) + P(S_t∩K²_t)

=AP(S_t−1∩K¹_t−1)

CP(L₀)_t−2

δ=0D^δE^t−2−δ

+DP(L_t−1∩K²_t−1) +CP(L_t−1∩K¹_t−1)

AP(S₀)_t−2

δ=0D^δE^t−2−δ

+EP(S_t−1∩K²_t−1)

=AC{P(L₀) + P(S₀)}

t−2

δ=0

D^δE^t−2−δ+DP(L_t−1 ∩K²_t−1) +EP(S_t−1∩K²_t−1)

=AC

{P(L₀) + P(S₀)}

t−2

δ=0

D^δE^t−2−δ+{DP(L₀) +EP(S₀)}

t−3

δ=0

D^δE^t−3−δ

+D²P(L_t−2 ∩K²_t−2) +E²P(S_t−2∩K²_t−2)

=AC

P(L₀) _t−2

δ=0

D^δE^t−2−δ+D

t−3

δ=0

D^δE^t−3−δ

+ P(S₀) _t−2

δ=0

D^δE^t−2−δ+E

t−3

δ=0

D^δE^t−3−δ

+D²P(L_t−2 ∩K²_t−2) +E²P(S_t−2∩K²_t−2)

=AC

P(L₀) _t−2

δ=0

D^δE^t−2−δ+D

t−3

δ=0

D^δE^t−3−δ+D²

t−4

δ=0

D^δE^t−4−δ+· · ·

+ P(S₀) _t−2

δ=0

D^δE^t−2−δ+E

t−3

δ=0

D^δE^t−3−δ+E²

t−4

δ=0

D^δE^t−4−δ+· · ·

=AC

P(L₀)

t−2

δ=0

D^δ

t−2−δ δ=0

D^δE^t−2−δ

+ P(S₀)

t−2

δ=0

E^δ

t−2−δ δ=0

D^δE^t−2−δ

. (A.37)

Let us simplify (A.37). We expand _δ=0{D _δ=0 D E } as follows:

E^t−2+DE^t−3+· · ·+D^t−3E+D^t−2 +DE^t−3+· · ·+D^t−3E+D^t−2

. .. ...

+D^t−3E+D^t−2 +D^t−2.

(A.38)

Then, the terms of (A.38) can be summed vertically as follows:

E^t−2+ 2DE^t−3+· · ·+ (t−2)D^t−3E+ (t−1)D^t−2

=

t−2

δ=0

(t−1−δ)D^t−2−δE^δ.

(A.39)

In the same way, _t−2

δ=0{E^δ_t−2−δ

δ=0 D^δE^t−2−δ} can be written as follows:

t−2

δ=0

E^δ

t−2−δ δ=0

D^δE^t−2−δ

=

t−2

δ=0

(t−1−δ)D^δE^t−2−δ. (A.40)

Therefore,

P(K²_t) =AC

P(L₀)

t−2

δ=0

(t−1−δ)D^t−2−δE^δ+ P(S₀)

t−2

δ=0

(t−1−δ)D^δE^t−2−δ

, (A.41) and

P(L_t∩K²_t) =ACP(L₀)

t−2

δ=0

(t−1−δ)D^t−2−δE^δ, (A.42)

P(S_t∩K²_t) =ACP(S₀)

t−2

δ=0

(t−1−δ)D^δE^t−2−δ. (A.43)

k= 3.

P(K³_t) = P(L_t∩K³_t) + P(S_t∩K³_t)

=AP(S_t−1∩K²_t−1)

ACP(S₀)_t−3

δ=0(t−2−δ)D^δE^t−3−δ

+DP(L_t−1∩K²_t−1) +CP(L_t−1∩K²_t−1)

ACP(L₀)_t−3

δ=0(t−2−δ)D^t−3−δE^δ

+EP(S_t−1∩K²_t−1)

=A²CP(S₀) _t−3

δ=0

(t−2−δ)D^δE^t−3−δ

+D

t−4

δ=0

(t−3−δ)D^δE^t−4−δ

+D²

t−5

δ=0

(t−4−δ)D^δE^t−5−δ

+· · ·

+D^t−3 0

δ=0

(1−δ)D^δE^−δ

+AC²P(L₀) t−3

δ=0

(t−2−δ)D^t−3−δE^δ

+E

t−4

δ=0

(t−3−δ)D^t−4−δE^δ

+E²

t−5

δ=0

(t−4−δ)D^t−5−δE^δ

+· · ·

+E^t−3 0

δ=0

(1−δ)D^−δE^δ

.

(A.44)

In the same way, we expand the formulas. First, we expand the first term of (A.44) in the bracket.

(t−2)E^t−3 + (t−3)DE^t−4 + · · · + 2D^t−4E + D^t−3 + (t−3)DE^t−4 + · · · + 2D^t−4E + D^t−3

. .. ...

+ 2D^t−4E + D^t−3 + D^t−3.

(A.45)

Then, the terms of (A.45) can be summed vertically as follows:

(t−2)E^t−3+ 2(t−3)DE^t−4 +· · ·+ (t−3)2D^t−4E+ (t−2)D^t−3

=

t−3

δ=0

(δ+ 1)(t−2−δ)D^δE^t−3−δ.

(A.46)

Next, the second term of (A.44) in the bracket is as follows:

(t−2)D^t−3+ 2(t−3)D^t−4E+· · ·+ (t−3)2DE^t−4+ (t−2)E^t−3

=

t−3

δ=0

(δ+ 1)(t−2−δ)D^t−3−δE^δ.

(A.47)

Therefore,

P(L_t∩K³_t) = A²CP(S₀)

t−3

δ=0

(δ+ 1)(t−2−δ)D^δE^t−3−δ, (A.48)

P(S_t∩K³_t) =AC²P(L₀)

t−3

δ=0

(δ+ 1)(t−2−δ)D^t−3−δE^δ. (A.49) We omit writing P(K³_t) because it can be calculated by (A.48) and (A.49).

k= 4.

P(L_t∩K⁴_t) = AP(S_t−1∩K³_t−1)

AC²P(L₀)_t−4

δ=0(δ+1)(t−3−δ)D^t−4−δE^δ

+DP(L_t−1∩K³_t−1)

=A²C²P(L₀)

t−4

δ=0

(δ+ 1)(t−3−δ)D^t−4−δE^δ+DP(L_t−1∩K³_t−1)

=A²C²P(L₀) t−4

δ=0

(δ+ 1)(t−3−δ)D^t−4−δE^δ

+D

t−5

δ=0

(δ+ 1)(t−4−δ)D^t−5−δE^δ

+· · ·

+D^t−4 0

δ=0

(δ+ 1)(1−δ)D^−δE^δ

(A.50) P(S_t∩K⁴_t) = CP(L_t−1∩K³_t−1)

A²CP(S₀)_t−4

δ=0(δ+1)(t−3−δ)D^δE^t−4−δ

+EP(S_t−1 ∩K³_t−1)

=A²C²P(S₀)

t−4

δ=0

(δ+ 1)(t−3−δ)D^δE^t−4−δ+EP(S_t−1∩K³_t−1)

=A²C²P(S₀) _t−4

δ=0

(δ+ 1)(t−3−δ)D^δE^t−4−δ

+E

t−5

δ=0

(δ+ 1)(t−4−δ)D^δE^t−5−δ

+· · ·

+E^t−4 0 δ=0

(δ+ 1)(1−δ)D^δE^−δ

(A.51)

In the same way,

(t−3)D^t−4 + 2(t−4)D^t−5E + · · · + (t−4)2DE^t−5 + (t−3)E^t−4+ (t−4)D^t−4 + 2(t−5)D^t−5E + · · · + (t−4)DE^t−5 +

... ...

2D^t−4 + 2D^t−5E + D^t−4.

(A.52) The vertical summation is

(t−3)(t−2)

2 D^t−4+2(t−4)(t−3)

2 D^t−5E+· · ·+(t−4)2×3

2 DE^t−5+(t−3)1×2 2 E^t−4. (A.53) Therefore,

P(L_t∩K⁴_t) =A²C²P(L₀)

t−4

δ=0

(δ+ 1)(t−3−δ)(t−2−δ)

2 D^t−4−δE^δ, (A.54)

P(S_t∩K⁴_t) =A²C²P(S₀)

t−4

δ=0

(δ+ 1)(t−3−δ)(t−2−δ)

2 D^δE^t−4−δ. (A.55) In the same way, we did these processes to k = 7 and found regularity, and it appears that there are different behaviors when k is odd and when k is even.

Thus, we divide it in two cases,k is odd andk is even.

In case k is odd.

P(L_t∩K¹_t) =AP(S₀)

t−1

δ=0

D^δE^t−1−δ,

P(L_t∩K³_t) =A²CP(S₀)

t−3

δ=0

(δ+ 1)(t−2−δ)D^δE^t−3−δ,

P(L_t∩K⁵_t) =A³C²P(S₀)

t−5

δ=0

(δ+ 1)(δ+ 2) 2

(t−4−δ)(t−3−δ)

2 D^δE^t−5−δ,

P(L_t∩K⁷_t) =A⁴C³P(S₀)

t−7

δ=0

(δ+ 1)(δ+ 2)(δ+ 3) 3×2

(t−6−δ)(t−5−δ)(t−4−δ)

3×2 D^δE^t−7−δ, ...

(A.56)

P(S_t∩K¹_t) = CP(L₀)

t−1

δ=0

D^t−1−δE^δ,

P(S_t∩K³_t) = AC²P(L₀)

t−3

δ=0

(δ+ 1)(t−2−δ)D^t−3−δE^δ,

P(S_t∩K⁵_t) = A²C³P(L₀)

t−5

δ=0

(δ+ 1)(δ+ 2) 2

(t−4−δ)(t−3−δ)

2 D^t−5−δE^δ,

P(S_t∩K⁷_t) = A³C⁴P(L₀)

t−7

δ=0

(δ+ 1)(δ+ 2)(δ+ 3) 3×2

(t−6−δ)(t−5−δ)(t−4−δ)

3×2 D^t−7−δE^δ, ...

(A.57)

In case k is even.

P(L_t∩K²_t) =ACP(L₀)

t−2

δ=0

(t−1−δ)D^t−2−δE^δ,

P(L_t∩K⁴_t) =A²C²P(L₀)

t−4

δ=0

(δ+ 1)(t−3−δ)(t−2−δ)

2 D^t−4−δE^δ,

P(L_t∩K⁶_t) =A³C³P(L₀)

t−6

δ=0

(δ+ 1)(δ+ 2) 2

(t−5−δ)(t−4−δ)(t−3−δ)

3×2 D^t−6−δE^δ, ...

(A.58)

P(S_t∩K²_t) = ACP(S₀)

t−2

δ=0

(t−1−δ)D^δE^t−2−δ,

P(S_t∩K⁴_t) = A²C²P(S₀)

t−4

δ=0

(δ+ 1)(t−3−δ)(t−2−δ)

2 D^δE^t−4−δ,

P(S_t∩K⁶_t) = A³C³P(L₀)

t−6

δ=0

(δ+ 1)(δ+ 2) 2

(t−5−δ)(t−4−δ)(t−3−δ)

3×2 D^δE^t−6−δ, ...

(A.59)

In (A.56)–(A.59), we can find the regularity obviously. Therefore, the general form of the recurrence relation is as follows:

• If k is odd,

P(L_t∩K^k_t) = A^k+12 C^k−12 P(S₀) t−k

δ=0

_k+1

2

δ

k+1 2

t−k−δ

D^δE^t−k−δ, (A.60)

P(S_t∩K^k_t) =A^k−12 C^k+12 P(L₀) t−k

δ=0

_k+1

2

δ

k+1 2

t−k−δ

D^t−k−δE^δ. (A.61)

• If k is even,

P(L_t∩K^k_t) =A^k2C^k2P(L₀) t−k δ=0

_k

2

δ

k 2 + 1 t−k−δ

D^t−k−δE^δ, (A.62)

P(S_t∩K^k_t) =A^k2C^k2P(S₀) t−k

δ=0

_k

2

δ

k 2 + 1 t−k−δ

D^δE^t−k−δ. (A.63) Note that _n

r

is the number of multisets of cardinality r with elements taken from a finite set of cardinality n, and is called multiset coefficient. It is also written as _nH_r. The value of multiset coefficients can be given explicitly as n

r

!!

=

r+n−1 r

=_r+n−1C_r = (r+n−1)!

r!(n−1)! = (r+ 1)(r+ 2)· · ·(r+n−1)

(n−1)! .

(A.64) Now we analyze the general form, i.e. (A.60)–(A.63), to know what they mean. As mentioned in (A.24)–(A.28), A and C mean that the mode change occurs, and D and E mean that the mode change does not occur. Let us see (A.60), for example. If the mode change occurred in an odd timeslot and the mode att is LC mode, the initial state of a subarea should be SC mode. That is why (A.60) has P(S₀). The chance of mode change is equal to t, so if the mode change occurs k times, then there are no mode changes t−k times. Because the mode change occurs on an odd timeslot and the final mode change should be S→L, the number of A is one more than C, and it means that A^k+12 C^k−12 . It

can be shown as follows:

EE· · ·EE

t−k−δ

ACAC· · ·ACA

k

DD· · ·DD

δ

.

Of course, E or D can be inserted between A and C, and the place of insertion is ^k+1₂ . For example, let us say the mode change occurs 5 times. Then,

ACACA.

E should be placed at and D should be place at , and the number of each possible place is 3, which is ^k+1₂ . The number of combination placing E is _k+1

t−k−δ2

!!

and placing D is _k+1

2δ

!!

. Therefore, the total number of combina-tions is

_k+1

2δ

!! _k+1

t−k−δ2

!!

, which is shown in (A.60).

In the same way, we can analyze other equations. From this analysis, we conclude that the general term means that we need to search for all combinations of mode state to calculate the probability of mode change, which is impossible if tis too large. Therefore, to calculate the probability of mode change, it is better to use recurrence relation than the general term.

Finally, we conclude that the reason that the graph of Fig. A.1 has a jagged shape is because the probability of mode change is affected by whether the number of mode changes is odd or even, according to (A.60)–(A.63).

We enclose the permissions that were used to write this dissertation. Please see the attached documents for a detailed description of the permissions.

• Yunseong Lee, Keisuke Miyanabe, Hiroki Nishiyama, Nei Kato, and Takashi Yamada, “Threshold-Based RRH Switching Scheme Considering Baseband Unit Aggregation for Power Saving in a Cloud Radio Access Network,”

IEEE Systems Journal, vol. 13, no. 3, pp. 2676–2687, Sept. 2019.

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Journals

[1] Yunseong Lee, Keisuke Miyanabe, Hiroki Nishiyama, Nei Kato, and Takashi Yamada, “Threshold-Based RRH Switching Scheme Considering Baseband Unit Aggregation for Power Saving in a Cloud Radio Access Network,”

IEEE Systems Journal, vol. 13, no. 3, pp. 2676–2687, Sept. 2019.

[2] Keisuke Miyanabe, Tiago Gama Rodrigues, Yunseong Lee, Hiroki Nishiyama, and Nei Kato, “An Internet of Things Traffic-Based Power Saving Scheme in Cloud-Radio Access Network,” IEEE Internet of Things Journal, vol. 6, no. 2, pp. 3087–3096, Apr. 2019.

Refereed Conference Papers

[3] Yunseong Lee, Katsuya Suto, Hiroki Nishiyama, Nei Kato, Hirotaka Ujikawa, and Ken-Ichi Suzuki, “A Novel Network Design and Operation for Reduc-ing Transmission Power in Cloud Radio Access Network with Power over Fiber,” in Proc. of IEEE/CIC International Conference on Communica-tions in China, pp. 283–287, Shenzhen, China, Nov. 2015. (Best Paper Award)

Non-Refereed Conference Papers

[4] Yunseong Lee, “A Study on the Power Saving Technology in Wireless Sen-sor Networks,” Annual Workshop on A3 Foresight Program, Pyeongchang, Korea, Jul. 2016. (Best Presentation Award)

[5] Yunseong Lee, “A Study on Wireless Access Network Construction Method Considering Power Supply,” Annual Workshop on A3 Foresight Program, Beijing, China, Jul. 2015.

Domestic Conference Papers

[6] Yunseong Lee, Hiroki Nishiyama, and Nei Kato, “Analysis of Power Con-sumption of Cloud-Radio Access Network using an M/M/c Queueing Model,”

IEICE Society Conference 2018, Kanazawa, Japan, Sep. 2018.

[7] Yunseong Lee, Hiroki Nishiyama, and Nei Kato, “A Study on Power Con-sumption of Cloud-Radio Access Network,” Technical Committee on Com-munication Systems (CS), Sendai, Japan, Sep. 2017.

[8] Yunseong Lee, Hiroki Nishiyama, and Nei Kato, “A Study on the Power Saving Technology in Cloud Radio Access Network with Power over Fiber,”

IEICE General Conference 2017, Nagoya, Japan, Mar. 2017.

[9] Yunseong Lee, Hiroki Nishiyama, Nei Kato, Hirotaka Ujikawa, and Ken-Ichi Suzuki, “A Study on the Power Saving Technology of Networks in Smart Factory,” IEICE Society Conference 2016, Sapporo, Japan, Sep. 2016.

[10] Yunseong Lee, Hiroki Nishiyama, and Nei Kato, “A Power Saving Tech-nology for Wireless Sensor Networks,” IEICE General Conference 2016, Fukuoka, Japan, Mar. 2016.

Awards

[11] A3 Foresight Program 2016: Research on Next Generation Networks and Network Security Best Presentation Award.

[12] Best Paper Award of 2015 IEEE/CIC International Conference on Com-munications in China (ICCC 2015).

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