Page 176 - Proceedings of the 2017 ITU Kaleidoscope
P. 176
2017 ITU Kaleidoscope Academic Conference
Finally, the optimal problem can be simplified as follows ∂Y
= 0 ⇔ p I = β I (26)
∂T I
max U w 1
∗
q(θ i ),T (θ i ) q (θ I ) = − (27)
c 1 θ I − c 2 (1 − θ I ) a I
I P
= max p i (T (θ i ) − c 1 θ i q (θ i ) − c 2 (1 − θ i ) q (θ i ))
i=1 If i = 1, the first order condition becomes
w ln 1 + 1 θ 1 q (θ 1 ) + 1 (1 − θ 1 ) q (θ 1 )
d 1 d 2 θ 1 1−θ 1 θ 2 1−θ 2
(β 1 + µ) + +
β 2
∂Y d 1 d 2 d 1 d 2
−T (θ 1 ) = 0
= ω −
w ln 1 + θ i q (θ i ) + (1 − θ i ) q (θ i ) − T (θ i ) q 1 q 1
1 1 ∂q 1 1 + θ 1 + 1−θ 1 1 + θ 2 + 1−θ 2
s.t. d 1 d 2 d 1 d 2 d 1 d 2
1 1
= w ln 1 + θ i q (θ i−1 ) + (1 − θ i ) q (θ i−1 ) = p 1 [c 1 θ 1 + c 2 (1 − θ 1 )]
d 1 d 2
(28)
−T (θ i−1 )
q(θ i ) ≥ q(θ i−1 ), i ∈ {2, · · · , I} ∂Y
(19) = 0 ⇔ p 1 − β 1 − µ + β 2 = 0 (29)
∂T 1
Step 2. Solve the Problem
Thus, we can obtain
Based on the Lagrangian function, we can obtain the follow-
ing function ∆ 1 = [(a 1 + a 2 ) (c 1 θ 1 + c 2 (1 − θ 1 )) − wa 1 a 2 ] p 1 −
2
2
c 1 θ 1 + p 1 (c 1 θ 1 + c 2 (1 − θ 1 )) −
p i (T i − c i θ i q i − c 2 (1 − θ i )q i )+ 4p 1 a 1 a 2
h i c 2 (1 − θ 1 ) w (p 1 + β 2 ) a 1 + wβ 2 a 2
1 1
β i w ln 1 + q i θ i + q i (1 − θ i ) − (30)
I P
Y = d 1 d 2 q (θ 1 ) =
∗
i=1 β i w ln 1 + 1 q i−1 θ i + 1 q i−1 (1 − θ i ) − ( √
d 1 d 2
wp 1 a 1 a 2 −(a 1 +a 2 )p 1 (c 1 θ 1 +c 2 (1−θ 1 ))+ ∆ 1
β i (T i + T i−1 ) 2p 1 (c 1 θ 1 +c 2 (1−θ 1 ))a 1 a 2
, ∆ 1 > 0
h i 0, o.w.
+µ ω ln(1 + 1 q 1 θ 1 + 1 q 1 (1 − θ 1 )) − T 1
(31)
d 1 d 2
(20)
where β i is the lagrangian multiplier corresponding to Local
Downward Incentive Constraints (LDICs) for user θ i . µ is 5. SIMULATION RESULTS
the coefficient of constraints. For 1 < i < I, the first order
condition can be derived by In this section, we carry out simulations to evaluate the pro-
posal by the comparison with two conventional schemes. In
θ i + 1−θ i θ i+1 + 1−θ i+1 Linear Pricing scheme, the operator only specifies a price
∂Y β i d 1 d 2 β i+1 d 1 d 2
=ω − P in the contract for a unit configuration. In Flat-contract
∂q i θ i 1−θ i θ i+1 1−θ i+1
1 + q i + 1 + q i + scheme, each user is provided with the same contract, which
d 1 d 2 d 1 d 2
also satisfies the IR constraint conditions. In the simulation,
− p i [c 1 θ i + c 2 (1 − θ i )] = 0
the modulation parameter w is equal to 2. The loss coeffi-
(21)
cient of caching contents in SBSs is 0.3, while the coeffi-
cient loss of providing contents in CCNs is 0.5. The delay to
∂Y
= p i − β i + β i+1 = 0 (22) achieve a unit content from SBSs is 0.2, while the delay to
∂T i
obtain a unit content from CCNs is 0.4.
Thus, Fig. 2 shows the utilities for operator U compared with Lin-
ear Pricing scheme and Flat-contract scheme. From Fig. 2,
∗
q (θ i ) =
√ the operator can obtain the maximum benefits with the pro-
(
, ∆ i > 0 posed scheme. In the conventional schemes, without pricing
wp i a i a i+1 −(a i +a i+1 )p i (c 1 θ i +c 2 (1−θ i ))+ ∆ i
2p i (c 1 θ i +c 2 (1−θ i ))a i a i+1
0, o.w. incentive scheme for the high type of users, the users’ desires
(23) to access contents from SBSs are reduced. The utilities for
∆ i = operator with these three schemes decrease when the delay
2 2 d 2 increases.
[(a i + a i+1 ) (c 1 θ i + c 2 (1 − θ i )) − wa i a i+1 ] p i − 4p i
c 1 θ i p i (c 1 θ i + c 2 (1 − θ i ))
a i a i+1
+c 2 (1 − θ i ) −wβ i a i + wβ i+1 a i+1 6. CONCLUSION
(24)
where a i = θ i + 1−θ i .
d 1 d 2 In this paper, we have proposed a contract-based caching
If i = I, the first order condition becomes scheme to improve the caching performance in CCNs. First-
ly, a novel model with two-layer heterogeneous wireless net-
θ I + 1−θ I
∂Y β I d 1 d 2 work is proposed to study the interaction between users and
= ω
∂q I θ I + 1−θ I (25) content operator. Secondly, based on the contract theory, the
1 + q I
d 1 d 2 optimal caching and pricing strategy for each user can be ob-
= p I [c 1 θ I − c 2 (1 − θ I )] tained in CCNs. Finally, simulation experiments are carried
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