1. Introduction

2. Materials and Methods

3. Results

3.1. Description of the proposed scenario

For understand the use of the Shapley value as a strategy for resource optimization under an IBOC broadcasting scheme in FM, a scenario is proposed with twelve (12) nodes or digital broadcasting stations perform IBOC hybrid transmission processes over FM, under an RF channel which for the particular case will be considered to offer a total bandwidth of 1150 kbps, due to conditions that affect the maximum performance of the channel.

Recorded in Table 6 each class of traffic per node, where it can be observed that the total BW required is 1387 kbps, which is higher than the total BW available in the RF channel, which indicates that the operation mode will be under a saturation state. To calculate the value of BW_T it to use the expression given by (7):

$${BW}_T=\frac{1}{T_s}\sum _{k=1}^{N_{sp}}{log}_2\left[1+\frac{{SNR}_k}{\Gamma }\right]$$

(7)

• N_sp: Number of subcarriers
• T_s: Time of an OFDM symbol (T_s = 2.902ms) for the particular case of IBOC FM.
• SNR_k : Signal to Noise Ratio present on the subcarrier k.
• Γ: This is known as SNR gap, which represents the loss in SNR incurred by using a specific discrete coding scheme. At [19] it is suggested that the value of Γ can be calculated for practical purposes by (8):

$$\mathsf{\Gamma }=-\frac{1}{\mathrm{1,6}}ln\left[\frac{{BER}_{obj}}{\mathrm{0,2}}\right]$$

(8)

Where the BER_obj, corresponds to the BER value to be sustained. For the particular case a value of 10⁻⁶. For the proposed scenario we have considered N=12 nodes that make up the RF system in saturated state (BW_T ≤ ∑^N_i=1 d_i) and a total available Bit-rate BW_T = E = 1150kbps

Table 6. BW required at each node according to service mode.

Table 6. BW required at each node according to service mode.

Node i	Service mode	Logic channels BW required [kbps].
		P1	P2	P3	P4	PIDS
1	MP1	98
2	MP2	98		12
3	MP3	98		25		1
4	MP3	98		25
5	MP5	25	74	25		1
6	MP1	98
7	MP4	98		25	25	1
8	MP6	50	49			1
9	MP2	98		12
10	MP5	25	74	25		1
11	MP5	25	74	25		1
12	MP6	50	49			1

As mentioned above, a bankrupt set will be considered consistent with the saturation state of the RF channel, in order to calculate the transferable utility value for each of the coalitions. The algorithm to perform the calculation of each of the transferable utility values is as follows:

Algorithm: CalculateCoalitions

1. Initialize Z as a vector containing numbers from 1 to Nj.2. Set n_coal to 0. 3. For each coalition size i from 1 to Nj:   a. Increment n_coal by n choose i, where n is Nj. 4. Initialize M_Coalitions as a matrix of zeros with dimensions (n_coal x Nj). 5. Initialize c (counter) to 0. 6. For each coalition size i from 1 to Nj:   a. Generate all combinations S of Z taken i at a time.   b. For each combination S:     i. Increment the counter.     ii. Initialize Suma_d to 0.     iii. For each member k in S:         - Suma_d += V(k)         - M_Coalitions(c, k) = k     iv. Calculate Suma_dT as BW_T minus (Total_V minus Suma_d).     v. Set V_Coalition(c) to the maximum of 0 and Suma_dT. End of Algorithm.

Table 7 presents the transferable utility value calculated for each of the possible coalitions in the saturation state, under the use of the bankruptcy game. The values v(S) consider that the principle of rational individuality must be satisfied (N, v) and the principle of efficiency $\sum _{i\in S}{\phi }_i\left(v\right)\ge v\left(S\right) \forall S\subseteq N,\sum _{i\in N}{\phi }_i\left(v\right)=v\left(N\right)$.

Therefore, the equations that describe each one of the imputations for the game are (N, v) are:

$$\begin{gathered} {\phi }_1\left(v\right)\ge 0 \\ {\phi }_2\left(v\right)\ge 0 \\ {\phi }_3\left(v\right)\ge 0 \\ {\phi }_4\left(v\right)\ge 0 \\ {\phi }_5\left(v\right)\ge 0 \\ {\phi }_6\left(v\right)\ge 0 \\ {\phi }_7\left(v\right)\ge 0 \\ {\phi }_8\left(v\right)\ge 0 \\ {\phi }_9\left(v\right)\ge 0 \\ {\phi }_{10}\left(v\right)\ge 0 \\ {\phi }_{11}\left(v\right)\ge 0 \\ {\phi }_{12}\left(v\right)\ge 0 \end{gathered}$$

(9)

The efficiency equation for the proposed scenario is:

$${\phi }_1\left(v\right)+{\phi }_2\left(v\right)+{\phi }_3\left(v\right)+{\phi }_4\left(v\right)+{\phi }_5\left(v\right)+{\phi }_6\left(v\right)+{\phi }_7\left(v\right)+{\phi }_8\left(v\right)+{\phi }_9\left(v\right)+{\phi }_{10}\left(v\right)+{\phi }_{11}\left(v\right)+{\phi }_{12}\left(v\right)=1150{\ast 10}^3$$

(10)

Based on the above expressions, the core of the game would be as follows:

$$C\left(v\right)=\left\{\begin{array}{c}\phi \left(v\right)\in {\mathbb{R}}^n|\sum _{i\in S}{\phi }_i\left(v\right)\ge v\left(S\right) \forall S\subseteq N, {\phi }_1\left(v\right)+{\phi }_2\left(v\right)+{\phi }_3\left(v\right)+{\phi }_4\left(v\right)\\ +{\phi }_5\left(v\right)+{\phi }_6\left(v\right)+{\phi }_7\left(v\right)+{\phi }_8\left(v\right)+{\phi }_9\left(v\right)+{\phi }_{10}\left(v\right)+{\phi }_{11}\left(v\right)+{\phi }_{12}\left(v\right)=v\left(N\right) \end{array}\right\}$$

(11)

The algorithm to calculate the Shapley value φ_k(v) ∀k ∈ N is as follows:

Algorithm: ShapleyValueCalculation

1. For each player k in [1, Nj]:   a. For each coalition size lg in [1, Nj]:     i. Initialize nMk to 0.     ii. For each coalition i in [1, n_coal]:        - For each player j in [1, Nj]:           * If M_Coalitions(i, j) is equal to k:             - Set N_ceros to 0.             - For each player g in [1, Nj]:                + If M_Coalitions(i, g) > 0:                   * Increment N_ceros by 1.             - If N_ceros is equal to lg:                * Increment nMk by 1.                * Copy the coalition M_Coalitions(i, :) to Mk_Coalitions(nMk, :).                * Copy the coalition value V_Coalition(i) to Vk_Coalition(nMk).                * Break the loop for j.     iii. Initialize Sub_coal as a matrix of zeros with dimensions (nMk, Nj).     iv. For each coalition i in [1, nMk]:        - Set j to 0.        - For each player g in [1, Nj]:           * If Mk_Coalitions(i, g) > 0 and Mk_Coalitions(i, g) is not equal to k:             + Increment j by 1.             + Set Sub_coal(i, j) to Mk_Coalitions(i, g).     v. For each coalition i in [1, nMk]:        - If lg is equal to 1:           * Set Vk_Sub_coal(i) to 0.        - Else:           * For each coalition j in [1, n_coal]:             + If Sub_coal(i, :) is equal to M_Coalitions(j, :):                * Set Vk_Sub_coal(i) to V_Coalition(j).     vi. Set M_Shapley(k, lg) to the sum of (Vk_Coalition - Vk_Sub_coal).     vii. Clear Vk_Coalition, Vk_Sub_coal, Mk_Coalitions, and Sub_coal. 2. For each player S in [1, Nj]:   - Set B(S) to (factorial(S-1) * factorial(Nj - S)) / factorial(Nj). 3. For each player i in [1, Nj]:   - Set Z to M_Shapley(i, :) element-wise multiplied by B.   - Set Peso(i) to the sum of Z.

Table 8 shows the result corresponding to the Shapley matrix in the saturation state for the proposed scenario. The last column shows the Shapley value calculated for each of the players. Additionally, it can be seen that $\sum _{i\in N}{\phi }_i\left(v\right)=v\left(N\right)={BW}_T$

Table 9 shows the values for each node corresponding to the bandwidth requested and the bandwidth assigned, considering that the total bandwidth demanded by the RF network is greater than the total bandwidth available in the channel and therefore, the values assigned for each node should be lower than the demanded value. The use of the Shapley value allocates in an equitable and proportional manner to the needs of each node, favoring the use of the Shapley value as a resource optimization strategy for an IBOC FM broadcasting system.

4. Conclusions

References

1. Maxson DP. The IBOC Handbook : Understanding HD Radio Technology. Focal Press; 2007. Accessed August 24, 2023. Available online: http://www.sciencedirect.com:5070/book/9780240808444/the-iboc-handbook.
2. Peyla PJ, Kroeger BW. New Service Modes for HD RadioTM Transmission. 2018 IEEE Broadcast Symp BTS 2018. Published online November 27, 2018. [CrossRef]
3. Zhuang Y, Cai C, Hu F, Hu F. A modified data rate limiting method in FM IBOC broadcasting based on dynamic spectrum allocation in real time. Int Conf Commun Technol Proceedings, ICCT. 2018;2017-October:1301-1307. [CrossRef]
4. Pérez J, Jimeno J, Tena E. Teoría de Juegos. 1 ed. Pearson-Prentice Hall; 2003.
5. Peleg B, Sudhölter P. Introduction to the Theory of Cooperative Games. Springer; 2007.
6. O'Neill, B. A problem of rights arbitration from the Talmud. Math. Soc. Sci. 1982, 2, 345–371. [Google Scholar] [CrossRef]
7. Moulin, H. Axiomatic Cost and Surplus-Sharing, Chapter 17. In: The Handbook of Social Choice and Welfare. ; 2001.
8. Aumann, R.J.; Maschler, M. Game theoretic analysis of a bankruptcy problem from the Talmud. J. Econ. Theory 1985, 36, 195–213. [Google Scholar] [CrossRef]
9. Curiel, I. Cooperative Game Theory and Applications: Cooperative Games Arising from Combinatorial Optimization Problems. Kluwer Academic Publishers; 1997.
10. Herrero, C.; Maschler, M.; Villar, A. Individual rights and collective responsibility: the rights–egalitarian solution. Math. Soc. Sci. 1999, 37, 59–77. [Google Scholar] [CrossRef]
11. Thomson, W. Axiomatic and Game-Theoretic Analysis of Bankruptcy and Taxation Problems: A Survey.; 2003.
12. Rodríguez, M. Contribuciones a La Teoría Del Valor En Juegos En Forma Estratégica y En Problemas de Bancarrota. (Científico S de P e I, ed.). Universidad Santiago de Compostela; 2005.
13. Shapley, L.S. “A value for n-persons games ” in Contributions to the Theory of Games II. Ann Math Stud. 1953, 307–317. [Google Scholar]
14. Magaña, A. Formación de Coaliciones En Los Juegos Cooperativos y Juegos Con Múltiples Alternativas. Thesis PhD. Universidad Politécnica de Cataluña; 1996.
15. Vesga-Ferreira, J.C.; Granados-Acuna, G.; Sierra-Carrillo, J.E. The Nucleolus as a Strategy for Resources Optimization in LANs on Power Line Communications. Indian J. Sci. Technol. 2016, 9, 1–14. [Google Scholar] [CrossRef]
16. Hu F, Cai C, Hu F, Zhuang Y. A modified real time dynamic spectrum adjustment scheme in FM IBOC broadcasting. Proc 2017 6th Int Conf Comput Sci Netw Technol ICCSNT 2017. 2018;2018-January:458-462. [CrossRef]
17. Dolgopyatova A V., Varlamov O V. Analysis of long-range VHF radio waves propagation to specify protection ratios between coexisting DRM+, RAVIS and IBOC systems. 2021 Syst Signal Synchronization, Gener Process Telecommun SYNCHROINFO 2021 - Conf Proc. Published online June 30, 2021. [CrossRef]
18. Fang W, Li J, Cai C, Sun R. Analysis of symbol mapping based on BICM-ID for FM IBOC. ICCASM 2010 - 2010 Int Conf Comput Appl Syst Model Proc. 2010;10. [CrossRef]
19. Cortés, J.A. Modulation and Multiple Access Techniques for Indoor Broadband Power-Line Communications. Universidad de Málaga; 2007.
20. Ferreira, J.C.V.; Acuna, G.G.; Barrera, J.A.V. Optimización del ancho de banda en redes BPL usando las técnicas nucleolus y max-min fairness. Rev Ing Univ Medellín. 2019, 18, 165–180. [Google Scholar] [CrossRef]