Spectrum Leasing for Micro-operators Using Blockchain Networks

Abstract

This paper introduces a spectrum sharing system for Micro Operators (MOs) using the blockchain network. In order to satisfy different network requirements for each service, the license for spectrum access should be dynamically allocated to the required spectrum bandwidth. We propose a spectrum lease contract for MOs to share spectrum with the Mobile Network Operator (MNO) is performed through the blockchain networks. Main reasons for applying the blockchain network to the spectrum sharing system are as follow. First, the blockchain networks share database with all participants. Second, networks have mutual trust among all participants. Third, it needs no central authority. Fourth, automated contract execution and transaction interactions are possible. The blockchain usage in the MO-based spectrum sharing system and the detailed process of spectrum lease contract are proposed. Then, the economic effects of spectrum sharing system for MOs is analyzed. The MO can be profitable by getting involved in the blockchain to take reward for a Proof of Work (PoW) and providing wireless service to its users.

Appendix

1.1 A. Proof of Proposition 1

To find the equilibrium price $p_{M}^{*} \left( {p_{s} } \right)$, it is verified that the objective function of (7) should be a concave function of $p_{M} (p_{s} )$. Differentiate (7) is as follow:

$$ \frac{{\partial^{2} \pi^{MO} }}{{p_{M} \left( {p_{s} } \right)^{2} }} = 1 - \frac{{p_{s} }}{{p_{M}^{3} }} < 0, if\,\, p_{M}^{3} < p_{s} . $$

(12)

Concavity of (7) for $p_{M} (p_{s} )$ is guaranteed in $p_{s} \in (p_{M}^{3} ,1]$. The equilibrium price can be obtained by solving the first order derivative of (7) as follows:

$$ \begin{aligned} \frac{{\partial \pi^{MO} }}{{\partial p_{M} \left( {p_{s} } \right)}} = p_{M} \left( {p_{s} } \right) - 1 - \frac{{p_{s} }}{2}\left( {1 - \frac{1}{{p_{M} \left( {p_{s} } \right)^{2} }}} \right) = 0 \hfill \\ 2p_{M} \left( {p_{s} } \right)^{3} - \left( {2 + p_{s} } \right)p_{M} \left( {p_{s} } \right)^{2} + p_{s} = 0 \hfill \\ \end{aligned} $$

(13)

Finally, three candidate solutions to maximize MO’s profit as follows:

$$ p_{M} \left( {p_{s} } \right) = \frac{1}{3}\left( {1 + \frac{{p_{s} }}{2}} \right)\left( {1 + 2cos\left( {\frac{\phi + 2n\pi }{3}} \right)} \right), where\,n \in \left\{ {1,2,3} \right\} . $$

(14)

There is a unique optimal solution when $n = 3$ because $p_{M} (p_{s} )$ is either 1 or negative when $n = 1$ or $n = 2$. ${\blacksquare}$

1.2 B. Proof of Proposition 2

The first order partial derivative of the objective function of (10) respect to $m$ is

$$ \frac{{\partial \pi^{MO} }}{\partial m} = p_{w} + \left( {p_{M} - p_{s} } \right)\left( {\frac{{p_{M} \left( {1 - \gamma } \right)^{2} }}{{\left( {1 - m} \right)^{3} }} - \frac{1 - \gamma }{{\left( {1 - m} \right)^{2} }}} \right) $$

(15)

If the Eq. (15) is greater than $0$ for all $m$, the objective function has the maximum value when $m$ is maximum, that is, when $m = 1$.

If $p_{w} > \frac{1}{{p_{M}^{2} (1 - \gamma )}}$, the Eq. (15) has following relationship:

$$ \frac{{\partial \pi^{MO} }}{\partial m} > \frac{{1 - (p_{M} - p_{s} )}}{{p_{M}^{2} (1 - \gamma )}} + (p_{M} - p_{s} )\left( {\frac{1}{{p_{M}^{2} \left( {1 - \gamma } \right)}} - \frac{1 - \gamma }{{\left( {1 - m} \right)^{2} }}} \right) \cdots (*) $$

(16)

Note that the function $\frac{1 - \gamma }{{\left( {1 - m} \right)^{2} }}$ is an increasing function for $m$. Substitute $m$ with the value $1 - p_{M} (1 - \gamma )$ which is maximum value of $m$ then the Eq. (16) is

$$ \begin{aligned} ( *) & > \frac{{1 - (p_{M} - p_{s} )}}{{p_{M}^{2} (1 - \gamma )}} + (p_{M} - p_{s} )\left( {\frac{1}{{p_{M}^{2} \left( {1 - \gamma } \right)}} - \frac{1 - \gamma }{{\left( {p_{M} \left( {1 - \gamma } \right)} \right)^{2} }}} \right) \\ & = \frac{{1 - \left( {p_{M} - p_{s} } \right)}}{{p_{M}^{2} \left( {1 - \gamma } \right)}} > 0 \\ \end{aligned} $$

(17)

Finally, $\frac{{\partial \pi^{MO} }}{\partial m} > 0$ for all $m$ when $p_{w} > \frac{1}{{p_{M}^{2} \left( {1 - \gamma } \right)}}$. ${\blacksquare}$

1.3 C. Proof of Proposition 3

The first order partial derivative of (10) respect to $m$ is (18).

$$ \begin{aligned} \frac{{\partial \pi^{MO} }}{\partial m} & = p_{w} + \left( {p_{M} - p_{s} } \right)\frac{1 - \gamma }{{\left( {1 - m} \right)^{2} }}\left( {\frac{1 - \gamma }{1 - m}p_{M} - 1} \right) \\ & < \left( {p_{M} - p_{s} } \right)\frac{1 - \gamma }{{\left( {1 - m} \right)^{2} }}\left( {\left( {1 - m} \right)^{2} + \left( {1 - \frac{1}{1 - m}} \right)\frac{1 - \gamma }{1 - m}p_{M} - 1} \right) \cdots (**) \\ \end{aligned} $$

(18)

The Eq. (18) is negative when $m \ge \gamma$ and $0 < m < 1$.

Finally, $\frac{{\partial \pi^{MO} }}{\partial m} < 0$ for all $m$ when $p_{w} < (1 - \gamma )(p_{M} - p_{s} )(1 - p_{M} )$. ${\blacksquare}$