Secondary Spectrum Market Under Supply Uncertainty

Abstract

This chapter studies the optimal investment and pricing decisions of a cognitive mobile virtual network operator (C-MVNO) under spectrum supply uncertainty. Compared with a traditional MVNO who is often stuck in long-term spectrum leasing contract, a C-MVNO can acquire spectrum dynamically in short-term by both sensing the empty “spectrum holes” of licensed bands and dynamically leasing from the spectrum owner. Compared to dynamic spectrum leasing, spectrum sensing is typically cheaper, but the obtained useful spectrum amount is random due to primary licensed users’ stochastic traffic. The C-MVNO needs to determine the optimal amounts of spectrum sensing and leasing by evaluating the trade-off between cost and uncertainty. The C-MVNO also needs to determine the optimal price to sell the spectrum to the secondary unlicensed users, taking into account wireless heterogeneity of users such as different maximum transmission power levels and channel gains. We model and analyze the interactions between the C-MVNO and secondary unlicensed users as a Stackelberg game, and show interesting properties of the network equilibrium, including threshold structures of the optimal investment and pricing decisions.

Appendix

2.1.1 Proof of Theorem 2.1

Given the total bandwidth $B_l+B_s\alpha$, the objective of Stage III is to solve the optimization problem (2.8), i.e., $\max _{\pi \ge 0}\min (D(\pi ),S(\pi ))$. First, by examining the derivative of $D(\pi )$, i.e., ${\partial D(\pi )}/{\partial \pi }=(1-\pi )Ge^{-(1+\pi )},$ we can see that the continuous function $D(\pi )$ is increasing in $\pi \in [0,1]$ and decreasing in $\pi \in [1,+\infty ]$, and $D(\pi )$ is maximized when $\pi =1$. Since $S(\pi )$ always increases in $\pi$ and $D(\pi )$ is concave over $\pi \in [0,1]$, $S(\pi )$ intersects with $D(\pi )$ if and only if $\frac{\partial D(\pi )}{\partial \pi }>\frac{\partial S(\pi )}{\partial \pi }$ at $\pi =0$, i.e., $B_l+B_s\alpha < Ge^{-1}$.

Next we divide our discussion into the intersection case and the non-intersection case:

• 1. Given $B_{l}+B_{s}\alpha \le Ge^{-1}$, $S(\pi )$ intersects with $D(\pi )$. By solving equation $S(\pi )=D(\pi )$ the intersection point is $\pi =\ln \left( \frac{G}{B_{l}+B_{s}\alpha }\right) -1$. There are two subcases:

  • when $B_{l}+B_{s}\alpha \le Ge^{-2}$, $S(\pi )$ intersects with $D(\pi )$, and $\min (D(\pi ),S(\pi ))$ is maximized at the intersection point, i.e., $\pi ^*=\ln \left( \frac{G}{B_{l}+B_{s}\alpha }\right) -1$. (See $S_3(\pi )$ in Fig. 2.3.)

  • when $B_{l}+B_{s}\alpha \ge Ge^{-2}$, $S(\pi )$ intersects with $D(\pi )$, and $\min (D(\pi ),S(\pi ))$ is maximized at the maximum value of $D(\pi )$, i.e., $\pi ^{*}=1$. (See $S_2(\pi )$ in Fig. 2.3.)

• 2. Given $B_{l}+B_{s}\alpha \ge Ge^{-1}$, $S(\pi )$ doesn’t intersect with $D(\pi )$. Then $\min (D(\pi ),S(\pi ))$ is maximized at the maximum value of $D(\pi )$, i.e., $\pi ^{*}=1$. (See $S_1(\pi )$ in Fig. 2.3.)

$\blacksquare$

2.1.2 Proof of Theorem 2.2

Given the sensing result $B_s\alpha$, the objective of Stage II is to solve the decomposed two subproblems (2.10) and (2.11), and select the best one with better optimal performance. Since $R_{III}^{ES}(B_s,\alpha ,B_l)$ in subproblem (2.10) is linearly decreasing in $B_l$, its optimal solution always lies at the lower boundary of the feasible set (i.e., $B_l^*=\max \{Ge^{-2}-B_s\alpha ,0\}$). We compare the optimal profits of two subproblems (i.e., $R_{II}^{ES}(B_s,\alpha )$ and $R_{II}^{CS}(B_s,\alpha )$) for different sensing results:

• 1. Given $B_s\alpha >Ge^{-2}$, the obtained bandwidth after Stage I is already in excessive supply regime. Thus it is optimal not to lease for subproblem (2.10) (i.e., $B_l^{ES3}=0$ of case (ES3) in Table 2.3).

• 2. Given $0\le B_s\alpha \le Ge^{-2}$, the optimal leasing decision for subproblem (2.11) is $B_l^*=Ge^{-2}-B_s\alpha$ and we have $R_{III}^{ES}(B_s,\alpha ,B_l)=R_{III}^{CS}(B_s,\alpha ,B_l)$ when $B_l=Ge^{-2}-B_s\alpha$, thus the optimal objective value of (2.10) is always no larger than that of (2.11) and it is enough to consider the conservative supply regime only. Since

  $$\begin{aligned} \frac{\partial ^2 R_{III}^{CS}(B_s,\alpha ,B_l)}{\partial B_l^2}=-\frac{1}{B_l+B_s\alpha }<0, \end{aligned}$$

  $R_{III}^{CS}(B_s,\alpha ,B_l)$ is concave in $0\le B_l\le G e^{-2}-B_s\alpha$. Thus it is enough to examine the first-order condition

  $$\begin{aligned} \frac{\partial R_{III}^{CS}(B_s,\alpha ,B_l)}{\partial B_l} =\ln \big (\frac{G}{B_l+B_s\alpha }\big )-2-C_l =0, \end{aligned}$$

  and the boundary condition $0\le B_l\le Ge^{-2}-B_s\alpha$. This results in optimal leasing decision $B_l^*=\max (G e^{-(2+C_l)}-B_s\alpha ,0)$ and leads to $B_l^{CS1}=G e^{-(2+C_l)}-B_s\alpha$ and $B_l^{CS2}=0$ of cases (CS1) and (CS2) in Table 2.3.

By substituting $B_l^{CS1}$ and $B_l^{CS2}$ into $R_{III}^{CS}(B_s,\alpha ,B_l)$ in Table 2.2, we derive the corresponding optimal profits $R_{II}^{CS1}(B_s,\alpha )$ and $R_{II}^{CS2}(B_s,\alpha )$ in Table 2.3. $R_{II}^{ES3}(B_s,\alpha )$ can also be obtained by substituting $B_l^{ES3}$ into $R_{III}^{ES}(B_s,\alpha ,B_l)$.$\blacksquare$

2.1.3 Supplementary Proof of Theorem 2.4

In this section, we prove that Observations 3 and 4 hold for the genera case (i.e., the general SNR regime and a general distributions of $\alpha$). We first show that Observation 4 holds for the general case.

2.1.3.1 Threshold Structure of Sensing

It is not difficult to show that if the sensing cost is much larger than the leasing cost, the operator has no incentive to sense but will directly lease. Thus the threshold structure on the sensing decision in Stage I still holds for the general case. We ignore the details due to space limitations.

2.1.3.2 Threshold Structure of Leasing

Next we show the threshold

structure on leasing in Stage II also holds. Similar as in the proof of Theorem 2.1, we define $D(\pi )=\pi \frac{G}{Q(\pi )}$ and $S(\pi )=\pi (B_s\alpha +B_l)$.

• We first show that $D(\pi )$ is increasing when $\pi \in [0,0.468]$ and decreasing when $\pi \in [0.468,+\infty )$. To see this, we take the first-order derivative of $D(\pi )$ over $\pi$,

  $$\begin{aligned} D'(\pi )=\frac{2Q(\pi )^2+Q(\pi )-(1+Q(\pi ))^2\ln (1+Q(\pi ))}{Q(\pi )^3}, \end{aligned}$$

  which is positive when $Q(\pi )\in [0,2.163)$ and negative when $Q(\pi )\in [2.163,+\infty )$. Since Eq. (2.16) shows that $Q(\pi )$ is increasing in $\pi$ and $\pi (Q)\mid _{Q=2.163}=0.468$, as a result $D(\pi )$ is increasing in $\pi \in [0,0.468]$ and decreasing in $\pi \in [0.468,+\infty )$. In other words, $D(\pi )$ is maximized at $\pi =0.468$.

• Next we derive the operator’s optimal pricing decision in Stage III. Figure 2.15 shows two possible intersection cases of $S(\pi )$ and $D(\pi )$. $B_{th1}$ is defined as the total bandwidth obtained in Stages I and II (i.e., $B_{s}\alpha +B_{l}$) such that $S(\pi )$ intersects with $D(\pi )$ at $\pi =0.468$. Here is how the optimal pricing is determined:

  • If $B_s\alpha +B_l\ge B_{th1}$ (e.g., $S_1(\pi )$ in Fig. 2.15), the optimal price is $\pi ^*=0.468$. The total supply is no smaller (and often exceeds) the total demand.

  • If $B_s\alpha +B_l< B_{th1}$ (e.g., $S_2(\pi )$ in Fig. 2.15), the optimal price occurs at the unique intersection point of $S(\pi )$ and $D(\pi )$ (where $D(\pi )$ has a negative first-order derivative). The total supply equals total demand.

• Now we are ready to show the threshold structure of the leasing decision.

  • If the sensing result from Stage I satisfies $B_s\alpha \ge B_{th1}$, then the operator will not lease. This is because leasing will only increase the total cost without increasing the revenue, since the optimal price is fixed at $\pi ^{*}=0.468$ and thus revenue is also fixed at $D(\pi ^{*})$.

  • Let us focus on the case where the sensing result from Stage I satisfies $B_s\alpha < B_{th1}$. Let us define $B=B_s\alpha +B_l$, then we have $B=G/Q(\pi )$ and $\pi =\ln (1+G/B)-G/(G+B)$. This enables us to rewrite $D(\pi )$ as a function of total resource $B$ only,

    $$\begin{aligned} D(B)=B\left[ \ln \left( 1+\frac{G}{B}\right) -\frac{G}{G+B}\right] . \end{aligned}$$

    The first-order derivative of $D(B)$ is

    $$\begin{aligned} D'(B)=\ln \left( 1+\frac{1}{B/G}\right) -\frac{1}{1+B/G}-\frac{1}{(1+B/G)^2}, \end{aligned}$$

    (2.20)

    which denotes the increase of revenue $D(B)$ due to unit increase in bandwidth $B$. Since obtaining each unit bandwidth has a cost of $C_{l}$ in Stage II, the operator will only lease positive amount of bandwidth if and only if $D'(B_{s}\alpha )>C_l$. To facilitate the discussions, we will plot the function of $D'(B/G)$ in Fig. 2.14, with the understanding that $D'(B/G)=D'(B)G$. The intersection point of $B/G=0.462$ in Fig. 2.14 corresponds to the point of $\pi =0.468$ in Fig. 2.15. The positive part of $D'(B)$ on the left side of $B/G=0.462$ in Fig. 2.14 corresponds to the part of $D(\pi )$ with a negative first-order derivative in Fig. 2.15. For any value $C_l$, Fig. 2.14 shows that there exists a unique threshold $B_{th2}(C_{l})$ such that $D'(B_{th2}(C_{l})/G)=C_{l}G$, i.e., $D'(B_{th2}(C_{l}))=C_{l}$. Then the optimal leasing amount will be $B_{th2}(C_{l})-B_{s}\alpha$ if the bandwidth obtained from sensing $B_{s}\alpha$ is less than $B_{th2}(C_{l})$, otherwise it will be zero.

Fig. 2.14

The relation between the normalized total bandwidth $B/G$ and the derivative of the revenue $D'(B/G)$

Fig. 2.15

Different intersection cases of $S(\pi )$ and $D(\pi )$ in the general SNR regime

2.1.3.3 Threshold Structure of Pricing and Observation 3

Based on the proofs above, we show that Observation 3 also holds for the general case as follows. Let us denote the optimal sensing decision as $B_s^*$, and consider two sensing realizations $\alpha _1$ and $\alpha _2$ in time slots 1 and 2, respectively. Without loss of generality, we assume that $\alpha _1<\alpha _2$.

• If $B_s^*\alpha _2\ge B_{th1}$, then the optimal price in time slot 2 is $\pi ^*=0.468$ (see Fig. 2.15). The optimal price in time slot 1 is always no smaller than $0.468$.

• If $B_s^*\alpha _1< B_s^*\alpha _2<B_{th1}$, then we need to consider three subcases:

  • If $B_s^*\alpha _1< B_s^*\alpha _2\le B_{th2}(C_l)$, then the operator will lease up to the threshold in both time slots, i.e., $B_l^*=B_{th2}(C_l)-B_s^*\alpha _1$ in time slot 1 and $B_l^*=B_{th2}(C_l)-B_s^*\alpha _2$ in time slot 2. Then optimal prices in both time slots are the same.

  • If $B_s^*\alpha _1\le B_{th2}(C_l)< B_s^*\alpha _2$, then the operator will lease $B_l^*=B_{th2}(C_l)-B_s^*\alpha _1$ in time slot 1 and will not lease in time slot 2. Thus the total bandwidth in time slot 1 is smaller than that of time slot 2, and the optimal price in time slot 1 is larger.

  • If $B_{th2}(C_l)\le B_s^*\alpha _1< B_s^*\alpha _2$, then the operator in both time slots will not lease and total bandwidth in time slot 1 is smaller, and the optimal price in time slot 1 is larger.

  To summarize, the optimal price $\pi ^*$ in Stage III is non-increasing in $\alpha$. And the operator will charge a constant price ($\pi ^*=0.468$) to the users as long as the total bandwidth obtained through sensing and leasing does not exceed the threshold $B_{th2}(C_l)$.

$\blacksquare$