Estimation of MVNO Type Distribution

After the MNO receives all the selections R⁽ⁱ⁾ at the end of the i^th round, let the total

number of MVNOs who chose the pair ^{* *} ~) ⁽⁾

by the MNO initially. We use Pearson's chi-square test [14] to determine if the distribution of the types of MVNOs based on R⁽ⁱ⁾ differs from the MNO’s current estimate. Let the value

of the test-statistic is





22 

frequency fits the estimated distribution and the MNO can proceed to solve the capacity

constraint problem. Otherwise, the MNO will use the observed data to re-estimate the

distribution of the types of the currently participating MVNOs.²

5.2 Estimation of F

⁽ⁱ⁾

(θ) from R

⁽ⁱ⁾

To derive an estimate of F⁽ⁱ⁾(θ) from R⁽ⁱ⁾, the maximum likelihood estimation (MLE)

method [15] is used. We first make an estimate of the underlying statistical model Fˆ⁽ⁱ⁾().

By differentiating the likelihood function with respect to each parameter and equating it to 0,

we can obtain the estimates of the parameters that govern the statistical model.

Since only the MVNOs that participated in the previous round are allowed to remain in

       

2  Note that to use the chi-square goodness of fit test properly, the expected frequency in each category should be at least five. If any frequency is less than 5, it should be combined with an adjacent category. However, combining categories may have unintended consequences, e.g., there may only be one category left.

23 

the interactive trading process, we know that the number of MVNOs participating in each

round will not increase as the process continues. This ensures that the population of

participating MVNOs will converge after a finite number of iterations.

Two‐phase Open Dynamic Bandwidth Trading  Process

1

Ground Knowledge: p(b; θ), F(θ), c

MNO

charge MVNOs subject to finite capacity constraint

Figure 2. Flowchart of the open dynamic bandwidth trading process

5.3 Capacity Constraint

According to the Pearson’s chi-square test when the observed frequency of the types of

the participating MVNOs fits the estimated distribution, the first phase terminates. The

bandwidth trading process proceeds to the second phase to resolve the finite capacity

24 

constraint. Note that the previous derivation of the price schedule was based on the

assumption that the amount of unused bandwidth for sale was infinite. However, in practice,

it is typically finite. Therefore, if the total bandwidth requested is greater than the total

capacity B, we need to decide how to allocate the bandwidth to the buyer MVNOs given the

finite capacity constraint. Otherwise, each MVNO who participated in the final round of the

phase one will be satisfied with the amount as stated in the submission; and the MNO will

charge them based on the discrete price schedule published in the final round.

If it is not possible to satisfy all the requests because of the finite capacity constraint,

we map the bandwidth allocation problem to the bounded knapsack problem, where the size

of the knapsack is the spectrum capacity B and the items are the MVNOs that remain in the

final round. The weights and values of the items are the quantities and returns in the final

discrete price schedule S_dis^*(final)= { ~)

Fig. 2 shows the flowchart of the trading model and the interaction between the MNO and

25 

the MVNOs.

5.4 Example

In this section, we provide an example to illustrate the key components of the proposed

open dynamic bandwidth trading process. First, we assume that the distribution of the types

of MVNOs follows a uniform distribution in the range [0, 1], i.e., F(θ) = θ, and the

demand price function is as follows:



Let the distribution of MVNOs that will join the open trading process follow a triangular

distribution F(θ | 0, 1, ω = 0.9), i.e.,

In addition, assume that ten MVNOs will join the process initially. The types of MVNOs are

generated by using the inversion method [16]. First, we randomly generate ten values from

the uniform distribution U(0, 1) denoted by Ui, i = 1, …10, and substitute them in the

following equation to obtain the types of MVNOs that follow the triangular distribution:



The distribution of the ten participating MVNOs is shown in Fig. 3.

26 

First, let us consider the case where the MNO has perfect information about the

MVNOs and its initial estimate of the type distribution function is exactly the true

distribution, i.e., a uniform distribution in the range [0, 1], which means F(θ) = θ. Note that

in the proposed process the MNO re-estimates the distribution of the types of the

participating MVNOs in each round based on the selection submissions regardless of the

initial assumption about the type of distribution. Here, it is assumed that the marginal cost c

is fixed (c = 10).

A. Optimal Price Schedule

Based on (3), (12), and (13), we first compute the I(b; θ) of the expected return as

27 

Figure 3. Distribution of the ten MVNOs that join the open dynamic bandwidth trading process initially

Then, we compute the optimal bandwidth allocation function b^*(θ)



Based on (20) and (21), the optimal price function T^*(θ) is derived as follows

500

In addition, the continuous optimal price schedule is computed as follows:



B. Discrete Price Schedule

Assume the size K of the discrete tabular price schedule is six. Using the method

described in Section 4.3, we derive the discrete price schedule from the continuous optimal

price schedule. The resulting discrete tabular price schedule is shown in Table I.

28 

TABLE I. T^HE D^ISCRETE P^RICE SCHEDULE AND T^HE T^YPE SUBINTERVALS IN T^HE F^IRST R^OUND

Bandwidth Price ^~_k k_a kb



0.0 0.00 0.5000 0.0000 0.5500

4.0 76.00 0.6000 0.5500 0.6375

7.0 127.75 0.6750 0.6375 0.7125

10.0 175.00 0.7500 0.7125 0.8000

14.0 231.00 0.8500 0.8000 0.9000

18.0 279.00 0.9500 0.9000 1.0000

C. Quantity-Price Selection

Based on the IR and IC constraints, each MVNO chooses the quantity-price pair that

maximizes its utility, as given in (1). For instance, for MVNO6 (θ = 0.72), the utilities

corresponding to the six pairs in the published price schedule table are 0, 13.6, 18.55, 19,

12.6 and -1.8, respectively. MVNO6 will choose the fourth pair <10, 175>, which is exactly

the one that the MNO designs for the types of MVNOs that fall in the range 0.7125 ≤ θ <

0.8. For MVNO1 (θ = 0.06), the utilities are 0, -39.2, -73.85, -113, -168.28, and -216.28

respectively. In this case, the pair <0, 0> will be chosen by MVNO1, which is again the one

designed for it. So do the remaining MVNOs.

D. Hypothesis Testing

We assume that all ten MVNOs respond in the first round. Table II shows the expected

29 

and observed frequencies of the six price pairs. The MNO performs hypothesis testing to

determine if the current estimate of the type distribution function needs to be revised. Here,

the chi-square value computed is 16.26, the degree of freedom is 5, and the critical

chi-square value is 11.07 with p-value = 0.05. Since 16.26 > 11.07, the null hypothesis "The

observed data and the estimated data are from the same distribution," is rejected, and the

type distribution function is re-computed.

TABLE II. The F^REQUNCY TABLE AFTER THE F^IRST R^OUND

Quantity 0 4 7 10 14 18

Price 0 76 127.75 175 231 279

Expected

frequency

5.5 0.875 0.75 0.875 1 1

Observed

frequency

3 0 2 4 0 1

E. Estimation of MVNO Type Distribution

After the computation, the likelihood function is formulated as [F(0.55)F(0)]³

)]2

6375 . 0 ( ) 7125 . 0 (

[F F [F(0.8)F(0.7125)]⁴[F(1)F(0.9)], where F(θ) is the triangular distribution F(θ

| 0, 1, ω). Differentiating the function with respect to c and equating it to 0, we have ω= 0.9.

The new estimate of the distribution is F(θ | 0, 1, ω), ω= 0.9. The MNO then computes the

new optimal price schedule as follows:

30 

The corresponding discrete price schedule as shown in Table III is announced and the

process enters the second round.

In the second round, MVNO1 (θ = 0.06), MVNO2 (θ = 0.37) and MVNO3 (θ = 0.48)

still choose the pair <0, 0>, which means they do not want to purchase any bandwidth.

Assume they therefore decide to leave the process. We also consider the situation where, for

no obvious reason, MVNO7 (θ = 0.73) decides to leave the process as well in this round.

Here, we wish to show that the proposed process and the associated schemes are able to

quickly adapt the estimate of the type distribution function according to the MVNOs who

remain in the process in the computation of the optimal price schedule that maximizes the

MNO’s expected return.

TABLE III. T^HE D^ISCRETE P^RICE SCHEDULE AND T^HE T^YPE SUBINTERVALS IN T^HE S^ECOND R^OUND

Bandwidth Price ^~_k k_a kb



0.0 0.00 0.5477 0.0000 0.5824

4.0 78.59 0.6184 0.5824 0.6571

8.0 147.16 0.6971 0.6571 0.7287

31 

11.0 192.38 0.7609 0.7287 0.7945

14.0 232.55 0.8287 0.7945 0.8641

17.0 267.89 0.9000 0.8641 1.0000

After the second round, six MVNOs remain in the process of the type values 0.65, 0.67,

0.72, 0.74, 0.75, and 0.92. According to the model, the 2^nd round submissions are as follows:

<4, 78.59>, <8, 147.16>, <8, 147.16>, <11, 192.38>, <11, 192.38>, and <17, 267.89>.

Table IV shows the expected and observed frequencies of the 6 sub-intervals.

TABLE IV. FREQUENCY TABLE AFTER THE SECOND ROUND

Quantity 0 4 8 11 14 17

Price 0 78.59 147.16 192.38 232.55 267.89

Expected

frequency

2.26 0.62 0.66 0.67 0.77 1.02

Observed

frequency

0 1 2 2 0 1

After hypothesis testing, the chi-square value is 8.63, which is less than the critical value

11.07. Thus, the observed data and the estimated data are deemed to be from the same

distribution. The iterative part of the process terminates and the process proceeds to the

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