CN102883294A - Segmental time interval billing method relevant to user behaviors - Google Patents

Abstract

The invention relates to a time-segment billing method associated with user behavior in an operation network, which utilizes the user network utility function and satisfaction function to establish the relationship between the network rate and the operator's revenue, and thereby obtains the operator's obtainable The network revenue limit and the conditions that the network rate should meet at this time. At the same time, through the optimization of the billing mechanism by time period, the operator's optimal segmentation method under different time divisions, the corresponding rate for each segment, and the revenue loss of the network revenue relative to the revenue limit at this time are obtained.

Description

A time-segment billing method associated with user behavior

technical field

The invention belongs to the technical field of network billing in a communication system, in particular to a time-division billing method for associated user behavior in an operating network. the

Background technique

The behavior of users using the network in a day often has time preference, which is the root cause of network traffic fluctuations in time. In recent years, with the rapid development of the mobile Internet, which aims to provide broadband access at any time and any place, the temporal fluctuation of network traffic has become more obvious. On the other hand, the billing mechanism of mobile operators mostly adopts usage-based billing and adopts a fixed rate. Under the effect of this billing mechanism, the network fee paid by the user is proportional to the total network traffic consumed by the user, regardless of when the traffic is used. In the case of limited network resources, the time preference of user network behavior will result in idle network resources idle and busy network congestion. Idle network resources will result in a waste of resources; while network congestion will affect the user's network experience and reduce the user's willingness to pay for network services (WTP). Whether it is idle network resources or network congestion, it will directly affect the network revenue of operators. The time-varying billing mechanism adapts to the user's time preference by dynamically changing the network rate, which largely alleviates the impact of the user's time preference on the operator's network revenue. However, frequent changes in rates will be resisted by users without their full understanding. As a semi-time-varying charging mechanism, time-divided charging can make a good compromise between network revenue and user acceptance, and is expected to be accepted by operators and users. Through the search of existing patents and related technologies, it is found that the existing time-based billing and network user behavior analysis methods include:

(1) A segmented charging method (CN03104823.4) realizes segmented charging by querying a preset temporary database table. A method of real-time segmented charging (CN03116111.1) provides a processing method for segmented charging when a user's use process spans several charging periods. The above two methods provide a specific implementation plan for segmented charging, but do not provide a method for determining the optimal segmented mode and the rate of each segment. the

(2) A real-time billing method and system (CN200910243211.7) predicts the user's bandwidth demand through the user's historical use records, and then controls the user's behavior of using the network according to the balance information of the user's account. It only formulates corresponding network service policies based on the user's network behavior, and does not involve billing policies. An intelligent pipe flow control method based on user behavior (CN201210010674.0) dynamically adjusts flow and billing control strategies according to changes in service characteristics, thereby solving the problem of uneven distribution of wireless network resources and flow to a certain extent. Neither of the above two methods aims at improving the operator's revenue. Most of the existing segmented billing methods only provide a simple implementation method, and do not give a criterion for formulating a segmented billing mechanism, and the methods of analyzing and utilizing user network behavior are mostly limited to controlling user behavior and There is no way to use the user's network behavior to formulate a segment charging strategy to maximize the operator's network revenue. the

Contents of the invention

The demand for mobile Internet access has shown a trend of rapid growth in recent years, which is both an opportunity and a challenge for mobile operators. On the one hand, the huge demand for mobile broadband access has created a huge industry, and on the other hand, it has also brought great challenges to operators' access networks. How to use the billing mechanism to guide users' network behavior and improve network revenue without increasing network resources and operating costs has become the focus of operators' attention. The present invention utilizes the action mechanism of the network rate on user behavior to determine the network revenue that the operator can obtain by using the time-varying charging mechanism and the time-segmented charging mechanism, and provides the operator's determination of the maximum cost when using the segmented charging mechanism. The optimal segmentation method and the corresponding rate method. At the same time, by comparing the income when using the time-varying charging mechanism and the time-based charging mechanism, the operator's revenue loss caused by the time-based charging mechanism can be obtained; the operator can comprehensively consider the revenue loss through this method And the user's acceptance of the time-division charging mechanism to select the optimal number of time-divisions, method of division and the rate corresponding to each period. the

The technical solution of the present invention to solve the above-mentioned problem is a time-segmented billing method for associating user behavior, which is characterized in that, in the mobile phone operating network, the billing platform uses the following means to associate mobile phone terminal users with network billing data:

(1) Obtain the relationship between user consumption bandwidth and network rate and the relationship between network rate and operator revenue through the network utility function and user satisfaction function S(x);

(2) Process the user's network behavior data in the basic billing cycle through the mechanism of the network rate on user behavior to obtain user network behavior characteristic parameters;

(3) Under the specific user behavior characteristics, obtain the optimal rate in the basic period and the operator's network revenue limit

(4) Obtain the network income, segmentation method and the rate corresponding to each time period of the time-segment billing mechanism through optimization when different segments are used;

(5) Under different segment numbers, compare the network revenue obtained by optimizing the time-based billing mechanism with the revenue limit R _lim to obtain the corresponding revenue loss rate η _mL .

Compared with existing technology, the beneficial effect of the present invention is:

Based on the effect mechanism of network rate on user behavior and the effect mechanism of network rate on operator revenue, an optimization criterion for formulating a time-based billing mechanism is provided for operators; this criterion can operate under different segments The merchant's network revenue is maximized. the

By comparing the revenue loss when the operator adopts the time-varying billing mechanism and the time-segmented billing mechanism, a method for estimating the revenue loss caused by the operator's adoption of the time-segmented billing mechanism is given. the

最大网络收益 

以及收益极限 

所述步骤(4)中运营商采用分时段计费机制时，通过解优化问题max(R′+R″+...)；变量：{p′，p″，...，n₁，n₂，...}获得最优的分段方式、对应的费率以及网络收益率。  In order to illustrate the embodiments of the present invention more clearly, the related diagrams and formula derivation needed to be used in the embodiments are briefly introduced below: in the described step (1), by calculating the satisfaction function S(x) of the user and solve the equation S'(x)=0 to obtain the relationship between the bandwidth that the user chooses to consume and the network rate and the relationship between the network rate and the operator's revenue; Select x(p _a ), x(p _b ) for the bandwidth under the rate p _a , p _b to obtain the user's network behavior characteristic parameters; in the step (3), through the relationship between the operator's network revenue and the rate R(p) derivation, and solve the equation R'(p)=0 to obtain the optimal rate

Maximum Network Gain

and profit limit

In the step (4), when the operator adopts the time-segmented charging mechanism, by solving the optimization problem max(R'+R"+...); variables: {p', p",..., n ₁ , n ₂ ,...} Obtain the optimal segmentation method, corresponding rate and network rate of return.

Apparently, the drawings in the following description are only some embodiments of the present invention, and those skilled in the art can obtain other drawings according to these drawings without any creative effort. the

The accompanying drawings are as follows:

Figure 1 Segmentation method of the two-segment time-segment billing mechanism

Figure 2 The bandwidth that users choose to consume when the fixed rate p _a =0.20 or p _b =0.25

Figure 3 User behavior characteristic parameter sequence Γ{σ _i ,η _i }

Figure 4 The optimization results of the time-varying billing mechanism

Figure 5. Optimization results of time-based billing mechanism

Figure 6 The impact of network rate changes on network revenue

Figure 7 Two-stage billing period distribution

Figure 8 The influence curve of network rate on revenue

Figure 9 is the growth rate curve of network revenue as the network rate changes. the

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the appended tables in the embodiments of the present invention. Obviously, what is described is only a part of the embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by persons of ordinary skill in the art without making creative efforts fall within the protection scope of the present invention. the

它表明了用户获得的网络效用u与其消费的带宽x间的关系。其中，η为用户带宽需求弹性，1/η越大，带宽需求越大；x₀为校正因子，表示需要QoS保障类业务的基本带宽需求。  1. Select user network utility function

It shows the relationship between the network utility u obtained by users and the bandwidth x they consume. Among them, η is the elasticity of user bandwidth demand, and the larger 1/η is, the greater the bandwidth demand is; x ₀ is a correction factor, indicating the basic bandwidth demand of services requiring QoS guarantee.

2. Obtain the satisfaction function S(x)=σu(x)-px of users using the network; among them, p is the network rate, and σ represents the network utility level, which is used to adjust the network utility and network rate to satisfy users The weight of degree influence. the

3. Take the derivative of S(x) and solve the equation S′(x)=0 to obtain the bandwidth that users with behavior characteristics {σ, η} choose to consume when the network rate is p $\hat{x}\left(p\right)=\frac{1}{\mathrm{\η}}(\ln \frac{p}{\mathrm{\σ\η}}-\ln (1-\frac{p}{\mathrm{\σ\η}}-\sqrt{1-\frac{2p}{\mathrm{\σ\η}}})).$

4. If the bandwidth that the user chooses to consume is x(p _a ) and x(p _b ) when the network rate is fixed at p _a or p _b , set {p _a , x(p _a )} and {p _b , x (p _b )} replace $\hat{x}\left(p\right)=\frac{1}{{\mathrm{\η}}_0}(\ln \frac{p}{{\mathrm{\σ}}_0{\mathrm{\η}}_0}-\ln (1-\frac{p}{{\mathrm{\σ}}_0\mathrm{\σ}{\mathrm{\η}}_0}-\sqrt{1-\frac{2p}{{\mathrm{\σ}}_0\mathrm{\σ}{\mathrm{\η}}_0}}))$ The characteristic parameters of user network behavior {σ ₀ , η ₀ } can be solved immediately in parallel. If the basic billing cycle (one day) is divided into 24 time periods, and the above process is repeated, the behavior characteristic parameter sequence Γ{σ _i , η _i } of the user in the basic cycle can be obtained; i=1, 2,..., twenty four.

最大网络收益 

以及收益极限  $R_{\lim }=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}R\left({\hat{p}}_i\right).$ 5. Under the specific user behavior characteristics {σ ₀ , η ₀ }, obtain the relationship between the operator's network revenue and the rate $R\left(p\right)=\frac{p}{{\mathrm{\η}}_0}(\ln \frac{p}{{\mathrm{\σ}}_0{\mathrm{\η}}_0}-\ln (1-\frac{p}{{\mathrm{\σ}}_0{\mathrm{\η}}_0}-\sqrt{1-\frac{2p}{{\mathrm{\σ}}_0{\mathrm{\η}}_0}}));$ Solve the equation R'(p)=0 to obtain the optimal rate

Maximum Network Gain

and profit limit $R_{\lim }=\underset{i=1}{\overset{\mathrm{no}}{\mathrm{\Σ}}}R\left({\hat{p}}_i\right).$

6. Since the operator charges for multiple basic periods [0, T] connected together. In the two-segment time-segment billing mechanism, the time periods corresponding to the rates p′ and p″ respectively should be {n ₂ +1,...,n,1,2,...,n ₁ }, {n ₁ +1,...,n ₂ }, as shown in Figure 1. The revenue of the operator in two billing periods can be expressed as $R^{\′}=\frac{T}{\mathrm{no}}(\underset{i=1}{\overset{{\mathrm{no}}_1}{\mathrm{\Σ}}}{\mathrm{\σ}}_{i}f\left(\frac{p^{\′}}{{\mathrm{\σ}}_i{\mathrm{\η}}_i}\right)+\underset{i={\mathrm{no}}_2+1}{\overset{\mathrm{no}}{\mathrm{\Σ}}}{\mathrm{\σ}}_{i}f\left(\frac{p^{\′}}{{\mathrm{\σ}}_i{\mathrm{\η}}_i}\right));$ $R^{\′\′}=\frac{T}{\mathrm{no}}\underset{i={\mathrm{no}}_1+1}{\overset{{\mathrm{no}}_2}{\mathrm{\Σ}}}{\mathrm{\σ}}_{i}f\left(\frac{p^{\′\′}}{{\mathrm{\σ}}_i{\mathrm{\η}}_i}\right).$

7. Solve the optimization problem: max(R′+R″), variables: {p′, p″, n ₁ , n ₂ }; the optimization result can be obtained At this point, the network revenue loss is expressed as: ${\mathrm{\η}}_{2L}=1-\frac{\max (R^{\′}+R^{\′\′})}{R},$

8. Similar to processes 3 and 4, when the number of segments m=3, 4, ..., 23 can be obtained, the corresponding optimal segmentation method, corresponding rate and η _3L , η _4L , ... η _23L can be obtained . (Specially, when the number of segments is 24, the period-based charging mechanism is equivalent to the time-varying charging mechanism, which has been obtained previously)

Simulation

1. When the fixed rates are p _a =0.20 and p _b =0.25 respectively, Figure 2 shows the bandwidth x(p _b ) and x(p _b ) consumed by the user, and the characteristic parameters of user behavior can be obtained by using these data Sequence Γ {σ _i , η _i }; i=1, 2, . . . , 24, as shown in FIG. 3 .

以及网络收益极限 $R_{\lim }=0.4477\underset{i=1}{\overset{24}{\mathrm{\Σ}}}{\mathrm{\σ}}_i,$ 如图4所示。  2. For users whose network behavior characteristic parameters are Γ{σ _i , η _i }, through optimization, it can be obtained that in the time-varying billing mechanism, the optimal rate corresponding to the user behavior characteristic parameters is

and network revenue limit $R_{\lim }=0.4477\underset{i=1}{\overset{\mathrm{twenty\; four}}{\mathrm{\Σ}}}{\mathrm{\σ}}_i,$ As shown in Figure 4.

3. Figure 5 shows the optimization results of the time-segment billing mechanism obtained by solving the multi-parameter optimization problem when the user behavior characteristic parameters are Γ{σ _i , η _i }; including the segment method and the rate corresponding to each segment and the corresponding profit loss ratio.

Appendix: Derivation of Formulas in the Text

1 User Network Behavior Modeling

1.1 User utility surplus

Utility is a concept in microeconomics, which is used to describe the degree of satisfaction that a specific quantity of goods or services brings to users. In network economics, the utility function u(x) is commonly used to describe the relationship between user network utility u and its consumption network bandwidth x. It is generally believed that the utility function has the following characteristics in its domain x ∈ [0, +∞]:

1) The network utility is non-negative, that is, u(x)≥0;

2) u(0)＝0, that is, when x→+∞, u(x)→1;

3) The utility function increases monotonously, that is, u′(x)≥0. the

The following utility function model is constructed here:

$\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&eta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&eta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Among them, x ₀ is a correction factor, representing the basic bandwidth demand of QoS guaranteed services; η represents the elasticity of user bandwidth demand, and the larger 1/η is, the greater the user's network bandwidth demand is, and vice versa.

Users meet their needs by using the network, and users also need to pay a certain network fee. The higher the user's network utility, the higher the user's satisfaction with the network; the higher the network fee, the lower the user's satisfaction. For this reason, we use the network utility surplus S to represent the satisfaction of users using the network, the expression is as follows:

S＝σu(x)-P (2)

Among them, P is the network price; σ represents the network utility level, which is used to adjust the weight of network utility and network price impact on user satisfaction. At the same time, the larger σ is, the higher the user is willing to pay for network services (WTP) when obtaining the same amount of network utility. The product ση is used to represent the user's bandwidth efficiency, and it will be proved later that it will directly affect the operator's network revenue. If and only if S≥0, the user will use the network service. In summary, the user's network behavior characteristics Γ can be uniquely determined by the parameter set {σ, η, x ₀ }.

1.2 User network behavior

In order to facilitate the analysis without loss of generality, the following assumptions are first made about the user's bandwidth requirements:

1) The user's behavior is affected by three factors, the user's network demand, the network fee that the user is willing to pay, and the network price;

2) The network adopts a linear billing mechanism, that is, P=px, where p is the network rate;

3) Only consider services that do not require QoS guarantee, that is, x ₀ =0, at this time, the user's network behavior characteristic Γ parameter group is simplified to {σ, η};

4) Users choose to consume a certain amount of bandwidth to maximize their satisfaction, that is,

$\underset{\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \}</span>}{\mathrm{<span\; class="notranslate">\; max</span>}}\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; arg</span>}<span\; class="notranslate">\; \{</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; \}</span>}{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&sigma;u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; px</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; )</span>$

应满足  According to the previous assumptions, it can be proved that S(x) is a convex function. Then, the bandwidth that users with specific network behavior characteristics Γ choose to consume

should meet

$S^{\&prime;}\left(\hat{x}\right)=0$ Γ{σ,η} (4)

If σ=σ0 and η=η ₀ , solving equation (4) can get:

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; p</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 5</span>}<span\; class="notranslate">\; )</span>$

If p=p ₀ , solve equation (4) to get:

$\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\sqrt{{\mathrm{<span\; class="notranslate">\; b</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; \&sigma;\&eta;</span>}}{{\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Equation (5) will be used to explore the impact of network tariffs on operator revenue, and Equation (6) will be used to explore the impact of user network behavior on operator revenue. User behavior characteristics Γ and bandwidth requirements can be established through formulas (5) and (6) As long as the bandwidth demand x _t of a certain user is given when the network rate is p; {σ _t , η _t }; t ∈ [0, T].

2 Time-varying billing mechanism optimization

2.1 Analysis of Operator Network Revenue

将a＝p/σ₀η₀及式(5)带入可得  Consider that the operator adopts a linear billing mechanism, that is, the network service fee paid by the user is proportional to its network traffic. For users with behavior characteristics Γ{σ ₀ , η ₀ }, the operator's network revenue can be expressed as

Substituting a=p/σ ₀ η ₀ and formula (5) into

$\mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; p</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

使得运营商获得最大收益。  According to the description of formula (7), Figure 8 shows the relationship between the network rate p and the network revenue R. It can be seen from the figure that as the network fee rate p increases, the network revenue R first increases and then decreases, and R will reach the maximum value during this period. At the same time, it can also be concluded that there is an optimal rate

Make the operator get the maximum benefit.

Derivation of formula (7) can get

${\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; +</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

时，网络取得最大收益：  Solve the equation R'(p)=0 to get

When , the network obtains the maximum benefit:

max R(p)＝0.4477σ ₀ (9)

At this time, the bandwidth consumed by the user is

$\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1.5433</span>}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 10</span>}<span\; class="notranslate">\; )</span>$

附近R′(p)变化缓慢，网络费率p的变化对运营商网络收益影响不大。例如，运营商要维持网络收益低于5％的收益损失，网络费率的变化范围只需要满足 即可。  According to the description of formula (8), Figure 9 shows the relationship between the revenue growth rate R'(p) and the network rate p. It can be seen from the figure that at the optimal rate

The nearby R′(p) changes slowly, and the change of the network rate p has little effect on the operator's network revenue. For example, if an operator wants to maintain a revenue loss of less than 5% of the network revenue, the range of network rate changes only needs to meet That's it.

The growth rate of network revenue with the change of network rate

下运营商获得收益的归一化差值。图6给出了费率相对偏移 

与R_L之间的关系。考虑到网络成本的因素，运营商在维持相同的收益损失的情况下，定价可以适当高于 

从而节省网络资源降低网络成本。  Define the operator's network revenue loss rate _RL as the non-optimal rate and the optimal rate

The normalized difference of revenue earned by the lower operator. Figure 6 shows the relative shift in rates

Relationship with _RL . Taking into account the factors of network cost, the operator can set the price appropriately higher than that while maintaining the same revenue loss

In this way, network resources are saved and network costs are reduced.

For the convenience of the following expression, let

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; -</span>\sqrt{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>$ $\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; p</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 11</span>}<span\; class="notranslate">\; )</span>$

At this point R(a) can be recorded as

R(a)＝σ ₀ f(a) $a=\frac{p}{{\mathrm{\&sigma;}}_0{\mathrm{\&eta;}}_0}\&Greater\; Equal;1/2---\left(12\right)$

i＝1，2，...，n，此时运营商的收益表示为  Divide the basic billing period [0, T] into n segments equally, and use i=1, 2, ..., n to number each segment. Assuming that users have different network behavior characteristics Γ{σ _i , η _i } in each segment; i=1, 2,..., n; then the optimal rate corresponding to each segment is

i=1, 2,..., n, the revenue of the operator at this time is expressed as

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; lim</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}\mathrm{<span\; class="notranslate">\; 0.4477</span>}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

来实现。网络费率变化对网络收益的影响如图6所示。  Equation (13) is the operator’s revenue limit under the network behavior characteristics Γ{σ _i , η _i }; i=1, 2, ..., n, it will pass to each period i in the basic cycle Set a corresponding rate

to fulfill. The impact of network rate changes on network revenue is shown in Figure 6.

3.2 Revenue loss of time-invariant billing mechanism

的相对偏移可表示为 它满足  If the operator adopts the fixed rate p of the constant billing mechanism, then the fixed rate is relative to the optimal rate

The relative offset of can be expressed as it satisfies

$<span\; class="notranslate">\; (</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; /</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; min</span>}<span\; class="notranslate">\; \{</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; max</span>}<span\; class="notranslate">\; \{</span>{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 14</span>}<span\; class="notranslate">\; )</span>$

方差越小，也即用户带宽效率{σ_iη_i}方差越小；采取固定费率所造成的网络收益损失也越小。  Comparing Equation (14) with Figure 6, it is possible to roughly estimate the network revenue loss caused by the operator using the time-invariant charging mechanism. It is not difficult to see that the time-varying optimal rate corresponding to user behavior characteristics Γ{σ _i , η _i }

The smaller the variance, that is, the smaller the variance of user bandwidth efficiency {σ _i η _i }; the smaller the loss of network revenue caused by adopting a fixed rate.

使得运营商的网络收益R(p_i)取得最大值。由凸优化理论可知，对于用户行为特征序列Γ{σ_i，η_i}；i＝1，2，...，n也必然存在最优固定费率 

使得运营商在基本计费周期内的收益取得最大值。此时，运营商的收益可表示为：  In 2.1, it has been proved that for specific user behavior parameters σ _i , η _i , there is always an optimal rate corresponding to it

Make the operator's network revenue R(p _i ) achieve the maximum value. According to the convex optimization theory, for the user behavior characteristic sequence Γ {σ _i , η _i }; i=1, 2, ..., n must also have an optimal fixed rate

It enables the operator to obtain the maximum benefit in the basic billing cycle. At this point, the revenue of the operator can be expressed as:

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 15</span>}<span\; class="notranslate">\; )</span>$

$\hat{P}\&Element;\left[0.2901\min \right\{{\mathrm{\&sigma;}}_i{\mathrm{\&eta;}}_i\},$ 0.2901max{σ _i η _i }]

获得的最大收益R_max必然小于收益极限R_lim，通过比较R_max和R_lim便可以计算出采用时不变计费机制的收益损失。因此，采用固定费率的收益损失率可表示为  In the following, a quantitative calculation will be made on the revenue loss when the time-invariant billing mechanism is adopted. Obviously, the operator adopts the optimal fixed rate

The maximum income R _max obtained must be smaller than the income limit R _lim , and the income loss of adopting the time-invariant billing mechanism can be calculated by comparing R _max and R _lim . Therefore, the rate of revenue loss with a fixed rate can be expressed as

${\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; max</span>}}}{{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; lim</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 16</span>}<span\; class="notranslate">\; )</span>$

Put equations (13) and (15) into equation (16) and simplify to get

${\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 0.4477</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 17</span>}<span\; class="notranslate">\; )</span>$

From formula (17), the revenue loss rate η _L of the operator can be calculated when the time-invariant billing mechanism is adopted.

2.3 Revenue loss of semi-time-varying billing mechanism

Because the operator charges for multiple basic periods [0, T] connected together. Under the two-stage semi-time-varying billing mechanism, the time periods corresponding to the tariff rates p′ and p″ respectively should be {n ₂ +1,...,n,1,2,...,n ₁ } , {n ₁ +1, ..., n ₂ }, as shown in Figure 7.

At this time, the network revenue of the operator can be expressed as

${\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; p</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 18</span>}<span\; class="notranslate">\; )</span>$

还要选取最优分段点 

此时的网络收益最大化问题可描述为  In order to maximize revenue, operators not only need to determine the optimal rate

Also select the optimal segmentation point

At this time, the network revenue maximization problem can be described as

max(R′+R″) (19)

Variables: {p′,p″, n ₁ , n ₂ }, $p^{\&prime;},p^{\&prime;\&prime;}\&Element;\left[\min \right\{{\hat{p}}_i\},\max \{{\hat{p}}_i\left\}\right]$

此时，运营商因为采取两段制固定费率的收益损失率可表示为  Simultaneous (18), (19) can get the optimal result

At this time, the revenue loss rate of the operator because of the two-stage fixed rate system can be expressed as

${\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; max</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; R</span>}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 20</span>}<span\; class="notranslate">\; )</span>$

Put max(R′+R″) and R _lim into formula (20) and simplify to get

${\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; L</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\frac{\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{{\mathrm{<span\; class="notranslate">\; no</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\frac{{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; p</span>}}}^{<span\; class="notranslate">\; \&prime;</span><span\; class="notranslate">\; \&prime;</span>}}{{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; \&eta;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; 0.4477</span>}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; twenty\; one</span>}<span\; class="notranslate">\; )</span>$

From formula (21), it can be calculated that for a specific user behavior characteristic Γ {σ _i , η _i }; i=1, 2, ..., n, the revenue loss of the operator due to the two-stage fixed rate .

The calculation method of the revenue loss rate η _mL of the fixed rate greater than two stages is similar to the billing mechanism of the two stage system, and will not be repeated here. It can be predicted that as the number of billing mechanism segments m approaches n, the operator's network revenue loss rate η _mL will approach zero. After obtaining {η _mL }; m=1, 2, ..., n, considering the user's resistance to frequent changes in network prices and the user's perception of network price changes, the operator can adjust the network revenue loss rate according to The requirements to choose the appropriate number of segments.

Claims (5)

1. A time-segment billing method associated with user behavior, characterized in that, in the mobile phone operating network, the billing platform adopts the following means to correlate mobile phone terminal users with network billing data: (1) Through the network utility function and user satisfaction function S(x) to obtain the relationship between user consumption bandwidth and network rate, as well as the relationship between network rate and operator revenue; (2) Process the user's network behavior data in the basic billing cycle through the mechanism of the network rate on user behavior to obtain user network behavior characteristic parameters; (3) Under the specific user behavior characteristics, obtain the optimal rate in the basic period and the operator's network revenue limit $R_{\lim }=\underset{i=1}{\overset{\mathrm{no}}{\mathrm{\&Sigma;}}}R\left({\hat{p}}_i\right);$ (4) Obtain the network income, segmentation method and the rate corresponding to each time period of the time-division charging mechanism through optimization when different number of divisions are used; (5) Under different segment numbers, compare the network revenue obtained by optimizing the time-based billing mechanism with the revenue limit R _lim to obtain the corresponding revenue loss rate η _mL . 2. the time-segment billing method of associating user behavior according to claim 1, is characterized in that, in the described step (1), by deriving the user's satisfaction function S (x), and solving the equation S'( x)=0 to obtain the relationship between the bandwidth that the user chooses to consume and the network rate, and the relationship between the network rate and the operator's revenue. _3. The _time -division billing method associated with user behavior according to claim 1, characterized in that, in the step (2), the user selects x(p _a ), x(p _b ) to obtain the user's network behavior characteristic parameters. 4. The time-segment billing method associated with user behavior according to claim 1, characterized in that, in the step (3), by deriving the relationship R(p) between the operator's network revenue and the rate, and Solve the equation R'(p)=0 to obtain the optimal rate

Maximum Network Gain

and profit limit $R_{\lim }=\underset{i=1}{\overset{\mathrm{no}}{\mathrm{\&Sigma;}}}R\left({\hat{p}}_i\right).$ 5. The time-division charging method associated with user behavior according to claim 1, characterized in that, when the operator adopts the time-division charging mechanism in the step (4), by solving the optimization problem max(R'+R ″+...); variables: {p′, p″, ..., n ₁ , n ₂ , ...} to obtain the optimal segmentation method, corresponding rate and network rate of return.

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