CN101111070B - Fair resource scheduling method based on resource scheduling in broadband wireless access system - Google Patents

Abstract

The present invention relates to the resource dispatch of a broadband wireless access system based on an IEEE802.16, the present invention is characterized in that the reserved bandwidth required by mutual necessity is calculated according to the Qos parameter of the different business, the admission and control is performed to the newly achieved business according to the reserved bandwidth, and the bandwidth is distributed by the means of a secondary dispatching mode according to bandwidth required by each business and the bandwidth required by a system, the reserved bandwidth is distributed according to each business firstly, the secondary dispatch is performed to the insufficient part of the bandwidth required by the means of the weighted recycle transfer method according to the residual bandwidth after a primary dispatch is performed to the system, the secondary dispatch distributes the different priority factors among the different businesses according to the business priority, meanwhile, the distribution performed according to the actual scale of the bandwidth required by the business residue accounting for the bandwidth required by residue also shall be taken into account. The present invention has the advantage of the equity, the high efficiency, the simplicity and the flexibility on the basis of ensuring the Qos quality.

Description

Fair resource scheduling method based on resource scheduling in broadband wireless access system

technical field

The invention designs a resource scheduling method in an IEEE 802.16 broadband wireless access (BWA) system, and belongs to the technical field of wireless communication.

Background technique

The IEEE 802.16 standard is the air interface specification of the wireless metropolitan area network, which has very important significance and broad application prospects in the future broadband wireless access. The 802.16 standard not only allows non-line-of-sight connections and supports large-capacity users, but also provides carrier-level QoS guarantees and fully supports the application of rich multimedia services such as voice, video, and data.

An effective QoS guarantee system must allocate wireless resources reasonably, especially for the complex environment of wireless communication environment, the QoS system of the wireless access system faces many challenges. The 802.16 standard implements a series of mechanisms at the MAC layer. The resource scheduling algorithm is an indispensable link in the QoS guarantee system of the broadband wireless access system. However, the 802.16 standard only proposes that the base station or the user perform resource scheduling according to the QoS request and priority of each service, without specifying a specific resource scheduling algorithm. However, the existing resource scheduling algorithms do not consider the fairness of the algorithm. When the high-priority business load is heavy, it is easy to cause the low-priority business to "starve to death" without bandwidth for a long time. This algorithm uses a certain balance mechanism to allocate resources reasonably on the premise of first ensuring the service quality of each business, so as to improve the fairness of the algorithm.

The IEEE 802.16 system has four different service levels of services, namely unsolicited grant service (unsolicited grant service, UGS), real-time polling service (real-time polling service, rtPS), non-real-time polling service (non-real-time polling service, nrtPS), best effort service (best effort service, BE).

The existing resource scheduling algorithm is a scheduling algorithm based on strict priority, and its specific implementation process is as follows:

First-level scheduling algorithm: among services of different priority types, the SPQ (strict priority queue: strict priority queue) strategy is adopted to satisfy the bandwidth requests of high-priority services first, and then when there is remaining bandwidth. Satisfy the bandwidth requests of low-priority services in turn;

Two-level scheduling algorithm: between services of the same priority type, different bandwidth allocation strategies are adopted according to the characteristics of the services: for rtPS services, the EDF (earliest deadline first) strategy is adopted; for nrtPS services, the WFQ (weighted fair queue) strategy is adopted ; For BE business, adopt RR (round robin) strategy or equal bandwidth strategy.

Because the first-level scheduling uses the SPQ policy, when high-priority services are heavily loaded, it is easy to cause high-priority services to occupy too many resources of low-priority services, so that low-priority services cannot obtain the necessary bandwidth. In the case of "starving to death", it cannot meet the service quality requirements of low-priority services, and loses fairness for low-priority services;

Resource reservation means that the system reserves a part of resources (such as bandwidth, buffer or processor capacity, etc.) during service access to ensure the QoS requirements of the service. Based on the idea of resource reservation, when a service is accessed, a part of its bandwidth is reserved, and the remaining bandwidth is flexibly and fairly allocated among various services according to a certain algorithm, so that the service can be effectively guaranteed. The QoS requirement also realizes the fairness of bandwidth allocation among various services.

Invention content:

The purpose of the present invention is to propose a fair resource scheduling algorithm based on resource reservation applied in the IEEE 802.16 broadband wireless access system, which can reduce the excessive occupation of high-priority services and low-priority services on the premise of ensuring the service quality of each business. In the case of high-level service bandwidth, fairness is improved and better system performance is obtained.

The characteristic of the present invention is that following steps are arranged successively:

Step (1), when a new service arrives, the base station based on the IEEE802.16 broadband wireless access system calculates the reserved bandwidth satisfying different QoS parameters according to the QoS parameters of different services, which is set to r _res , which includes the The bandwidth reserved for establishing a connection and the reserved bandwidth required for a newly established connection. If ∑r _res ≤ r _total , access is allowed, otherwise, access is denied, and r _total is the total bandwidth of the system;

For non-application grant services, r _res = r _max , where r _max is the maximum input data rate, a set value;

For best-effort traffic, r _res = 0;

For real-time polling services, $r_{\mathrm{res}}=\frac{r_{\max }+r_{\min }}{2}{P}_0{e}^{-\mathrm{\λ}(t-T_S)},$

Among them, r _min is the minimum input data rate, the set value;

P ₀ is the packet loss rate threshold, a set value;

λ is the arrival rate of packets implementing the polling service, a known value;

is the maximum delay, and T _S is the time of one frame, both of which are set values;

For non-real-time polling service, r _res = r _min ;

Step (2), the base station allocates the system bandwidth periodically according to the bandwidth required by each service and the existing bandwidth of the system according to the media access control layer mechanism stipulated in IEEE802.16, and sends the allocation result in the downlink subframe Each user station is notified in the media access control information UL-MAP:

Step 1: Level 1 Scheduling

First, the reserved bandwidth of the system is used to meet the bandwidth requests of services with priority from high to low. If the bandwidth r _req required by the service is smaller than the reserved bandwidth, then the bandwidth r _{allo_1} of the first-level scheduling = r _req , otherwise, r _{allo_1} = r _res , the business with the highest priority refers to non-application grant business, and the rest are real-time polling, non-real-time polling and best-effort business;

Step Two: Secondary Scheduling

Then allocate the remaining bandwidth after the first-level scheduling to each service according to the weighted cyclic transfer method, and the weight factor of each service is: k _i =λ _i ·d _i ;

Among them, _λi is the ratio of the remaining required bandwidth of each service to the total remaining required bandwidth: namely ${\mathrm{\&lambda;}}_i=(r_{\mathrm{req},i}-r_{\mathrm{res},i})/\underset{j=1}{\overset{\mathrm{no}}{\mathrm{\&Sigma;}}}(r_{\mathrm{req},j}-r_{\mathrm{res},j}),$ If r _{req, i} < r _{res, i} , let r _{req, i} - r _{res, i} = 0;

j=1, 2,..., n, is the number of services requiring allocation of remaining bandwidth,

d _i is a priority factor, set value, the priority factor of real-time business should be greater than the priority factor of non-real-time business, should set different priority factors for each non-real-time business;

The bandwidth obtained by the second-level scheduling is: r _{allo_2} = (r _total -∑r _{allo_1} ) · k _i /∑ k _i ,

After two-level allocation, the bandwidth obtained by each service is r _allo =r _{allo_1} +r _{allo_2} .

This scheduling method based on resource reservation has the following advantages:

1. QoS guarantee: the strategy of allocating reserved bandwidth in admission control and first-level scheduling, so that the QoS parameters of the service can be guaranteed;

2. Fairness: The WRR algorithm is adopted in the second-level scheduling to improve the fairness of scheduling;

3. Efficiency: When calculating the reserved bandwidth r _res of the rtPS service, the time delay requirement of the rtPS service is also taken into account instead of the maximum data input rate r _max as the judgment standard, so under the condition of guaranteeing QoS, it reduces The waste of bandwidth allows access to more services, thereby improving bandwidth utilization;

4. Simplicity: Compared with algorithms such as deficit fair priority queue DFPQ, this algorithm is more intuitive, and it is easier to understand and operate;

5. Flexibility: Different priority factors in the WRR algorithm can be set for different services according to the load and other conditions, which reflects the flexibility of the algorithm.

Description of drawings

Figure 1. QoS guarantee system of IEEE 802.16 system;

Figure 2. The flow chart of the existing resource scheduling algorithm;

Fig. 3. The flow chart of the resource scheduling algorithm proposed by the present invention;

Figure 4. Changes in the bandwidth obtained by each service when the rtPS service grows;

Figure 5. Changes in the bandwidth obtained by each service when the nrtPS service grows;

Fig. 6. Comparison chart of fairness index before and after adopting the present invention.

Detailed ways

In order to realize the technical problem to be solved by the present invention, the present invention adopts a resource scheduling algorithm that introduces the idea of resource reservation and combines it with an admission control algorithm, including the following steps:

(1) When applying for access to each service, the subscriber station SS reports the QoS parameters required by the service to the base station BS, and the BS determines the reserved bandwidth required by the service according to these parameters. If the sum of the reserved bandwidth of the online application for access service is less than the total bandwidth of the system, the service access is allowed, otherwise the access is refused;

(2) The BS dynamically allocates the system bandwidth before the start of each frame, and the allocation method is:

First-level scheduling: allocate reserved bandwidth to each business first, so that the service quality of each business is guaranteed;

Second-level scheduling: The remaining bandwidth of the system is allocated among various services according to the WRR algorithm, and the weight factor of WRR is determined by the bandwidth required by the service and the characteristics of the service itself;

Specific embodiments of the present invention will be further described below.

Figure 3 shows the flow chart of the new algorithm, which is described in detail as follows:

(1) After the arrival of a new service, first calculate the reserved bandwidth of the service according to the QoS parameters of the service

Our algorithm is: when a service requires to establish a connection, the BS calculates the reserved bandwidth to meet these parameters according to its QoS parameters, and set it as r _res , if all the bandwidth reserved for the established connection plus the bandwidth for the new connection If the sum of the required reserved bandwidth is less than the total bandwidth r _total , that is, ∑r _res ≤ r _total , access is allowed. r _res is calculated as follows:

1. For UGS services, since all QoS parameters need to be satisfied, r _res = r _max ;

2. For the rtPS service, r _res should be a value between r _min and r _max . When the BS provides the bandwidth of this service r _res , the delay and packet loss rate of this service can meet the requirements; Calculated by a certain mathematical method, it can also be measured according to the actual system;

3. For nrtPS services, since there is no delay requirement, r _res = r _min ;

4. For the BE service, since there is no Qos requirement, r _res =0;

(2) According to the MAC layer mechanism specified in the IEEE 802.16 system, the BS will periodically allocate the system bandwidth according to the bandwidth required by each service and the existing bandwidth of the system, and record the allocation result in the MAC control information UL_MAP of the downlink subframe Notify each SS;

The bandwidth allocation method is:

a. First-level scheduling:

First allocate the reserved bandwidth to each business in order of priority from high to low, but if the bandwidth r _req required by the business is less than the reserved bandwidth r _res , then only the reserved bandwidth r _res needs to be allocated; that is, if r _req < r _res , then the bandwidth r _{allo_1} = r _req obtained by level one scheduling; otherwise r _{allo_1} = r _res ;

b. Secondary scheduling:

Allocate the remaining bandwidth after the first-level scheduling to each service according to the WRR algorithm;

The weighting factors for each business are:

k _i =λ _i · d _i ;

Among them, _λi is the ratio of the remaining required bandwidth of each service to the total remaining required bandwidth: namely ${\mathrm{\&lambda;}}_i=(r_{\mathrm{req},i}-r_{\mathrm{res},i})/\underset{j=1}{\overset{\mathrm{no}}{\mathrm{\&Sigma;}}}(r_{\mathrm{req},j}-r_{\mathrm{res},j})$ (If r _{req, i} <r _{res, i} , let r _{req, i} -r _{res, i} = 0; r _{req, i} and r _{res, i} respectively represent the required bandwidth and reserved bandwidth of the i-th business; r _{req, j} and r _{res, j} represent the required bandwidth and reserved bandwidth of the jth business respectively)

d _i is the priority factor, the specific value can be determined according to the actual business volume of each business flow in the system, and the following two principles should be met at the same time:

1. The priority factor of real-time services should be greater than that of non-real-time services to reduce the probability of packet loss due to delay exceeding QoS requirements;

2. For non-real-time services, different priority factors can also be set for different services, so as to ensure that services with high priority can obtain greater bandwidth.

The bandwidth obtained by the second-level scheduling is: r _{allo_2} = (r _total -∑r _{allo_1} )·k _i /∑k _i

Therefore, after two-level allocation, the bandwidth obtained by each service is: r _{allo, i} = r _{allo_1, i} + r _{allo_2, i} , where r _{allo_1, i} is the bandwidth obtained by the i-th service through one-level scheduling, r _{allo_2, i} is the bandwidth obtained by the i-th service through two-level scheduling.

In order to compare the effects before and after the implementation of the present invention, the algorithm of the present invention has been simulated, and the simulation parameters are as follows: total bandwidth B=10Mbps, frame length t _f =10ms

business type average bandwidth Maximum bandwidth Minimum bandwidth Reserved bandwidth UGS \ 1.6Mbps \ 1.6Mbps rtPS 4.3Mbps 5.2Mbps 3.4Mbps 4Mbps nrtPS 2.8Mbps 3.6Mbps 2Mbps 2Mbps BE \ \ 1.6Mbps 0

Fig. 4 has shown the condition of the bandwidth obtained by each service flow under the condition that the rtPS service volume increases. Here, we call the original algorithm SPQ (Strict Priority Queue Algorithm), and the new algorithm RFPQ (Reserved Fair Priority Queue Algorithm). It can be seen from the figure that when the total traffic volume does not exceed the total system bandwidth, the results obtained by the two allocation schemes are the same; when the rtPS traffic volume reaches a certain level, using the existing algorithm, the rtPS business will occupy the bandwidth of the BE business, resulting in BE The business decreases linearly, so that the QoS requirement cannot be met in the end; and in the present invention, the bandwidth obtained by the rtPS service no longer increases linearly, and its growth trend slows down; while the bandwidth of the nrtPS and BE services decreases slowly. Compared with the existing algorithm, the bandwidth allocation of the three services tends to be balanced, so that even if the traffic of the rtPS service is heavy, the system can accommodate more low-priority services under the premise of ensuring QoS.

Fig. 4 has shown the situation of the bandwidth obtained by each business flow under the condition that the nrtPS business volume increases. It can be seen from the figure that using the existing algorithm, the increase of nrtPS services will not affect the rtPS services, but will cause a significant drop in BE services; using this algorithm, when the nrtPS traffic increases, the bandwidth obtained by rtPS will decrease slightly, However, within an acceptable range, the service quality assurance of the business will not be reduced; the bandwidth decline trend of the BE business is significantly slowed down compared with the existing algorithm, which shows that this algorithm can take care of low-priority services to a certain extent.

Figure 5 shows the change of the fairness index when the rtPS traffic increases. Referring to the literature, the fairness index is defined as the mean square error of the bandwidth satisfaction of each service:

$\mathrm{<span\; class="notranslate">\; fairness</span>}<span\; class="notranslate">\; =</span>\sqrt{{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; UGS</span>}}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; rtPS</span>}}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; wxya</span>}}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; BE</span>}}<span\; class="notranslate">\; -</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}$

Among them, S=r _allo /r _req is the bandwidth satisfaction degree of the service; S=(S _UGS +S _rtPS +S _nrtPS +S _BE )/4 the average value of the bandwidth satisfaction degree of each service; it can be seen from the figure that the existing algorithm is adopted , which can effectively reduce the mean square error of the bandwidth satisfaction degree of each service, and make the bandwidth obtained by each service tend to be consistent, thereby improving the fairness of the resource scheduling algorithm for each service.

The method for calculating the reserved bandwidth for rtPS services is as follows:

First we build an approximate mathematical model. Considering the most conservative situation, it is assumed that the bandwidth allocated to a certain rtPS service in each frame of the system is r _res , the data input rate is r _in , r _min ≤ r _in ≤ r _max , the data output rate is r _res , and the rtPS service The arrival of data packets is a Poisson process, and the arrival rate is λ, so the number of bytes arriving within a period of time [t ₀ , t ₀ +t] is $x\left(t\right)=\underset{i=1}{\overset{N\left(t\right)}{\mathrm{\&Sigma;}}}{l}_i,$ Where N(t) is the number of packets arriving within [t ₀ , t ₀ +t], and l _i is the length of the packet. Since l _i is independent and identically distributed and statistically independent from N(t), X(t) is a composite Poisson process. Let the generating function of l _i be F(S), and the Poisson process {N(t)} The generating function is G(S), G(S)=e ^λt[S-1] , where t is the time parameter and S is the Laplacian operator, then: the generating function of X(t) is G(F( S))=e ^λt[F(S)-1] , thus the expectation and variance of X(t) can be obtained: E(X(t))=λtE(l _i ), D(X(t))= λtE(l _i ² ); when t is large enough, N(t) is also large enough, because l _i is independent and identically distributed, and by the law of large numbers, X(t) obeys normal distribution: X(t)～N(λtE (l _i ), λtE(l _i ² )). average data rate $r_{\mathrm{in}}\left(t\right)=\frac{x\left(t\right)}{t},$ Also obey the normal distribution: r _in (t) ~ N (λE (l _i ), λE (l _i ² )/t), r _max and r _min as the upper and lower limits of r _in (t), should have: ( r _max +r _min )/2=E(r _in (t))=λE(l _i );

In fact, we can regard the packet arrival, waiting, and service process as a batch processing system. The packet arrival is a Poisson process, and the server processes it every other frame (T _S ), and processes a batch of data packets at a time. The number of packages is set as Y(t), which should satisfy the following relationship: $\underset{i=1}{\overset{Y\left(i\right)}{\mathrm{\&Sigma;}}}{l}_i<r_{\mathrm{res}}\&Center\; Dot;T_S;$ The processing time is T _S ; now we consider the distribution of the waiting time W(t) of the package: in a batch of packages processed at the same time, we consider the package with the longest waiting time; since the queue is a first-come-first-served FCFS system , so the first packet in this batch arrives first, so the waiting time is also the longest. In order to make the waiting time not exceed the specified QoS parameters, we need to get the distribution of the waiting time W(t) of the first packet. When this packet arrives, the preceding packets are processed batch by batch, and the processing time is T _S . In order to find the distribution of the waiting time W(t), we can regard the previous packets as being processed one by one. , in one frame time, the number of processed bytes is T _S r _res , so the number of packets is T _S r _res /E(l _i ), and the average processing time of each packet is T _S /(T _S r _res /E(l _i ))=E(l _i )/r _res , so this is equivalent to a classical M/D/1 queuing system, which can be obtained: P(W<t)=1-ρe ^λt , where, ρ=λh=λE(l _i )/r _res , and from (1), it can be seen that ρ=(r _max +r _min )/2r _res . It can be known from ρ<1, $r_{\mathrm{res}}>\frac{r_{\max }+r_{\min }}{2}.$ When the delay exceeds the maximum delay t _d specified at the time of access, the data packet will be discarded, so the probability of packet loss due to delay can be obtained: $P_d=P(W+T_S>t)={\mathrm{\&rho;e}}^{\mathrm{\&lambda;}(t-T_S)};$ Since there is a regulation on the packet loss rate of the real-time service during access: P _d < P ₀ , where P ₀ is the packet loss rate threshold of the real-time service. Thus one can get: $r_{\mathrm{res}}>\frac{r_{\max }+r_{\min }}{2}{P}_0{e}^{-\mathrm{\&lambda;}(t-T_S)},$ but at the same time $\frac{r_{\max }+r_{\min }}{2}<r_{\mathrm{res}}\&le;r_{\max },$ So the minimum bandwidth reserved for rtPS is $\min (\frac{r_{\max }+r_{\min }}{2}{e}^{-\mathrm{\&lambda;}(t-T_S)},r_{\max }).$ Available $r_{\mathrm{res}}=\frac{r_{\max }+r_{\min }}{2}{P}_0{e}^{-\mathrm{\&lambda;}(t-T_S)}.$

Among them, r _max and r _min are the maximum bandwidth and minimum bandwidth respectively, λ is the packet arrival rate of the RtPS service, P ₀ is the packet loss rate threshold, t is the maximum delay, and T _S is the time of one frame. The reserved bandwidth of the RtPS service can be calculated by the above formula according to each QoS parameter of the service.

Claims (1)

1. The fairness resource scheduling algorithm based on resource reservation in the broadband wireless access system is characterized by the following steps in sequence: Step (1), when a new service arrives, the base station based on the IEEE802.16 broadband wireless access system calculates the reserved bandwidth satisfying different QoS parameters according to the QoS parameters of different services, which is set to r _res , including the established If ∑r _res ≤ r _total , the reserved bandwidth for the connection and the reserved bandwidth required for the newly established connection are allowed, otherwise, the access is denied, and r _total is the total bandwidth of the system; For non-application grant services, r _res = r _max , where r _max is the maximum input data rate, a set value; For best-effort traffic, r _res = 0; For real-time polling services,

Among them, r _min is the minimum input data rate, the set value; P ₀ is the packet loss rate threshold, a set value; λ is the arrival rate of the packets of the real-time polling service, a known value; t is the maximum delay, and T _S is the time of one frame, both of which are set values; For non-real-time polling service, r _res = r _min ; Step (2), the base station periodically allocates the system bandwidth according to the bandwidth required by each service and the existing bandwidth of the system according to the media access control layer mechanism stipulated in IEEE802.16, and puts the allocation result in the UL- Each user station is notified in the MAP message: Step 1: Level 1 Scheduling First, the reserved bandwidth of the system is used to meet the bandwidth requests of services with priority from high to low. If the bandwidth r _req required by the service is smaller than the reserved bandwidth, then the bandwidth r _{allo_1} of the first-level scheduling = r _req , otherwise, r _{allo_1} = r _res , the business with the highest priority refers to non-application grant business, and the rest are real-time polling, non-real-time polling and best-effort business; Step Two: Secondary Scheduling Then allocate the remaining bandwidth after the first-level scheduling to each service according to the weighted cyclic transfer method, and the weight factor of each service is: k _i =λ _i ·d _i ; Among them, _λi is the ratio of the remaining required bandwidth of each service to the total remaining required bandwidth: namely

If r _{req, i} < r _{res, i} , let r _{req, i} - r _{res, i} = 0; j=1, 2,..., n, is the number of services requiring allocation of remaining bandwidth, r _{req, i} and r _{res, i} are the required bandwidth and reserved bandwidth of the i-th service respectively, r _{req, j} and r _{res, j} are the required bandwidth and reserved bandwidth of the jth service respectively, d _i is a priority factor, set value, the priority factor of real-time business should be greater than the priority factor of non-real-time business, should set different priority factors for each non-real-time business; The bandwidth obtained by the second-level scheduling is: r _{allo_2} = (r _total -∑r _{allo_1} ) · k _i /∑ k _i , After two-level allocation, the bandwidth obtained by each service is r _allo =r _{allo_1} +r _{allo_2} .

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