In this section, the average throughput performance of the band-AMC system is evaluated. It is assumed that a full buffer traffic model is used, that is, infinite traffic waiting for each user. Depending on the channel quality, it is assumed that each user belongs to one of groups. The channel quality of all users in the same group is identically distributed. Let and represent the total bandwidth and coherence bandwidth, respectively. Then, the total number of independent subbands can be given approximately by . Note that the optimal number of subbands may be greater than or equal to . For example, it has been demonstrated in [5] that the optimum contribution to performance improvement is found for , where denotes the bandwidth of subband. Nevertheless, can be still fixed to the minimum number of independent subbands, that is, is just large enough to warrant the independence of channel qualities between the adjacent subbands. Determining a proper is beyond the scope of this paper.

Now let a vector represent the sampled values of a channel quality for user in group . Note that is not always necessarily equal to . Therefore, we consider two different cases: and For the case of there is no correlation between those samples, that is, the channel quality for each subband is independent of each other. Denoting as the expected value of CIR for band in group , then the following probability density function (PDF) for CIR of the corresponding band under the condition that can be obtained:

For the diversity channel, meanwhile, CIR for each user in group is given by taking average of CIRs for all subbands, that is, . In the case that are identically distributed over a whole bandwidth, turns out to be the normalized random variables.

The design of band-AMC system depends on the bandwidth of each subband, channel characteristics, the number of users served by band-AMC mode, the feedback overhead to report the CQI of subbands, and so on. Consider the situation that the bandwidth of subband, , chosen by band-AMC system is subject to the nonflat fading characteristics. This particular situation can be specified by for , which corresponds to the case of Then, the observed channel quality for each subband cannot be represented by (4). When the bandwidth is divided into several adjacent segments, each with the bandwidth of , it can be now approximated as , . Then, (4) is replaced with the following PDF:

where denotes the convolution operation: .

### 3.1. CQI Report for Band-AMC Mode

Suppose that every band-AMC user feedbacks -best CQI reports to the base station in every scheduling interval. To represent the chance that each subband is selected for feedback, define a band selection vector for user as follows:

where is the probability that user in group has a preference to the band within chances, that is, . In the case that samples in the subband are independent and identically distributed, it is obvious that However, consideration must be taken, that the dependence assumption is retained when the 's are no longer identically distributed, that is, for the *inid* case.

Let denote the CDF of -order statistics, exclusive of band within the entire band pool, where represents a band set of the system, that is, . Hence, the probability that the user in group selects the band is given by

The CDF of the -order statistic is generalized to

where the summation extends over all permutations of for which and [10]. For the distribution of order statistics in the *inid* case, however, the density of every possible order must be found out separately on a case-by-case basis, which makes (8) involve the complicated and tedious calculation, especially as the number of bands increases. Fortunately, an alternative method for computing has been provided by Cao and West [7]. It is acceptable to have results and recurrence relations valid in the *iid* case, requiring only simple modification to hold quite generally. For convenience of notation, let denote the . Starting with

they prove the following relation:

where , and

with

Now from (7) and (9)–(12), the band selection vector can be directly determined. It is obvious that is obtained with , which corresponds to the case of full CQI feedback.

### 3.2. Maximum System Throughput in Band-AMC Mode

Let denote the total number of users in group . The probability that band is simultaneously selected by users can be written as follows:

where .

Similarly, a vector is defined to represent the distribution of order statistics in the corresponding band *j*. By means of the max C/I-scheduling scheme, the received signal quality is then expressed as

In general, the optimum signal quality in band is expected as the number of users selecting the corresponding band increases. By order statistics, the conditional CDF of the received CIR in band given can be calculated as

where and

Therefore, the CDF of the received CIR in band can be expressed as

When the existing cellular systems are considered, in which multi-path fading is dominant, the rate function of the Shannon type with the log-based linear relationship between rate and CIR may not be valid. In practice, a link-level simulation is performed in order to determine the required CIR for a given data rate, so as to meet the target frame error rate (FER). Let denote a set of MCS levels with the corresponding data rates , with the data rate for MCS level *m* defined by . To meet the given level of FER, a range of CIR is prescribed for each data rate . More specifically, the CIR required for is prescribed as . For the given target FER, the average system throughout of band is defined as follows:

Considering overall bandwidth, therefore, the average throughput of band-AMC system is provided by .