Base-Station Selections for QoS Provisioning Over Distributed Multi-User MIMO Links in Wireless Networks

# Base-Station Selections for QoS Provisioning Over Distributed Multi-User MIMO Links in Wireless Networks

\authorblockNQinghe Du and Xi Zhang
\authorblockANetworking and Information Systems Laboratory
Department of Electrical and Computer Engineering
Texas A&M University, College Station, TX 77843, USA
Email: {duqinghe@tamu.edu, xizhang@ece.tamu.edu}
The research reported in this paper was supported in part by the U.S. National Science Foundation CAREER Award under Grant ECS-0348694.
###### Abstract

We propose the QoS-aware BS-selection and the corresponding resource-allocation schemes for downlink multi-user transmissions over the distributed multiple-input-multiple-output (MIMO) links, where multiple location-independent base-stations (BS), controlled by a central server, cooperatively transmit data to multiple mobile users. Our proposed schemes aim at minimizing the BS usages and reducing the interfering range of the distributed MIMO transmissions, while satisfying diverse statistical delay-QoS requirements for all users, which are characterized by the delay-bound violation probability and the effective capacity technique. Specifically, we propose two BS-usage minimization frameworks to develop the QoS-aware BS-selection schemes and the corresponding wireless resource-allocation algorithms across multiple mobile users. The first framework applies the joint block-diagonalization (BD) and probabilistic transmission (PT) to implement multiple access over multiple mobile users, while the second one employs time-division multiple access (TDMA) approach to control multiple users’ links. We then derive the optimal BS-selection schemes for these two frameworks, respectively. In addition, we further discuss the PT-only based BS-selection scheme. Also conducted is a set of simulation evaluations to comparatively study the average BS-usage and interfering range of our proposed schemes and to analyze the impact of QoS constraints on the BS selections for distributed MIMO transmissions.

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## I Introduction

\PARstart

To increase the coverage of broadband wireless networks, distributed multiple-input-multiple-output (MIMO) techniques, where multiple location-independent base stations (BS) cooperatively transmit data to mobile users, have attracted more and more research attentions [2, 4, 3]. In particular, the distributed MIMO techniques can effectively organize multiple location-independent BS’s to form the distributed MIMO links connecting with mobile users Like the conventional centralized MIMO system [5, 7, 6], the distributed MIMO system can significantly enhance the capability of the broadband wireless networks in terms of the quality-of-service (QoS) provisioning as compared to the single antenna system. However, the distributed nature for cooperative multi-BS transmissions also imposes many new challenges in wide-band wireless communications, which are not encountered in the centralized MIMO systems. First, the cooperative distributed transmissions cause the severe difficulty for synchronization among multiple location-independent BS transmitters. Second, as the number of cooperative BS’s increases, the computational complexity for MIMO signal processing and coding also grow rapidly. Third, because the coordinated BS’s are located at different geographical positions, the cooperative communications in fact enlarge the interfering areas for the used spectrum, thus drastically degrading the frequency-reuse efficiency in the spatial domain. Finally, many wide-band transmissions are sensitive to the delay, and thus we need to design QoS-aware distributed MIMO techniques, such that the scarce wireless resources can be more efficiently utilized.

Towards the above issues, many research works on distributed MIMO transmissions have been proposed recently. The feasibility of transmit beamforming with efficient synchronization techniques over distributed MIMO link has been demonstrated through experimental tests [3], suggesting that complicated MIMO signal processing techniques are promising to implement in realistic systems. For the centralized MIMO system, the antenna selection [6, 7] is an effective technique to reduce the complexity, which clearly can be also extended to distributed MIMO systems for the BS selection. It can be expected that the BS-selection techniques can significantly decrease the processing complexity, while still achieving high throughput gain over the single BS transmission. Also, it is desirable to minimize the number of selected BS’s through BS-selection techniques, which can effectively decrease the interfering range and thus improve the frequency-reuse efficiency of the entire wireless network. Most previous research works for BS/antennas selections mainly focused on the scenarios of selecting a subset of BS’s/antennas with the fixed cardinality [6, 7, 4]. However, it is evident that based on the wireless-channel status, BS-subset selections with dynamically adjusted cardinality can further decrease the BS usage. More importantly, how to efficiently support diverse delay-QoS requirements through BS-selection in distributed MIMO systems sill remains a widely cited open problem.

To overcome the aforementioned problems, we propose the QoS-aware BS-selection schemes for the distributed wireless MIMO links, which aim at minimizing the BS usages and reducing the interfering range, while satisfying diverse statistical delay-QoS constraints. In particular, we develop two BS-usage minimization frameworks for distributed multi-suer MIMO transmissions. The first framework uses the joint block-diagonalization (BD) and probabilistic transmission (PT) for multiple access of multi-user over distributed MIMO links, while the second framework employs time-division multiple access (TDMA) techniques. We derive the optimal QoS-aware BS-selection and the corresponding resource allocation schemes for these two frameworks, respectively. We also discuss the PT-only based BS-selection scheme. Simulations are conducted for comparative analyses among the above BS-selection schemes.

The rest of this paper is organized as follows. Section II describes the system model for distributed MIMO transmissions. Section III introduces the statistical QoS guarantees and the concept of effective capacity. Section IV develops the joint BD and PT (BD-PT) optimization frameworks for QoS-aware BS-selections over multi-user distributed MIMO links and derives the corresponding optimal solution. Section V derives TDMA-based QoS-aware BS-selection scheme. Section VI conducts simulations to perform comparative analyses for our proposed schemes. The paper concludes with Section VII.

Notations: The operator used on a real or complex number generates the absolute value; the operator used for a set represents the cardinality of this set. We use boldface to denote matrices and vectors. For an matrix , we denote by the element on the th row and th column; denotes the Frobenius norm of , where . The operators and generate the transpose and conjugate transpose, respectively. The operator is the indication function. If the statement in the subscript is true, we have ; otherwise, .

## Ii System Model

### Ii-a System Architecture

We concentrate on the wireless distributed MIMO system for downlink transmissions depicted in Fig. 1, which consists of distributed BS’s, mobile users, and one central server. The th BS has transmit antennas for and the th mobile user has receive antennas for . All distributed BS’s are connected to the central server through high-speed optical connections. The data to be delivered to the th mobile user, , arrives at the central server with a constant rate, which is denoted by . Then, the central server dynamically controls these distributed BS’s to cooperatively transmit data to the corresponding mobile users under the specified delay-QoS requirements.

For multi-user downlink transmissions, the distributed BS’s and the mobile users form the broadcast MIMO link for data transmissions. The wireless fading channels between the th BS and the th mobile user is modeled by an matrix , where is the complex channel gain between the th receive antenna of th mobile user and the th transmit antenna of the th BS. All elements of are independent and circularly symmetric complex Gaussian random variables with zero mean and the variance equal to , implying that has continuous cumulative distribution function (CDF). Also, the instantaneous aggregate power gain of the MIMO link between the th mobile user and the th BS, denoted by , is defined by

 γn,m≜1Mm∥Hn,m∥2F (1)

Since the Frobenius norm of the channel matrix can effectively characterize the channel quality in terms of achieving high throughput [6], the aggregate power gain given in Eq. (1) will play an important role in our BS selection design. We further define as the CSI for the th mobile user for . The matrix follows the independent block-fading model, where does not change within a time period with the fixed length , called a time frame, but varies independently from one frame to the other frame. Furthermore, we define , representing a fading state of the entire distributed MIMO system.

In order to decrease the complexity and suppress the interfering range of the distributed MIMO transmission, the cental server dynamically selects a subset of BS’s to construct the distributed MIMO link. Then, our design target is to minimize the average number of needed BS’s subject to the specified QoS constraints. We suppose that each mobile user can perfectly estimate its CSI at the beginning of every time frame and reliably feed CSI back to the central server through dedicated control channels. Based on CSI and QoS requirements, the central server then adaptively selects the subset of BS’s and organizes them to transmit data to mobile users through the distributed MIMO links.

### Ii-B The Delay QoS Requirements

The central data server maintains a queue for the incoming traffic to each mobile user. We mainly focus on the queueing delay in this paper because the wireless channel is the major bottleneck for high-rate wireless transmissions. Since it is usually unrealistic to guarantee the hard delay bound over the highly time-varying wireless channels, we employ the statistical metric, namely, the delay-bound violation probability, to characterize the diverse delay QoS requirements. Specifically, for the th mobile user, the probability of violating a specified delay bound, denoted by , cannot exceed a given threshold . That is, the inequality

 Pr{Dn>D(n)th}≤ξn,n=1,2,…,Nmu, (2)

needs to hold, where denotes the queueing delay in the th mobile user’s queueing system.

### Ii-C Performance Metrics and Design Objective

We denote by the cardinality of the selected BS subset (the number of selected BS’s) for the distributed MIMO transmission in a fading state. Then, we denote the expectation of by and call it the average BS usage. As mentioned in Section II-A, our major objective is to minimize through dynamic BS selection while guaranteeing the delay QoS constraint specified by Eq. (2). We will also evaluate the average interfering range affected by the distributed MIMO transmission. The instantaneous interfering range, denoted by , is defined as the area of the region where the average received power under the current MIMO transmission is larger than then certain threshold denoted by . The average interfering area is then defined as the expectation over all . Clearly, minimizing can not only reduce implementation complexity, but also decrease the average interfering range affected by the transmit power.

### Ii-D The Power Control Strategy

The transmit power of our distributed MIMO system varies with the number of selected BS’s. In particular, given the number of selected BS’s, the total instantaneous transmitted power used for distributed MIMO transmissions is set as a constant equal to . Furthermore, linearly increases with by using the strategy as follows:

 PL=Pref+κ(L−1),L=1,2,…,Kbs, (3)

where is called the reference power and describes the power increasing rate against . Also, we define for . The above power adaptation strategy is simple to implement, while the average transmit power can be effectively decreased through minimizing the average number of used BS’s. In addition, Eq. (3) can upper-bound the instantaneous interferences and the interfering range over the entire network.

## Iii Effective Capacity Approach for Statistical Delay-QoS Guarantees

In this paper, we apply the effective capacity approach [9, 12, 10, 18] to integrate the constraint on delay-bound violation probability given by Eq. (2) into our BS selection design. Consider a stable discrete-time queueing system with the stationary time-varying arrival-rate and departure-rate (service-rate) processes. The asymptotic analyses based on the large deviation principal [9, 8] show that under the sufficient conditions, the probability that the queue-length, denoted by , exceeding a given bound can be approximated by

 Pr{Q>Qth}≈e−θQth, (4)

where is a constant called QoS exponent. It is clear that the larger (smaller) implies the lower (higher) queue-length-bound violation probability.

By using , the delay-bound violation probability can be approximated [9] by

 Pr{D>Dth}≈e−θ¯¯¯¯CDth. (5)

for the constant rate . When the arrival rate is not time-varying, the approximation in Eq. (5) needs to replace with effective bandwidth [9, 8] function of the arrival rate process, which is defined as the minimum constant service rate required to guarantee QoS exponent .

Then, to upper-bound with a threshold , using Eq. (5), we get the minimum required QoS exponent as follows:

 θ=−log(ξ)¯¯¯¯CDth. (6)

Consider a discrete-time arrival process with constant rate and a discrete-time time-varying stationary departure process, denoted by , where is the time index. In order to guarantee the desired determined by Eq. (6), the statistical QoS theory [8, 9] shows that the effective capacity of the service-rate process needs to satisfy

 C(θ)=¯¯¯¯C, (7)

given the QoS exponent . The effective capacity function is defined in [9] as the maximum constant arrival rate which can be supported by the service rate to guarantee the specified QoS exponent . If the service-rate sequence is time uncorrelated, the effective capacity can be written [12] as

 C(θ)≜−1θlog(E{e−θR[k]}), (8)

where denotes the expectation.

In our distributed MIMO system, the BS selection result is designed as the function determined by the current CSI. Thus, the corresponding transmission rate (service rate) is time independent under the independent block-fading model (see Section II-A). Then, applying Eqs. (6)-(7), the delay QoS constraints given by Eq. (2) can be equivalently converted to:

 EH{e−θnRn−e−θn¯¯¯¯Cn}≤0,,n=1,2,…,Nmu, (9)

where and denotes the expectation over all .

## Iv Joint Block-Diagonalization and Probabilistic Transmission Based Base-Station Selection

As discussed in Section II, based on CSI and QoS requirements, the central server will adaptively select the subset of BS’s and organizes them to transmit data to mobile users through the distributed MIMO links. Given the cardinality of the desired BS subset in a fading state, we denote by the set of indices of selected BS’s, where and for . Note that once a BS is selected, we use all its transmit antennas for data transmissions. For the specified , we use to denote the set of active users, picked by the central server, which can receive the data in this fading state, where is the cardinality of . For presentation convenience, we use to represent a specific transmission mode (or mode in short). Moreover, we term the pairs with for as multi-user modes, and term the pairs with as single-user modes.

Given , and , the channel matrix of the th mobile user for , modeled by , is determined by

 H(n)ΩL≜[Hn,iL,1 Hn,iL,2 ⋯ Hn,iL,L], (10)

where is an matrix. Furthermore, we use to denote the power gain matrix for under the given , where

 ¯¯¯¯¯Υ(n)ΩL(i,j)=EH{∣∣H(n)ΩL(i,j)∣∣2∣∣∣fixing ΩL}. (11)

The physical-layer signal transmissions can be modeled by

 y(n)ML=H(n)ΩL∑i∈ULs(i)ML+ς(n),n∈UL,

where represents the th user’s input signal vector for the MIMO channel , is the signal vector received by the th user, and is the complex additive white Gaussian noise (AWGN) vector with unit power for each element of this vector. In this section, we employ the block-diagonalization (BD) technique [15] to implement multiple access for multi-user modes in our QoS-aware BS-selection framework.

For dynamic BS selections in distributed MIMO transmissions, and are both functions of CSI and QoS requirements. Then, we need to answer the following questions: (i) Given , which transmission mode will be used for single-user and multi-user modes, respectively? (ii) When do we use single-user or multi-user modes? (iii) For a specific multi-user mode, how do we quantitatively allocate the wireless resources across multiple mobile users under the BD based transmissions? (iv) Which will be selected for distributed MIMO transmissions in each fading state to decrease the average BS-usage while satisfying the QoS requirements?

Clearly, we can not examine all combinations of to minimize the BS usage due to the too high complexity. Then, in Section IV-A, we develop the heuristic algorithms to efficiently select for the specified in multi-user transmission modes. In Section IV-B, we determine how to select in single-user transmission mode. Based on schemes developed in Sections IV-A and IV-B, we further answer questions (iii) and (iv) through formulating and solving the joint BD-PT based BS-usage minimization problem in Sections IV-C and IV-D.

### Iv-a Selection of ML in Multi-User Transmission Modes

In each fading state, we pick multi-user transmission modes as candidates for distributed MIMO transmissions. These transmission modes corresponds to , respectively, representing different levels of BS usages. As mentioned previously, the derivation of global optimal selection strategy in terms of minimizing the average BS usage is intractable, since the complexity of examining all possible is too high. Therefore, for a given , we determine through a two-step method. We first propose the priority BS-selection to determine the BS subset . Then, based on the selected , we derive through a joint channel-priority user-selection process.

A.1. Priority BS-Selection to Determine

Consider any fading state . The th user’s global maximum achievable transmission rate is attained when all BS’s are used and all the other users do not transmit. In this case, we have and . Moreover, all BS’s and the th user builds a single-user MIMO channel . Then, the maximum achievable rate is equal to the capacity for the MIMO channel with power , which is given by [5]

 R(n)max = maxΞ(n):Tr(Ξ(n))=PKbs{BTlog[det(I+HnΞ(n)H†n)]}

where generates the determinant of a matrix, evaluates the trace of a matrix, and is the covariance matrix of . Correspondingly, we get the maximum achievable effective capacity of the th user, denoted by , as follows:

 C(n)max=−1θnlog(EH{e−θnR(n)max}), (12)

for . Furthermore, we define the effective-capacity fraction for the th user as the ratio between the traffic loads and the maximum achievable effective capacity. Denoting the effective-capacity fraction by , we define . Note that can be readily obtained off-line based on the statistical information of wireless channels, and thus can be used to design the BS selection algorithm during the data transmission process. For presentation convenience, we sort in the decreasing order and denote the permuted version by , where indicates the order from the higher priority to the lower priority. In the rest of this paper, we use the term of user to denote the user associated with the th largest effective-capacity fraction.

Clearly, for a higher , the th user needs more wireless resources to meet its QoS requirements. Thus, in order to satisfy the QoS requirements for all users, we assign higher BS-selection priority to the user with larger . Following this principle, we design the priority BS-selection algorithm to determine in each fading state and provide the pseudo code in Fig. 2. In the pseudo code given by Fig. 2, we use temporary variables and to denote the subsets of BS’s which have been selected and which have not been selected, respectively.

As shown in Fig. 2, in each fading state the BS-selection procedure starts with the selection for user , who has the highest priority. After picking one BS for user , we select one different BS for user . More generally, after selecting for user , we choose one BS for user from the BS-subset , which consists of the BS’s that have not been selected. This procedure repeats until BS’s are selected. For user-’s selection, we choose the BS with the maximum aggregate power gain over the subset , where denotes the instantaneous aggregate power gain between user and the th BS (see Eq. (1) for its definition). In addition, after user-’s selection, if the number of selected BS’s is still smaller than , we continue selecting one more BS for user , as shown in line 06 in Fig. 2, and repeat this iterative selection procedure until having selected BS’s. Clearly, users with higher priorities benefitted more from the above algorithm. Also note that the mobile users’ priority order is determined by the effective-capacity fraction, which adapts to the mobile users’ QoS requirements.

A.2. The Principle of the Block Diagonalization Technique

The block-diagonalization (BD) precoding techniques [15] have been widely used for MIMO transmissions because of its low complexity. In this section, we also apply the BD technique for our QoS-aware BS selection framework. For completeness of this paper, the principles of the BD technique are summarized as follows.

Given transmission mode , the idea of block diagonalization [15] is to use a precoding matrix, denoted by , for the th user’s transmitted signal vector, where for some , such that for all satisfying and . By setting , where is the th user’s data vector to be precoded by , we can rewrite the received signal as

 y(n)ML=H(n)ΩL∑i∈ULΓ(i)MLˆs(i)ML+ς(n)=ˆΓ(n)MLˆs(n)ML+ς(n),

where . Under this strategy, the th user’s signal will not cause interferences to other active users. Accordingly, the MIMO broadcast transmissions are virtually converted to orthogonal MIMO channels with channel matrices . Thus, the th user’s maximum achievable rate, denoted by , is equal to the capacity of the equivalent MIMO channel , as follows:

 (13)

subject to for , where is the covariance matrix of and denotes the power allocated for the th user under mode . Correspondingly, we will set the service rate of the th user equal to . Note that may not exist, which then results in a service rate equal to 0. Also, we set for or . For the procedures to determine the precoding matrix of the th user, where for some , please refer to [15].

A.3. Derivation of Active-User Set

Note that given we may not be able to accommodate all users, because of the limited number transmit antennas. Although several algorithms for selecting active-user set have been proposed [16, 17], they cannot be applied in the framework of this paper, because the QoS provisioning for mobile users are not addressed those in these algorithms. Next, we determine through a joint channel-priority method for active user selections. The pseudo code of this algorithm is provided in Fig. 3.

In the joint channel-priority algorithm provided by Fig. 3, we iteratively select users one by one into the set . In particular, we use variables and to represent the temporary sets of users which have and have not been selected, respectively. As shown in Fig. 3, lines 02 through 10 describe loops for iterative user selection, where we pick one user in each loop until all users are selected (i.e., ) or no more user can be accommodated (examined by line 12). Within each loop, given the existing active-user set we examine the channel quality of each user after BD. Specifically, we first get the BD precoding matrix of the th user. Then, we derive , which is average channel-power-gain after BD over all transmit antennas, representing the average channel quality, and also obtain line 06, which characterizes the instantaneous channel quality. We further define a variable , as shown in line 07, where and indicate that the channel quality is above and below the average level, respectively. Obtaining , our selection criteria are as follows. First, we desire to select the user with higher , implying that this user’s current channel is better compared with its statistical channel qualities, which will more efficiently use the system resources towards this user’s QoS requirement. Second, if two users have the same , we will select the user with higher priority. Following this criterion, in line 08, we pick the unique user from in the current loop, whose index is denoted . However, if , implying the maximum achievable rate equal 0. As a result, no more user can be admitted, including the th user. We will then terminate the loop, as shown in line 09, to finish the selection process.

### Iv-B BS Selection in Single-User Transmission Modes

For single-user transmission modes, we have , . Thus, at any time instant, there is only one user receiving data from multiple BS’s through a single-user MIMO channel . Accordingly, the maximum achievable rate for the th user is equal to the capacity of with power , which is denoted by . However, even for the single-user case, the complexity of high of choosing to maximize the achievable data rate is too high, since we need to examine all combinations. Norm-based antenna selections have been demonstrated to be effective in achieving high system throughput with low complexity in centralized MIMO system [6, 7], which can be also extended to BS-selection in distributed MIMO system. Specifically, for the th user with specified in our framework, we select BS’s with largest aggregate channel power gain. As a result, in each fading state we have single-user modes as candidates for distributed MIMO transmissions. Given the transmission mode with and the active user , we will set the service rate equal to .

### Iv-C The Optimization Framework for Transmission Mode Selection and Resource Allocation

We have derived candidate in multi-user and single-user transmissions modes, respectively. We still need to answer how to allocate power over different mobile users in multi-user modes and which transmission mode will be eventually used for distributed MIMO transmissions. In this section, we employ the probabilistic transmission to determine finally selecting which transmission mode. Specifically, we use multi-user mode determined through algorithms given in Figs. 2 and 3 with a probability denoted by , ; also, we use single-user mode with BS-subset cardinality and with a probability denoted by , , . Note that is the probability of the case that nothing is transmitted. Clearly, the sum over all and must be equal to 1. For multi-user mode, we denote power allocated to the th user in transmission mode by , while the total power constraint is given by Eq. (3). For presentation convenience, we further define and with to describe the probabilistic transmission policy; we also define with to characterize the power allocation policy in multi-user modes. Then, we formulate the following optimization problem to derive the efficient transmission-mode selection and the corresponding power allocation policy:

: Joint BD-PT based BS-usage minimization

 min(ϕ,q,P){EH{Kbs∑L=1L(ϕL+Kmu∑n=1qL,n)}} s.t.: 1). Kbs∑L=0ϕL+Kbs∑L=1Kmu∑n=1qL,n=1,∀H (14) (15) 3). EH{Kbs∑L=0(ϕLe−θnR(n)ML(P(n)L)+qL,ne−θnR(n)ΩL) +Kbs∑L=0∑j,j≠nqL,j}−e−θn¯¯¯¯Cn≤0, ∀n. (16)

### Iv-D Derivations of the Optimal Solution of Problem A1

D.1. The Properties of

Before solving , we first summarize the properties of determined by Eq. (13). Based on results in [5], the th user’s MIMO channel (after BD) can be converted to parallel Gaussian sub-channels, where is the rank of . Correspondingly, the th sub-channel’s SNR is equal to , where the square root of is the th largest nonzero singular value of . The optimal power allocated to the th sub-channel follows the water-filling allocation, which is equal to , where and is selected such that . Since has only non-zero singular values, we define for and for . We can further show that is a strictly concave function and

 dR(n)ML(P(n)L)dP(n)L=BTμ(n)L (17)

holds. Moreover, if for , we get:

 P(n)L = [iμ(n)L−∑ij=11ε(n)L,j]+; (18) R(n)ML(P(n)L) = BTlog(∏ij=1ε(n)L,j)+BTilogμ(n)L. (19)

D.2. The Optimal Solution to

###### Theorem 1

The optimal solution for optimization problem , if existing, is given by

 (P(n)L)∗=⎧⎪ ⎪⎨⎪ ⎪⎩[i∗μ(n)L−∑i∗j=11ε(n)L,j]+,if n∈UL;0,if n∉UL; (20)

for all , , and , where is the square of ’s th largest singular value; is the unique solution satisfying the following conditions:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩μ(n)L=(ζ∗H,LBTθnλ∗n)−11+i∗BTθn∏i∗j=1(ε(n)L,j)−BTθn1+i∗BTθn;μ(n)L∈[1ε(n)L,i∗,1ε(n)L,i∗+1),∀n,L,H. (21)

The corresponding optimal PT policy is determined by

 {ϕ∗L=1(ψL=ψ∗);q∗L,n=1(ψL,n=ψ∗), (22)

where is the indication function and is defined as

 ψ∗≜min{minL{ψL},min(L,n){ψL,n}} (23)

with

 ⎧⎪ ⎪⎨⎪ ⎪⎩ψL≜L+∑Kmun=1λ∗ne−θnR(n)ML((P(n)L)∗),  0≤L≤Kbs;ψL,n≜L+λ∗ne−θnR(n)ΩL+∑j,j≠nλ∗j,  L∈[1,Kbs], ∀n;

if multiple ’s and/or ’s all equal to , the corresponding transmission modes will be allocated equal probability with the sum probability equal to 1. The variables are constants over all fading state; given , is selected to satisfy the equation for all and ; accordingly need to be selected such that the equality of Eq. (16) holds.

{proof}

We construct ’s Lagrangian function, denoted by , as

 JA1(ϕ,q,P;λ,ζH)=EH{JA1(ϕ,q,P;λ,ζH)} (24)

subject to , where

 JA1(ϕ,q,P;λ,ζH) +Kmu∑n=1λn[Kbs∑L=0(ϕLe−θnR(n)ΩL,UL(P(n)L)+qL,ne−θnR(n)ΩL) +(Kbs∑L=0∑j,j≠nqL,j)−e−θn¯¯¯¯Cn]. (25)

In Eqs. (24)-(25), and ’s are the Lagrangian multipliers associated with Eq. (16), which are constants over all fading states and satisfies ; are the Lagrangian multipliers associated with Eq. (15) in each fading state, and .

The optimization problem ’s Lagrangian dual function [11], denoted by , is determined by

 JA1(λ,ζH) ≜ min(ϕ,q,P){JA1(ϕ,q,P;λ,ζH)} (26) = EH{min(ϕ,q,P){JA1(ϕ,q,P;λ,ζH)}}.

subject to for all . Lagrangian duality theory shows [11] that is always a concave function, whose maximizer is upper-bounded by the optimum of . We then denote the maximizer of by . Also, we denote by the minimizer to Eq. (26), which varies with . Then, given , we have

 (ϕ∗,q∗) = argmin(ϕ,q){JA1(ϕ,q,P∗;λ∗,ζ∗H)} (27) \lx@stackrel(a)= argmin(ϕ,q){Kbs∑L=0ϕLψL+Kbs∑L=1Kmu∑n=1qL,nψL,n},

for all , where and