Game Theoretic QoS Modeling for Joint Resource ... - IEEE Xplore

2012 IEEE Wireless Communications and Networking Conference: MAC and Cross-Layer Design

Game Theoretic QoS Modeling for Joint Resource Allocation in Multi-User MIMO Cellular Networks Lei Zhong, and Yusheng Ji National Institute of Informatics (NII), Tokyo, Japan Email: {lei, kei}@nii.ac.jp Abstract—This study addresses the resource allocation problem for multi-user MIMO systems with consideration of real-time services. Specifically, we focused on the delay constraints modeling in this paper, since it is a fundamental QoS requirement for all real-time services. To simultaneously meet the delay constraints of all users, we first modeled this resource allocation problem with specific delay restrictions as a bargaining game. Based on the Nash bargaining solution, we derived the system utility function that achieves delay constraints in a long term. Then, we formulated the resource allocation problem as an optimization problem through the weighted sum rate maximization, which can be solved by a modified iterative water-filling algorithm. Simulation results show that our proposed resource allocation algorithm not only achieves delay requirements for different realtime services, but also obtains the Pareto optimal efficiency with acceptable complexity.

I. I NTRODUCTION Over the past decade, the Multiple-Input Multiple-Output (MIMO) communications has drawn vast attentions for its significant improvement in channel capacity [1], and widely adopted by emerging wireless broadband systems. For a MultiUser MIMO (MU-MIMO) system, even there is only one antenna equipped at each user, multiple antennas of several users still can be multiplexed in space and share the same resources in frequency and time domains to significantly enhance the system capacity [2]. Therefore, in the downlink, the Base Station (BS) forms multiple spatial channels by transmit beamforming and allocates powers on each channel, which, however, has brought some new challenges in resource allocation. Firstly, spatial channels are inherently not orthogonal between users, which causes inter-user interference and thus introduces a complex beamforming processing for interference control. Secondly, since the power allocation is coupled with beamforming, the joint resource allocation in an MU-MIMO system that achieves system capacity is usually a NP-hard problem. [3]. Several sub-optimal algorithms able to solve this resource allocation problem with low complexity have been proposed in many recent studies [3]–[5]. All of them are based on the approach that heuristically select a group of users with not greater number of antennas than that at BS, which further simplified the resource allocation within each time slot into parallel independent channels using zero-forcing beamforming. In [3], [4], for example, the proposed algorithms reduced the computational complexity by decoupling the user selection and power allocation into independent two steps, in

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which the selection of users are simply based on their spatial correlations and allocation of powers are based on the zeroforcing orthogonalized channels. However, the objectives of all these studies are system capacity, and more importantly, based on the assumption of independent channel statuses between any two time slots, greedy resource allocation in a long term still guarantees optimal. While, it does not hold when we consider the effect of real-time traffic on the resource allocation. Not only the system capacity, as the demand for various services with diverse Quality-of-Service (QoS) requirements grows dramatically in wireless systems, another challenge for resource allocation is to guarantee the QoS for all these services, particularly, the real-time services with specific delay bounds. A few studies have paid attention to this problem in MU-MIMO systems. In [6], the authors proposed an opportunistic scheduling with proportional fairness (PF), which technically only suitable for best-effort traffic in a long term performance. The study in [7] proposed a general system utility model that including the PF algorithm and other ones such as Max Weight (MW) algorithm as its special case. However, this algorithm performs badly with real-time traffic, for which the algorithm can not guarantee specific delay bounds. The study in [8] proposed a prioritized QoS management with different treatments to different kinds of services. Unfortunately, this algorithm is not best for dynamic services in real systems due to the static prioritization. Thus, we propose a new utility function suitable for both real-time and non-real-time services based on the cooperative game theory. In specific, we modeled the delay as the cost of each user, and derived the system utility function using Nash Bargaining Solution (NBS) [9] to minimize all users’ costs. Moreover, as our utility function is in accordance with the form of maximization weighted sumrate problem [10], we then modified the well-known Iterative Multiuser Water-Filling (IMWF) algorithm to maximize the system utility, which achieving optimal system capacity in a long term with delay bounds. The rest of the paper is organized as follows. We describe the system model in Section II, delay-bounded utility model based on Nash bargain solution is presented in Section III, and formulation of the resource allocation problem and the modified iterative solution are described in Section IV. In Section V, we present our simulation results and conclude the paper in Section VI.

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Assuming ideal adaptive modulation and coding schemes are adopted, users have the ability to match the rate they can achieve according to their channel conditions. As discussed in [11], rate rk of user k that can be approximated as a function of bit error ratio (BER) and SINR by

User 1

  

Joint Beamforming and Power Control

  

User 2

   User K

Fig. 1.

rk = log2 (1 + δ · SIN Rk ),

Downlink of a multi-user MIMO system

II. S YSTEM M ODEL Before proceeding, some notations are given as follows. Matrices are denoted by bold capital letters and vectors by bold lowercase letters. Notation · denotes the Frobenius norm, and E[·] is the expected value of the random variable. Subscripts (·)T , (·)H , and (·)† correspond to the transpose, conjugate transpose and pseudo inverse, respectively. The set of complex numbers is denoted by C. We consider the downlink of an MU-MIMO system with one BS serving K users, which is illustrated in Fig. 1. Let K denote the set of all users and k ∈ K = {1, . . . , K} represents the kth user. Without loss of generality, we assume that each user has only one service flow and equips with one antenna. At the BS side, queue Qk buffers the data intended for user k with an arrival rate λk . We only consider the significant queuing delay for simplicity, the data in each queue should not exceeds it because of the delay requirement. Let vector s = (s1 , . . . , sK ) denote the data flows streaming out from the queues, and the bit energy of each flow is assumed to be E{sk sH k } = 1 for any user k. These data flows are then power weighted and overlapped, i.e. joint beamforming and power control processing, and transmitted as signal x = (x1 , . . . , xN ) supposing there are N transmit antennas (usually N < K) at BS. To track the fast variation of wireless channels, the operation of resource allocation should be repeated at each time slot as long as its duration is smaller than the channel coherent time, which means the channel gains can be treated as constant within each time slot. As a result, for any time slot, the receiving signal of a user k can be given by  yk = hk x + h k x + nk j=k

 √ √ = hk w k p k s k + hk wj pj sj +nk .    j=k desired signal   

(1)

interference

where, for user k, vector hk ∈ C1×N is the channel gains, and wk ∈ CN ×1 is the normalized beamforming weights. pk is allocated power, and nk is the Gaussian noise with variance 2 E[nk nH k ] = σ . The second term in (1) is the interference from the other users. Therefore, the Signal-to-Interferenceplus-Noise Ratio (SINR) for user k can be given by SIN Rk = 

pk |hk wk | j=k

2 2

pj |hk wj | + σ 2

.

(2)

(3)

where δ = −1.5/ ln (5 · BERk ) is a parameter to bridge the gap between Shannon capacity and the practical modulation and coding scheme. Based on above system model of a single cell downlink with real-time services, our objective is to find an optimal resource allocation in a long term that achieves optimal system capacity with delay constraints by joint beamforming and power control. III. BARGAINING FOR D ELAY R EQUIREMENTS In this section, we explain the modeling of delay bounds in the utility function using game theory. In fact, other QoS requirements can also be modeled in a similar way. As our problem is theoretically a MIMO broadcast channel capacity problem, a popular method to achieve a point, also known as a Pareto-optimal point, on the boundary of the capacity region of the MIMO broadcast channel is to solve a weighted sum rate maximization [12]. In the long term, we seek the best point on the boundary that can meet delay bounds of all users. By different weights, different points on the boundary can be achieved, which shows, in the following, how we model the delay bounds into the weights through NBS. Obviously, K users with different real-time traffic share the downlink channels can be seen as a K-person bargaining game. Each of them has a delay cost dk , which is defined as: Definition 1. Once a packet has arrived at queue Qk at time t, for a given queue length qk (t), the expected queuing delay can be expressed by: dk (t) = qk (t)/rk (t),

∀k,

(4)

where rk (t) is the instantaneous data rate of flow sk in time t. We dropped the variant t, since the following analysis focused with each time slot. Let D = (D1 , . . . , DK ) represents the maximum cost (delay bounds) that users can afford and the baseline for bargaining. In each time slot, the resource allocation try to decrease the expected delays within the delay bounds. Otherwise, until the violation of the delay bounds, some of the users are not satisfied by the allocation, which should be controlled by a admission mechanism. Hence, we mainly consider the case that the maximum costs are always feasible. Defining R as all feasible rate sets, as the queue length is seen as a constant within each time slot, our objective ∗ ) ∈ R here is then to find the optimal r ∗ = (r1∗ , . . . , rK that minimize the cost of all users. We give the following formal Pareto-optimal rate allocation definition as a criterion to choose optimal r ∗ .

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Definition 2. The allocation point r ∗ is said to be Pareto optimal, if and only if there is no other allocation r  , such that d k < d∗k , ∃k, and d k ≤ d∗k , ∀k, in other words, there exists no other allocation that leads to lower delays for some users without increasing delays for other users. Among the Pareto-optimal points, we can derive our intended point r ∗ by Nash bargaining axioms. For simplicity, more details about NBS refer to [9]. Here, we only give the existence and uniqueness theorem without proof, which also can be found in [9]. Theorem 1. Existence and Uniqueness of Nash bargaining solution: There is unique solution function r ∗ = ϕ(R, D) that satisfies all axioms of Nash bargaining game, and it satisfies r ∗ = arg max q

rk ≥ dk ∗ k

K 

(Dk −

k=1

qk ). rk

(5)

Maximize (5) leads to minimize dk , and not violate the delay bounds D. Note that the well known PF algorithm that only suitable for best-effort traffic, can be derived from the Nash bargaining solution as well with no specific delay requirement in (5). It shows that our model is also applicable to best-effort traffic. A. Problem Formulation As we described above, in traditional method [3]–[5], there is no correlation between two time slots because of no traffic requirements consideration. However, based on our system model, we have dynamically considered the delay bounds into the utility function, which are adaptively optimized among many time slots. Another difference is our model do not have the limitation on the number of users be served at each time slot, which simplifies the following algorithm design. Note that we omit all the constraints for some intermediate mathematical derivation, but we give them in detail in the final expression in (8). Hence, the problem can be formulated as max rk

K  k=1

qk (Dk − ), rk

max rk

K

k=1

qk ln (Dk − ), rk

subject to C1: C2:

K

k=1 pk ≤ Ptotal rk ≤ qk /Ts , ∀k

1 1 )¯ rk (t − 1) + rk (t), tc tc

(9)

where tc is the exponential moving average weight factor. Meanwhile, the delay requirement vector is updated by D(t) = D(t − 1) − Ts . To sum up, the bargaining game model for a MU-MIMO downlink system can be defined as follows: the system has weighted sum-rate with vector r as its system objection function, where all entries of r are bounded and have a nonempty feasible set. The goal is to minimize all dk simultaneously from an initial agreement point D. IV. M ODIFIED I TERATIVE WATER -F ILLING A LGORITHM In this section, we give an modified iterative water-filling algorithm based on the duality transformation method [10]. Unlike traditional algorithms for system capacity that need exhaustive search for all possible subgroup of users, our proposed algorithm automatically optimize for users that avoids the factorial search space. The problem in (8) is easier to solve with the Lagrange Multiplier. The KKT condition in [13] is utilized to reformulate the optimal scheduling policy. We have the following theorem. Theorem 2. Let pk be the power allocation both for user k. Then, they satisfy the conditions: pk = [

2

∇Ur¯(t − 1) σ 2 wk  + − ] μ ηk

(10)

Proof: The following Lagrangian is obtained with the Lagrange multiplier: L=

(7)

Define Ur = k=1 ln(Dk − qk /rk ). To find the maximum value of Ur (t) defined on variable r(t) = (r1 (t), . . . , rK (t)), we can find the maximum value of ∇Ur¯ (t − 1) · r(t) for the game, such that K  ∂Ur max |r¯ rk (8) rk ∂rk k k=1

r¯k (t) = (1 −

(6)

The problem in (6) is difficult to solve directly. Actually, the following expression is equivalent to (6) by K 

where Ptotal and Ts correspond to the total transmitting power constraint and time slot duration. Constraints C1 and C2 define the feasible set of r(t). In each time slot, the iterative algorithm can be viewed as selecting rate vector r(t) = (r1 , . . . , rK ) that has maximum projection onto the gradient of the function at optimal point r¯(t − 1), where ∇Ur (t − 1) is a concave gradient function of user’s average throughput r¯(t − 1) up to time t − 1, which can be updated by using an exponentially weighted low-pass time window with filtering factor tc , given as

K  ∂Ur

∂rk

k=1 K 

−

|r¯k rk − μ(

K 

pk − Ptotal )

k=1

(11)

ηk (rk − qk /Ts )

k=1

where μ and ξk are non-negative Lagrange multipliers. Differentiate (11) with respect to pk , by using the KKT optimality conditions,  ∂Ur ∂rk  ∂rk ∂L = |r¯k −μ− ηk = ∂pk ∂rk ∂pk ∂pk

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< 0 pk = 0 = 0 pk > 0 (12)

TABLE I S IMULATION PARAMETERS

That is, we meet the KKT conditions by having some users K r with the largest k=1 ∂U ∂rk |r¯k rk at each time slot. The solution is the global maximum because the problem is quasi-convex. From (12), the power allocation to user k can be obtained pk = [

2

2

∇Ur¯(t − 1) σ wk  + − ] μ η

(13)

V. S IMULATION R ESULTS We compared the performance of our proposed Delay Bounded Resource Allocation (DBRA), PF, and MW algorithms with respect to system throughput, packet loss ratio, and average delay under the condition of services with different delay requirements.

Parameters

Values

Cell radius

1000 m

System bandwidth

200kHz

Noise density

-174dBm/Hz

Path loss

128.1+37.6log10(d) (d in km)

No. of transmitting antennas (Nt )

4

No. of users

10

Exponential moving average

100

weight factor (tc ) 10−4

BER

A. Simulation Layout

B. Simulation Results We evaluated the performance of our proposed DBRA algorithm with MW and PF algorithms, which is two typical case in [7]. The utility functions and their objectives for these algorithms are listed as follows. qk r DBRA: U = ∂U 2 −q r ∂rk |r¯k = Dk r¯k k ¯k Objective: achieving the Pareto-optimal point with delay guarantee. PF: U = r¯1k Objective: achieving the Pareto-optimal point with proportional fairness. MW: U = qk Objective: achieving minimum average delay.

Utility (absolute value)

10

In our simulation, a single BS with multiple antennas serves multiple users. We assumed all the users would be fed by Poisson arrival traffic as illustrated in Figure 1. Arrival rate λk with different mean values was used to simulate different system loads. For simplicity, the packets size were exponentially distributed around a mean of 128 Bytes, and any packet larger than 576 Bytes was set to a maximum of 576 Bytes. The users were uniformly distributed within a cell and moved according to the random walk mobility model. All users in each time slot had a probability of 12 of holding still, otherwise, they moved randomly at a uniformly chosen speed of up to 2 m/s. Since the users were also randomly distributed, we modeled the multi-user MIMO channel including both large-scale and small-scale fading. lk and θk are defined as the distance from BS to user k and shadow fading,

respectively. Thus, the channel vector for user k is hk = βθk /lkα hw , where hw is a classical frequency-flat Rayleigh fading channel vector in which the entries are independent and identically distributed complex Gaussian random variables with zero mean and unit variance. β is a constant and α is the path-loss exponent. Shadow fading was modeled as a normal distribution with a mean value of 0 and a variance of 8 dB. The correlation distance for shadow fading was set to 10 m. The other simulation parameters are listed in Table I.

D