The Massive MIMO is one of the core technologies in future 5G wireless systems in which base stations (BSs) operate with a large number of antennas. Using many antennas at the BS provides very high beamforming gain and the capability of serving multiple users simultaneously via spacial multiplexing at the same time and frequency resources. Consequently, enormous spectral efficiency can be achieved , .
One of the main challenges for a TDD based Massive MIMO system is the accurate estimation of the channel of users. The estimation is realized through special uplink signals called aspilot signal or Sounding Reference Signal (SRS)111 In the rest of the paper, we use these terms interchangeably. allocated to users. Multiplexing and beamforming gain can be achieved only by the users with pilot signals, and the gain and the system performance degrade when those signals are not carefully assigned to the users. For instance, the users with high amount of traffic may be preferred to have pilot signals to maximize the network throughput. On the other hand, the practical problems that obstruct the pilot allocation must also be taken into account. Specifically, pilot signals are configured through RRC signaling  and the frequent reconfiguration of the pilots can cause an intolerable signaling overhead in practice. Therefore, in addition to a smart SRS allocation, the minimization of the frequency of SRS configuration needs attention.
Although the impact of reusing the same pilot signals on different cells (i.e., pilot contamination) has been well investigated in the literature, the allocation of these signals with the consideration of varying user’s traffic and channel conditions has received little attention. In , ,  and  the pilot allocation and scheduling are considered without the impact of the network traffic. In , the pilot allocation is done for only special messages and the results cannot be generalized.
Our contributions are summarized as follows: i-) We formulate the problem of the scheduling and pilot allocation for Massive MIMO networks with the associated signaling cost as a stochastic optimization problem under Lyapunov optimization framework; ii-) We develop an optimal algorithm that operates at two different time-scale and does not need the future knowledge of the system. We also derive analytical bounds on the performance of the optimal algorithm in terms of average queue size and the signaling cost; iii-) We implement a realistic network setting, and demonstrate the performance of our algorithm and depict the tradeoff between the signaling cost and the average queue size.
Ii System Model and Problem Formulation
We consider a Massive MIMO capable cellular network where there is a BS with antennas serving users. Let be the set of users with single antenna. The BS can transmit simultaneously up to users via its Multi-User (MU)-MIMO capability and . The system schedules users on time-slot fashion at each (e.g., regular scheduling decision that LTE performs at every 1 ms) and let be the number of users served simultaneously at time , . In this case, the transmit power for a scheduled user is , and is the total transmit power. The transmission rate for a user at time slot , for all and , is given by,
is channel vector of userin the downlink direction, and represents the complex transmit symbol for user . Also, is the normalized precoding vector of user and is zero mean complex Gaussian additive noise with power . Furthermore, B is the system bandwidth and is the fraction of the total time/frequency resource used for data transmission, and the fraction is donated for obtaining SRS. We note that depends on the length of the coherence block and pilot signals .
Let be the scheduling decision given for user at time slot . If user is scheduled then , else . We assume there are number of pilot signals that can be allocated among users, where . The decision for SRS allocation is taken at a larger time-scale denoted as . In every time slot of the form where , the decision denoted as is taken to decide whether user , , should have SRS or not. if , the system allocates SRS to user and it can benefit from the multiplexing and beam forming gain between . If , user cannot have a SRS till the next time . After deciding on for every user, a SRS configuration flag is set. If for all users (i.e. the same SRS allocation as previous time), then the decision on the reconfiguration of SRS denoted as is set to 0, otherwise . That is to say when is given for all users, is completely determined.
At each time slot , data randomly arrives to the queue of each users. Let be the amount of data (bits or packets) arriving into the queue of user at time slot . We assume that is a stationary process and it is independent across users and time slots, and for all and . We denote the arrival rate vector as , where . Let denote the vector of queue sizes, where is the queue length of user at time slot . A queue is strongly stable if . Moreover, if every queue in the network is stable then the network is called stable. The dynamics of the queue of user is
Let denote the capacity region of the system, which is the largest possible set of rates that can be supported by a joint scheduling and SRS allocation algorithm with ensuring the network stability.
We recall that configuring SRS allocation frequently (i.e., at every ) causes signaling overhead and costly but the SRS allocation should be also sufficiently adaptive and dynamic. Let be the cost when SRS allocation is reconfigured, i.e., . Then, the average cost is given . The control decisions of the system are for user . Let be the set of all possible control decisions. We consider the following optimization problem:
The problem (2)-(3)-(4) aims at minimizing the average cost by taking the scheduling and SRS allocation decisions optimally. The problem constitutes a stochastic optimization problem and we next propose a solution based on Lyapunov optimization technique.
Iii Joint SRS Configuration, Allocation and Scheduling
In our work, we use Lyapunov drift and optimization tools . The advantage of this tool is the ability to deal with performance optimization and queue stability problems simultaneously in a unified framework. We first define quadratic Lyapunov function as measuring the total queue size in the system. We then define the conditional T-slot Lyapunov drift that is the expected variation in the Lyapunov function over slots as follows:
where is system parameter that characterizes a tradeoff between performance optimization and delay in the queues. According to the Lyapunov optimization theory, the problem (2)-(3)-(4) can be reinterpreted as the minimization of (6) which can be done by first deriving an upper bound for (6) in the following Lemma.
Given , and at time , for any feasible decision, we have
The proof is given in Appendix A.
Now, our aim is to find a method that minimizes the right hand side of (7), and this is realized by maximizing the following term in (8) given in the following problem.
Opt 1: Maximize over :
It is clear that the solution of Opt 1 requires the prior knowledge of the future queue sizes and the data rates which depends on the future channel conditions over , and this knowledge cannot be obtained in practice. In order to overcome this issue, first we follow the idea in  and approximate the future queue sizes as the current observation, i.e., for all and . Then, we obtain a looser but more relaxed bound as follows.
Given , and at time , for any feasible decision, we have the following bound,
The proof uses the fact that for every
and uses these upper and lower bounds for on the R.H.S of (7) and the rest of the proof is similar to the proof of Lemma 1 and omitted here. Lemma 2 reveals that now the optimal control actions can be taken by maximizing the third term in the R.H.S of (9), which yields the optimal solution but with a higher average queue delay. However, we still need the future channel information (i.e., transmission rates ) to maximize the third term in the R.H.S of (9) optimally. Here, we exploit one of key benefits of a Massive MIMO system: under certain condition (i.e., Rayleigh fading channel) the fluctuation over the transmission rates due to small scale fading can be neglected and it only depends on the large-scale fading such as path-loss and shadowing. This is known as channel hardening effect  that occurs when the number of antennas at the BS is sufficiency large, which is the case for Massive MIMO systems. Hence, the user rates222 The large-scale fading can be measured by the BS over a longer time interval. become nearly deterministic and simplifies the scheduling and resource allocation problem. This channel characteristic and the result in Lemma 2 reduce Opt 1 to the following simple optimization problem:
In Opt 2, and are replaced by and due to the Lemma 2 and the hardening effect, respectively, and Opt 2 becomes a deterministic problem. We define as the set of users with SRS at the beginning of time and derive the following algorithm that solves Opt 2 optimally.
Joint Scheduling and SRS Allocation (JSSA):
Input: , , , , .
Step 1.1 (SRS Allocation): At every , among all users, for each set with size , do:
Set the transmit power to
where choose the first elements of a given set of numbers sorted in decreasing order. is called as the best set with size .
Find the weight
Step 1.2: Find
Step 1.3: Determine the set and . Set and the maximum weight .
Step 1.4: Repeat Step 1.1, Step 1.2 and Step 1.3 only for the users and determine the best and the maximum weight denoted as and as Step 1.2 and Step 1.3, respectively.
Step 1.5: if , then reconfigure SRS allocation and assign SRS to the users in and update . Otherwise, do not change the SRS configuration.
Step 2 (Scheduling): If the condition in Step 1.5 is satisfied, at every , schedule the users in , and otherwise, schedule the users in .
JSSA decides on how many and which users must be chosen among all users and determines that it is worth to reconfigure SRS by comparing the performance achieved among the users that have already SRS. As JSSA optimally minimizes the R.H.S of (9) with the hardening effect, we have the following Theorem that shows performance of JSSA.
(Lyapunov Optimization) Suppose is an interior point in , and there exits such that . Then, under JSSA, we have the following bounds:
Where is the optimal solution of problem (2)-(3)-(4).
To avoid redundancy with existing literature, we omit the details here. The sketch of the proof is as follows: it follows similar steps in Theorem 5.4 of  by first showing the existence of a stationary randomized algorithm that is optimal and achieves the minimum time average cost by choosing the control actions independently from
but according to a fixed probability distribution known to the system. Then, it is shown that JSSA is better than the randomized algorithm in minimizing the R.H.S of (9) and thus it is also optimal. Theorem 1 implies that the average cost under JSSA approaches to the optimal costas increase, while the average queue sizes also increase.
Iv Numerical Analysis
In our simulations, there is a single cell covering a square of 250 m x 250 m area with a Massive MIMO capable BS. We set and there are users uniformly and independently distributed in the cell. We adapt the same channel models and take the related parameters given in  for large-scale and NLOS fading. We apply MMSE precoding, and set MHz, and Watts. We assume that millisecond, and at each time slot users send their SRS to the BS, and . 3GPP FTP Model 3 is considered, where user traffic follows Poisson arrival process with a payload size 0f 0.2 MB and different mean arrival rate varying between 0.5 and 2 seconds.
We first show the performance of JSSA when ms. Figure 1 depicts the time average (every 100 ms) total network throughput achieved by JSSA with different values of . Modified JSSA (M-JSSA) configures SRS without any cost at every and it constitutes a benchmark to the performance of JSSA. When , JSSA achieves almost the same throughput as that of M-JSSA and the average number of SRS configuration is approximately 0.94, which implies that more than 90% of the time JSSA attemps to reconfigure SRS and hence the cost is high. We observe that when the average total queue size approaches to a fixed point as shown in Figure 2, which means the network is stabilized .
As increases to , the throughput achieved by JSSA and the benachmark approach to the same value, which reveals that the network is still stable. However, the average number of SRS reconfiguration is reduced to 0.3, so the cost decreases. We also observe from Figure 2 that the average total queue size is higher when compared to the case with , that is aligned with the theoretical result found in Theorem 1.
We have investigated the problem of SRS allocation and scheduling problem in a single cell Massive MIMO network. By applying Lyapunov optimization tool, we have developed a joint scheduling and SRS allocation algorithm that can perform well under random traffic and channel conditions. In simulation results, we show that the average signaling cost can be reduced at the expense of an increase in the average queue delay. The problems of the SRS allocation with different objectives (i.e., reducing delay) with fairness can be other research directions. Also, SRS allocation in a multi-cell setup with pilot contamination would be an interesting research problem.
Appendix A Proof of Lemma 1
The proof starts with finding an upper bound for the Lyapunov drift given in (5) by using the following fact: for user , the following inequality holds.
By summing (12) over and knowing that and for all and we obtain,
Then, by taking the conditional expectation of (13) with respect to and summing over all users, and dividing by 1/2 we have,
Finally, we add the penalty term to both sides of above inequality and rearranging the resulting terms, we have the bound in Lemma 1. This completes the proof.
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