SparsityCognizant MultipleAccess Schemes for Large Wireless Networks With Node Buffers
Abstract
This paper proposes efficient multipleaccess schemes for large wireless networks based on the transmitters’ buffer state information and their transceivers’ duplex transmission capability. First, we investigate the case of halfduplex nodes where a node can either transmit or receive in a given time instant. In this case, for a given frame, the transmitters send their buffer states to the destination which assigns the available time duration in the frame for data transmission among the transmitters based on their buffer state information. The network is said to be naturally sparse if the number of nonemptyqueue transmitters in a given frame is much smaller than the number of users, which is the case when the arrival rates to the queues are very small and the number of users is large. If the network is not naturally sparse, we design the user requests to be sparse such that only few requests are sent to the destination. We refer to the detected nonemptyqueue transmitters in a given frame as frame owners. Our design goal is to minimize the nodes’ total transmit power in a given frame. In the case of unslottedtime data transmission, the optimization problem is shown to be a convex optimization program. We propose an approximate formulation to simplify the problem and obtain a closedform expression for the assigned time durations to the nodes. The solution of the approximate optimization problem demonstrates that the time duration assigned to a node in the set of frame owners is the ratio of the squareroot of the buffer occupancy of that node to the sum of the squareroots of each occupancy of all the frame owners. We then investigate the slottedtime data transmission scenario, where the time durations assigned for data transmission are slotted. In addition, we show that the fullduplex capability of a node increases the data transmission portion of the frame and enables a distributed implementation of the proposed schemes. Our numerical results demonstrate that the proposed schemes achieve higher average bits per unit power than the fixedassignment scheme where each node is assigned a predetermined fraction of the frame duration.
I Introduction
Motivated by the goal of increasing spectrum efficiency, we propose a set of efficient multipleaccess schemes for large wireless networks. When the number of transmitters is large, the communication between the nodes consumes most of the available time and spectrum resources. Therefore, designing an efficient scheme for such scenarios is of great interest.
The compressive sensing (CS) paradigm is an efficient approach that enables the receiver to reconstruct the original sparse signal from few noisy observations [1, 2]. The authors in [3, 4] use phasecoherent analog transmission of randomlyweighted data from sensor nodes to the fusion center. The additive property of the multipleaccess channel enables projections of data onto an appropriate basis at the fusion center. The authors of [5] investigated a randomaccess CSaided scheme for underwater sensor networks. Their goal was to design a powerefficient random data collection scheme. A CSaided medium access control (CSMAC) scheme is proposed in [6] where the access point (AP) allocates a random sequence to each user. All user requests for gaining uplink transmissions access are sent simultaneously in a synchronous manner. The authors of [7] studied the asynchronous scenario of [6]. The data arrival dynamics and queue states were not considered in [6, 7].
In this paper, we design efficient sparsityaware multipleaccess schemes for large wireless networks with node buffers. The contributions of this paper are summarized as follows. We use the CS techniques to realize our proposed MAC schemes. We investigate the scenarios when the system is naturally sparse and when it is nonsparse. We investigate the case of unslottedtime data transmission. Based on the buffer state information (BSI) at the transmitters, the destination assigns equivalent time durations to each transmitter. We investigate the case of slottedtime data transmission system, where data are sent to the destination in a slotted manner, and design the system and its sparsity level based on the available number of data time slots per frame. To avoid wasting the data time slots, we propose two solutions for the case when the number of detected users per frame is lower than the available data time slots. We investigate the cases of half/fullduplex nodes and show that the fullduplex capability enables the nodes to implement the proposed schemes in a distributed manner and reduces the needed time for control signals overhead.
Notation: denotes the set of complex vectors of size . denotes the absolute value. The operators , , denote norm, norm, and norm of a vector, respectively. In addition, , , and denote the Hermitian (i.e. complexconjugate transpose), inverse, and transposition operations, respectively. A summary of the key variables adopted in this paper is provided in Table I.
Ii CS Background
In this section, we introduce the CS principles. A comprehensive treatment of the subject is given in [8] and the references therein. CS approximations were originally stimulated by the idea that natural signals are often sparse in some domain. That is, a signal can be represented by a few significant coefficients in some representative domain. For instance, natural images are sparse in the frequency domain. On one hand, traditional compression can exploit this sparsity to reduce the signal size, however, this requires statistical analysis of the entire signal to find the significant coefficients. On the other hand, the CS approach removes this requirement. For a given sparse signal of sparsity level , i.e., it has at most significant (nonzero) coefficients, we can capture the coefficients without the need to collect the complete data set of some large size .
Consider the system of equations where is a known measurement vector, is an unknown vector, denotes the measurement matrix, and is a bounded noise (error) vector. If the measurement matrix satisfies the socalled restricted isometry property (RIP) condition [8, 9], the sparse vector can be recovered with a number of measurements on the order of measurements, where is the sparsity level of . To obtain the sparsest solution to this system of equations, the following problem is solved
(1) 
where is chosen such that it bounds the amount of noise in the measurements and is the number of nonzero entries in . In general, finding the optimal solution to this problem is not computationally efficient. Hence, two main approaches have been proposed in the literature to efficiently compute a sparse suboptimal solution to this system of equations; specifically, norm minimization and greedy algorithms. As shown in the CS literature, we can recover the sparse signal using the following constrained norm minimization problem [9]
(2) 
However, the solution of the optimization problem is not exactly sparse because many nonzero entries with nonsignificant values can typically appear in Ë. An additional heuristic optimization steps can be applied to enforce a finite number of nonzero entries as in [10]. Alternatively, the greedy algorithms provide more control on the set of nonzero elements whose indices and values are determined iteratively. We describe the orthogonal matching pursuit (OMP) algorithm [11] as one of the widelyused greedy algorithms. OMP takes , , and a certain stopping criterion as its inputs and computes a sparse solution for the unknown vector as its output. Hence, we denote the OMP operation by . The stopping criterion can be a predefined sparsity level (number of nonzero entries) of or an upper bound on the norm of the residual error term .
Iii System Model and Proposed Schemes
We assume an uplink scenario where a set of buffered transmitters wishes to communicate with a common destination (basestation) as shown in Fig. 1. The set of transmitting nodes is labeled as and the destination is denoted by . We consider buffered nodes where node has a finitelength buffer (queue), denoted by , whose maximum size is packets. The size of a packet is bits. Each source node transmits its buffer state (i.e. the number of packets stored in its queue) to the destination. We assume Gaussian channels
Assuming that the average transmit power of transmitter per unit frequency is Watts/Hz, the channel capacity of link (i.e. the link connecting transmitter and the basestation) is then given by
(3) 
If transmitter wishes to send bits, the time required for such transmission, given a channel bandwidth of Hz, is given by
(4) 
Each transmitter sends its buffer state as a single signal with an amplitude corresponding to its value in packets. The used constellation points are , where is the maximum number of data packets that can be stored at any of the queues. The time needed for bits transmission, , is obtained at the basestation upon demodulating the signal representing the number of bits stored in using the expression in (4). If the system is sparse or can be made sparse using a request probability as will be explained later, the basestation then knows the time duration needed for each node using CS approximations. In this case, the communication of the buffer state information requires only measurements. If the system can not be made sparse, orthogonal signals should be used by the transmitters for their communication with the basestation.
If the system is sparse, the buffer state signal sent by a user is modulated (multiplied) by a sequence (e.g. Bernoulli, Gaussian, or Gold codes) of length bits. The sequences of all transmitters represent the measurement matrix which is used by the destination to resolve the transmitter requests. The transmitters request vector received at the destination is
(5) 
where is the measurement vector received at the destination, is the user buffer states vector with representing the buffer state of transmitter , is the measurement matrix adopted by the system, and is the additive noise vector at the destination whose size is . We assume that the transmitters send their requests with sufficient powers such that the requests are decoded perfectly at the destination as long as the requests are less than or equal to the system sparsity level. Thus, if the number of requests is less than or equal to the sparsity level, the nonzero components are decodable with probability .
It should be mentioned here that the output of the CS solver is used as follows. The location of the nonzero element represents the transmitter index while the demodulated value represents the number of packets (or bits) stored at the sender’s buffer. Since the output may contain errors, we assume that the output for the th user is bits (i.e. estimated value of bits), where is the output corresponding to the th user in packets and is the packets size in bits. Note that must be one of the constellation points used to represent . If user has no data to send or decides not to access the channel (i.e. decides not to send a request which occurs with probability ), and, hence, . The corresponding time for sending the bits is . The time needed by all transmitters, from the basestation viewpoint, is . In a given frame, we refer to the detected nonemptyqueue transmitters as the frame owners. The destination sends the time durations to the frame owners. If the received average signaltonoise ratio (SNR) at the destination is high, we can assume that the destination knows the buffer states at the transmitters perfectly. Hence, and .
Let denote the number of packets that arrive at the buffer of transmitter at the end of frame , which can follow any distribution, with average rate packets per frame (packets/frame). At the end of frame (i.e. beginning of frame ), the number of packets remaining at the th transmitter’s queue is
(6) 
where is the buffer size at the beginning of frame and is the buffer maximum capacity in packets. Since the queue states are detected accurately at the destination, .
It should be noted that the random characteristics of the arrivals at transmitters increase the sparsity of the system. This means that the sporadic nature of the arrivals increases the possibility of having many emptyqueue transmitters and hence lower requests. Assume Bernoulli arrivals with mean for all queues,
(7) 
where denotes choose . The system is sparse with sparsity level if this probability is very small which is the case as long as is very small and is large. In case of other arrival distributions (i.e. nonBernoulli arrivals), in Eqn. (7) is replaced with the probability that the queue has at least one packet and is the probability of having no packets (i.e. the queue is empty). If the system is sparse due to large and very small , all users can attempt to send their queue states to the destination in every frame.
When the system is not naturally sparse, exploiting the request probability is necessary to ensure sparsity of the system. In this case, the system sparsity condition is
(8) 
where is the sparsity level and is a very small positive number. This condition is based on the worst case scenario when all transmitters have data to send. Designing the system based on this criterion guarantees the sparsity of the system in all other cases of queue occupancies (i.e. when a set of nodes is empty and the other sets are not empty). The optimal is the highest feasible value of that satisfies the sparsitylevel constraint in (8). This is because as increases, the number of served users increases. However, this also decreases the ability of the destination to decode the users request vector using only measurements.
The sequences used by the transmitters can be seen as randomaccess codes that are used to distinguish between the users at the destination. Our goal is to accommodate as many transmitters as possible; hence, we increase the length of the sequences. However, longer sequences lead to longer communication time, i.e., longer time needed to capture more observations at the destination to determine all the nonemptyqueue transmitters, and consequently a lower data transmission time is achieved.
Iiia UnslottedTime Data Transmission
In this scenario, we assume that the communication between the transmitters and the destination is divided into three stages. In the first stage, the transmitters send their buffer states to the destination. In the second stage, the destination assigns the transmission time to each user. In the third stage, the frame owners transmit their data to destination. The frame structure under this scenario is shown in Fig. 2. Our proposed communication protocol during a frame for the unslottedtime data transmission scenario is described as follows:

If the system is naturally sparse, i.e., satisfies the condition in Eqn. (7), the users send their BSI to the destination in each frame using CS approximations without using the request probability.

If the system is not naturally sparse, the transmitters probabilistically send their requests to the destination. The access probability is .

Upon decoding the buffer state vector, the destination sends feedback signals to the transmitters which contain the time assignments among the transmitters. This can be implemented as follows. The destination sends an acknowledgement (ACK) signal with the identification (ID) of the transmitter which is assigned to the next dedicated time duration. Then, the length of the time duration assigned to that user is broadcasted. Each of these activities requires seconds to be implemented; hence, the total time spent in identifying the next transmitter and its transmission time declaration is time units. We assume that the errors in decoding these values at all nodes are negligible.
^{3} 
Each transmitter sends its data during its assigned transmission time.
After sending the transmitter requests and the feedback signals, the remaining time for data transmission is , where is the frame duration, is the total number of nonemptyqueue transmitters during the current frame (i.e. number of frame owners), and is the total time used to inform the frame owners about their dedicated time durations.
The optimization problem that minimizes the total transmit power of nodes under the constraint that the sum of the users’ transmission times is at most is stated as follows:
(9) 
The objective function is linear in . The second derivative of the term is
(10) 
The second derivative is always positive for ; hence, is convex in . Since the positive weighted sum of convex functions is also convex, the constraint is convex. Since the objective function is linear and the constraint is convex, our optimization problem is a convex program and can be solved efficiently using any convex optimization solver [12]. To gain further insights and derive closedform expressions for the time assignments, we seek an approximation for the optimization problem in (9). Letting , where , . Thus, the optimization problem is rewritten as
(11) 
Eliminating the constant terms from the objective function, the optimization problem is rewritten as
(12) 
Taking the logarithm of base for the objective function, and using Jensen’s inequality [13], the objective function of (12) can be upper bounded as follows:
(13) 
The approximate optimization problem is stated as follows:
(14) 
This approximation is tight when is very small; which is the case when the buffer capacity is much less than the bandwidth, i.e., . Noting that increasing the time assigned for data transmission decreases the objective function, the constraint in (14) holds with equality. That is, . Using the Lagrange multipliers, we get
(15) 
where is the Lagrange function and is the Lagrange multiplier. Taking the first derivative of with respect to , we get
(16) 
Equating this value to zero and setting , we get
(17) 
Using the constraint that , we calculate as follows:
(18) 
Thus,
(19) 
Substituting (19) into (17), the optimal that minimizes the approximate optimization problem is given by
(20) 
The solution of the approximate formulation illustrates the fact that the users with the higher queue occupancies should be assigned longer time durations. It should be noted here that the achievable rate under this scheme is bits/frame. If all frame owners have the same queue occupancy, the optimal solution will be , i.e., splitting the remaining time in the frame, , equally among the frame owners.
Using the optimal solution in (20), the total transmit power in this case is
(21) 
The value is an upper bound on the total transmit power. Since the queues are limited in size, assuming that all queues are full of data packets, the maximum value of , denoted by , is given by
(22) 
where is the number of active transmitters.
If we assume a virtual buffer with a total number of bits in frame , the power needed to send the bits during the data transmission time of the frame, , is given by
(23) 
where is the minimum power level needed to send all bits. It should be noted that this scenario, where a virtual buffer maintains a total number of bits equal to the number of bits in the frame owners’ buffers, is an upper bound on what can be achieved in the original scenario where we have a set of nodes (i.e. frame owners) each of which has its own data packets stored at its own buffer and wishes to send them to its destination. Hence, the transmit power in this virtual system is a lower bound on the transmit power of the original scenario. To demonstrate this point, we study the following example. Assume that we have only two frame owners, say nodes and , where each of them has only one packet of size bits, i.e., bits. In this case, and since we have only two nodes with equal buffer sizes, the time duration is divided equally between them. Hence, the total transmit power is
(24) 
If we assume that the two packets are stored in a single buffer (i.e. virtual buffer of a total size equal to bits), the needed power to transmit bits over time units is
(25) 
Based on the above argument, the total transmit power is lower bounded as
(26) 
The total transmit power obtained from solving the original optimization problem in (9) may take any value between and . That is, .
IiiB SlottedTime Data Transmission
In this scenario, we assume that the communication between the transmitters and the destination is divided into three stages. In the first stage, the transmitters attempt to send their buffer states to the destination. In the second stage, upon getting the compressed signal of the buffer states, the transmitters are assigned to a set of data time slots each with duration time units. In the third stage, the users assigned to the available data time slots send their data to the destination. The frame structure under this scenario is shown in Fig. 3. The assignment of time slots to transmitters depends on the number of frame owners and the number of data time slots per frame. That is,

If the system is designed such that the number of requests is always lower than or equal to the number of data time slots, i.e., , then all decoded requests will be satisfied, i.e., each transmitter gets one time slot. However, in this case, we may have some unused data time slots if the number of frame owners is lower than the number of data time slots, .

If the number of requests (i.e. frame owners) exceeds the number of available data time slots, the frame owners with highest queues occupancy will be selected for data transmission.
In this system, the destination communicates with the users during the control signals exchange in an unslotted manner. Note that increasing allows selection of more transmitters such that the users with high queues occupancy can be detected adequately. The access probability can be increased and hence the detected users (requests). Adding one additional user to the problem results in time loss that increases the sparsity level to instead of . Since the users probabilistically send their BSI to the destination in each frame, we may have a number of frame owners less than as explained in the abovementioned cases. Hence, some time slots may be wasted. We propose two solutions to this problem.

Since losing a time slot with duration can be worse than losing a time duration of when , where is a positive integer and is the time needed to increase the sparsity level from to for accepting additional users per frame, we can increase the number of users per frame by increasing the request probability, . However, since we have only data time slots per frame, more requests result in less time for data transmission per time slot, i.e., the data time slot duration, , decreases as number of requests increases. Since the number of frame owners can be less than , some time slots may be unutilized in a given frame. Hence, the cardinality of the set of users that will use the available time slots in a given frame is , where . In this approach, the destination does not need to send the transmission times and instead it sends the value of in the current frame and broadcasts the IDs of the users that will use the available time slots. Accordingly, the communication of the control signals between the destination and the users requires time units, where is the time consumed for announcing the value of and is the time needed for announcing the IDs of the group of users that will use the data time slots. Consequently, the data time slot duration is . Note that the control signal exchange takes place over the time interval from instant to instant and data transmission takes place over the time interval from instant to instant . Using this approach, the complexity of the system is reduced and the time slots loss and time durations broadcasting is reduced as well. One may optimize both the number of time slots per frame, , and the system sparsity level, , to maximize the system throughput. This approach is concerned with maximizing the number of transmitted bits and serving as many users as possible.

An alternative solution, which is concerned with the total transmit power per frame, is to minimize the transmit power by assigning the time slots to the frame owners such that the total transmit power is minimized.
Specifically, if the number of decoded requests is less than the sparsity level , we assume that the destination assigns multiple time slots to some users based on their queue states. Hence, users use lower transmit powers for data transmissions. For this powersaving approach, the nodes can be assigned as follows. Each participating node gets a slot; afterwards, the queues with the highest occupancy among the frame owners are assigned to the remaining time slots and so on until the end of all time slots. In this approach, if , the destination announces the value of , the users’ IDs, and the number of time slot assigned to each user. Hence, the communication of the control signals between the destination and the users requires time units. Consequently, the duration of a data time slot is . If , which means that each user will be assigned to only one time slot, the destination does not need to send the number of time slots assigned to each user; hence, the communication of the control signals between the destination and the users requires only time units. Consequently, the duration of a data time slot in this case is .
In the second abovementioned approach, the transmit power of transmitter when it sends bits over its assigned data time slots, which can be time slots, is given by
(27) 
The total transmit power per frame is thus given by
(28) 
It is clear that the transmit power is exponentially decreasing with . For a given set of frame owners, and are fixed; hence, is the only term that controls the value of the transmit powers. To maintain the transmit power low, the destination increases the number of time slots assigned to the users with high queues occupancy over the users with low queues occupancy. Thus, we conclude that users with higher queue states must be allocated more time slots than other users. If the number of requests is greater than , the destination cannot assign more than one time slot to any user. However, if the number of requests is less than , we can assign more time slots to one or more users based on their buffer states.
To maximize the performances of the proposed schemes, we employ the following algorithm. For each request access probability, , we obtain the sparsity level by solving Eqn. (8). Then, for each sparsity level, we choose the minimum cardinality of the observations, i.e., , such that the probability of users request decoding is close to . Using the resultant sparsitymeasurements pairs, we can optimize the system performance efficiently.
In what follows, we investigate the case of fullduplex nodes [14, 15, 16] which enables a timeefficient communication between the users and their destination.
In this case, each node has two radiofrequency (RF) chains and is equipped with one antenna for each RF chain.

For the unslottedtime data transmission scenario, we let transmitter be the one which uses the spectrum during the time duration from instant to instant , transmitter is the one which uses the spectrum from instant to instant , and so on until the end of the frame. Each transmitter starts and ends its data transmission according to the predefined transmissions sequence.

For the slottedtime data transmission scenario, given that the queue states are known at all nodes, the nodes access the channel based on a predefined order for channel accessing until each transmitter uses only one slot. Afterwards, if the number of frame owners is lower than the number of time slots per frame, the assignment of transmitters to the remaining time slots is based on the queues occupancy until the end of the frame. Using these approaches for the unslotted and slottedtime data transmission scenarios, there is no need for the feedback signals from the destination to the users which consume a large portion of the frame time, especially when the number of users is large.
We emphasize here that the residual selfinterference caused by the transmission and reception of data at the same time and over the same frequency band does not have any effect on our analysis. That is, let us assume that the selfinterference is modeled as a fading coefficient with value where is a circularlysymmetric Gaussian random variable that remains constant during a frame but changes from one frame to another [14, 15, 16]. If the user sends zero, which is the case when its queue is empty or it decides not to access, the signal received at the user due to selfinterference is zero. Hence, the selfinterference has no impact on the received signal at the user’s receiver. If user sends a request, the received signal at user is given by
(29) 
where is the measurement vector received at the th user’s receiver, is the user buffer state vector at the th user, is the measurement matrix adopted by the system, and is the additive noise vector at the th user’s receiver. The OMP output is the same as in the case of no selfinterference but the th index is instead of . Given that the user knows its own state, i.e., knows that its transmitted signal is , it does not need to know the value of . Note that all users and the destination solve the same problem in (29) with different values of measurements, noise, and residual selfinterference coefficient.
Iv Simulations
In this section, we evaluate the performances of our proposed schemes. For convenience, we denote the CSaided schemes for the unslottedtime data transmission and slottedtime data transmission by and , respectively. The case where each user sends its queue state individually for seconds under the unslotted and slotted scenarios are denoted by and , respectively. We model the arrivals at the queues as Bernoulli distributed random variables with mean for all queues. For the measurement matrix, we assume that Bernoulli sequences are used by the transmitters.
In Fig. 4, we plot the request probability, , for users versus both the sparsity level and minimum measurements needed for a given sparsity level when the probability of decoding one of the elements in the request vector is less than . The sparsity level, , is obtained from Eqn. (8) with . From the figures, and since the sparsity level must be much lower than the measurements length as well as the number of users, the maximum request probability must be less than , i.e., , otherwise, we need observations (measurements) to solve the compressed signal. For the parameters used, we find that the ratio between and is around . Thus, using , where , is a very reasonable approximation of the optimal for a given sparsity level. This matches the observations in the literature that the number of measurements is empirically given by , where [6].
Unless otherwise explicitly stated, we assume that Figs. 5, 6, 7, and 8 are generated using a probability of decoding one of the elements in the buffer state vector to be less than , , users, Watts/Hz, maximum buffer size packets, number of bits per packet bits, MHz, and ms.
The case of unslottedtime data transmission with halfduplex nodes are considered in Figs. 5 and 6. Fig. 5 demonstrates the impact of the queues arrival rates on the sum of the average powers used by the transmitters. The total power increases with since the queues occupancy increase and hence the power needed to deliver the bits to the destination. The fixedassignment scheme has the highest transmit power since each transmitter has only of the frame duration to transmit its bits. Thus, its bits per unit power is small. The figure reveals that the analytical solution of our approximate formulation is close to the optimal solution of the total power minimization problem.
In Fig. 6, we plot the number of transmitted bits per unit power for the proposed schemes. Our proposed schemes have much higher bits per unit power than the fixedassignment scheme. The CSaided scheme, where we send requests and exploit the system sparsity, outperforms the case when each user sends its buffer state individually. The fixedassignment scheme has the lowest average number of bits/Watts which is bits/Watts. For high arrival rate at the queues, the sum of the average powers per frame for is almost equal to that of the fixedassignment. This is because at high almost all queues have the same occupancy and therefore the optimal solution is to assign of the remaining time in the frame to each buffer. Accordingly, the performance of is equal to the fixedassignment scheme.
Figs. 7 and 8 study the case of slotted transmissions for fullduplex nodes. Fig. 7 shows the average number of transmitted bits per unit power for the proposed schemes versus the frame duration, . The figure considers the case when the system is naturally sparse, i.e., when the arrival rates to the queues are small and the number of users is large. The mean arrival rate of is packets/frame. Since packets/frame and , the system is naturally sparse with sparsity level and the needed measurements is samples. The case of CSaided scheme outperforms the case when each user sends its queue state separately. The number of bits per seconds under the parameters used in Fig. 7 is shown in Fig. 8. We note that as the frame duration increases, the average number of bits per second decreases. This is because, for given queue states detected by the destination, all bits will be delivered to the destination during the frame time regardless of the frame length. The only difference is the transmit power needed to deliver the bits. As the frame length increases, the needed power decreases. For Figs. 7 and 8, we assume that the number of data time slots per frame is .
V Conclusions
In this paper, we have proposed new multipleaccess schemes for large wireless networks when the nodes are equipped with data buffers. We showed that the system is naturally sparse when the arrival rates to the queues are small and the number of users is large. In this case, we can directly apply the CS sparse signal recovery techniques to obtain the queue states of the transmitters at the destination with few measurements (i.e. signaling time to inform the destination regarding the state of each buffer). We designed the system for the nonsparse case and enforced the user requests to be sparse using the appropriate request probability. For the total transmit power minimization problem, we showed that the problem is convex. Moreover, we proposed an approximate formulation which has a closedform solution and showed that it achieves a minimum total transmit power per frame close to that of the original optimization problem. Our numerical results demonstrated that the proposed schemes achieve higher average bits per unit power than the fixedassignment scheme, where each node is assigned to of the frame duration. In addition, the average number of bits/unit power increases with increasing the frame time and/or transmit bandwidth. However, the number of delivered bits per unit time to the destination decreases with increasing frame duration, .
Footnotes
 For the case of fading channels, the only difference is that the transmit powers will be scaled by the fading coefficients. That is, if the power used by user is without fading, the power with fading is , where is the fading coefficient between node and the destination. To avoid very large transmit powers specially at very low fading gain, i.e., very low , we suggest two modifications to the system: (1) we make the users with very low fading gains OFF/silent during the current frame. This increases the system sparsity level. We can adjust the request probability based on the fading value, (2) we put a threshold on the maximum transmit power such that the users either transmit data corresponding to the maximum transmit power or remain silent if the needed power is higher than the maximum level.
 If all queues have different average arrival rates, we replace with the maximum among all arrival rates. That is, .
 In practice, the probability of decoding error is very small. For example, using QAM, the bit error rate is when the SNR is dB.
 The implementation of fullduplex communications using a single antenna for both data transmission and reception can be performed as in [16].
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