Interference-Aware Scheduling for Connectivity in MIMO Ad Hoc Multicast Networks
We consider a multicast scenario involving an ad hoc network of co-channel MIMO nodes in which a source node attempts to share a streaming message with all nodes in the network via some pre-defined multi-hop routing tree. The message is assumed to be broken down into packets, and the transmission is conducted over multiple frames. Each frame is divided into time slots, and each link in the routing tree is assigned one time slot in which to transmit its current packet. We present an algorithm for determining the number of time slots and the scheduling of the links in these time slots in order to optimize the connectivity of the network, which we define to be the probability that all links can achieve the required throughput. In addition to time multiplexing, the MIMO nodes also employ beamforming to manage interference when links are simultaneously active, and the beamformers are designed with the maximum connectivity metric in mind. The effects of outdated channel state information (CSI) are taken into account in both the scheduling and the beamforming designs. We also derive bounds on the network connectivity and sum transmit power in order to illustrate the impact of interference on network performance. Our simulation results demonstrate that the choice of the number of time slots is critical in optimizing network performance, and illustrate the significant advantage provided by multiple antennas in improving network connectivity.
Ad hoc networks, MIMO networks, interference networks, scheduling, beamforming, connectivity
Interference management in co-channel ad hoc networks is a challenging problem. Simple time-division multiple access (TDMA)-based designs are inefficient and usually result in relatively poor performance. While multiple antennas are useful in enhancing channel gain and reducing interference, incorporating the extra degrees of freedom offered by MIMO (multi-input, multi-output) nodes into the design of the network further complicates matters. Designs based purely on spatial-division multiple access (SDMA) are not appropriate for large networks, since the number of available antennas is usually insufficient for cancelling all of the co-channel interference. Consequently, space-time (STDMA) solutions must generally be employed, in which multiple network links are scheduled into each time slot, and beamforming techniques are used within each slot to mitigate the resulting interference.
Prior work that has addressed STDMA scheduling for ad hoc networks has typically focused on finding solutions that maximize the sum throughput of the network [1, 2, 3, 4]. However, such solutions inevitably lead to links with poor performance and localized network congestion, which cannot be tolerated in applications where the network must perform multicast streaming . An alternative is of course to use techniques that ensure fairness (i.e., max-min rate, proportional-fair scheduling, etc.) [6, 7, 8], but such techniques typically do not directly address network reliability. Performance may be fair, but how likely are the links able to achieve this performance?
The goal of this paper is to suggest methods for addressing the above issues using physical (PHY) layer techniques in combination with interference-aware scheduling. We introduce a novel definition of network connectivity that quantifies the probability that all links in the ad hoc network are able to achieve a certain pre-specified throughput. The PHY-layer parameters (beamforming vectors, transmit powers) and the scheduling decisions are then made to maximize this connectivity metric, taking into account the interference produced by links that are simultaneously active. The scenario we have in mind is ad hoc multicast network streaming, in which a source node attempts to transmit a continuous data stream to all other nodes in the ad hoc network with maximum reliability. Applications of this problem include tactical military networks (source is the unit commander, other nodes are teams or individuals under its direction), sensor surveillance networks (source is a sensor streaming data from a detected event, network nodes are monitoring stations attempting to form a coordinated response), vehicular ad hoc networks (VANETs), etc. For example, a typical VANET scenario involves a vehicle in the network that detects an unsafe road condition that must be reported to all other network nodes. In such a case, throughput is not the most important aspect of the network, since the actual message may be relatively short. Instead, the reliability of the network in sharing the message from the source with all the vehicles in the network is the key to achieving safety [9, 10]. Mobile sensing with distributed platforms (e.g., ground-based robots or UAVs) is another VANET application where connectivity is more critical than throughput, at least until a detection occurs. In sensing mode, such a network must continually be connected in order to arrive at consensus regarding the parameter(s) being sensed, or until one of them detects an object of interest, in which case the network may reconfigure itself for high-throughput data gathering.
Connectivity is fundamentally different from both throughput and energy consumption, which are the metrics most commonly used to quantify the performance of a wireless network. Connectivity performance is more relevant for applications where robustness or reliability is the most critical factor, applications (e.g., such as in military or emergency response scenarios) where the overriding concern is ensuring that information is shared with all nodes in the network at some pre-determined minimum rate. Throughput and energy consumption do not address network robustness. Instead, we argue that the connectivity of the network, as defined in the paper, is a reasonable way to measure the network’s robustness, and thus we set about choosing the network parameters to maximize the connectivity metric. A solution that achieved a higher throughput or lower energy consumption than our solution would necessarily have poorer connectivity and hence less robustness, and thus be a less desirable operating condition for the type of applications we assume.
Our focus is on how the use of multiple antennas in the network can lead to dramatic improvements in connectivity. The additional spatial degrees of freedom offered by the antennas reduce interference between the links, they allow a “denser” scheduling of users for a given set of resources, and they reduce the amount of transmit power (and hence interference to other links) to achieve a given throughput. The combined gain of these effects can only be observed by considering the joint optimization of the transmit beamformers, scheduling and power control. Unlike individuals with hand-held communication devices, VANET nodes are typically not power constrained and are large enough to support the use of multiple antennas. Thus, vehicular communication systems are a natural platform on which to consider the performance of multiple-antenna communications. Since mobility is a defining aspect of VANETs, it is important to account for the impact of the resulting time-varying wireless channels. Unlike most work in multiple-antenna ad hoc networks, we explicitly take the time-varying channels into account and quantify their impact on the reliability (connectivity) of the network.
This work draws on ideas and techniques that have been studied by many others, but in different contexts, including connectivity, beamforming for interference networks, and interference-based scheduling. Relevant prior work in these areas is briefly discussed below.
Investigations of radio network connectivity have been conducted by researchers over the last several decades. The original connectivity paradigm was expressed in terms of the so-called “geometric disk” model and percolation theory. In the geometric disk model, two nodes are assumed to be directly connected if their distance is smaller than some minimum transmission radius. This results in a simple binary description of connectivity (i.e., the network either is or is not connected) but lacks an indication of the quality of information flow. Percolation theory revolves around finding node density conditions under which a given node belongs to an unbounded cluster of connected nodes [11, 12]. However, neither of these approaches is reasonable for realistic networks, where one must consider the effects of fading and interference. The impact of fading on network connectivity has been addressed in [13, 14, 15, 16]. Of particular interest to this paper is , which showed that multiple antennas can significantly reduce the node isolation probability in an interference-free network, and , whose simulation results demonstrated that multiple antennas can significantly enhance the connectivity of an ad hoc network, measured as the number of links that meet a given requirement on the outage capacity, or the symbol error rate of orthogonal space-time block coding. Interference aspects of the problem have been studied in , which investigated the connectivity of sensor networks with regular topologies. The network connectivity was defined as the probability that a path exits between any two pairs of nodes in the network, and simulation results illustrated how an increase in node density led to decreased connectivity due to interference effects.
To reduce the impact of interference in ad hoc networks, interference graph and coloring techniques have been used in the design of scheduling or routing algorithms [18, 19, 20]. In , a linear programming (LP) method was proposed for computing lower and upper bounds for the maximum throughput that can be supported by a multi-hop network. A conflict (interference) graph was used to find the constraint conditions for the LP formulation. In , the authors proposed the construction of a link-directional interference graph to account for the directional traffic over each network link. They investigated a coloring algorithm with two colors on the interference graph to schedule transmissions in ad hoc networks employing TDMA or frequency division multiple access (FDMA). In , active links in a multi-hop network are scheduled in an STDMA scheme where a frame is divided into equal length time slots and each time slot can be allocated to several links simultaneously. Utilizing the interference graph, a heuristic algorithm is proposed to minimize the frame length under a constraint that each link’s minimum data rate requirement is satisfied. An earlier version of this paper  discusses the use of interference graphs together with MIMO for improved connectivity.
The use of multiple antennas to improve the performance of ad hoc networks has been a topic of considerable recent research . For example, in , the transmitter for each link uses the principal singular vector of the channel matrix as the transmit beamformer and each receiver uses MMSE beamforming to mitigate the inter-link interference, and it was shown that there exists an optimal number of active links that maximize the network throughput. Beyond this value, the network becomes interference-limited, and performance is degraded. The results in  show if each link transmits a single data stream without CSI, while the receiver uses partial-zero-forcing interference cancellation, the capacity lower bound increases linearly with the number of antennas. In , an optimal scheduling policy is proposed that can maximize the average sum rate of the MIMO ad hoc network under the constraint that the average data rate of each link is larger than a certain threshold. In , two optimization problems are considered: one is to maximize the sum rate under sum power and proportional fairness constraints, and the other is to minimize the sum transmit power under a constraint on the minimum data rate of each link. In , several distributed scheduling methods were proposed for MIMO ad hoc networks. In these methods, the links whose channel condition satisfy a pre-defined threshold are divided into groups for simultaneous data transmission to maximize the overall network throughput. Also relevant is the recent research on MIMO interference networks, where techniques based on interference alignment  and game theory  have been proposed.
I-C General Approach and Contributions
As mentioned earlier, in this paper we consider PHY-layer optimization and scheduling for an ad hoc MIMO network in multicast streaming mode. In particular, we assume a source node desires to stream data to all other nodes in the network via a pre-determined multi-hop routing table (for example, a minimum spanning tree). Our emphasis will be on obtaining high reliability and low congestion for the network by maximizing the network connectivity, which we define as the probability that all links are able to support the desired throughput. The contributions of this work are as follows:
We examine the multi-hop multicast streaming problem using a detailed PHY-layer model, and demonstrate the significant benefit that multiple antennas, power control and proper scheduling can have on network robustness.
We propose a new definition of ad hoc network connectivity that approximates the probability that all links in the network can achieve a certain average throughput. The metric provides a continuous measure of connectivity performance that is more descriptive than the simple binary metrics often used in wireless networks. In addition, this metric leads to a more robust solution for fast fading channels than techniques based on requiring that only the average rate of each link be above some threshold [3, 7].
Since optimal connectivity will require that network links be simultaneously active, we develop beamforming and power control algorithms for the MIMO interference channels in each time slot that maximize the network connectivity metric.
Together with the beamforming algorithm, we propose an approach for STDMA scheduling based on greedy coloring of the interference graph that finds near-optimal assignments of links to each time slot.
Unlike nearly all prior work in this area, both the beamforming and scheduling algorithms take into account the fact that the channel state information (CSI) may be outdated when it is used, and we illustrate the impact of the outdated CSI on connectivity. This is particularly important for VANET applications, where the network nodes are mobile.
We illustrate via a number of simulations the dramatic improvement in connectivity that can be obtained when the network nodes are equipped with multiple antennas. By jointly considering the problems of beamformer design, scheduling and power control, we observe that the use of multiple antennas provides a “multiplicative” benefit” that exceeds what one would expect from their use in addressing the problems individually. Furthermore, our simulations indicate that adding antennas to the network nodes actually reduces the relative performance loss due to outdated CSI.
We derive analytic expressions for an upper bound on the network connectivity and a lower bound on the sum transmit power of the network assuming interference-free transmissions and no CSI errors. These optimistic bounds provide benchmarks to determine the robustness of the proposed approach.
We note here that a key limitation of the proposed approach is that the optimization described above takes place assuming that the multicast routing tree is pre-determined and remains fixed. This is clearly suboptimal since the routing decisions determine the links, which in turn create the interference environment to be mitigated. In principle, a complete cross-layer solution would jointly address the routing, scheduling and PHY-layer issues all at once, but such a problem is very complex and remains a topic of future investigation.
The rest of the paper is organized as follows. Section II describes the assumed network-level model, introduces the definition of the network connectivity metric and formulates the general optimization problem to be solved. Section III then presents the link-level MIMO model, which includes a description of the time-evolution of the MIMO channels and the outdated CSI. Section IV describes the max-connectivity beamforming solution, and Section V puts everything together in the scheduling algorithm, with an appropriate power control iteration to reduce the transmit power on each link to its minimum possible value. Performance bounds on connectivity and sum transmit power are derived in Section VI, and results from a collection of simulation studies are presented in Section VII to illustrate the performance of the algorithm. Section VIII then concludes the paper and summarizes our results. The notation used in the paper is summarized in Table I.
|Number of nodes in the network|
|Number of antennas per node|
|Number of links in the network spanning tree|
|Data rate requirement for active links|
|Number of time slots in a frame|
|The signal-to-interference-plus-noise ratio of the th link|
|Threshold for to guarantee the data rate requirement|
|Outage probability for the th link|
|Network connectivity metric|
|Data symbol transmitted over the th link|
|Transmit beamformer for the th link|
|Transmit power of the th link|
|Distance between the transmitter of link and the receiver of link|
|Path loss exponent|
|Channel matrix between the transmitter of link and the receiver of link|
|Estimate of the channel between the transmitter of link and the receiver of link|
|Largest singular value of|
|Channel perturbation for the link between the transmitter of link and the receiver of link|
|Channel temporal correlation coefficient|
|Additive noise vector at receiver of link|
|Receive beamformer at receiver of link|
|Scheduling priority of link|
Ii Network Model and Connectivity Definition
In this section, we provide a model for the network configuration and the physical layer channel assumed in this paper. We also define a connectivity metric for the network, which is the performance objective we wish to optimize.
Ii-a Network Configuration and Assumptions
We consider a multi-hop wireless network with a set of nodes, each of which is equipped with antennas (the assumption of an equal number of antennas is not strictly necessary, but simplifies the presentation). We assume that a source node wants to share a streaming message with all other nodes in the network through a pre-defined routing tree, as depicted for example in Figure 1. To avoid congestion and maintain a constant average data flow from the source to all nodes, each link must achieve a certain minimum data rate with high probability. Performance beyond that achievable with simple TDMA-based protocols is possible with the availability of multiple antennas, but co-channel interference must be managed through appropriate scheduling, power control and transmit/receive beamforming. The ability of the network to achieve these goals depends heavily on the accuracy of the CSI available to the scheduler, as well as that available to each link of the network.
The scheduling and transmit parameter design are centralized, and are based on outdated CSI fed back to the scheduler from the individual links. Consequently, we avoid the use of spatial multiplexing and assume that the data on each link is transmitted via a single data stream using a single transmit beamformer. The signal-to-interference-plus-noise ratio (SINR) of this data stream determines the rate of the link. Design of the receive beamformers and power control is performed by the receivers on each link, who are assumed to be aware of the statistics of the interference and the instantaneous effective CSI (channel times transmit beamformer). The source message is assumed to be broken down into packets, and the transmission is conducted over multiple frames. Each frame is divided into time slots, and each link in the routing tree is assigned one time slot in which to transmit its current packet. The scheduler determines the number of time slots and which links are active in each time slot, in order to optimize the connectivity of the network (defined below). The problem with dividing the frame into too many slots (emphasizing time rather than space-time multiplexing of the links) is that, for a fixed frame duration, the slot length is shorter, thus requiring a higher SINR to achieve the same throughput over the frame. This is the fundamental trade-off: fewer slots means more interference, but a lower SINR requirement; many slots means less interference, but a higher SINR requirement. We want to find the optimal number of slots for the best performance.
Ii-B Link Throughput Requirement
For a given frame, we assume that each of the active links will be allocated a single time slot, and during this time slot, each transmitter can occupy the entire available system bandwidth to send its data packet to the intended receiver. Define to be the minimum rate at which a link can be considered connected, and let the number of slots in a frame be . For link to meet the rate requirement, its SINR should satisfy the following condition:
where . Due to fading and co-channel interference, the actual may be smaller than . A link is said to experience an outage when the SINR at the receiver is smaller than the threshold . The outage probability for link is thus
Ii-C Definition of Network Connectivity
We assume that the main goal of the network is robustness rather than throughput. We want to allocate the resources of the network (transmit power, beamformers, scheduling) such that the network remains connected with the highest possible probability, where the term “connected” implies that each link is able to communicate at or above the minimum acceptable rate . This in turn requires that an active link must have a certain minimum SINR. We define the connectivity as the likelihood that all the links in the network can achieve a SINR that allows transmission at or above the desired minimum data rate. Note that this is significantly different from simply requiring that the average rate be above some threshold, which is the approach taken in prior work on fast fading channels.
If the network interference has been properly managed so that its impact on a given link is negligible, then the probability of a successful transmission on a given link is independent of the other links, and the connectivity metric can be defined as
According to this definition, the network connectivity is equal to the probability that none of the links in the network experiences an outage during the transmission frame. Assuming a network of links and a frame with slots, can be expressed as a general function of the transmission parameters:
where and denotes the link beamformers, is the transmit power allocated to each link, and represents the channels between the transmitter of link and receiver of link , indicates the link scheduling scheme, is the slot number allocated to link , and . Based on (4), the connectivity can thus be expressed as
where denotes the probability density function (PDF) of and is the indicator function defined as
Under this definition, when the connectivity of the network is high (near one), the total network throughput would be approximately bounded below by the number of network links times . To achieve a higher total throughput, could be increased, but this would likely reduce the connectivity. A desirable trade-off between connectivity and throughput could conceivably be reached by appropriately adjusting the minimum link rate .
The optimization problem associated with maximizing the connectivity of the network could thus be expressed as
where denotes an upper bound on the transmit power. Direct optimization of (7) is intractable, since one cannot find a closed-form expression for the probability of (6) in terms of the parameters of interest. Instead, we choose to solve the following related optimization problem:
The relationship between this simplified problem and the original optimization is analogous to the relationship between the geometric and arithmetic means. Instead of maximizing the product of the link connection probabilities, we maximize its sum.
We divide the solution of (8) into two sub-problems: (1) scheduling and (2) beamformer design and power allocation. We solve sub-problem (2) for different results of sub-problem (1) to determine which scheduling result is best. The transmit beamforming problem is discussed in the next section, and the scheduling algorithm is described in Section V.
Iii Interference Channel Model with Outdated CSI
Iii-a Interference Channel Model
Assume without loss of generality that links are simultaneously active with link during a given time slot (). The signal at link ’s receiver can be expressed as
where the transmitted symbol is a unit-magnitude data symbol, is the distance between the transmitter of link and receiver of link , is the path loss exponent, and is an additive, spatially white noise vector with covariance given by and is the identity matrix.
Assuming receiver employs a beamformer , the SINR for link is given by
where . Assuming the receiver knows the covariance matrix and the channel vector , the optimal that maximizes is given by :
and the resulting SINR for link can be expressed as
Iii-B Outdated CSI
The CSI at the scheduler will be outdated due to the time required for this information to be fed back from the network nodes. To quantify the CSI uncertainty due to the feedback delay, we adopt a first order Markov model to characterize the time variation of the channel 
where denotes the channel matrix during the data transmission period, represents the channel feedback, and is a perturbation matrix. The elements of and are assumed to be i.i.d complex Gaussian random variables with distribution . The coefficient is used to determine the level of uncertainty in the CSI at the scheduler. In effect, under this model the scheduler assumes a Gaussian distribution for , with mean and independent entries with variance .
where It is observed that is a function of the channel set and the channel perturbation set . Given the channel set , is a random variable, the distribution of which depends on the elements in . The conditional expectation of with respect to is given by
where step (a) is due to the fact that perturbation matrices are independent of each other and denotes the trace operator. Note that the use of the expected value in (III-B) is due to the fact that the CSI in may not be precisely known. According to (13), the channel may have changed from the time it was reported since the network nodes may be mobile. If precise CSI is available, then the expectation can be dropped and the instantaneous value of can be used instead. Calculation of the term is very complicated, so instead we use the following lower bound based on Jensens inequality[29, Lemma 4]:
where denotes that is a positive semidefinite matrix. In the above calculation, we use the fact that is a complex Gaussian random vector with distribution . Substituting (16) into (III-B), the conditional expectation of the is lower bounded by
Iv Transmit Beamforming for Connectivity
In this section we introduce how the transmit beamformer and power are calculated for a given link schedule. Consider a scenario in which links are transmitting simultaneously. In this case, the number of links that can meet the desired rate requirement depends on the beamformer and transmit power that each link adopts. Define as the right hand side of (17). The problem of finding the optimal beamformer and transmit power based on the outdated CSI can be stated as:
where is the approximation parameter, the problem can be converted to finding the maximum value of a constrained continuous nonlinear multivariable function. Replacing with , the optimization problem in (18) can be approximated as
Note that when is small, the sigmoid function is smooth. As , the sigmoid function approaches the indicator function. Starting with relatively small values for and then increasing it in several steps, the solution involving the indicator function can be found. For each fixed value of , the problem in (20) can be solved numerically using, for example, the active-set method. The algorithm can be initialized with arbitrary power allocations, and by setting the transmit beamformer equal to the principal singular vector of .
V Scheduling for Maximum Connectivity
In this section, we propose a scheduling algorithm that maximizes the connectivity metric defined earlier using the concept of coloring from graph theory[19, 20]. The algorithm is based on the use of the interference and collision graph (ICG) of the network[3, 19, 20, 18].
V-a The Interference Graph and Greedy Coloring
We use an ICG to model the relationships between the active links. In the ICG, each vertex represents a directional link in the transmission graph. We define two types of neighbors in the ICG. The first are interfering neighbors, which represent links that could be simultaneously active and hence interfere with one another, and the second are colliding neighbors, which represent links that cannot be active at the same time. In our application, colliding links include those that share the same transmitter, or those where a transmitter in one is a receiver in the other. All links that are not colliding are considered to be interfering, although the amount of interference between two given links could be low if they are far apart. In the paragraphs below, we more precisely define the ICG and concepts related to our scheduling algorithm.
Interference and collision graph: The interference and collision graph can be defined based on the transmission graph . A given link in is represented by a vertex in . Suppose for link that node is the transmitter and is the receiver, and suppose for link that is the transmitter and is the receiver. Links and are colliding links if any of the following are true: or (note that for the multicast tree we assume, will never occur). An edge between two vertices in represents that the two corresponding links are colliding links and they could not be assigned to the same time slot; if two vertices in do not share an edge, they represent interfering links which can be assigned to the same time slot, provided that the resulting interference could be managed. As an illustration, Fig. 2 represents a partial ICG for the network of Fig. 1, where the colliding links and interfering links are connected with solid and dashed edges, respectively (for the sake of clarity, only edges associated with and are plotted; the remainder of the edges can be generated in a similar fashion). For example, links and share the same transmitter node 1, so they are colliding links in Fig. 2; the receiver of link is the same as the transmitter of link , so and are also colliding links.
Coloring: In our application, “coloring” refers to the process of assigning time slots to the network links, or equivalently, to the nodes in the interference graph. Given a set of colors in the discrete set (colors can be considered as distinct non-negative integers), a coloring of the graph is an assignment of the elements (or colors) in to the vertices of , one color for each vertex, such that no adjacent vertices occupy the same color. A greedy coloring enumerates the vertices in a specific order and assigns to the smallest color that is not occupied by the neighbors of among . The vertices can be ordered according to their edge degree, which is the number of edges incident to the vertex [31, chap. 5]. To apply coloring to the ICG, we need to define an order for the vertices. Before we proceed to the scheduling order, some related definitions are necessary.
Scheduling freedom: For a link , the scheduling freedom is the number of available colors that can be allocated to this link. The higher the value of , the higher the possibility that link will be allocated to a “good” color (one that leads to low interference).
Collision degree: Given a link , the collision degree is the number of its colliding neighbors in the ICG.
Constraint and free color sets: For link , the interfering color set includes colors occupied by the neighbors that interfere with . The unavailable color set includes the colors occupied by the neighbors that collide with . The free color set is defined as , and corresponds to the set of colors that could be assigned to link without causing any interference (or collisions). The constraint color set is defined as . The colors in this set can also possibly be assigned to link , but with some additional interference that would have to be mitigated through beamforming.
Scheduling priority: The scheduling order is determined using the largest singular values of the channel matrices . The higher the value of or the smaller the channel gain , the more likely it is that link will be affected by interference. Such a link will have fewer colors it could be assigned to, and hence a smaller value of . To increase the likelihood that links with low scheduling freedom can be allocated a good color, the scheduling priority of link is defined as
where is a constant larger than .
V-B Scheduling Algorithm for Connectivity
Based on the above definitions, we propose here a scheduling algorithm for optimizing connectivity. The algorithm assumes a particular value for the number of slots , and is repeated until a value of is found that maximizes the connectivity metric. The minimum possible value for is the maximum collision degree over all vertices in the ICG. The algorithm begins by ordering the links according to their scheduling priority, and then assigns a color to them one-by-one, from highest to lowest priority. Consider the link at position in the priority ordering. If link can be added to a color that already has had other links assigned to it, and if the beamformers and power levels for these links can be adjusted to accommodate link without causing any of them to drop below , then link is added to this color. If there are multiple colors for which this is true, it is added to the color that requires the smallest increase in transmit power to accommodate it. If the addition of link to any of these colors causes one of the links (including possibly link ) to drop below the SINR threshold, and if there exist free colors that have not had any links assigned to them, link is assigned to one of the free colors. If no free colors exist, then link is assigned to the color that caused the smallest number of links to drop below the threshold (and with the smallest increase in power, in case of a tie). In this latter case, it is hoped that the power control algorithm described in the next section will reduce the interference sufficiently so that all links assigned to the color will end up being active. A more precise mathematical description of the algorithm is given below.
Let be the right singular vector of corresponding to the largest singular value, and let be the initial transmit power allocated to link . Assuming a value for , initialize the active link set . Links that do not qualify for cannot meet the desired target rate for the given value of . Initialize the transmit beamformers as , the transmit powers as and the color set . Let denote the sum transmit power of the active links in each color, and let represent the number of links that are unable to meet the the target for each color. Initialize these sets to contain all zeros. The initial schedule is also set to zeros. Construct the ICG based on the relationship between the active links. Compute the scheduling priority of the vertices for .
Select the link with the highest scheduling priority: , and construct the free color set and the constraint color set for link .
Assign link to color , set , , and
skip to step 5.
else if and
There aren’t enough colors to avoid collisions between the active links, so the
algorithm must be restarted with a larger value for .
For , construct the link set , which
contains the links currently assigned to color .
For each , assume the links in the set are transmitting simultaneously, and use the transmit beamforming algorithm of Section IV to find the new beamformer and transmit power sets , . For link set , based on , , calculate the expected number of links that will be unable to meet the SINR threshold with link added: , and calculate the updated sum transmit power due to the addition of link : . Set to be the number of links that will drop below the SINR threshold if link is added to color , and define to be the additional transmit power required to add link to color .
Find . If more than one color corresponds to the minimum , select the color with minimum .
Assign link to color , set , , . Use ,
to update the components of and which correspond to links in the set
else if and
Assign link to color and set , .
Set , and repeat step 2 until each vertex , , is allocated a color.
Once the scheduling is complete, the active links will transmit data according to the scheduling result using the beamformers in and, at least initially, the transmit powers in . As explained below, the actual transmit power for each link will be fine-tuned based on feedback from the receivers.
V-C Local Power Control for Active Links
Since the nodes are energy limited, to extend the lifetime of the network and to reduce the mutual interference caused by the co-channel links, the transmit power of each link should be minimized under the constraint of the QoS requirement. Due to the approximation in (19), the use of the lower bound in (17), and the presence of outdated CSI, the actual based on and will not be exactly equal to the threshold . In most instances it will be greater than due to the use of the lower bound in (17), but in some rare cases it can be below the threshold. To remedy this latter situation, we reduce the transmit power of any links whose SINR exceeds the threshold, which reduces co-channel interference and the transmit power consumed by the network. For a given time slot , the network schedule assigns links in set to transmit simultaneously. The power control algorithm steps through each link in the time slot, reducing power for the link if its SINR exceeds the threshold. A given link may be revisited several times, since reductions in transmit power for other links reduces the overall interference, and may allow further reductions in transmit power for the link. This process is assumed to repeat a maximum of times. If, after all iterations, there are any links whose SINR is below the threshold, these links are declared to be in outage, their power is reduced to zero, and an additional iterations are performed to reduce the transmit power even further. A mathematical description of the power control algorithm for time slot is described below.
Iterative Power Control Algorithm
Initialize the transmit power and beamformer of link with and , set maximum iteration lengths and .
Link ’s receiver calculates based on (12). If , the receiver for link informs the transmitter to reduce to .
, if , go to step 2.
If for any , set and repeat step 2 for another iterations to further reduce the transmit power.
Vi Performance Bounds
In this section, we derive two performance bounds that can be used to evaluate the limiting behavior of the proposed algorithms. The first is an upper bound on the network connectivity metric, and the second is a lower bound on the average sum transmit power. Both bounds are derived under the assumption that the CSI is perfect, and each active link is free of interference. When is small, the connectivity bound should match the performance of the proposed approach if the network interference has been properly accounted for. This is not the case for the bound on transmit power, however, since the interference mitigation results in beamformers that require excess power to achieve the rate threshold. The difference between the required transmit power and the lower bound represents the price paid for the enhanced connectivity that results from operating the system as an interference network. Due to the complexity of calculating the bounds, expressions are derived only for the cases .
Vi-a Upper Bound on Network Connectivity
The connectivity bound is derived assuming the absence of interference, and perfect CSI at the scheduler (). Each transmitter uses maximum power and selects the principal singular vector of the channel matrix as its beamformer. When the receiver is free of co-channel interference, the resulting signal-to-noise-ratio (SNR) for link is given by
If we define , then the upper bound on connectivity can be expressed as
The squared singular value corresponds to the largest eigenvalue of the central Wishart matrix . Define , and note that the cumulative density function (CDF) of is given by [32, eq. (6)]:
where we have dropped the subscript since the distribution is assumed to be identical for each link, is the gamma function, is an matrix defined by , and the lower incomplete gamma function has the following series expansion:
For , (24) reduces to:
where is the minimum value of the channel gain that can guarantee .
For , after some cumbersome algebra, the CDF of the largest eigenvalue can be expressed as:
where , , , are defined in the appendix. The network connectivity upper bound is then computed as
For the single antenna case (), the interference-free at the receiver is given by
where is a Rayleigh random variable. Define , so that . The probability of a successful transmission for the link can be expressed as
and thus the network connectivity is given by
It is easy to verify that both and are monotonically increasing functions of and larger than 1. and represent the connectivity gain provided by the use of multiple antennas. Since is proportional to , the larger the link distance, the greater the connectivity gain offered by the use of multiple antennas.
Vi-B Lower Bound on Average Sum Transmit Power
A lower bound for the average sum transmit power of the network can be obtained under the assumption that each active link selects its beamformer as the right singular vector of the channel with largest singular value, and assuming there is no co-channel interference between the links.
For link , given , the power allocation at the transmitter is:
The average transmit power of link can be obtained by averaging over the random variable . Denote the probability density function (PDF) of as , then can be explicitly obtained by taking the derivative of (26) or (28) with respect to .
When , is given by
The average transmit power for link is calculated as111In principle, in this equation, but to numerically evaluate the integral, we simply choose a large enough value such that the integrand is essentially zero.