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Wenjian Zhang, Senlin Jiang, "Effect of Node Mobility on MUMIMO Transmissions in Mobile Ad Hoc Networks", Wireless Communications and Mobile Computing, vol. 2021, Article ID 9954940, 9 pages, 2021. https://doi.org/10.1155/2021/9954940
Effect of Node Mobility on MUMIMO Transmissions in Mobile Ad Hoc Networks
Abstract
In this paper, we investigate the expected outage probability and expected throughput of a multiuser multipleinput multipleoutput (MUMIMO) transmission in mobile ad hoc networks (MANET) in the presence of cochannel interference and unpredictable interbeam interference. In order to achieve multiuser diversity gain, the receiving nodes are required to report measured channel information to the transmitting node. During the time gap between channel measurement and data transmission, the channel may change with the location of moving nodes. The unpredictable behavior may cause a mismatch between the weight of beams and the instantaneous channel and the interbeam interference in the data transmission phase. In order to obtain the closed form expected outage probability, we categorize the behavior of nodes according to whether the interbeam interference exists or not and the number of received interference beams. The probability of each category and the closed form outage probability of an instantaneous MUMIMO transmission of each category are derived. Additionally, the expected throughput of an MUMIMO transmission which changes with the number of receiving nodes is obtained, and the optimal value of receiving nodes to maximize the expected throughput is discussed. Numeric results show the unpredictable interbeam interference degrades the outage probability performance. Reducing the duration of the time gap could improve the expected outage probability and expected throughput.
1. Introduction
The diversity of applications gives rise to the requirement on the flexibility of wireless networks. A conventional cellular network or wireless local area network (WLAN) supplies the inability to satisfy the requirements. Due to the agility network structure, MANET has attracted a lot of attention in different application areas, such as military battlefield [1], postdisaster reconstruction [2], emergency mission, and vehicular communication [3].
A MANET is composed of a collection of mobile nodes equipped with a wireless transceiver. The nodes communicate with each other directly or by multihop link with the help of intermediate nodes [4]. Because of the lack of coordination of infrastructure or central node, multiple transmissions may occur on the same channel simultaneously and interfere with each other. Inevitably, the cochannel interference impairs the performance of transmissions. In prior works, the transmission performance in the presence of cochannel interference was investigated [5, 6], and some interference coordination schemes were proposed [7–9].
In order to reduce cochannel interference and improve the aggregate throughput of MANET, multipleinput multipleoutput (MIMO) has been exploited by many researchers [10–12]. Especially, MUMIMO, due to its extremely high spectral efficiency, receives a significant attention [13–16]. In the 3rd Generation Partnership Project (3GPP) long term evolutionadvanced (LTEA) [17] and IEEE 802.11 ac [18], the MUMIMO is adopted. An MUMIMO transmission is performed in two phases: beam selection phase and data transmission phase. In the beam selection phase, the transmitting node selects a set of orthogonal beams according to the reported channel measurement information from candidate receiving nodes. In the data transmission phase, the transmitting node delivers data packets on the same channel to receiving nodes simultaneously, using the selected orthogonal beams.
The orthogonal beams are proven to maximize the received SINR of a receiving node. However, achieving the performance requires the matching between the weight of a beam and instantaneous wireless channel. As shown in Figure 1, a time gap for reporting channel measurement information is inevitable in the beam selection phase, whose duration is related to the number of the candidate receiving nodes. Due to the randomness of nodes’ movement, the behavior of nodes is unpredictable during the time gap. If a node moves randomly during the time gap, the wireless channel between the moving node and the transmitting node or a receiving node changes with the location of the moving node. In the data transmission phase, the instantaneous wireless channel mismatches with the weight of beams derived from measured channel information. As a result, the beams for each receiving node may interfere with each other. The unpredicted interbeam interference will degrade the performance of an MUMIMO transmission. In previous work [13], the influence of the interbeam interference on the outage probability performance was exploited, but the randomness of interference was not taken into account.
The expected throughput of an MUMIMO transmission is the sum of the expected throughput of each receiving node. Intuitively, the increase of receiving nodes could bring more expected throughput of an MUMIMO transmission. However, with the increase of receiving nodes, the probability of that multiple nodes keep motionless simultaneous during the time gap decreases, but the probability of interbeam interference increases. Moreover, the increase of receiving nodes reduces the power of the desired signal and potentially increases the power of interbeam interference. For a receiving node, the instantaneous received SINR and the expected throughput are reduced. Therefore, the expected throughput of an MUMIMO transmission may not be improved with the increase of receiving nodes. How to balance the expected throughput and the number of receiving nodes is worth discussing.
Motivated by the discussion above, in the paper, we investigate the outage probability performance and expected throughput of an MUMIMO transmission in MANET in the presence of cochannel interference and unpredictable interbeam interference. Summarily, our main contributions are given as follows:(i)We categorize the mobility behavior of nodes according to whether the interbeam interference exists or not and the number of received interference beams. The probability of each category is derived.(ii)We derive the closed form outage probability of an instantaneous MUMIMO transmission considering the cochannel interference and unpredictable interbeam interference.(iii)We explore the expected throughput of an MUMIMO transmission and discuss the optimal number of receiving nodes to maximize the expected throughput.
The remainder of the paper is structured as follows. Section 2 gives the system model. Section 3 derives the instantaneous and expected outage probability of an MUMIMO transmission in the presence of cochannel interference and unpredictable interbeam interference. Section 4 explores the expected throughput of an MUMIMO transmission and discusses the optimal number of receiving nodes to maximize the expected throughput. Section 5 describes the numeric results. Finally, Section 6 gives the conclusions.
2. System Model
We assume a MANET consisting of a collection of halfduplex nodes adopts slotted ALOHA protocol [19] and MUMIMO transmissions. In each slot, a subset of nodes transmits data on the same channel, with the same transmit power . Considering the nodes in MANET are placed arbitrarily, the transmitting nodes in each slot are assumed to be distributed according to a Spatial Poisson Point Process (SPPP) with intensity [20]. The model has been used and the validity has been confirmed in prior works. Hence, the set of transmitting nodes is denoted as , where is the position of transmitting node at some time instant.
In the beam selection phase of an MUMIMO transmission, the transmitting node equipped with transmit antennas selects orthogonal beams using reported channel information. In the data transmission phase, the transmitting node delivers data packets to receiving nodes equipped with a single receive antenna simultaneously, using orthogonal beams , where satisfies and .We assume the duration of the time gap is , and the wireless channel does not change with time within the time gap. At the start time of the time gap, the nodes in an MUMIMO system are assumed to be motionless with a probability . After a time interval , each node changes to the opposite state independently, i.e., the motionless node becomes moving or vice versa. We assume the number of state change in a period of time obeys a Poisson distribution. Hence, the time interval follows an exponential distribution with mean , i.e.,
Further, we assume a node moves in a very short time could result in an obvious variation of location, and the instantaneous wireless channel changes with the location.
3. Outage Probability Evaluation
We assume a receiving node locates at the origin. In the data transmission phase, the received signal of the receiving node is denoted aswhere is the channel gain vector between the receiving node and the transmitting node , whose entries follow complex Gaussian distribution . Similarly, represents the channel gain between the receiving node and another transmitting node . is the symbol vector transmitted from the transmitting node. Additionally, represents the additive white Gaussian noise (AWGN) with zero mean and variance of .
For the receiving node , whether the interbeam interference signal exists or not depends on whether the wireless channels change with the location of nodes during the time gap or not. In case all nodes keep motionless during the time gap, the orthogonality of beams could be achieved, and the interbeam interference is eliminated. In case the transmitting node moves during the time gap, the beams for all receiving nodes interfere with each other. Otherwise, the beam for a motionless receiving node is interfered with the beams for the moving receiving nodes, but the beam for a moving receiving node is interfered with the other beams. With the goal of evaluating the performance of an MUMIMO transmission, the behavior of nodes is categorized according to whether the interbeam interference exists or not and the number of received interference beams. In order to simplify the calculations, we define several events to depict the behavior of nodes as follows.When the event or the event with occurs, the beam for a receiving node is interfered with the beams for receiving nodes. When the event with occurs, the beam for a motionless receiving node is interfered with the beams for moving receiving nodes, but the beam of a moving receiving node is interfered with the beams for receiving nodes. With this in mind, we further define the measurement of instantaneous received SINR for motionless and moving receiving nodes as event and , respectively.
Definition 1. An event that the transmitting node and the receiving nodes of an MUMIMO system both keep motionless during the time gap is denoted as event .
Definition 2. An event that the transmitting node moves during the time gap, independent of the states of the receiving nodes, is denoted as event .
Definition 3. An event that the transmitting node is motionless but receiving nodes move during the time gap is denoted as event .
The outage probability of an MUMIMO transmission is defined as the probability that instantaneous received SINR does not exceed the required SINR threshold. The expected outage probability is the sum of outage probability weighted by the probability of defined events. Assuming the required SINR threshold is , hence, the expected outage probability is expressed aswhere is the probability of event and represents the instantaneous received SINR of the receiving node when the event occurs.
3.1. Probability of the Events
In terms of Definition 1, the event means that all nodes are motionless at the start time of the time gap and keep motionless during a time interval not less than . Using the probability density function (PDF) in equation (1), the probability of event is denoted as
When the event occurs, no matter whether the receiving nodes move or not, the transmitting node is moved during the time gap. Hence, the probability of event is denoted as
As described in Definition 3, the transmitting node and receiving nodes are motionless at the start time of the time gap and keep motionless within a time interval not less than . But receiving nodes move during the time gap. Hence, the probability of event is denoted as
Furthermore, the probabilities of event and are and , respectively.
3.2. Outage Probability of Instantaneous MUMIMO Transmission
In an MUMIMO transmission, the instantaneous received SINR of the receiving node is denoted aswhere if receiving node k is a motionless node, or if receiving node k is a moving node.
For defined events above, the instantaneous received SINRs could be obtained from by setting or .(i)When , the instantaneous received SINR (ii)When and , the instantaneous received SINR (iii)When and , the instantaneous received SINR Neglecting the effect of AWGN in the interferencelimit system for analytical simplicity, the outage probability of an instantaneous MUMIMO transmission is denoted aswhere and . and represent chisquared random variables with and degrees of freedom, respectively. Using the PDF of chisquared random variable with degrees of freedom, the term is derived aswhere is complementary incomplete Gamma function with . Inserting the result into equation (8), the outage probability is expressed as
3.2.1. Outage Probability when
In case , the effect of interbeam interference is eliminated. The outage probability in equation (10) is rewritten aswhere is the Laplace transform of . The transform of is employed and is the th order differential of the Laplace transform of . Note that when , the outage probability is simplified as
According to the system model, the transmitting nodes in each slot are distributed according to an SPPP with intensity . Consequently, the cochannel interference could be modeled as a general Poisson shot noise process [21]. The Laplace transform of random variable is given aswhere and . Especially, in case , we have . Hence, when , the outage probability is expressed as
In order to achieve a high probability of successful reception, it is reasonable that a very strict quality of service (QoS) is required, resulting in that the density of transmitting nodes shall be very small. Consequently, the th order differential of could be evaluated using the firstorder Taylor series around . For all , the th order differential of is given approximately as
In case , combining the equation (11) with (15) and discarding the term , the outage probability of an instantaneous transmission when the event occurs is derived as
3.2.2. Outage Probability when
In this section, in order to simplify the derivation of outage probability, we first give a definition and a proposition as follows [22].
Definition 4. Given a Beta distributed random variable with two positive shape parameters and , its cumulative distribution function (CDF) is a regularized incomplete beta function, denoted aswhere is the Gamma function.
Proposition 1. For any , and nonnegative integers , , and where ,
In case , the performance of MUMIMO transmission is limited by interbeam interference and cochannel interference simultaneously. The outage probability in equation (10) is rewritten as
When , the outage probability is simplified as . Otherwise, the outage probability is further denoted as
Note that and its th order differential were derived in equations (13) and (14). Using the PDF of a chisquared random variable with degrees of freedom, the Laplace transform and its th order differential are, respectively, denoted as
Hence, when , the outage probability is expressed as
In case , the outage probability could be obtained by combing equation (19) with the results in equations (15) and (22). With the goal of simplifying the expression, we use Definition 4 and Proposition 1 above and perform the firstorder Taylor series around with discarding the term . As a result, the outage probability of an instantaneous transmission is denoted aswhere .
The outage probability could be obtained with when the event occurs, or with when the event or occurs.
4. Throughput of MUMIMO Transmission
Assume adaptive QAM modulation with discrete rates is applied in an MUMIMO transmission. When an event occurs, if the instantaneous received SINR lies in , the instantaneous rate is given aswhere is Shannon Gap with variable rate QAM transmission [23]. Hence, the longterm expected throughput of receiving node when the event occurs is denoted as
The expected total throughput of an MUMIMO transmission is given as
Intuitively, the increase of receiving nodes could bring more expected total throughput. However, with the increase of the number of receiving nodes, the probability that multiple nodes keep motionless during the time gap decreases, causing the probability that the event or occurs decreases. In addition, the increase of receiving nodes reduces the power of the desired signal and potentially increases the power of interbeam interference. For a receiving node, the instantaneous received SINR and the longterm expected throughput are reduced. Hence, there is an optimal number of receiving nodes which could maximize the expected total throughput of an MUMIMO transmission, i.e.,
5. Numeric Results
This section reports the results of computer simulations. In the simulation, the MANET nodes are assumed to be located randomly in a circle region with 1 km radius. The simulation configurations are listed in Table 1.

Figure 2 shows the probabilities of the defined events. With the increase of the duration of the time gap, the probability that all the nodes keep motionless decreases, i.e., the probability of the event decreases. But the probability that the transmitting node is moving increases, causing the increase of the probability of the event . When the event occurs, the probability is dependent on the number of moving nodes. If the moving nodes are less than the receiving nodes, the probability that the motionless nodes keep the same state decreases; hence, the probability of the event decreases. Otherwise, the probability that all receiving nodes are moving increases with the duration of the time gap.
The outage probabilities of instantaneous MUMIMO transmissions are given in Figure 3. On the one hand, with the increase of the intensity of the MANET nodes, a receiving node experiences more severe cochannel interference signals. On the other hand, the increasing receiving nodes reduce the power of the desired signal but enhance the power of interbeam interference. As a result, the outage probabilities of instantaneous MUMIMO transmissions increase with the intensity of the MANET nodes and the number of receiving nodes. Further, because of the moving receiving nodes in the event and , i.e., , the receiving nodes will receive the interbeam interference signal. Hence, the outage probabilities of the event and () are higher than that of the event (). And the outage probability increase with the number of moving receiving nodes.
Figure 4 describes the relationship between the expected outage probability and the duration of the time gap. In terms of the definition of the events, the probability of the event decreases but the probabilities of the event and increase with the duration of the time gap. As a result, the expected outage probability increases with the probabilities of the event and . However, the increase of the expected outage probability is related to the number of receiving nodes. For instance, the increase of is faster than that of . On the other hand, with the increase of the number of moving receiving nodes, the probability of event and further the expected outage probability increase.
Figure 5 reflects the impact of the number of receiving nodes on the expected throughput of an MUMIMO transmission. When the number of the receiving nodes is less than a threshold, the increasing receiving nodes reduce the expected throughput of each receiving node and the total expected throughput. While the number of receiving nodes excesses the threshold, the total expected throughput begins to grow with the increase of the receiving nodes. As a result, an optimal value of equal to 2 or the number of the transmit antennas could maximize the total expected throughput. On the other hand, it is obvious that the increase of the duration of the time gap reduces the total expected throughput. In case of a fixed duration of the time gap, the total expected throughput increases with the number of moving receiving nodes.
6. Conclusions
This paper evaluates the expected outage probability and expected throughput of an MUMIMO transmission in MANET in the presence of cochannel interference and unpredictable interbeam interference. To derive the closed form results, the unpredictable behavior of nodes is categorized. Based on the categories, the closed form outage probability of instantaneous MUMIMO transmissions and further expected outage probability are obtained. The analytical and numeric results indicate the unpredictable interbeam interference degrades the outage probability performance. Reducing the duration of the time gap could improve the expected outage probability and expected throughput.
As a future work, we will explore the performance optimization by interference management or resource assignment based on the derived results. At the same time, we believe that the application of integrated cognitioncontrolcommunication technology to make network intelligent is an important direction.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
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Copyright
Copyright © 2021 Wenjian Zhang and Senlin Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.