I Introduction
The Internet of Things (IoT) is one of the most attractive concepts in the area of information and communication technology. IoT is expected to play an important role in our daily life by supporting massive connectivity with seamless service. It involves the interconnection of different, and possibly heterogeneous objects through the Internet using different communication technologies. The objects are equipped with communications capabilities and can vary from sensors, smart objects, etc. [1, 2, 3].
The total number of IoT connections is expected to grow tremendously the next years. The deployment of random access protocols in the IoT networks can potentially mitigate the congestion caused by massive amount of IoT devices with low signaling overhead [4]. To support the massive connectivity in future IoT networks, practical techniques are required to collect data from a large set of devices and the traditional orthogonal multiaccess schemes are not sufficient. The works in [5, 6, 7, 8, 9, 10] have considered data aggregation under different scenarios and setups and have evaluated the benefits of such technique.
Networklevel cooperation introduced in [11] and [12] can be an effective alternative method for data aggregation. It is plain relaying without any physical layer considerations and it has been shown to provide large gain in terms of throughput and delay performance. Recently, several works have investigated relaying at the network level [13, 14, 15, 16, 17, 18, 19, 20, 21]. The deployment of aggregators under networklevel cooperation can improve the throughput and delay in IoT networks [22]. However, due to the queueing delay, the stability conditions at the aggregator queues need to be considered.
In this work, we consider a random access IoT network with two aggregators which can receive and forward data in form of packets from the network nodes. The nodes are transmitting in a random access manner under slotted time which is a common assumption in IoT technologies such as LoRa. The aggregators use networklevel cooperation and their receiving and transmitting modules are operating in different bands, thus, they have outofband full duplex capabilities. Furthermore, we assume multipacket reception (MPR) capabilities at the receivers. MPR is suitable to capture the SINR (SignaltoInterferenceplusNoiseRatio) based model and is more realistic to model the wireless transmission. Similar assumptions to our work can be found in [5, 9, 22].
The contributions of our work can be summarized as follows. Our primary goal is to address the problem of providing support for data collection in IoT networks by applying networklevel cooperation. We provide the throughput analysis of the IoT network consisting of sensors that are assisted by two aggregators, from which we can gain insights on the scalability of the considered network. In addition, we study the stability conditions for the queues at the aggregators, which guarantee finite queueing delay. Furthermore, we study the average delay of the packets possibly received and forwarded by the aggregators. Our system is modeled as a twodimensional discrete time Markov chain, and we show that the generating function of the stationary joint queue length distribution can be obtained by solving a fundamental functional equation with the theory of boundary value problems. The analysis in this work can act as a framework for other research directions that involve multiple aggregators with interacting queues.
The remainder of the paper is organized as follows: Section II describes the system model and in Section III we present the analysis related to the networkwide throughput. In Sections IV and V, we provide the analysis for the average delay per packet. The numerical evaluation of the theoretical results is presented in Section VI, and Section VII concludes the paper.
Ii System Model
Iia Network Layer Model
We consider a wireless network consisting of IoT nodes/sensors/objects, which intend to communicate to a common destination/sink , and two aggregators, denoted by and , which can help aggregating and relaying messages from the IoT nodes to . The network model is depicted in Fig. 1. The nodes are located in two nonoverlapping regions. In the first region there are nodes, which are within the service range of the aggregator . In the second region there are nodes within the service range of the aggregator . Note that the transmissions in the first region cannot be overheard by the nodes in the second region and vice versa. This is a common assumption since there is planning for the placement of the aggregators in order to increase the coverage area without interfering with each other. In the following, we will use the terms nodes, sensors, and objects interchangeably.
The sensors intend to transmit packets to the destination node and they are assumed to be saturated, i.e., they always have packets to transmit. In case the transmission from a sensor to fails, the aggregator can help relaying the message to and the aggregators do not generate their own traffic. We consider using networklevel cooperation at the aggregators [11, 12], which means that the aggregators are cooperating as relays in a decodeandforward manner. The packets are assumed to have equal length and the time is divided into slots, which corresponds to the transmission time of a packet. We assume that the sensors access the wireless channel randomly without any coordination among them. We consider a full multipacket reception (MPR) channel model, which allows the receivers to successfully receive more than one packets when there are multiple transmissions in the same slot [23]. As the result, the sink node can receive information simultaneously from the sensors and the aggregators. Note that when all nodes are transmitting, we can have in total up to interfering devices at the sink. This assumption in the literature [5, 22, 10].
We assume that different frequency bands are allocated to the sensors and the aggregators, thus, there is no interference between them. On the other hand, the transmission of a node creates interference to the other nodes of the same kind, i.e., there is interference between the sensors, and between the aggregators. The transmitting and receiving units of the aggregators are operating in different channels/frequency bands to avoid selfinterference, which can be considered as outofband full duplex mode [24]. The aggregators are equipped with queues that store possible packets from the sensors that failed to reach the destination. The queues are assumed to have infinite capacity, thus there is no packet dropping.^{1}^{1}1In practice, the buffers have limited size, which is usually quite large. Our analysis based on the infinite buffer size assumption can capture this scenario with minor modifications. The arrival and service rates for the queues are defined in Section III. This is a common assumption in the literature, in the IoT context [22].
IiB Physical Layer Model
At the beginning of a timeslot, sensor nodes that belong to the coverage area of
, attempt to transmit with probability
, . The aggregator will attempt to transmit a packet with probability if it has a nonempty queue. Note that we assume that all the sensors in the same area transmit with the same probability. Our analysis can be easily extended to handle the general case where each node has different transmit probabilities.The success probability between a sensor in the first coverage area and its aggregator is denoted by , when there are sensors from the same area transmitting in a timeslot. this success probability is the probability that the received SINR is greater than a threshold. The expression for this probability is omitted due to space limitation, and it can be found in [13]. Note that transmitting sensors from the other coverage area do not create interference at the aggregator. However, the concurrently transmitting sensors from both areas interfere with each other at the destination . Denote by the success probability to the sink from a sensor in coverage area when there are active transmitters from area 1 and active transmitters from area 2. Similarly we can define .
A packet transmission from a sensor in the first area fails to reach the destination with probability , when there are active sensors in the first area and active sensors in the second area. In this case, that packet will be stored in the queue of aggregator with probability . Otherwise, with probability , the aggregator fails to decode that packet and it has to be retransmitted by the sensor in a future time slot.
Recall that if there are stored packets in the queues of the aggregator , , then transmits a packet with probability . If only one aggregator is active, then the packet will be successfully transmitted to with probability . If both aggregators transmit simultaneously, then with probability the packet from is successfully received by node . If a transmitted packet from an aggregator fails to reach the , that packet remains in the queue and will be retransmitted in a later time slot.
Iii Throughput and Stability Analysis
In this section, we characterize the network throughput performance and provide the stability conditions for the queues at the aggregators.
Iiia Throughput Analysis
The throughput per node consists of the direct throughput from each sensor to the destination and the throughput contributed by the aggregator. Recall that the devices that are in coverage from the first aggregator cannot cause interference at the receiver of the second aggregator. Moreover, the devices that are covered by the th aggregator, , are transmitting with probability , for .
The direct throughput^{2}^{2}2The direct throughput in this setup is equivalent to the throughput in the network without aggregators. from a sensor in the first coverage area to the sink is given by
(1) 
The contributed throughput from a sensor to the aggregator in the first coverage area is given by
(2) 
The total throughput seen by a sensor in the first coverage area is . Similarly, we can obtain the throughput seen by a sensor located in the second coverage area which is assisted by the second aggregator .
Then, we need to characterize the average arrival rates at the aggregators, denoted by and . Since we assume full MPR capability at the receivers, the th aggregator can receive up to packets in a timeslot. We define as the probability that packets will arrive in a timeslot at the th aggregator. The average arrival rate at the th aggregator is given by
(3) 
The probability where is given by
(4)  
Similarly we can obtain . The networkwide throughput is . The previous expressions for the throughput are valid when the queues at the aggregators are stable.^{3}^{3}3Here we will not consider the case that the queues are not stable, but in order to obtain the throughput in this case one needs to replace the sum of expressions with the service rate of the aggregator. In this case the networkwide throughput will be given by .
IiiB Stability Analysis at the Aggregators
The average service rate for the aggregator is given by
(5) 
where is the queue size at queue . The notation for the success probabilities used in (5) is the one introduced in Section II. We can easily see from (5) that the service rate of one queue depends on the status of the other queue. Thus, the queues are coupled. In order to bypass this difficulty we deploy the stochastic dominance technique introduced in [25]. The proof is along the lines of [20, 26] and is omitted due to space limitations. The stability conditions for the queues at the aggregators are described by the region where is given by (6).
(6)  
Iv Preliminary Analysis
Let be the number of packets in the buffer of aggregator , , at the beginning of the th slot. Then, is a discrete time Markov chain with state space . The queues of both aggregators evolve as follows:
(7) 
where is the the difference of the number of packets that enter the buffer of the th aggregator at the beginning of slot ( equals or ), and .
Before proceeding with the analysis, we will slightly modify the notation for the success probabilities presented in Section II in order to be more convenient for the delay analysis. If two sensors transmit a packet simultaneously, represents the probability that the packet from sensor is successfully received by node and the packet from sensor failed to be received by . represents the probability that the packets from both nodes are successfully received by . Then we have ,which denotes the probability that both packets fail to be received by the node . If both aggregators transmit simultaneously, then with probability the packet from is successfully received by node , with probability the packets from both aggregators are successfully received by , while with probability , both of them failed to be received by .
Let be the generating function of the joint stationary queue process,
Then, by exploiting (7) we obtain after lengthy calculations
(8) 
where,
is the kernel of the functional equation (8) and
Clearly, for every fixed with , it is regular in for , and continuous in for ; similar observation hold for the variable .
Some interesting relations are directly obtained using (8). In particular, by setting in (8) , dividing with , and taking the limit , by using the L’Hospital rule, and viceversa we obtain the following conservation of flow relations:
(9) 
where, ,
Note that the previous equations are the same with the ones obtained in the previous section, but here we use the more convenient notation for the delay analysis regarding the success probabilities. In order to facilitate the presentation we present the case of two nodes, but clearly the analysis holds for the general case of nodes, just by replacing the right parts of and .
Equations (9), equate the flow of jobs into an aggregator, with the flow of jobs out of the aggregator. Looking carefully at (9) it is readily seen that the following analysis is distinguished in two cases:

. Then,
Iva Kernel analysis
The kernel plays a crucial role in the following analysis and here we provide some important properties. For convenience, assume in the following that . It is readily seen that
where, for ,
The roots of are , , where , .
Lemma IV.1.
For , , has exactly one root such that . When , . Similarly, has exactly one root , such that , for .
Proof.
See Appendix A. ∎
The lemma below provides information about the location of the branch points of the twovalued functions , , its proof is based on algebraic arguments, thus it is omitted.
Lemma IV.2.
The algebraic function , defined by , has four real branch points . Moreover, , and , . Similarly, , is defined by , it has four real branch points , and , and , .
Let , , where , the complex planes of , , respectively. In (resp. ), denote by (resp. ) the root of (resp. ) with the smallest modulus, and (resp. ) the other one. Define the image contours, , , where stands for the contour traversed from to along the upper edge of the slit and then back to along the lower edge of the slit. In the following lemma we provide exact characterization for the smooth and closed contours , respectively:
Lemma IV.3.
The algebraic function , lies on a closed contour , which is symmetric with respect to the real line and defined by
where, , and
Moreover, , are the extreme right and left points of , respectively. Similarly, , lies on a closed contour . Its exact representation is derived as for , and further details are omitted.
Proof.
See Appendix B. ∎
IvB The boundary value problems
Here, we proceed with the derivation of the probability generating function of the joint stationary queue length distribution at relays. The analysis is distinguished in two cases according to the values of the parameters.
IvB1 A Dirichlet boundary value problem
Let . It can be easily seen that
Therefore, for ,
(10) 
Both , and , where , are analytic functions. Using analytic continuation arguments we consider (10) for
(11) 
By noticing that is real for , i.e., , we have
(12) 
where . Clearly, some technical requirements are needed to be everything well defined. In particular, we have to investigate the possible poles of , , where be the interior domain bounded by , and , , . This is equivalent with the investigation of the zeros of , . Moreover, in order to solve (12) we first transform the problem from to the unit circle; see Appendix C for details. Let the conformal mapping, , and its inverse .
By applying the transformation, the problem is reduced to the determination of function regular for , continuous for such that, , . The solution of the Dirichlet problem with boundary condition (12) is:
(13) 
where , a constant to be defined by setting in (13) and using the fact that , (In case has a pole, say , we still have a Dirichlet problem for the function ). Setting , , we obtain after some algebra,
which is an odd function of
, andThus, . Substituting in (13) we obtain after simple calculations an integral representation of on a real interval for , i.e.,
(14) 
Similarly, we can determine by solving another Dirichlet boundary value problem on the closed contour . Then, using the fundamental functional equation (8) we uniquely obtain .
IvB2 A homogeneous RiemannHilbert boundary value problem
In case , consider the following transformation:
Then, for ,
(15) 
Using similar arguments as in previous subsection, we have for
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