Modeling of Time and Frequency Random Access Network and Throughput Capacity Analysis

In this paper, we model the random multi-user multi-channel access network by using the occupancy problem from probability theory, and we combine this with a network interference model in order to derive the achievable throughput capacity of such networks. Furthermore, we compare the randommulti-channel access with a cognitive radio system in which the users are able to minimize the channels occupancy. Besides, we show that the sampling rate can be reduced under the Nyquist rate if the use of the spectrum resource is known at the gateway side. This scenario is referred as "cognitive radio" context. The mathematical developments and results are illustrated through various simulations results. The proposed model is particularly relevant in analyzing the performance of networks where the users are not synchronized neither in time nor in frequency as it is often the case in various Internet of Things (IoT) applications. Received on 16 November 2016; accepted on 07 May 2017; published on 31 May 2017


Introduction
The issue of transmitting asynchronous signals on a single channel has been studied for several decades [1] and it led to some protocols such as ALOHA (ALOHAnet) proposed by N. Abramson in 1985 [2]. Since then, many solutions have been proposed in the literature to overcome the distortions induced by the collisions among the transmitted signals. Ref. [3] and references therein provide an extensive overview of solutions based on packet retransmissions and [4] deal with solutions based on signal coding. More recently, the model has been extended to the case of random frequency channel access besides random time channel access [5]. The authors proposed to model the interferences induced by the collisions of ultra narrow-band signals featuring a random frequency channel access in a context of the Internet-of-Things (IoT) applications. Collisions occur at the gateway side (i.e. the node between the UEs and the network) in many applications, since the latter do not coordinate nor schedule the uplink transmissions from user (e.g. objects).
In this paper, we propose to further analyze the issue of signal collisions in the network interference problem [6], [7] by considering random multi-channel and multi-user accesses. In other words we extend the analysis to account for random behavior along the frequency dimension as well where each user of the network transmits not only at random times but also on randomly chosen channels within the band. The random frequency access of N u users into N c channels can be modeled by the occupancy problem used in probability theory [13]. From this model we subsequently derive an analytical expression of the achievable throughput capacity of such a network, i.e. the probability that randomly transmitted signals are properly decoded at the receiver. To the best of our knowledge, the occupancy problem is an original approach for modeling a multi-channel and multi-user access network.
Besides the proposed approach, we compare the performance of random channel access with a cognitive radio scenario where the users are able to choose the channel to transmit in, such that the channel occupancy shall be optimized. In this cognitive radiooriented scenario, the gateway is able to analyze the real-time occupancy of the channels in order to better schedule the transmission of the users, through a downlink signal carrying information. Such a scheme is referred as "fair access", as all the channels tend to be uniformly used. Various simulations illustrate the obtained theoretical results, and they reveal that a weak improvement is achieved by the fair channel access compared with the random one.
In addition to the throughput capacity analysis, we show that the sampling rate at the gateway can be reduced in random multi-channel access, as soon as the spectrum resource is known (at least partially thanks to spectrum sensing for instance) at the gateway. Under this assumption of cognitive gateway, we derive an upper bound of the achievable sub-Nyquist sampling rate, based on the occupancy problem.
The remainder of the paper is organized as follows: Section 2 presents the model of the considered network, and provides a reminder of the occupancy problem. The expression of the throughput capacity is derived in Section 3, and Section 4 is devoted to the sub-Nyquist sampling at the gateway. Simulations are provided in Section 5, and Section 6 concludes this paper.

Network Model
In this paper, it is assumed that all the transmitters (called users) of the network are distributed in the two-dimensional plane according to an homogeneous Poisson point process with a given intensity λ (in users per unit area), whose distribution is given as follows: As depicted in Fig. 1, gateways are also located in the plane, in order to pick up the signals from users. For instance considering the gateway at the center of Fig. 1, we assume it services a cell of radius R c that contains N u users, indicated as small black circles. The users outside this cell (white circles in Fig. 1) are potential interfering users. It should be noted, however, that the users may interfere with each other as well. According to the defined parameters, the intensity λ is equal to N u /(πR 2 c ). The role of a gateway consists in scanning and sampling the band B of bandwidth B which is regularly subdivided into N c channels {B 0 , B 1 , .., B N c −1 } of width B/N c . The considered random multi-channel multi-user transmission model can be formalized as follows: • Each user can randomly access, i.e. select, one of the N c channels with a probability P a = 1 N c . In that way, a reuse of one or more channels may occur 1 , as shown in Fig. 2-(a). In the following, we denote by R the reuse factor of the channels, i.e. the number of users sharing the same channel.
• A slotted traffic scheme is assumed for each user (see Fig. 2-(b)). Therefore, an asynchronous slotted traffic is considered for each channel at the gateway, due to the independence between users. The probability of transmission of a signal (or duty cycle 2 ) is denoted by q ∈ [0, 1], and for simplicity, it is supposed that q is the same for each user. As depicted in Fig. 2-(b), collisions may occur between packets when at least two users transmit at the same time on the same channel.
According to the model in [6,10], the received power at the gateway from a user at a distance R can be defined as where P t denotes the transmitted power, and G is a random variable that depends of the propagation environment. Several models can be used to describe G, depending on the shadowing (mainly due to large obstacles), and the multipath fading (mainly due to the constructive or destructive combinations of the replicas of the transmitted signal). The term 1/R 2c is defined 1 meaning that a channel can be selected by more than one user. 2 Note that duty cycle is sometimes used to refer to the overlapping factor between to packets, as in [6].   as the far-field path loss, where c depends on the propagation environment and is in the range of 0.8 to 4 [6]. It should be emphasized that the far-field path loss fits the considered model, in which the users are supposed to be at least several meters away from the gateway.

Occupancy Problem
The random access to N c channels by N u users is an instance of the occupancy problem in probability. This theory provides useful tools to deal with the time-frequency use of the band B in the considered transmission model. In particular, two theorems will be used in this paper. Theorem 1. Let N u users randomly accessing N c channel with a probability P a = 1 N c . Then the probability that b channels are used (at least by one user) is Proof. see [13], Chapter 4.

Theorem 2.
Under the same assumptions as in Theorem 1, the probability that R users access the same channel (0 ≤ R ≤ N u ) is Proof. Let P(j) be the probability of the event "the j-th channel is chosen", and P(X j = R) the probability that R users use the j-th channel. As P(j) = 1 N c and P(X j = R) are equal for any j, then we obtain The reuse factor R can be defined as the sum X j = R = k x j,k , where x j,k is a random variable which counts the number of users in the channel j. Since the users access to the channels independently of each other, x j,k are the results of Bernoulli trials with probability 1/N c to access the channel j. Therefore, the sum X j = k x j,k obeys the binomial distribution defined in (4).

Deriving the Throughput Capacity
In this section, we derive the expression of the throughput capacity, which is defined in [6] as the probability that a signal is transmitted and successfully decoded at the receiver side. We hereby derive the throughput capacity of the multi-user multi-channel system considering two cases: 1. The users randomly access the channels. In this case, the aforementioned occupancy problem shall be used.
2. The users fairly access the channels, in the sense that the number of users per channel is minimized, therefore minimizing the probability of collision in each channel. This case could be reached in a cognitive radio context where the gateways know the actual occupancy of the channels and schedule the users. Fig. 3 illustrates the two considered cases, where N u = 8, and N C = 4. Each user in the band is represented by a grey square. It can be noted that in the random access scenario, the reuse factor R varies from 0 to 4, whereas it is equal to 2 for any channel in the case of fair channel access.

Throughput Capacity in Random Channel Access
This section deals with the analysis of the throughput capacity, whose definition is given hereafter. In the rest of the paper, we use the so-called physical interference model [7,8], in which the following condition is set: • a message from any user in the cell is successfully decoded if the corresponding signalto-interference-plus-noise ratio (SINR) exceeds a given threshold γ p . Thus, successful decoding for a user k in a given channel n (with reuse factor R, and 1 ≤ n ≤ N C ) requires that: where P r,k,n is the received power of the k-th user in the n-th channel as defined in (2), and σ 2 n is the noise power in the n-th channel. The term I k,n , using the general definition in [6], is the interference power that can be written as where ∆ i,n is the overlapping factor between the i-th interfering signal and the proper signal from user k. It is assumed that ∆ i,n obeys a uniform distribution in [0, 1]. By convention, we consider that the R − 1 first terms of the sum in (7) correspond to the users, and the indexes i > R point out the interfering users that are located outside the considered cell. It has been shown in [9] that the interference defined as the superposition of numerous signals from users distributed according to a homogeneous Poisson process on a plane can be modeled as a αstable distribution [11,12], which can be seen of a generalization of the Gaussian distribution. No analytic expression of the probability density function (pdf) of the α-stable law can be derived, but its characteristic function is expressed as where and According to [6], the parameters α, β, and γ are defined as with Γ(.) the gamma function. It is worth noting that λ R is now a variable function of R, namely λ R = R/(πR 2 c ). The throughput capacity denoted by T R,n,R k,n for a given reuse factor R in the n-th channel, and located at a distance R k,n from the gateway, can be defined as where P(transmit) = q R = 1 − (1 − q) R is the probability that a channel is occupied, and P(no outage) = P(SIN R k,n ≥ γ p ). The probability P(SIN R k,n ≥ γ p ) can be rewritten by substituting (7) into (6) as and hence where F I k,n is the cumulative distribution function (cdf) of I k,n . Several closed-form of (14) corresponding to different types of shadowing and fading have been derived in [6,10]. In particular, it should be noted that the expectation in (13) The Rayleigh fading leads to the following expression: Since the users are homogeneously distributed in the cell, then the distance R k,n obeys the uniform distribution denoted by U (0, R c ), and defined as: The throughput capacity is then obtained by averaging P(SIN R k,n ≥ γ p ) on the interval [0, R c ] as: Note that the bound 0 of the integral in (18) should be replaced by a positive value according to the far-field model (typically >1 meter). Besides, this also avoids the division by zero if the path-loss only model in (15) is used. Since we consider a slotted-asynchronous packet transmission, then the value E{∆ 1/b k,n } can be derived by following the developments in [10], which lead to: Since the users are independent and the N c channels have the same probability of access P a , the throughput capacity for the given n-th channel with reuse factor R is given by: and the overall achieved throughput capacity considering all the users of the cell and all the channels is defined as the following weighted sum: where P(R) is given in (4). It can be noticed in the developments from (6) to (21) that at least five parameters have an influence on the throughput capacity value T : the radius of the cell R c , the number of users N u , the number of channels N c , the duty cycle q, the threshold γ p , and the noise level σ 2 n . Note that we assume σ 2 n = σ 2 for any channel 1 ≤ n ≤ N c .

Throughput Capacity in Fair Channel Access
In the scenario of fair channel access (i.e. the one related to the cognitive radio scheme), the number of users per channel is minimized. Thus, the average reuse factor can be expressed as R ′ = N u /N c , and P(R ′ ) = 1. As a consequence, the throughput capacity in (21) can be simplified as It is worth mentioning that this scenario inevitably induces that the users are aware of the occupancy of the medium. This condition can be achieved if the users share information, or by means on a feedback on the channels occupancy from the gateway to the users. These solutions are especially deployed in cognitive radio systems. It must be emphasized that, although a fair channel access shall reduce the interference between users, it necessitates the deployment of additional processing at the gateway and users sides, such as spectrum sensors, and it may reduce the spectrum resource, if gateways and users share information. Furthermore, such solutions may be not unavailable, since the users are independently deployed, e.g. in IoT networks. In Section 5, simulation results will show the gain in throughput capacity of the fair channel access achievable in a cognitive radio, compared with the random channel access, where users are independent from one to another.

Sub-Nyquist Sampling
In the previous section, we characterized the throughput capacity in cases where users of the networks are aware of the medium state or not. In this section, we focus on the reduction of the sampling rate, which can be achievable at the gateway side. Let assume that the gateway is aware of the spectrum resource, e.g. by means of spectrum sensing, or by using a predefined model such as the daily traffic model described in EARTH project [14]. Note that in this section, the users are supposed to feature a random channel access.
The Nyquist-Shannon theorem would impose to sample the band B at what is called the Nyquist sampling rate f s > f N y = 2B with B corresponding to the bandwidth of B. According to the channel access model, the signals may be sparse in the band (as shown for instance in Fig. 3), and therefore a very large amount of samples will correspond to noise if B is sampled at the Nyquist frequency. Should the sample data need to be transferred and/or stored, this represents an unnecessary cost in bandwidth/memory since in such a case most of the data will be thrown away. However, it is known that sparse signals in the frequency domain can be sampled at a sub-Nyquist rate [15,16]. As a consequence, a cognitive gateway may help to reduce the aggregate needed bandwidth and/or memory.
From the gateway standpoint, the aggregate signals from the N u users can be seen as a multi-band signal with frequency support B b ⊂ B corresponding to the b channels that are randomly accessed. We define λ(B b ) as the Lebesgue measure of B b , namely λ(B b ) = bB N c . In theory, [15], it is possible to sample the band B at the average Sub-Nyquist Sampling (SNS) frequency f SN S = 2λ(B b ) ≤ 2B, and operate a perfect reconstruction of the original signal, provided that the occupied channels are known.
Performing the SNS and the reconstruction is beyond the scope of this paper (see [15,16] for details on uniform and nonuniform sampling). Nevertheless, in this line of work our aim is to provide a condition that guarantees a sufficient SNS frequency f SN S (i.e. 2B ≥ f SN S ≥ 2λ(B b )) with probability γ. In other words, f SN S is defined in such a way that the reconstruction could be perfect in a least 100 × γ% of cases. To achieve this, we express F(b) the cumulative distribution function corresponding to the probability P(b) in (3) as: and we define It is worth noting that the proposed definition of f SN S is not a tight upper bound of the theoretical achievable SNS frequency since (23) only considers the frequency access to channel without taking into account the time traffic of the signals (i.e. it is equivalent to consider q = 1). However, it provides a general idea on how much the amount of samples can be reduced thanks to the SNS, using the formulation of occupancy problem in (3).

Throughput Capacity
T versus R c . Fig. 4 depicts the throughput capacity T versus radius of the cell R c , for different N u values. N c = 5 channels are used in Fig.-(a), and N c = 10 in Fig.-(b). The duty cycle is set as q = 0.1, and the threshold is arbitrary set equal to γ p = 1. It can be observed that the maximum of throughput capacity is reached at R c ≤ 10 m for any considered N u values. This result demonstrates that the throughput capacity is maximized in small cells. Furthermore, the behavior for N c = 5 and N u = 100 is very similar to N c = 10 and N u = 200. As a rule of thumb, this result can be generalized as: the throughput capacity T only depends on the density N c /N u independently of N c and N u values, when N u >> 1. To prove this assertion, let (N c1 , N u1 ) a couple of parameters leading to the binomial probability mass function P 1 (R), and (N c2 , N u2 ) another couple of parameters such that N c2 = 2N c1 and N u2 = 2N u1 leading to the binomial probability mass function P 2 (R). We now express the expectation, the mode, and the second order moment of both P 1 (R) and P 2 (R): • The expectation E of the binomial distribution is given by: • The mode M of the binomial distributions can be expressed as: • Finally, the second-order moment µ ′ 2 of the binomial distributions can be expressed as: It can be straightforwardly shown that if N u >> 1 and N c1 > 1, then dominates in (25), and dominates in (26). Therefore, we can conclude that the distribution P 2 (R) is very similar to P 1 (R).
T versus N u . Fig. 5 depicts the throughput capacity T versus the number of users N u , for R c = 10m in Fig. 5-(a), and R c = 40m in Fig. 5-(b). The throughput capacity behaviors are compared with the theoretical "fair" case in which the channels are fairly chosen by the users, i.e. the N u users are uniformly distributed in the N c channels. It must emphasized that the difference of the achieved throughput capacity values between the random and the fair channel access is very small. Indeed, the throughput capacity in the fair channel access case is 2% higher than the random channel access case. This shows that the fair channel access (e.g. using a cognitive radio approach) does not lead to a large gain in terms of throughput capacity. Furthermore, it must be reminded that such systems require more complexity and/or a higher amount of the spectrum resource. However, this conclusion holds since N u >> N c , in fact it is known that a cognitive radio approach improves performance when N u ≈ N c .
T versus γ p . Another parameter that plays a key role in the throughput capacity T value is the threshold γ p . The value of γ p reflects the robustness of the modulated signal against the different distortions, namely the interferences, including the propagation channel, and the noise. Thus, for a fixed SINR level, a low γ p value  may correspond to a strongly coded signal, while higher γ p could correspond to a signal that is very sensitive to distortions. Fig. 6 shows the throughput capacity T versus the threshold γ p , and can be verified that the lowest the γ p value, the highest the throughput capacity. Moreover it should be noticed that T achieves higher values for N c = 5 than for N c = 10 when q = 0.01 and q = 0.1, whereas the opposite can be observed when q = 0.5.
T versus q. Fig. 7 depicts the throughput capacity T versus the duty cycle q. Furthermore, the random channel access case is compared with the fair channel access. It can be observed that when N u = 10, the maximum T value is achieved when q = 1, i.e. when the users transmit continuously. However, when the density of user per channel increases the throughput  capacity increases from zero to its maximum value and then decreases, as it can be observed in [6] for instance. Moreover, the difference between the random channel access behavior and the fair channel access reduces when N u increases. Besides, it can also be noticed that the T value in the former case can be larger than T in the latter case (and therefore the term "fair" has no sense anymore). This is clearly observable in the case N u = 100 for q values larger than 0.5.

Reduction of the Sampling Rate
In order to analyze the achievable upper bound of sampling frequency f s , we define D as the density of users with respect to the number of channels, namely D = N u /N c . Fig. 8 depicts the obtained f s given as a   ratio of f N y = 2B versus density D, using three values of threshold γ ∈ {0.8, 0.9, 0.99}. The case γ = 0.8 relaxes the constraint on f s , i.e. potential aliasing phenomenon is allowed, whereas γ = 0.99 corresponds to a case where almost no aliasing is expected. Fig. 8-(a) shows the achieved results for N c = 10, and Fig. 8-(b) for N c = 20.
It can be observed on both Figs. 8-(a) and (b) that the f s values increases with the density, and the lower γ the lower f s . Furthermore, it can be noticed that for a given density value, the achievable f s is lower for N c = 20 than N c = 10, which reflects the lower negative skew of P(b) in (3) when N c increases. More generally, it is worth noting that a large reduction of the average sampling frequency can be achieved at D 1.2, e.g. up to 30% at

Conclusion
In this paper, we provided an analysis of the random multi-user multi-channel access in terms of throughput capacity. To to so, we modeled the random channel access by means of the occupancy problem. Furthermore, we compared the random channel access with a cognitive radio approach at the users side, where the users are able to access the channel in an fair way, i.e. by minimizing the number of users per channel. Simulations results revealed that such an approach provides only a slight throughput capacity gain. Besides, a cognitive radio system requires more 8 EAI Endorsed Transactions on Cognitive Communications 02 2017 -05 2017 | Volume 3 | Issue 11 | e2 complexity and more spectrum resource. Therefore, the use of cognitive users may be discussed in future networks, such as IoT for instance.
In addition to the throughput capacity analysis, we examined the advantage of using cognitive gateways, and we proved that a reduction of the sampling rate may be possible, based on the occupancy problem. Thus, if the gateway is able to estimate the occupancy of the channel, then the amount of data to be sampled and stored can be reduced. Thus, the paper provides an analysis of multi-user multi-channel system which could be useful for the design and the performance evaluation of future networks.