Random Access Analysis for Massive IoT Networks Under a New Spatio-Temporal Model: A Stochastic Geometry Approach

Massive Internet of Things (mIoT) has provided an auspicious opportunity to build powerful and ubiquitous connections that face a plethora of new challenges, where cellular networks are potential solutions due to their high scalability, reliability, and efficiency. The random access channel (RACH) procedure is the first step of connection establishment between IoT devices and base stations in the cellular-based mIoT network, where modeling the interactions between static properties of the physical layer network and dynamic properties of queue evolving in each IoT device are challenging. To tackle this, we provide a novel traffic-aware spatio-temporal model to analyze RACH in cellular-based mIoT networks, where the physical layer network is modeled and analyzed based on stochastic geometry in the spatial domain, and the queue evolution is analyzed based on probability theory in the time domain. For performance evaluation, we derive the exact expressions for the preamble transmission success probabilities of a randomly chosen IoT device with different RACH schemes in each time slot, which offer insights into the effectiveness of each RACH scheme. Our derived analytical results are verified by the realistic simulations capturing the evolution of packets in each IoT device. This mathematical model and the analytical framework can be applied to evaluate the performance of other types of RACH schemes in the cellular-based networks by simply integrating its preamble transmission principle.


I. INTRODUCTION
Massive Internet of Things (mIoT) is deemed to connect billions of miscellaneous mobile devices or IoT devices that empowers individuals and industries to achieve their full potential. This issue has been regarded as one of key differences between mIoT and human-to-human (H2H) wireless communication networks, such that the conventional H2H communication architecture needs to be adjusted to support the mIoT networks.
Previously, cellular network (e.g, Long Term Evolution (LTE)) and short-range transmission technologies (e.g, ZigBee, Bluetooth) were considered as potential solutions to support mIoT networks, however none of them can achieve all wide coverage, low power consumption and supporting massive IoT devices at the same time [1][2][3][4]. To solve this, Low-Power Wide Area Networks (LPWANs) is proposed as an alternative solution for mIoT networks that enables the operation in the unlicensed band (e.g, LoRa, Sigfox) and licensed band (e.g, extended coverage GSM-IoT, enhanced machine type communication, and narrow band IoT (NB-IoT)). According to the Third Generation Partnership Project (3GPP), the IoT technologies are suggested to be developed based on the existing cellular infrastructure, due to its low additional hardware deployment cost as well as high-level of security by operating on the licensed band [3][4][5][6][7][8][9].
In the cellular-based mIoT network, connections between IoT device with BS are provided by incorporating these IoT devices in existing cellular networks directly or via IoT gateways. In this network, the number of IoT devices is expected to raise up to more than thirty thousands per cell and such IoT devices may request access simultaneously for their small size data packets uplink transmission [5,10,11]. As such improving the access mechanisms of current cellular systems is one of key challenges for the cellular-based mIoT network [3][4][5][6][7]12]. In LTE, a device performs Random Access CHannel (RACH) procedure when it needs to establish or re-establish a data connection with its associated BS, and the first step of RACH is that the device transmits a preamble via physical random access channel (PRACH) [13]. Two ways exist for accessing to the network: 1) the contention-free RACH for delayed-constrained access requests (e.g, handover), where the BS distributes one of the reserved dedicated preamble to a device, and then the device uses its dedicated preamble to initiate a contention-free RACH; 2) the contention-based RACH for delay-tolerant access requests (e.g, data transmission), where an IoT device randomly chooses a preamble from non-dedicated preambles to transmit to its associated BS [13]. Generally, the contention-based RACH is much more sensitive to IoT traffic [3,4,12], such that most works have analyzed its scalability characteristics in supporting massive concurrent access requests [14][15][16][17][18].
The works on the analysis of contention-based RACH in cellular-based IoT networks, have focused on addressing the following two problems: 1) modelling preamble success transmission impacted by physical channel propagation characteristics (e.g., the fading, noise and interference) [19]; 2) modelling time-varying queues and RACH schemes in MAC layer [14,16,17]. In [14,16,17], the preamble transmission failure impacted by the physical channel propagation characteristics is not modelled. This is especially important in the uplink transmission of largescale cellular-based mIoT networks, where the received signal-to-noise-plus-interference ratio (SINR) at the BS is negatively influenced by the mutual interference generated by other IoT devices due to massive concurrent access requests. In this scenario, the random positions of the transmitters make accurate modelling and analysis of the this interference even more complicated.
Stochastic geometry has been regarded as a powerful tool to model and analyze mutual interference between transceivers in the wireless networks, such as conventional cellular networks [20][21][22], wireless sensor networks [23], cognitive radio networks [24,25], and heterogenous cellular networks [26][27][28]. However, there are two aspects that limit the application of conventional stochastic geometry analysis to the RACH analysis of the cellular-based mIoT networks: 1) conventional stochastic geometry works focused on analyzing normal uplink and downlink data transmission channel, where the intra-cell interference is not considered, due to the ideal assumption that each orthogonal sub-channel is not reused in a cell, whereas massive IoT devices in a cell may randomly choose and transmit the same preamble using the same subchannel; 2) these conventional stochastic geometry works only modelled the spatial distribution of transceivers, and ignored the interactions between static properties of physical layer network and the dynamic properties of queue evolving in each transmitter due to the assumptions of backlogged network with saturated queues. [29,30].
To model these aforementioned interactions, recent works have studied the stability of spatially spread interacting queues in the network based on stochastic geometry and queuing theory [28][29][30]. The work in [29] is the first paper applying the stochastic geometry and queueing theory to analyze the performance of RACH in distributed networks, where each transmitter is composed of an infinite buffer, and its location is changed following a high mobility random walk. The work in [30] investigated the stable packet arrival rate region of a discrete-time slotted RACH network, where the transceivers are static and distributed as independent Poisson point processes (PPPs). The work in [28] analyzed the delay in the heterogeneous cellular networks with spatiotemporal random arrival of traffic, where the traffic of each device is modelled by a marked Poisson process, and the statistics of such traffic with different offloading policies are compared.
In this paper, we develop a novel spatio-temporal mathematical framework for cellular-based mIoT network using stochastic geometry and probability theory, where the BSs and IoT devices are modelled as independent PPPs in the spatial domain. In the time domain, the new arrival packets of each IoT device are modelled by independent Poisson arrival processes [15,28,31,32].
The packets status in each IoT device that are jointly populated by the new Poisson arrival packets and the accumulated packets in the previous time slots according to its stochastic geometry analysis, determines the aggregate interference at the received SINR in the current time slot, which then determines the active probability of IoT device (i.e., IoT device have available packets and permission to transmit currently) in the current time slot. The contributions of this paper can be summarized in the following points: • We present a novel spatio-temporal mathematical framework for analyzing contention-based RACH of the mIoT network. Assuming the independent Poisson arrival, the packets accumulation and preamble transmission of a typical IoT device in each time slot is accurately modelled.
• With single time slot, we derive the exact expressions for the preamble detection probability of a randomly chosen BS, the preamble transmission success probability of a randomly chosen IoT device, and the number of received packets per BS in the cellular-based mIoT networks.
• With multiple time slots, the queue statuses are firstly analyzed based on probability theory, and then approximated by their corresponding Poisson arrival distributions, which facilitates the queueing analysis. By doing so, we derive the exact expressions for the preamble transmission success probability of a randomly chosen IoT device in each time slot with the baseline, the ACB, and the back-off schemes for their performance comparison.
• We develop a realistic simulation framework to capture the randomness location, preamble tranmission, and the real packets arrival, accumulation, and departure of each IoT device in each time slot, where the queue evolution as well as the stochastic geometry analysis are all verified by our proposed realistic simulation framework.
• The analytical model presented in this paper can also be applied for the performance evaluation of other types of RACH schemes in the cellular-based networks by substituting its preamble transmission principle.
The rest of the paper is organized as follows. Section II presents the network model. Sections III derives preamble detection probability of a randomly chosen BS and the preamble transmission success probability of a randomly chosen IoT device in single time slot. Section IV derives the preamble transmission success probabilities of a randomly chosen IoT device in each time slot with different schemes. Finally, Section V concludes the paper.

II. SYSTEM MODEL
We consider an uplink model for cellular-based mIoT network consists of a single class of base stations (BSs) and IoT devices, which are spatially distributed in R 2 following two independent homogeneous Poisson point process (PPP), Φ B and Φ D , with intensities λ B and λ D , respectively.
Same as [20,27,33], we assume each IoT device associates to its geographically closest BS, and thus forms a Voronoi tesselation, where the BSs are uniformly distributed in the Voronoi cell.
Same as [28,30], the time is slotted into discrete time slots, and the number and locations of BSs and IoT devices are fixed all time once they are deployed.

A. Network Description
We consider a standard power-law path-loss model, where the signal power decays at a rate r −α with the propagation distance r, and the path-loss exponent α. We consider Rayleigh fading channel, where the channel power gains h(x, y) between two generic locations x, y ∈ R 2 is assumed to be exponentially distributed random variables with unit mean. All the channel gains are independent of each other, independent of the spatial locations, and identically distributed (i.i.d.). For the brevity of exposition, the spatial indices (x, y) are dropped.
Uplink power control has been an essential technique in cellular network [20,27,34]. We assume that a full path-loss inversion power control is applied at all IoT devices, where each IoT device compensates for its own path-loss to keep the average received signal power equal to a same threshold ρ. By doing so, as a user moves closer to the desired base station, the transmit power required to maintain the same received signal power decreases, which saves energy for battery-powered IoT devices. More importantly, it helps to solve the "near-far" problem, where a BS cannot decode the signals from cell-edge due to high aggregate interference from other nearby IoT devices. The transmit power of ith IoT device P i depends on the distance from its associated BS, and the defined threshold ρ, where P i = ρr i α . In order to successfully transmit a signal from the IoT device, the maximum transmit power should be high enough for its pathloss inversion, otherwise, it does not transmit the signal and goes into a truncation outage. Here, we assume that the density of BSs is high enough and none of the IoT device suffers from truncation outage (i.e., the transmit power of IoT device is large enough for uplink path-loss inversion, while not violating its own maximum transmit power constraint).

B. Contention-Based Random Access Procedure
In LTE, the first step to establish an air interface connection is delivering requests to the associated BS via RACH [13], where the contention-based RACH is favored by mIoT network for the initial association to the network, the transmission resources request, and the connection re-establishment during failure [3,4,6,12]. The contention-based RACH has four steps: In step 1, each device randomly chooses a preamble from avalaible preamble pool, and send to its associated BS via PRACH. In step 2, the IoT device sets a random access response (RAR) window and waits for the BS to response with an uplink grant in the RAR. In step 3, the IoT device that successfully receives its RAR transmits a radio resource control connection request with identity information to BS. In step 4, the BS transmits a RRC Connection Setup message to the IoT device. Note that, only within the step 1 preamble is transmitted via PRACH, but within other steps signals are transmitted via normal uplink and downlink data transmission channel.
Further details on the RACH can be found in [13].
LTE defines prime-length Zadoff-Chu sequences as the random multiple access codes (also called preambles) for the step 1 of RACH, where different preambles are orthogonal [13]. In the mIoT network, each IoT device requests for access in the first available opportunity leading to a huge number of IoT devices transmit preambles simultaneously, such that the network performance might degrade due to that the preambles cannot be detected or decoded by the BS [3,4]. Such preamble transmission failures occur in the step 1 of contention-based RACH, which has been analysed in many previous works [16,17,35,36]. This preamble transmission can be failed due to the following two reasons: 1) a signal cannot be recognized by the received BS, due to its low received SINR; 2) the BS successfully detected more than two signals using same preamble simultaneously, such that the collision occurs, and the BS cannot decode any collided signals. The 3GPP and organization members have investigated the preamble transmission failure problem of the RACH in mIoT network [5,10,11]. In this work, we limit ourselves to single preamble transmission fail, and leave the collision for our future work, thus we assume that a RACH procedure is always successful if the IoT device successfully transmits the preamble to its associated BS. Without loss of generality, we assume that each BS has an available preamble pool with same number of different preambles, known by its associated IoT devices, where N p denotes the number of preambles. Each preamble has an equal probability (1/N p ) to be chosen by an IoT device, and the average density of the IoT devices using the same preamble is λ Dp = λ D /N p , where the λ Dp is measured with unit devices/preamble/km 2 .

C. Physical Random Access CHannel and Traffic Model
PRACHs is formed by sequences of allocated time-frequency resources, which are reserved in the uplink channel, and repeated in the system with a certain period that specified by the BS.
For instance, the uplink resource reserved for RACH preamble transmission has a bandwidth corresponding to six resource blocks (1.08 MHz), and the RACH opportunity is repeated with a periodicity varies between every 1ms to 20 ms [4,13]. Without loss of generality, we assume that each time slot consists of a gap interval duration τ g and a RACH duration τ c , where the RACH duration τ c is shorter than the gap interval duration τ g as shown in the Fig. 1.
We model the arrival of new packets in mth time slot at each IoT device as independent Poisson arrival process, Λ m New with same arrival rate ε m New as [15,31,32]. It is assumed that the new packets arrival only takes place within the gap duration τ g of each time slot, such that the number of new arrival packets N m New in duration τ g is described by the Poisson distribution with N m New ∼ Pois(τ g ε m New ), where µ m New = τ g ε m New . Within the duration of τ g , the new arrival packets are stored in their buffers immediately, and then each device with non-empty buffer try to request uplink data transmission channel resources for its head-of-line packet in the following RACH within the duration of τ c . Note that the data transmission after a successful RACH can be easily extended following the analysis of preamble transmission success probability in RACH.
Due to the main focus of this paper is analyzing the time-slotted contention-based RACH in the mIoT network, we assume that the actual intended packet transmission is always successful if the corresponding RACH succeeds.
Each device is assumed to have an infinite buffer, where the arrived packets are stored in the buffer until successful transmission, and none of the packets will be dropped off. In each device, the packets are scheduled as a queue for transmission, where each packet has same priority, and the BSs are unaware of the queue status of their associated IoT devices. Each IoT deivce transmits packets via a First Come First Serve (FCFS) packets scheduling scheme -the basic and the most simplest packet scheduling scheme, where all packets are treated equally by placing them at the end of the queue once they arrive [37].
In the temporal domain, the queue status of each device is evolved following transmission condition over time. A packet is successfully transmitted and removed from the buffer once the RACH succeeds, otherwise, this packet will be still in the first place of the queue, and the IoT device will try to request channel resources for the packet in the next available RACH. The number of accumulated packets in the buffer in mth time slot is denoted as N m Cum , where the buffer status N m Cum is recorded (Recording N m Cum in Fig.1) at the begin of the mth time slot. Note that the buffer of each IoT device is set empty at the beginning of the first time slot (m = 1).
The evolving of queue status in an IoT device is described in Table I

D. Transmission Schemes
In the cellular-based mIoT network, a huge number of IoT devices are expected to request for access frequently, such that network congestion may occur due to mass concurrent data and signaling transmission [5]. To solve this, efficient RACH control mechanisms are needed for congestion reduction. In the following, we listed three schemes to meet this requirement: • Baseline scheme: each IoT device attempt RACH immediately when there exists packet in the buffer. The baseline scheme is the simplest scheme without any control of traffic.
Due to RACH attempts are not be alleviated at the IoT devices, the baseline scheme can contributes to the relatively faster buffer flushing in non-overloaded network scenarios.
However, once the network is overloaded, high delays and service unavailability appear due to mass simultaneous access request.
• Access Class Barring (ACB) scheme: each non-empty IoT device draws a random number q ∈ [0, 1], and attempts to RACH only when q ≤ P ACB , here P ACB is the ACB factor specified by the BS according to the network condition [5,13]. ACB scheme is a basic congestion control method that reduces RACH attempts from the side of IoT devices based on the ACB factor. It is known that a suitable ACB factor can keep the allowable access in a reasonable density, and assure a relative high data transmission rate when the network is overloaded.
• Back-off scheme: each non-empty IoT device transmits packets same as baseline scheme, when there exists packet in the buffer. However, when RACH fails, the IoT device automatically defers the RACH re-attempt and waits for t BO time slots until it trys again.
Back-off scheme is another basic congestion control method, where each IoT device can automatically alleviate congestion and requires less control massage from BS than that of ACB scheme [3].

E. Signal to Noise plus Interference Ratio
As we mentioned earlier, each IoT device transmits a randomly chosen preamble to its associated BS to request for channel resources, where different preambles represent orthogonal sub-channels, and thus only IoT devices choosing same preamble have correlations. Note that IoT devices belonging to a same BS may choose same preamble, such that the intra-cell interference is considered. A preamble can be successfully received at the associated BS, if its SINR is above the threshold. Based on Slivnyak's theorem [38], we formulate the SINR of a typical BS located at the origin as where ρ is full path-loss inversion power control threshold, h o is channel power gain from the typical IoT device to its associated BS, σ 2 is noise power, I inter is aggregate inter-cell interference, and I intra is aggregate intra-cell interference. It is noted that only the non-empty IoT devices are active and generate interference, where the active probability of each IoT device can be treated using the thinning process. The active probability of each IoT device in the mth time slot T m can be defined as where N m New is the number of new arrived packets in the mth time slot, and N m Cum is the number of accumulated packets in the mth time slot.
where Z in is the set of intra-cell interfering IoT devices, 1 {·} is the indicator function that takes the value 1 if the statement 1 {·} is true, and zero otherwise, N m New j is the number of new arrived packets of jth device in the mth time slot, N m Cum j is the number of accumulated packets of jth device in the buffer in the mth time slot, h j is channel power gain from the jth intra-cell interfering IoT device to the typical BS. The aggregate inter-cell interference is expressed as where Z out is the set of inter-cell interfering IoT devices, · is the Euclidean norm, N m New i is the number of new arrived packets of ith device in the mth time slot, N m Cum i is the number of accumulated packets of ith device in the buffer in the mth time slot, h i is channel power gain from the ith inter-cell interfering IoT device to the typical BS, u i is the distance between the ith inter-cell IoT device and the typical BS, and P i is the actual transmit power of the ith inter-cell IoT device, and P i depends on the power control threshold ρ and the distance between the ith inter-cell typical IoT device and its associated BS r i with P i =ρr i α .

III. ANALYSIS UNDER THE SINGLE TIME SLOT MODEL
In this section, we provide a single time slot model (1st time slot), where the queue status (number of packets in buffer) of each IoT device only depends on the new packets arrival process Λ 1 New . Note that inactive IoT devices (those with no packet in buffer) do not attempt RACH, such that they do not generating interference. Due to that the preamble (sub-channel) has an equal probability to be chosen, the analysis performed on a randomly chosen preamble can represent the whole network. The probability that the received SINR at the BS exceeds a certain threshold γ th is written as where L I (·) denotes the Laplace Transform of the PDF of the aggregate interference I. Note that whether an IoT device is active only depends on the new packets arrival process Λ 1 New in the 1st time slot, such that the active probability (the thinning factor) of each IoT device T 1 in the 1st time slot is expressed as where µ 1 New = τ g ε 1 New , ε 1 New is the new packets arrival rate of each IoT device in the 1st time slot, and the probability of N 1 New = 0 during τ g in the 1st time slot is equal to e −τgµ 1 New .
We perform the analysis on a randomly chosen BS and a BS associating with a randomly chosen IoT device in terms of the preamble detection probability and the preamble transmission success probability. The probability that the received SINR at a randomly chosen BS exceeds a certain threshold γ th has been studied in many stochastic geometry works [20,22,27,33]. Those analyses focus on the uplink transmission channel of a cellular networks, without considering intra-cell interference due to TDMA or FDMA assumptions, and only considered inter-cell interference. In their models, the aggregate interference is the same, no matter if the tagged BS is randomly chosen, or is determined by a randomly chosen device via association, thus the probability that the received SINR exceeds a threshold γ th at a randomly chosen BS is equally same as the probability of a BS associating with a randomly chosen uplink device.
Different from the conventional stochastic geometry works in [20,22,27,33] with no intracell interference, we take into account the intra-cell interference due to the same preamble reuse among many IoT devices in a cell during their uplink RACH. We will derive the preamble detection probability from the view of a randomly chosen BS (i.e., each BS has an equal probability to be chosen), and the preamble transmission success probability from the view of a BS that a randomly chosen IoT device belongs to (i.e., the probability of a BS being chosen is determined by the number of its associated IoT devices). The difference between these two characteristics comes from the fact that a cell, that a randomly chosen IoT device belonging to, has chance to cover more IoT devices than a randomly chosen cell [22,39].

A. Preamble Detection Probability
We first perform analysis on a randomly chosen BS, and one of its active associated IoT where T 1 is given in (6). Remind that λ Dp is the intensity of IoT devices using same preamble.
Proof. See Appendix A.
Since the interference generating by each intra-cell IoT device is strictly equal to ρ, such that the aggregate intra-cell interference only depends on the number of active interfering IoT devices in the Voronoi cell. We assume Z in denotes the number of active IoT device in a specific a Voronoi cell has been derived by the Monte Carlo method in [40], and the Probability Mass function (PMF) of the number of IoT devices Z in in a randomly chosen BS has been clearly introduced in [22], which is expressed as where c = 3.575 is a constant related to the approximate PMF of the PPP Voronoi cell, and Γ (·) is gamma function. For the Voronoi cell with at least one active IoT device, the PMF of the number of active interfering intra-cell IoT devices Z B in a randomly chosen Voronoi cell (BS) is given by The Laplace Transform of aggregate intra-cell interference is conditioned on the number of interfering intra-cell IoT devices Z B , which is derived in the following Lemma.
Lemma 2. The Laplace Transform of aggregate intra-cell interference at the typical BS in cellular-based mIoT network is given by Proof. See Appendix B.
Substituting (7) and (10) into (5), we derive the preamble detection probability of the typical BS in the 1st time slot P 1 d in the following theorem.
Theorem 1. The preamble detection probability of a randomly chosen BS in the 1st time slot of the cellular-based mIoT network is given by Proof. Following the proofs of Lemma 1 and Lemma 2.

B. Preamble Transmission Success Probability
Next, we move to the preamble transmission success probability that is performed on a BS in which a randomly chosen IoT device belongs to. Conditioned on a randomly chosen IoT device in its cell, the PMF of the number of interfering intra-cell IoT devices in that cell Z D is expressed as [22] P {Z D = n} = c (c+1) Γ(n + c + 1)( The difference between (9) and (12) is clearly explained in [39]. Briefly speaking, in (9), each Voronoi cell has an equal probability to be chosen, whilst in (12), a Voronoi cell with more IoT devices has a higher probability to be chosen. Following similar approach in the proof of Lemma 2, and with the help of (12), the Laplace Transform of aggregate intra-cell interference is given in the following lemma.
Lemma 3. The Laplace Transform of aggregate intra-cell interference at the BS to which a randomly chosen IoT device belongs in the cellular-based mIoT network is given by Substituting (7) and (13) into (5), we derive the preamble transmission success probability of the 1st time slot P 1 t in the following theorem.
Theorem 2. In the depicted cellular-based mIoT network, the preamble transmission success probability of a randomly chosen IoT device of the 1st time slot is given by Proof. The proof follows from Theorem 1.
In Theorem 1, the preamble detection probability at a randomly chosen BS in mIoT network is analyzed based on the number of active interfering intra-cell IoT devices in that randomly chosen Voronoi cell (BS) in (9), whereas in Theorem 2, the preamble transmission success probability at a randomly chosen IoT device is described by the number of interfering intra-cell IoT devices in that cell, where that randomly chosen IoT device belongs to in (12).

C. The Number of Received Packets per BS
Next, we analyze the number of received packets per BS of cellular-based mIoT networks as a function of the densities of IoT devices using same preamble and BS, which reflects the density of successfully RACH IoT devices using same preamble per BS ( [22], e.q. (6)). In our model, the number of received packets per BS in the 1st time slot C 1 t is defined as Substituting (14) into (15), we derive the number of received packets per BS C 1 t in the following proposition.
Proposition 1. The number of received packets per BS in the 1st time slot is To obtain the optimal number of received packets per BS in proposed IoT-enabled cellular network, we take the first derivative on C 1 t , and obtain the density of BSs achieving the maximum number of received packets per BS λ * B as . (17) It is easily observed from (14) that the preamble transmission success probability P 1 t is negatively proportional to the density ratio λ Dp /λ B . Undoubtedly, increasing the BS intensity λ B improves the preamble transmission success probability, whereas it does not strictly improve the number of received packets per BS, which can be found in (16). Such phenomenon occurs due to that the number of received packets per BS is jointly determined by two opposite factors: 1) the average received SINR of each BS, 2) the average number of associated IoT devices of each BS. More specifically, increasing the BS intensity increases the received SINR, but decreases the average number of associated IoT devices, which contributes to higher number of received packets per BS in the scenario of overloaded network, but decreases the that in the scenario of non-overloaded network due to low utilization of channel resources. Therefore, there exists a optimal BS density deployment which enables the maximum number of received packets per BS as shown in (17).

IV. ANALYSIS UNDER THE PROPOSED SPATIO-TEMPORAL MODEL
In this section, we analyze the performance of the cellular-based mIoT network in each time slot with different schemes. We first introduce how to analyze the queue evolution with the baseline scheme, and then extend to other two schemes.

A. Baseline Scheme
We start by studying the preamble transmission success probability of the baseline scheme at any time slot. The queue status and the preamble transmission success probability are interdependent, and imposes a causality problem. More specifically, the preamble transmission success probability of current time slot depends on the aggregate interference from those active IoT devices with packets in their buffer in that time slot, thus we need to know the current queue status, which is decided by the previous queue statuses, as well as the preamble transmission success probabilities of previous time slots. Recall that the evolution of queue status follows Table   I  Cum is expressed as where µ 1 New = τ g ε 1 New , ε 1 New is the new packets arrival rate of each IoT device in the 1st time slot, and P 1 t is the preamble transmission success probability of the IoT device in the 1st time slot given in (14). The reason for (18)  Based on (18), we derive the CDF of the number of cumulated packets in the 2nd time slot We are interested in the zero-cumulated packets probability in the 2nd time slot, since it determines the density of active IoT devices (with more than one packet in the buffer) in that time slot, and the activity probability of IoT devices. Based on the probabilistic statistics and (18), we present the active probability of IoT devices in the 2nd time slot as Following the proof of Theorem 2 and substituting (20) into (A.1), we derive the preamble transmission success probability of a randomly chosen IoT device in the 2nd time slot as Similar as (18) and (19), we can derive the PMF and the CDF of the number of cumulated packets in the 3rd time slot N 3 Cum as and respectively. In (22) and (23), f N 2 Cum (x) is given in (18), and P 2 t is given in (21). The PMF and CDF of N m Cum in the mth time slot can be calculated by the iteration process, however, as m increases, it becomes more complicated, and hard to analysis. As such, we approximate the number of cumulated packets in the mth time slot as a Poisson distribution (m > 1) in the following subsection, which largely simplifies the derivations. Cum is equal to zero, due to the buffer of each IoT device is set as empty at the beginning of the first time slot (N 1 b = 0). In the 2nd time slot, µ 2 Cum depends on the new packets arrival rate µ 1 New and the preamble transmission success probability P 1 t of an IoT device in the 1st time slot, which is given by where µ 1 n = τ g ε 1 n , ε 1 n is the new packets arrival rate of each device in the 1st time slot, f N 1 Cum (·) is the PMF of the number of new arrived packets N 1 Cum , P 1 t is given in (14) of Theorem 1. In (24), (a) is the density of the cumulated packets in the 2nd time slot due to the success transmission in the 1st time slot, and (b) is the density of the cumulated packets in the 2nd time slot due to the unsuccess transmission in the 1st time slot.
According to Poisson approximation and (24), the CDF of the number of packets in the 2nd time slot due to previous cumulated packets N 2 Cum is approximated as and the active probability of an IoT devices in the 2nd time slot is approximated as where µ 2 N ew is given in (24). Similarly, the intensity of the number of accumulated packets in the 3rd time slot µ 3 Cum is where µ 2 Cum is given in (24), and P 2 t is given in (21). Thus, we approximate the CDF of the number of accumulated packets in the 3nd time slot N 3 Cum as where µ 3 Cum is given in (27).  We have derived the preamble transmission success probability of an IoT device in the 1st time slot given in (14), and that of the following M time slots can be derived based on the iteration process. In the mth time slot (m = 2, 3, · · · , M ), we first derive the number of accumulated packets µ m Cum , based on the proposed Poisson approximation approach (i.e., following equation (24). Second, we derive the active probability of an IoT device T m using µ m Cum . Finally, we derive the preamble transmission success probability of an IoT device P m t using T m . The preamble transmission success probability of a randomly chosen IoT device in the mth time slot is derived in the following Theorem.
where the active probability of each IoT device is In (30), the intensity of number of accumulated packets µ m Cum is The number of received packets per BS in the mth time slot C m t is derived by using The mean of preamble transmission success probabilities of a randomly chosen IoT device over M time slots is where T m is active probability of each IoT device in the mth time slot, which is given in (30).

B. Access Class Barring Scheme
In the ACB scheme, the BS first broadcasts the ACB factor P ACB , then each IoT device can access the network with probability P ACB or defer its access with probability (1 − P ACB ).
Transmission failure occurs not only due to a low SINR, but also resulting from the failure of passing the ACB check (q < P ACB ). In the derivation, the Laplace Transform of aggregate interference is jointly determined by the ACB probability (P ACB ) and the active probability in mth time slot T m ACB of each IoT device. The preamble transmission success probability of a randomly chosen IoT device with the ACB scheme in the mth time slot is also derived based on the iteration process, which is presented in the following Proposition.
Proposition 3. The preamble transmission success probability of a randomly chosen IoT device in mth time slot in the spatio-temporal model with the ACB probability P ACB is derived as where the active probability of each IoT device with ACB scheme is In (36), the intensity of number of accumulated packets µ m Cum,ACB is The mean of the preamble transmission success probabilities of a randomly chosen IoT device and the mean of the transmission capacities per BS per preamble with ACB transmission scheme over M time slots are presented in the following proposition.

Proposition 4. The mean of the preamble transmission success probability of a randomly chosen
IoT device with ACB scheme over m time slots is The mean of numbers of received packets per BS with ACB scheme over m time slots is (39)

C. Back-Off Scheme
We assume that each IoT device defers its access and waits for t BO time slots, when such IoT devices failed to transmit a packet in the last time slot. The analysis of the back-off scheme is similar to the ACB scheme, due to the back-off procedure can be visualised as a group of IoT devices are completely barred for a time slot. In the 1st time slot, none of IoT device defers the access attempt, such that the transmission procedure is same as the baseline scheme.
After the 1st time slot, the back-off procedure starts to execute, an active IoT device defers its access attempt if the back-off being trigged. The preamble transmission success probability of a randomly chosen IoT device with the back-off scheme in the mth time slot is also derived based on the iteration process, which is presented in the following Proposition.
Proposition 5. The preamble transmission success probability of a randomly chosen IoT device with back-off scheme is where the active probability of each IoT device with back-off scheme is In (41), the intensity of the number of accumulated packets with back-off scheme µ m Cum,BO is Cum,BO , m > 2.

(42)
Due to the back-off mechanism, only active IoT devices without RACH attempt failures in the last t BO time slots can attempt to transmit a preamble, and only those IoT devices generate interference that determine the preamble transmission success probability in the mth time slot. In (40), whether an IoT device generating interference is jointly determined by the active probability (41) and the probability of that active IoT device does not deferring its access attempt B m (ie., due to back-off mechanism). We derive B m using where (a) is the probability that an randomly chosen IoT device fails to transmit a preamble in the (m − j)th time slot, and thus this IoT device would defer its RACH request in the mth time slot due to the back-off mechanism.
The mean of the preamble transmission success probabilities of a randomly chosen IoT device and the mean of the transmission capacities per BS per preamble with back-off scheme over M time slots are presented in the following proposition.
Proposition 6. The mean of the preamble transmission success probability of a randomly chosen IoT device with back-off scheme over M time slots is (44) The mean of numbers of received packets per BS with back-off scheme over M time slots is (45)

V. NUMERICAL RESULTS
In this section, we validate our analysis via independent system level simulations, where the BSs and IoT devices are deployed via independent PPPs in a 100 km 2 area. Each IoT device employs the channel inversion power control, and associated with its nearest BS. Importantly, the real buffer at each IoT device is simulated to capture the packets arrival and accumulation process evolved along the time. The received SINR of each active and non-deferred IoT device (i.e., IoT devices with packets and do not deferred by the ACB or the back-off mechanism) in each time slot is captured, and compared with the SINR threshold γ th to determine the success or failure of each RACH attempt. Furthermore, in the ACB scheme, we also simulate that each IoT device generates a random number q ∈ [0, 1] and compares with the ACB factor P ACB to determine whether the current RACH is deferred, and in the back-off scheme, we capture all RACH failures and practically defer RACH attempts of these IoT devices for the next t BO time slots. In all figures of this section, we use "Ana." and "Sim." to abbreviate "Analytical" and "Simulation", respectively. Unless otherwise stated, we set the same new packets arrival rate for each time slot (ε 1 New = ε 2 New = ε 3 New = · · · = ε m New = 0.1 packets/ms), ρ = −90 dBm, σ 2 = −90 dBm, λ B = 10 BS/km 2 , λ Dp = 100 IoT deivces/preamble/km 2 , α = 4, and γ th = −10 dB.
In the back-off scheme, we set that failure transmission IoT device waits 1 time slot before retransmission in the back-off scheme. Fig. 3 plots the preamble detection probability P 1 d and the preamble transmission success probability P 1 t versus the SINR threshold γ th for the 1st single time slot. We set the duration of gap interval between two RACHs as τ g = 1 ms, which is the duration of gap interval between two RACHs, and µ 1 New = τ g · ε 1 New = 0.1. The analytical curves of preamble detection probability and the preamble transmission success probability are plotted using (11) and (14), respectively.
We first see the well match between the analysis and the simulation results, which validates the accuracy of developed single time slot mathematical framework. As expected, both the preamble detection probability and the preamble transmission success probability increases with decreasing the SINR threshold. The preamble transmission success probability of a randomly chosen IoT device is always lower than the preamble detection probability of a randomly chosen BS, due to that a randomly chosen IoT device has higher chance to associate with a BS with large number of intra-cell interfering IoT devices as shown in (9) and (12), which leads to relatively low average received SINR. Interestingly, increasing the density ratio between the IoT devices and the BSs increases the gap between the preamble transmission success probability and the preamble detection probability, due to that it improves the probability of such randomly chosen IoT device associating with a BS with large number of intra-cell interfering IoT device, but each BS is equally treated in the analysis of the preamble detection probability without influence of density ratio.   t is calculated using (14). We observe that increasing the density ratio between the IoT devices and the BSs decreases the preamble transmission success probability of the 1st time slot, due to the increasing aggregate interference from more IoT devices transmitting signals simultaneously. We also notice that increasing the interval duration between RACHs decreases the preamble transmission success probability. This can be explained by the reason that the number of new arrival packets during longer interval duration increases, and leads to higher active probability of IoT devices as shown in (6).  Fig. 5 plots the number of received packets per BS C 1 t versus the density of BSs λ B for various SINR threshold γ th . We set λ Dp = 500 IoT deivces/preamble/km 2 . The analytical curves for the number of received packets per BS are plotted using (16), and the optimal BSs densities that achieve the maximum number of received packets per BS are plotted using (17). We can see that the calculated optimal BS densities well predict the optimal density points achieving the maximum number of received packets per BS. The first increasing trend of the number of received packets per BS is mainly due to the improvement of the average received SINR, whereas the decreasing trend after λ * B is mainly due to the decreased average number of associated IoT devices of each BS leading to the reduction in channel resources utilization. caused by the increasing average number of accumulated packets. For each scheme, its preamble transmission success probability with γ th = −5 dB decreases faster than that with γ th = −10 dB, due to the higher chance of the accumulated packets being reduced for γ th = −10 dB leading to relatively lower average active probability of each IoT device. Interestingly, we observe that the preamble transmission success probabilities of a random IoT device at each time slot always follow ACB(P ACB = 0.5)>back-off>ACB(P ACB = 0.9)>baseline scheme (except the 1st time slot, where the back-off procedure is not executed), this is because more strict congestion control schemes reduce more access requests from the side of IoT devices, which decrease the aggregate interference in the network.
We also notice that for γ th = −10 dB case, the preamble transmission success probabilities with the P ACB = 0.5 slightly outperform that of ACB scheme (P ACB = 0.9), and the gap between them reduces with increasing time, whilst for γ th = −5 dB, the preamble transmission success probabilities with P ACB = 0.5 is much greater than that with P ACB = 0.9, and such gap increases with increasing time. This is because for γ th = −5 dB, the ACB scheme P ACB = 0.5 is more efficient than P ACB = 0.9, in terms of providing higher average SINR by reducing the probability of queue flushing, but reversely for γ th = −10 dB, the ACB scheme (P ACB = 0.5) has less access requests leading to lower utilization of channel resources. The preamble transmission success probability of a randomly chosen IoT device with back-off scheme is fluctuated, due to the alternation of high load and low load network condition in each time slot. Furthermore, for γ th = −10 dB case, the fluctuation become stable quickly, due to the accumulated packets can be handled much quicker.  . Note that this simulation method with new arrival traffics happen in first several time slots is to examine how well the network can handle bursty traffic, where similar practical simulations has been tested in [10,11]. In both Fig. 7(a) and Fig. 7(b), the preamble transmission success probabilities decrease in the first 10 time slots, due to increasing traffic (new packets arrived) leading to increasing active probabilities of IoT devices. After first 10 time slots, these probabilities increase with time, due to decreasing traffic (i.e., no new packets arrive) leading to decreased active probabilities of IoT devices. After most of the accumulated packets are delivered with time, the preamble transmission success probabilities reaches the stable ceiling. Interestingly, we see that the preamble transmission success probabilities in Fig.   7(a) (γ th = −8 dB) become stable earlier than that in Fig. 7(b) (γ th = −6 dB), due to that the higher chance of the accumulated packets being reduced in lower threshold case.
The preamble transmission success probability of the baseline scheme increases rapidly after first 10 time slots and outperforms other two schemes after first 12th time slots in Fig. 7(a), but it increases relatively slowly after first 10 time slots and only outperforms that of the ACB scheme after first 25 time slots in Fig. 7(b), due to that the baseline scheme provide faster buffer flushing, which leads to lower chance of the accumulated packets being reduced in relatively higher loaded network condition due to the high aggregate interference. The back-off scheme performs better than the baseline scheme in the first 10 time slots (except 1st time slot where back-off is not executed), due to that it automatically defers the retransmission requests and control the congestion in the overloaded network condition. Interestingly, it gradually outperforms the ACB scheme with strictly ACB factor P ACB = 0.3 after the first 10 time slots, due to that the back-off scheme automatically release the blocking of packets and provide faster buffer flushing than the ACB scheme in the non-overloaded network condition. In Fig. 8(a) and Fig. 8(b), we plot the mean of preamble transmission success probabilities and the mean of numbers of received packets per BS over 10 time slots with each scheme, respectively. We set τ g = 1 ms and ACB factor P ACB = 0.3. Note that the new traffics arrival happen in every time slot. In Fig. 8(a), the ACB scheme always outperforms the other two schemes, and the mean of probabilities of the back-off scheme is slightly higher than that of the baseline scheme before γ th = −25 dB, and then such gap between the back-off scheme and the baseline scheme increase with increasing γ th , which is due to that the back-off scheme blocks more packets.
In Fig. 8(b) we observe that 1) For −40 ≤ γ th ≤ −25 dB, the mean of numbers of received packets per BS with the back-off scheme is slightly lower than the baseline scheme, but nearly double that of the ACB scheme, due to the preamble transmission success probability is close to 1 as shown in Fig. 8(a), and thus less packets are blocked in the IoT device in the back-off scheme. 2) For −25 < γ th ≤ −15 dB, the mean of numbers of received packets per BS with the baseline and the back-off schemes decrease dramatically and reduce to same level with the ACB scheme. The back-off scheme gradually outperforms the baseline scheme around the γ th = −20 and −15 dB, because the back-off scheme gradually blocks more IoT devices, and provides better network condition as well as higher probabilities of removing packets from the queue. 3) For −15 < γ th ≤ −5 dB, the ACB scheme outperforms the other schemes, which showcases that the ACB scheme with a relatively strict ACB factor can provide improved successful transmission in overloaded network. VI. CONCLUSION In this paper, we developed a spatio-temporal mathematical model to analyze the RACH of cellular-based mIoT networks. We first analyzed RACH in the single time slot, and provide the preamble detection probability performed on a randomly chosen BS, preamble transmission success probability performed on a BS associated with a randomly chosen IoT device. We then derived the preamble transmission success probabilities of a randomly chosen IoT device with baseline, ACB, and back-off schemes by modelling the queue evolution over different time slot.
Our numerical results show that the ACB and back-off schemes outperform the baseline scheme in terms of the preamble transmission success probability. We also show that the baseline scheme outperforms the ACB and back-off schemes in terms of the number of received packets per BS for light traffic, and the back-off scheme performs closing to the optimal performing scheme in both light and heavy traffic conditions.

APPENDIX A
A PROOF OF LEMMA 1 The Laplace Transform of aggregate inter-cell interference can be derived as where γ(a, b) = b 0 t a−1 e −t dt is the lower incomplete gamma function. As mentioned earlier, the transmit power of IoT device is large enough for uplink path-loss inversion, while not violating its own maximum transmit power constraint, and thus The moments of the transmit power is obtained as E P [P