I Introduction
The Internetofthings (IoT) is penetrating to different vertical sectors (e.g., smart cities, public safety, healthcare, autonomous deriving, etc.) which will bring billions of new devices to the already congested wireless spectrum. A recent report from ABI Research predicts that 75% of the growth in wireless connections between today and the end of the decade will come from nonhub devices, e.g., sensor nodes [1]. Scalability is one of the major challenges for the next frontier of IoT networks [2]. Such scalability is essential to accommodate the surging machinetype communications within the proliferating IoT applications. Accordingly, new wireless technologies should be developed to serve such unprecedented traffic, which is essentially a blend of humantype and machinetype communications. The challenge is more acute in the uplink (UL) direction since most of the IoT applications are ULcentric by nature [3]. This underlines the utter need for increasing the UL data transmission capacities and the efficiency of medium access schemes [3, 4, 5]. It is a matter of fact that the contention over scarce resources for medium access escalates substantially as the number of devices and traffic intensity grow. This can cause excessive access delays leading to frequent packet drops [6].
In the realm of Release13, Release14, and beyond, the 3GPP provisions few IoTspecific technologies (e.g., ECGSMIoT, LTEeMTC, and NBIoT) in order to accommodate IoT traffic [7, 8]. The 3GPP solutions adopt a scheduled UL (SCUL) approach for the sake of interference management and guaranteed QoS provisioning. The SCUL involves a random access scheduling request (RASR) prior to resource allocations. This is because the BS should be first notified upon data generation at the device buffer. The RASRs are not supervised by the BS and are subject to intracell and intercell interference. The BS then provides exclusive access transmission (EATx) resource blocks for successful RASRs, and hence, EATx transmission does not experiences intracell interference. Note that a single successful RASR can secure EATx over several subsequent frames. Given the sporadic traffic of the IoT devices, such twophase SCUL (i.e., RASR then EATx) scheme may impose an unnecessary delay. The RASR overhead becomes significant for shorter EATx transmission periods. Furthermore, the multiphase handshaking processes (i.e., scheduling request, scheduling response, resource allocation, and EATx transmissions) impose longer wakeup time and data processing for the IoT devices, which shortens the battery lifetime of the IoT devices [9]. To alleviate such delay, signaling, and power consumption overhead, several lowpowerwidearea (LPWA) networks (e.g., LoRa and Sigfox) adopt a single phase random access UL (RAUL) data transmission scheme [9, 4]
. Each IoT device persistently transmits the data from its buffer over a randomly selected resource block. Relying on the sporadic data pattern at each of the IoT devices, prompt RAUL data transmission is expected to help devices to flush their buffer soon after data generation. Consequently, RAUL scheme has the potential to reduce transmission delay, however, due to the probability that more than one IoT device may choose the same resource block, the data transmission is exposed to intracell interference. As such, the SCUL scheme experience intracell interference in the RASR phase only whilst the RAUL scheme may suffer intracell interference in every data transmission. Since each transmission scheme has its own intuitive merits, there is a pressing need for a mathematical framework that characterizes the tradeoff between both transmission schemes and identifies the effective operational scenario of each scheme.
Ia Related Work
In this context, stochastic geometry serves as the baseline model for massive and interference limited networks (see [10, 11, 12, 13] and the references therein). However, standalone stochastic geometry analysis fails to account for the temporal aspects of the IoT such as traffic intensity per device and data accumulation in buffers. The temporal aspects within scheduling problems are usually captured via interacting queueing models [14, 15, 16], however, the work in [14, 15, 16] does not account for the spatial aspects (e.g., node density and mutual interference) that govern the interactions between the queues. Recently, spatiotemporal models that integrate stochastic geometry and queueing theory are proposed to jointly account for perdevice traffic intensity, spatial device density, medium access control (MAC) scheme, devices’ buffer states, and the mutual interference between devices[17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Thus, the IoT network can be abstracted by a network of spatially interacting queues, where the interactions among the devices are governed by a signaltointerferenceplusnoiseratio (SINR) capture model. However, [17, 18, 19, 20, 21, 22] focus on ad hoc networks. Specifically, [18, 17, 19] discus the stability of ad hoc networks. The work in [20] investigates throughput optimization in ad hoc networks with spatial randomness. [21] assess the performance of ad hoc networks with unsaturated traffic in terms of the outage probability and the average packet delay. The authors in [22] analyze the tradeoff between the delay and the security performance in ad hoc networks. The work in [23] addresses downlink scheduling, which does not involve an RASR phase because the BS encloses the data queue and is responsible for scheduling. The work in [24, 25] analyzes the Random Access CHannel (RACH) performance in cellularbased IoT networks which does not involve EATx phase for data transmission. The UL scenario where the data queue is at the device side is considered in [26]. However, only RAUL data transmission with power ramping and transmission deferral are considered. To the best of the authors’ knowledge, the 3GPP based SCUL, with joint RASR then EATx phases, for IoT UL traffic is never addressed in the literature. Furthermore, there are no existing studies that assess and compare the 3GPP based SCUL (i.e., grantbased access) and the LPWA based RAUL (i.e., grantfree access) schemes for IoT networks.
IB Contributions
This paper presents a novel spatiotemporal model for IoT UL communications to characterize the SCUL and RAUL transmission schemes.^{1}^{1}1This work is presented in part in [27]. From the spatial perspective, the BSs and devices are modeled via two independent Poisson point processes (PPPs). Besides its simplicity and tractability, the PPP model is validated via several experimental and theoretical studies [10, 12, 13]
. From the temporal perspective, each communication link is modeled via a discrete time Markov chain (DTMC) to track packets generation and departure from devices’ buffers. Consequently, the overall developed spatiotemporal framework models the IoT devices as a network of spatially interacting DTMCs, where the interactions are governed by the SINR capture model. To this end, the developed mathematical framework is used to characterize the scalability of SCUL and RAUL transmission schemes. The scalability of the network is investigated via the spatiotemporal traffic demand along with the notion of queueing stability. Within this context, a stable buffer (i.e., queue) is the one that has packet departure probability greater than the packet arrival probability
[28]. Otherwise, the number of packets in the devices’ buffers would grow indefinitely driving these buffers (i.e., the network) to a state of “instability”. As such, an IoT network can scale in terms of devices and/or traffic rates as long as it still falls within the stability region. In this paper, scalability is characterized by the Paretofrontier of all pairs of devices density and perdevice traffic intensity that keeps the devices’ buffers stable. Consequently, the terms scalability and stability are hereafter interchangeable. The contributions of the paper can be summarized as follows:
SCUL transmission with the joint RASR and EATx phases of the 3GPP is modeled via a tandem queueing approach. An SINR capture model is adopted, where the SINR has to exceed a given threshold for packet departure. Since several devices served by a given BS may simultaneously select the same resource for RASR, only the device with the highest SINR succeeds if its SINR exceeds the RASR SINR threshold. The EATx phase enforces a single device per channel per BS, and hence, the UL SINR threshold is the only constraint for transmission success.

The RAUL scheme is modeled via a baseline DTMC. An SINR capture model is adopted, where the SINR has to exceed a given threshold for packet departure. Since several devices served by a given BS may simultaneously utilize the same resource for RAUL, only the device with the highest SINR succeeds if its SINR exceeds the UL SINR threshold. It is important to note that the developed RAUL model in this paper is different from [26], which constrains packet departure with the UL SINR threshold only. Hence, the RAUL model in [26] is more optimistic as all intracell interfering devices satisfying the SINR threshold simultaneously deliver their packets to the serving BS.

The SCUL and RAUL techniques are compared in terms of transmission success probability, delay, average queue size, and scalability. The effective operational scenario of each transmission scheme is identified. For instance, RAUL transmission is effective for lower device intensity with high traffic demand per each device. However, when the devices intensity grows, intracell interference becomes overwhelming and scheduling is necessary to maintain stability.
IC Notation & Organization
Throughout the paper, we use the math italic font for scalars, e.g.,
. Vectors are denoted by lowercase math bold font, e.g.,
, while matrices are denoted by uppercase math bold font, e.g., . We use the calligraphic font, e.g.,to represent a random variable (RV) while the math typewriter font, e.g.,
is used to represent its instantiation. Moreover, and denote, respectively, the expectation and the Laplace Transform (LT) of the PDF of the random variable . We use to denote the probability and the overbar, e.g, to denote the probabilistic complement operator. is the indicator function which has value of one if the statement is true and zero otherwise. indicates the Gamma function and is the Gaussian hypergeometric function. Finally, denotes the value at the iteration.The rest of the paper is organized as follows. Section II presents the system model, the approximations, and methodology of analysis. Section III presents the spatial, temporal, and spatiotemporal analysis of the depicted IoT network for both the SCUL and RAUL schemes. Section IV presents and discusses some numerical results. Finally, Section V summaries and concludes the paper.
Ii System Model and Approximations
Iia Spatial & Physical Layer Parameters
We consider a singletier network where the BSs are spatially distributed in according to a homogeneous Poisson point process (PPP) with intensity . The devices are spatially distributed in via an independent PPP with intensity . Without loss of generality, each device is assumed to associate to its nearest BS. Hence, the average number of devices associated to each BS is denoted as . A powerlaw pathloss model is considered where the signal power decays at a rate with the propagation distance , where is the pathloss exponent. In addition to the pathloss attenuation, Rayleigh flat fading is assumed, in which all the channel power gains (
) are exponentially distributed with unity power gain. All channel gains are assumed to be independent of each other, independent of the spatial locations, and are identically distributed (i.i.d). It is also assumed that the channel power gains are independent across different time slots for all the devices. Fig.
1 shows a netwrok realization for within an area of km.All UL transmissions utilize full pathloss inversion power control with threshold . That is, each device controls its transmit power such that the average signal power received at its serving BS is equal to a predefined value . It is assumed that the BSs are dense enough such that each of the devices can invert its path loss towards the closest BS almost surely, and hence the maximum transmit power of the IoT devices is not a binding constraint for packet transmission. Extension to fractional power control and/or adding a maximum power constraint can be done by following the methodologies in [29] and [30].
IiB MAC Layer Parameters
We consider a discrete time slotted network in which time is discretized into slots with equal durations (). Each time slot can be used for a single transmission attempt (e.g., RASR or EATx for the SCUL). Moreover, we assume that a single packet arrival and/or departure can take place per time slot via a First Come First Served (FCFS) discipline. A geometric interarrival packet generation, with parameter (packet/slot), is assumed at each device. The arrived packets at each device are stored in a buffer (i.e., queue) with infinite length until successful transmission using a UL resource block.^{2}^{2}2 Infinite buffers are assumed for generality. In the numerical results, it is shown that small buffer sizes are sufficient as long as the network is stable. Different from the arrival process, the departure process (i.e., successful packet transmission) cannot be assumed. Instead, the departure process has to be characterized according to the UL transmission protocol and SINR distribution.
IiC SCUL Scheme
The data is generated at the devices’ buffers and the BS is unaware of the devices’ buffer status.^{3}^{3}3IoT devices have sporadic traffic patterns and can remain idle (i.e., with empty buffers) and go to sleep mode for long periods of time to save battery. Buffer state updates lead to unnecessary wakeups that deplete batteries [31]. Furthermore, buffer state updates from billions of devices would impose overwhelming signaling overhead. Devices with nonempty buffers should send an RASR to the serving BS [32, 8]. A device that experiences successful RASR is scheduled by the BS for the EATx phase. To model such twophase UL traffic in 3GPP networks, a tandem (i.e., consecutive) queueing approach with heterogenous departure probabilities is introduced. The tandem queueing model for the SCUL transmission scheme with joint RASR and EATx phases is depicted in Fig.2. The first queue (colored in red in Fig.2) represents the RASR process that occurs prior to resource allocations. The latter queue (colored in cyan in Fig.2) represents EATx data transmission after resource allocation. A detailed description of the two phases for the SCUL transmission is given in the sequel.
IiC1 RASR Phase
To request a UL resource block, each device randomly and independently transmits its request on one of the available primelength orthogonal ZadoffChu (ZC) sequences defined by the LTE physical random access channel (PRACH) preamble [32]. Since the number of the ZC sequences is finite, it is possible for more than one device to select the same ZC sequence for RASR, which leads to mutual intracell and intercell interference. Without loss of generality, we assume that all BSs have the same number () of ZC sequences generated from a single root sequence.^{4}^{4}4This implies that the BSs are dense enough such that all the sequences within each BS are generated from cyclic shifts of a single root sequence [32].^{5}^{5}5 If the neighboring BSs use different root sequences, a thinned PPP with a thinning factor of can be used, where is the total number of available root sequences. We assume a power capture model for successful RASR, in which the signal can be decoded if and only if the SINR at the serving BS is greater than the RASR threshold . Moreover, each BS can only decode one RASR request per ZC code per time slot. That is, across the intracell interfering devices over the same ZC code within the same BS, only the device that has the highest SINR can succeed, i.e., the BSs have a capturing capability [33]. We also assume that the response of each RASR attempt is instantaneous and errorfree. Upon RASR failure, the ZC code random selection is repeated. It is worth to highlight that the depicted model is consistent with PRACH configuration index14 in which there is a PRACH resource in each time slot [32]. It is assumed that the RASR phases are timesynchronized for all the BSs in the network.
IiC2 EATx Phase
Each BS has resource blocks that are dedicated for UL transmission. A device that succeeds in RASR is granted EATx (i.e., intracell interference free) for the next subsequent time slots on a randomly selected free resource block by its serving BS.^{6}^{6}6The serving BS randomizes the channel allocations for the scheduled devices at each time slot to decorrelate interference across time slots. If all the resource blocks are occupied by other UL transmissions, the device has to reperform the RASR phase. If a device flushes its buffer before completing the EATx transmission slots, it immediately goes back to the idle state and releases the allocated channel. Hence, setting requires a successful RASR prior to each EATx packet transmission attempt. As such, is a design parameter for the SCUL scheme. We assume a power capture model for successful EATx transmission, where a data packet departs from the device buffer if and only if the SINR at the serving BS is greater than the UL threshold . We also assume that the feedback of each transmission attempt is instantaneous and errorfree. It is assumed that the EATX phases are timesynchronized in for all the BSs in the network.
IiD RAUL Transmission
In the RAUL scheme, each device directly sends the data packets to its closest BS without scheduling. To diversify mutual interference, each device randomly and independently selects a resource block among the available resource blocks for each transmission/retransmission. Since the number of the resource blocks is finite, it is possible for more than one device to utilize the same frequency for RAUL, which may lead to both intracell and intercell interference. Without loss of generality, we assume that all BSs have the same number of the frequency channels and the transmission slots are timesynchronized for all the BSs in the network. We assume a power capture model for successful RAUL transmission, where a data packet departs from the device buffer if and only if the SINR at the serving BS is greater than the UL threshold . Moreover, in the case of intracell interference, the BS can successfully decode the packet from the device with the highest SINR only, i.e., the BSs have a capturing capability [33]. We also assume that the feedback of each transmission attempt is instantaneous and errorfree. The queueing model for a typical device is shown in Fig. 3, in which the device keeps transmitting as long as it has a nonempty buffer.
IiE Methodology of Analysis
The DTMC in Fig.2 and Fig.3 model the temporal evolution of the number of packets in the system as well as the service phase (i.e., RASR and EATx transmission slots) at each device. Such queueing systems are categorized within the QuasiBirthDeath (QBD) processes because the population of the system (i.e., the packets in the buffer) can only be incremented or decremented by one in each time slot [34]. If the arrival and departure processes are known, the steadystate solution of such QBDs can be characterized. However, the departure process depends on the SINR distribution, which should be derived using stochastic geometry analysis. Meanwhile, the SINR distributions depend on the interfering devices states (e.g., idle or active), which require the steadystate solution of the queueing models. Hence, the stochastic geometry analysis for the SINR distribution and the queueing theory analysis for the devices’ steadystate probabilities are interdependent, which necessitates an iterative solution. In this paper, we adopt the following methodology to characterize the performance of the depicted system model with such spatiotemporal interdependence. First, the stochastic geometry analysis is conducted in Section IIIA, where expressions for the RASR, EATx, and the RAUL success probabilities are obtained as functions queueing theory parameters. Then, the queueing theory analysis is presented in Section IIIB, where the steadystate solutions of the queueing models are obtained as functions of the stochastic geometry parameters. The iterative algorithm that simultaneously solves the stochastic geometry and queueing theory set of equations is then presented in Section IIIC.
Iii Performance Analysis
While some IoT devices might be free to move, it is not expected to have tangible variation in terms of the network geometry over consecutive time slots. This is because the locations of the BSs are fixed and that the time slots are too short (e.g., in the scale of milliseconds.) for a tangible geographical displacement. Hence, it is reasonable to consider an arbitrary but fixed network realization that does not change over time. In such static network setup, different devices generally have location dependent performance according to the number and relative locations of proximate interferers [35, 19]. However, the location dependent performance discrepancies are negligible in the depicted system model due to the employed full pathloss inversion power control along with the randomized channel selection per transmission [36, 37, 26]. Consequently, the success probabilities of the typical device (i.e., after spatial averaging) is representative to the performance of all devices, which leads to the following approximations:
Approximation 1.
The departure rates (i.e., transmission success probabilities) of all queues (i.e., devices)^{7}^{7}7Hereafter, the terms devices and queues will be used interchangeably. in the network are assumed to be memoryless and are characterized by the departure rate of the typical queue.
Remark 1.
The full path loss inversion makes the received UL signal power at the serving BSs independent from the service distance (i.e., the distance between the device and the serving BS). Hence, the different realizations of the service distance across the devices do not affect the SINR. Furthermore, the random channel selection randomizes the set of interfering devices over different time slots, which decorrelate the interference across time. Hence, all devices in the network tend to have memoryless departure rates and perform as a typical device as shown in [36, 37, 26]. The accuracy of such approximation is validated in Section IV.
Utilizing Approximation 1, the performance is assessed for a test BS located at an arbitrary origin in , which becomes a typical BS under spatial averaging. Before delving into the analysis, we state the following two important approximations that will be utilized in this paper.
Approximation 2.
The spatial correlations between proximate devices, in terms of transmission power and buffer states, can be ignored.
Remark 2.
It is well known that the sizes of adjacent Voronoi cells are correlated. Such correlation affects the number of devices, as well as, the service distance realizations in adjacent Voronoi cells. Consequently, the transmission powers and devices states (e.g., active or idle) at adjacent cells are correlated. Accounting for such spatial correlation would impede the model tractability. Hence, we follow the common approach in the literature and ignore such spatial correlations when characterizing the aggregate interference [37, 36, 38, 39, 29, 26, 40]. However, all spatial correlations are intrinsically accounted for in the Monte Carlo simulations that are used to validate our model in Section IV.
Approximation 3.
For EATx, the point processes of scheduled intercell interfering devices seen at the test BS is modeled by a nonhomogenous PPP.
Remark 3.
Despite that a PPP is used to model the complete set of devices, the subset of UL scheduled devices over a given channel is not a PPP. The constraint of scheduling one device per Voronoi cell per channel leads to a Voronoiperturbed point process for the set of mutually interfering devices. Approximation 3 is commonly used in the literature to maintain tractability [37, 38, 39, 29, 26, 40].
It is worth mentioning that Approximations 13 are mandatory for tractability, regularly used in the literature, and are all validated in Section IV via independent MonteCarlo simulations. Based on these approximations, the spatial and temporal analysis are presented in, respectively, Section IIIA and Section IIIB. The spatiotemporal model that combines both stochastic geometry and queueing theory analysis is then presented in Section IIIC.
Iiia Stochastic Geometry Analysis
This section presents the stochastic geometry analysis for the departure rate probabilities for the SCUL and RAUL schemes. As mentioned before, the analysis is focused on a test BS located at the origin and a randomly selected device that it serves. For notational convince, let be the set of channel gains between the test BS and all devices that it serves and let be the channel gain between the test BS and the selected device for the analysis. For organized presentation, the analysis for each of the SCUL and RAUL schemes is provided in a separate subsection.
IiiA1 SCUL Scheme
Referring to Fig. 2, let be the probability of being in the idle state (i.e., empty buffer) and be the substochastic vector containing the probabilities of being in the RASR phase and each of the phases dedicated for EATx for nonempty buffer. For a given and , this section characterizes the probabilities , , and using stochastic geometry.
First we characterize the RASR success probability . To evaluate the interference experienced by the test device, we find the Laplace transforms (LT) of the aggregate intracell and intercell interferences. The probability of successful RASR for the test device is given by
(1) 
which follows from the adopted SINR capture model along with the fact that the BS can only decode the RASR with the highest SINR among the intracell interfering devices. denotes intercell interference for RASR while denotes the intracell interference for RASR, and is the noise power. The RASR success probability in (IIIA1) is characterized the following lemma.
Lemma 1.
The RASR success probability in a PPP singletier network where each device employs full pathloss inversion power control is given by
(2) 
with
(3)  
(4) 
where , is the Gaussian hypergeometric function, and (3) is not exact due to Approximation 2 mentioned earlier in this section. The expectation
is over the Probability Density Function (PDF) of the number of neighbors
which is found in [41] as:(5) 
where is the probability of being in the RASR state and is a constant related to the approximate PDF of the PPP Voronoi cell area in .
Proof.
See Appendix A. ∎
After a successful RASR, the device proceeds to the EATx phase for data transmission if there is an available (i.e., not used by other devices) resource block at the serving BS. Let be the probability that a device is using one of the available resource blocks for data transmission and is in the transmission time slot. Let be a random variable that represents the number of transmitting devices at the transmission time slot. As a result, the PDF of the number of devices within the same cell that have reserved resource blocks for the next time slot is given by
(6) 
where (6) follows from the superposition property of the PPP. And hence, the intensity of the device that have reserved resource blocks for the next time slot equals to . Consequently, the probability that a device proceeds from the RASR phase to EATx phase, i.e., the probability of finding an available resource block at the next time slot for EATx transmission can be expressed as follows:
(7) 
where (7) follows from the definition of the CDF for a random variable. It is worth to mention that (6) and (7) exclude the devices in the transmission slot since those devices have to resend RASR and will not be allocated EATx resources in the next time slot.
Once the device enters the EATx phase, it operates over an exclusive channel for subsequent time slots. Exploiting Approximation 1 and the fact that the channel allocation randomly changes from one time slot to another, the packet transmission success probability is independent from one time slot to another. Hence, the probability of EATx packet transmission success in an arbitrarily selected time slot is
(8) 
Comparing (8) and (IIIA1), it is clear the EATx does not experience intracell interference, and hence, the condition of highest SINR within the cell does not exits. Furthermore, it is important to note that statistically dominates . This is because EATx allows at most one intercell interferer per BS per channel as opposed to RASR which permits multiple intercell interferers per BS per channel. The packet transmission success probability in the EATx phase is characterized in the following lemma.
Lemma 2.
The probability of successful data transmission in a PPP singletier network where each device employs full pathloss inversion power control, can be expressed as
(9) 
IiiA2 RAUL Scheme
Referring to Fig. 3, let be the probability of empty buffer. This section characterizes the successful packet transmission probability for a given using stochastic geometry.
In the RAUL scheme, the devices directly transmit UL data packets over a randomly selected channel. Hence, transmissions in the RAUL scheme are subject to intracell and intercell interference. Considering the fact that only one of the intracell interfering devices can succeed at a given time slot, the transmission success probability for the RAUL scheme is characterized via the following lemma.
Lemma 3.
The transmission success probability in the depicted PPP network with RAUL scheme, where the message with the highest SINR is decodable if its SINR is greater than the detection threshold , is given by
(10) 
with
(11)  
(12) 
where and (11) is not exact due to Approximation 2 mention earlier in this section. The expectation is over the PDF of the number of neighbors as:
(13) 
where is the probability of being in the idle state and is a constant related to the approximate PDF of the PPP Voronoi cell area in [41].
Proof.
Similar to Lemma 1. ∎
IiiB Queueing Theory Analysis
This section develops the queueing theory analysis to track the temporal packet accumulation/departure at the devices’ buffers. As stated in Approximation 1 and Remark 1, the steadystate distribution of the test device is representative to all devices in the system and that the queue departures are memoryless. Since only one packet arrival and/or departure can occur in each time slot, the temporal evolution of the number of packets in the test device buffer can be traced via a QBD queueing model with the following general probability transition matrix
(14) 
where , and are the substochastic matrices that capture the transitions between the queue levels (i.e., number of packets in the buffer). Particularly, the substochastic matrices , and capture, respectively, one step increasing, unchanged, and one step decreasing number of packets int the buffer. Furthermore, , , and are the boundary transition vectors between the idle state and Level 1 (i.e., empty buffer and nonempty buffer with only one packet). The transmission matrix in (14) will be populated according to the utilized transmission scheme (i.e., SCUL or RAUL) as shown in the sequel.
IiiB1 SCUL Scheme
To analyze the QBD queueing model for the SCUL, shown in Fig.2, we employ the matrixanalyticmethod (MAM) [42, 34]. Particularly, the departure process is modeled via a Phase (PH) type distribution that captures all transition phases the queue can experience until a packet departure.^{8}^{8}8Interested readers may refer to [26] for full technical details on how to combine MAM and stochastic geometry into a unified framework. Referring to the SCUL queueing model shown in Fig.2 and utilizing the PH distribution for the departure process, the transmission matrix in (14) is populated as follows: , , , , , and where is a row vector of zeros of length , , and is used in this paper to represent a column vector of the proper length. The substochastic matrices and are of size and are given by
(15) 
where , , and are derived, respectively in (2), (7), and (2) via stochastic geometry analysis. It is worth mentioning that the probability transition matrix of the QBD queueing model is irreducible, aperiodic, and positive recurrent (i.e., ergodic DTMC. Hence, the steadystate probabilities exist [34].
By Leoynes theorem [28], the queueing model in (14) is stable if the average arrival rate is less than the average service rate. Following [34, 42], let and the vector be the unique solution of the system of equations given by and . Then, the sufficient stability condition for the SCUL is
(16) 
Let be the stationary distribution, where represents the probability of being in the idle state and is the probability vector of being in any service phases at the level of the queue. The steadystate solution for the SCUL queueing model is obtained by solving the following system of equations
(17) 
An explicit expression for the steadystate solution vector can be obtained as highlighted in the following lemma.
Lemma 4.
Proof.
Remark 4.
Since neither nor is a rank one matrix, the MAM matrix cannot be expressed via an explicit expression and is determined via the numerical algorithm given in Algorithm 1.
IiiB2 RAUL Scheme
For the RAUL scheme shown in Fig. 3, the queueing model in (14) is populated as follows: , , , , , and . Hence, the RAUL queueing model reduces to a Geo/Geo/1 queue with the following transition matrix
(20) 
The sufficient stability condition for the RAUL scheme with the transition matrix in (20) reduces from (16) to
(21) 
Given that the stability condition in (21) is satisfied, let be the steadystate distribution of RAUL queueing model, where represents the probability of having packets in the buffer. Following the same procedure in Lemma 4, solving the system of equations and with the transition matrix in (20) yields to the following steadystate solution
(22) 
IiiC Iterative Solution & Performance Assessment
Sections IIIA and IIIB show clear interdependence between stochastic geometry (i.e., spatial) and queueing theory (i.e., temporal) analysis. Particularly, the steadystate solution and are required in Section IIIA to characterize the interference and derive the packet departure rates via stochastic geometry. Meanwhile, the packet departure rates are required to derive the DTMC steadystate solution in Section IIIB via queueing theory. Hence, we employ an iterative solution to simultaneously solve the system of equations obtained via stochastic geometry and queueing theory analysis. By virtue of the fixed point theorem, such iterative solutions are shown to converge to a unique solution [43, 21, 26, 19, 44, 45]. The output of the iterative spatiotemporal algorithm provides the steadystate probabilities and packet departure rates for the considered transmission scheme, which are then used to define the following performance metrics.

Average buffer Size : Let be the instantaneous buffer size of the test device, then the average buffer size in given by
(23) where denotes the probability of being in level (i.e., the buffer contain packets) and phase .

Waiting Time in the Queue: Let be the queueing delay (i.e., the number of time slots spent in the buffer until uplink scheduling) experienced by a given packet, then the average delay (
), the variance (
), and the index of dispersion () can be evaluated, respectively, as:(24) (25) (26) 
Stability Region: Defines the system parameters where the stability conditions (16) and (21) are satisfied for, respectively, the SCUL and RAUL schemes. Operating within the stability region guarantees bounded average delay. Otherwise, the network fails to satisfy the spatiotemporal traffic requirements of the IoT devices, in which the average delay and the average number of packets in the buffers become infinite.
The spatiotemporal iterative solution and performance of each transmission scheme are presented in the sequel.
IiiC1 SCUL Scheme
The spatiotemporal iterative algorithm for the SCUL scheme is given in the following theorem.
Theorem 1.
The probability of being idle and the steadystate substochastic vector for the transmission phases for a generic device in the SCUL scheme is obtained via Algorithm 2.
Using the output of Algorithm 2, the average queue length for a stable device is given by
(27) 
where (IIIC1) follows from the fact that has a spectral radius less than one [34].
Moreover, the waiting time in the queue is given by [34]:
(28) 
The average value, the variance, and the index of dispersion can be computed by substituting (IIIC1) in (24), (25), and (26), respectively.
IiiC2 RAUL Scheme
The spatiotemporal iterative algorithm for the SCUL scheme is given in the following corollary.
Corollary 1.
The probability of being idle for the RAUL scheme is obtained via Algorithm 3.
Proof.
Similar to Theorem 1. ∎
Iv Numerical Results
Notation  Description  Value 

devicestoBS ratio  device/BS  
pathloss exponent  
geometric arrival parameter  
power control threshold  dBm  
noise power  dBm  
detection threshold for RASR  dB  
detection threshold for EATx  dB  
number of ZC codes dedicated for RASR  code per BS  
number of allocated time slots for EATx  and  
number of resource blocks for EATx  and  
detection threshold for RAUL  dB  
number of resource blocks RAUL  and 
This section validates the developed spatiotemporal model via independent Monte Carlo simulation and presents some numerical results to assess and compare the performance of the SCUL and RAUL schemes. It is important to note that the simulation is used to verify the stochastic geometry analysis for the transmission success probabilities, i.e., to validate Approximations 13 as well as the approximation of PDF of the Voronoi cell area while calculating the distribution of the number of users in the cell. On the other hand, the queueing analysis is exact, and hence, is embedded into the simulation. In each simulation run, the BSs and IoT devices are realized over a 100 km area via independent PPPs according to the steadystate distribution. Each IoT device is associated to its nearest BS and employs channel inversion power control. The collected statistics are taken for devices located within 1 km from the origin to avoid the edge effects. The received SINR for each device is measured and a successful transmission is reported if the SINR is greater than the detection threshold for EATx. On the other hand, only the device with the highest SINR succeeds if its SINR exceeds the UL SINR threshold and for RASR and RAUL transmissions, respectively. Without loss in generality, the system parameters used for this section are reported in Table I. We consider two operating scenarios for the total available bandwidth, namely, MHz and MHz. For the SCUL scheme, the RASR takes place over MHz and each resource block occupies 180 kHz. As a result, the number of available recourse blocks are for the MHz and for the MHz. On the other hand, all the available spectrum can be utilized for data communications in the RAUL scheme which makes the available number of resource blocks for the MHz and for the MHz, which is 10 more than the EATx resource blocks in the SCUL scheme.


Fig. 4 shows the transmission success probabilities for the SCUL scheme at steadystate versus the devicestoBS ratio . It is important to note the close match between the analysis and simulation results which validates the developed mathematical framework. Obviously, by comparing Fig. 4(a) with Fig. 4(b), when the total bandwidth increases the RASR success probability increases, this is mainly because of the probability of available resources increase when there are more UL frequency channels, which in turn reliefs the RASR intracell interference by accommodating more devices after RASR success. Moreover, Fig. 4 also shows the EATx transmission success probabilities. Note that the steadystate value of the EATx scheme is less than that of the RASR scheme at low device density because . However, as the device density in the RASR increase, the EATx success probability outperforms that of the RASR scheme despite that fact that . Hence, Fig. 4 shows that EATx enforces a constant despite the value of by alleviating intracell interference and allowing only one intercell interferer per BS. Fig. 4(a) shows that the queue will fall into instability when the devices intensity, or equivalently , goes beyond because of the limited resource blocks. Note that the results in Fig. 4 is consistent with eq. (2). It is important to highlight that the instability point in Fig. 4(a) is due to the instability of the RASR in Fig. 4(a). Hence, despite that the EATx provision a constant success probability for the scheduled devices, the SCUL bottleneck is in the SARA process. Hence, Fig. 4 highlights the benefit/drawback of the SCUL scheme that can provision a certain QoS for scheduled UL transmission upon RASR success.
Fig. 5 shows the RAUL transmission success probabilities at steadystate versus the devicestoBS ratio . It is important to note the close match between the analysis and simulation results which validates the developed mathematical framework. The figure shows that the performance of the RAUL transmission is affected by the system load. Hence, the RAUL scheme cannot provide QoS guarantee for data transmission when compared to the EATx. The figure also shows that the performance of RAUL can be improved by increasing the number of channels, which diversifies interference and can be used to avoid system instability. By comparing Fig. 4 and Fig. 5, the RAUL shows a better performance than the EATx at low device density for . This is mainly due to the 10 higher number of resource blocks at the RAUL scheme, and hence, limited intracel interference at low device density. However, as the density of the devices increase, the success probability for the EATx scheme outperforms that of RAUL scheme. It is also worth noting that the success probability for the RASR scheme is better than that of the RAUL scheme because .








Fig. 6 and Fig. 7 show, respectively, the steadystate average queue length and the average waiting time at stable network operation for the SCUL and the RAUL schemes. Comparing both transmission schemes, the figures show that the prompt transmission of the RAUL scheme offers lower average queue size and delay as long as the network is stable. Hence, Figs. 6 and 7 support the intuition that prompt transmission of the packets, even without scheduling, expedite packet delivery and helps devices to flush their buffers soon after packets generation. However, as the devices density increases, the interference becomes overwhelming and scheduling is necessary. Hence, the SCUL scheme extends the system stability for higher devices density for scarce resources scenario (i.e., the 10 MHz scenario). Comparing Fig. 6 and Fig. 7 also reveals the effect of the EATX transmission slots () in the SCUL scheme. The figures show that the higher the the SCUL scheme has better average performance in terms of queue length and waiting time. However, this improved performance comes at the expense of a higher index of dispersion for waiting time as depicted in Fig. 8. I.e., the waiting time for the packets will have higher deviation from the mean increases. Fig. 8 also shows that RAUL scheme generally has higher index of dispersion for the waiting time than the SCUL. Moreover, the figure shows that the variance decreases as the intensity increases. This is mainly due to the severe interference level at high intensities, and hence, the packets experience significantly large waiting time and low index of dispersion.
To better compare the scalability of the SCUL and RAUL scenarios, the Paretofrontier of the stability regions for both schemes are shown in Fig. 9. The stability Paretofrontier identifies the system parameters that guarantee stable system performance. Operating beyond the stability region lead to unstable queues and unbounded delay. For instance, the instability point in Figs. 6 and 7 occurs at and for the RAUL in the 10 MHz scenario. This point is located at the stability Paretofrontiers of the RAUL scheme in Fig 9(a). Similarly, Figs. 6 and 7 show that the SCUL scheme with and become unstable at devices densities of, respectively, and , at in the 10 MHz scenario. Such information can also be extracted from the Paretofrontiers of the SCUL scheme for and in Fig 9(a). Hence, the stability region in Fig 9(a) offers insightful information for the scalability, and identifies the effective operational scenario, of each transmission scheme. Having said that, Fig 9 shows that RAUL offers more scalability in terms of traffic intensity and that SCUL offers more scalability in terms of the devices density. Particularly, the RAUL succeeds to support higher traffic intensity for [] for the 10 MHz [20 MHz] scenario. In this case, the RASR would cause unnecessary delay and it is better to promptly transmit UL data packets without scheduling. When the devices intensity increases, RAUL would lead to overwhelming interference and scheduling becomes a necessity. Consequently, the SCUL scheme succeeds to support high devices density that cannot be supported by the RAUL.
V Conclusions
One of the key challenges associated with the IoT is tremendous growth in the number of uplink connections. The 3GPP community is seemingly set to continue to pursue a scheduled uplink (SCUL) transmission paradigm. On the other hand, the LPWA community (e.g., the likes of Sigfox and LoRa) have adopted a random access uplink transmission (RAUL) paradigm. A legitimate question is: which one of those two paradigms is better in the context of the IoT? Such a dilemma has been faintly tackled in the literature using dominantly qualitative arguments. This paper, however, provides a concrete framework for tackling the question in hand. The paper develops an integrated model featuring the use of stochastic geometry and queueing theory for uplink transmissions. We capture the mutual interference between the IoT devices by utilizing spatially interacting tandem queues model. The performance of both schemes is analyzed in terms of 4 key parameters: transmission success probability, buffer queue length, access delay time, and scalability. The latter is expressed in terms of the IoT device intensity and traffic arrival rates. The key takeaway is that RAUL is the best choice for lower device intensities and traffic volumes. Beyond that, SCUL features a more robust performance. The rationale for that stems from the failure of RAUL to handle the escalation in intracell interference with growing device intensities and traffic rates. Interestingly, this means that LPWA technologies today are actually being operated in their own “sweet spot”, i.e., where the number of IoT devices per base station is relatively lower than what LTE base stations are engineered for. As such, RAUL and SCUL ought to be perceived as complementary paradigms rather than contentious.
a Proof of Lemma 1
Let the number of neighbors in the cell equals to
, then the Complementary Cumulative Distribution Function (CCDF) of the maximum channel gain between
independent Rayleigh fading channel gains is given by:(32) 
Substituting (32) in (IIIA1) and noting that yield to:
(33) 
Because of the independency of the PPP in different regions [46] and after applying the binomial expansion for the numerator of (33), we get:
(34) 
Note that the nearest BS association and the employed power control enforce the following two conditions; (i) the intracell interference from an interfering device is equal to , and (ii) the intercell interference from any interfering device is strictly less that . The aggregate intercell interference received at the serving BS of the test device is obtained as:
(35) 
Ignoring the correlations between the transmission powers of the devices in the same and adjacent Voronoi cells, the LT of (35) can be approximated as:
(36) 
The LT is obtained by using the probability generating function (PGFL) of the PPP [46] and following [38], substituting the value of from [Lemma 1,[38]] and evaluating the integral gives:
(37) 
Let the signal from all the devices in the test cell to be , and let be the maximum signal among devices in the test cell, hence to calculate the Intracell interference conditioned on the number of neighbors , we first find the LT of the truncated exponential PDF as follows:
(38) 
By deconditioning on the PDF of which has the form of we get:
(39) 
After applying the law of total probability, (
2) in Lemma 1 is obtained.References
 [1] C. Drubin, “The Internet of things will drive wireless connected devices to. 40.9 billion in 2020.” Microwave Journal, vol. 57(10), no. 51, Aug 2014.
 [2] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE Journal on Selected Areas in Communications, vol. 32, no. 6, pp. 1065–1082, June 2014.
 [3] A. Bader, H. ElSawy, M. Gharbieh, M. S. Alouini, A. Adinoyi, and F. Alshaalan, “First mile challenges for largescale IoT,” IEEE Communications Magazine, vol. 55, no. 3, pp. 138–144, March 2017.
 [4] A. Laya, C. Kalalas, F. VazquezGallego, L. Alonso, and J. AlonsoZarate, “Goodbye, aloha!” IEEE Access, vol. 4, pp. 2029–2044, 2016.
 [5] M. S. Ali, E. Hossain, and D. I. Kim, “LTE/LTEA random access for massive machinetype communications in smart cities,” IEEE Communications Magazine, vol. 55, no. 1, pp. 76–83, January 2017.
 [6] M. Polese, M. Centenaro, A. Zanella, and M. Zorzi, “On the evaluation of LTE random access channel overload in a smart city scenario,” in Proc. of the 2016 IEEE International Conference on Communications (ICC), Kuala Lumpur , Malaysia, May 2016.
 [7] X. Lin, J. G. Andrews, A. Ghosh, and R. Ratasuk, “An overview of 3GPP devicetodevice proximity services,” IEEE Communications Magazine, vol. 52, no. 4, pp. 40–48, April 2014.
 [8] Y. P. E. Wang, X. Lin, A. Adhikary, A. Grovlen, Y. Sui, Y. Blankenship, J. Bergman, and H. S. Razaghi, “A primer on 3GPP narrowband Internet of things,” IEEE Communications Magazine, vol. 55, no. 3, pp. 117–123, March 2017.
 [9] R. S. Sinha, Y. Wei, and S.H. Hwang, “A survey on LPWA technology: LoRa and NBIoT,” ICT Express, vol. 3, no. 1, pp. 14 – 21, March 2017.
 [10] M. Z. Win, P. C. Pinto, and L. A. Shepp, “A mathematical theory of network interference and its applications,” vol. 97, no. 2, pp. 205–230, Feb. 2009, special issue on UltraWide Bandwidth (UWB) Technology & Emerging Applications.
 [11] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic geometry and random graphs for the analysis and design of wireless networks,” IEEE Journal on Selected Areas in Communications, vol. 27, no. 7, pp. 1029–1046, September 2009.
 [12] H. ElSawy, E. Hossain, and M. Haenggi, “Stochastic geometry for modeling, analysis, and design of multitier and cognitive cellular wireless networks: A survey,” IEEE Communications Surveys Tutorials, vol. 15, no. 3, pp. 996–1019, Third 2013.
 [13] H. ElSawy, A. SultanSalem, M. S. Alouini, and M. Z. Win, “Modeling and analysis of cellular networks using stochastic geometry: A tutorial,” IEEE Communications Surveys Tutorials, vol. 19, no. 1, pp. 167–203, Firstquarter 2017.
 [14] M. J. Neely, “Order optimal delay for opportunistic scheduling in multiuser wireless uplinks and downlinks,” IEEE/ACM Transactions on Networking, vol. 16, no. 5, pp. 1188–1199, Oct 2008.
 [15] L. Tassiulas and A. Ephremides, “Dynamic server allocation to parallel queues with randomly varying connectivity,” IEEE Transactions on Information Theory, vol. 39, no. 2, pp. 466–478, Mar 1993.
 [16] G. D. Çelik and E. Modiano, “Scheduling in networks with timevarying channels and reconfiguration delay,” IEEE/ACM Transactions on Networking, vol. 23, no. 1, pp. 99–113, Feb 2015.
 [17] Y. Zhong, M. Haenggi, F. C. Zheng, W. Zhang, T. Q. S. Quek, and W. Nie, “Toward a tractable delay analysis in ultradense networks,” IEEE Communications Magazine, vol. 55, no. 12, pp. 103–109, DECEMBER 2017.
 [18] Y. Zhong, M. Haenggi, T. Q. S. Quek, and W. Zhang, “On the stability of static poisson networks under random access,” IEEE Transactions on Communications, vol. 64, no. 7, pp. 2985–2998, July 2016.
 [19] G. Chisci, H. ElSawy, A. Conti, M. S. Alouini, and M. Z. Win, “On the scalability of uncoordinated multiple access for the Internet of Things,” in 2017 International Symposium on Wireless Communication Systems (ISWCS), Aug 2017, pp. 402–407.
 [20] P. H. J. Nardelli, M. Kountouris, P. Cardieri, and M. Latvaaho, “Throughput optimization in wireless networks under stability and packet loss constraints,” IEEE Transactions on Mobile Computing, vol. 13, no. 8, pp. 1883–1895, Aug 2014.
 [21] Y. Zhou and W. Zhuang, “Performance analysis of cooperative communication in decentralized wireless networks with unsaturated traffic,” IEEE Transactions on Wireless Communications, vol. 15, no. 5, pp. 3518–3530, May 2016.
 [22] Y. Zhong, X. Ge, T. Han, Q. Li, and J. Zhang, “Tradeoff between delay and physical layer security in wireless networks,” IEEE Journal on Selected Areas in Communications, pp. 1–1, 2018.
 [23] Y. Zhong, T. Q. S. Quek, and X. Ge, “Heterogeneous cellular networks with spatiotemporal traffic: Delay analysis and scheduling,” IEEE Journal on Selected Areas in Communications, vol. 35, no. 6, pp. 1373–1386, June 2017.
 [24] N. Jiang, Y. Deng, X. Kang, and A. Nallanathan, “Random access analysis for massive IoT networks under a new spatiotemporal model: A stochastic geometry approach,” IEEE Transactions on Communications, pp. 1–1, 2018.
 [25] N. Jiang, Y. Deng, A. Nallanathan, X. Kang, and T. Q. S. Quek, “Analyzing random access collisions in massive IoT networks,” IEEE Transactions on Wireless Communications, pp. 1–1, 2018.
 [26] M. Gharbieh, H. ElSawy, A. Bader, and M. S. Alouini, “Spatiotemporal stochastic modeling of IoT enabled cellular networks: Scalability and stability analysis,” IEEE Transactions on Communications, vol. 65, no. 8, pp. 3585–3600, Aug 2017.
 [27] ——, “A spatiotemporal model for the LTE uplink: Spatially interacting tandem queues approach,” in 2017 IEEE International Conference on Communications (ICC), May 2017, pp. 1–7.
 [28] R. Loyens, “The stability of a queue with nonindependent interarrival and service times,” in Proc. Cambridge Philos. Soc, vol. 58, no. 3, pp. 470–520, 1962.
 [29] S. Singh, X. Zhang, and J. G. Andrews, “Joint rate and SINR coverage analysis for decoupled uplinkdownlink biased cell associations in HetNets,” IEEE Transactions on Wireless Communications, vol. 14, no. 10, pp. 5360–5373, Oct 2015.
 [30] A. AlAmmouri, H. ElSawy, and M.S. Alouini, “Loadaware modeling for uplink cellular networks in a multichannel environment,” in Proc. of the 25 IEEE Personal Indoor and Mobile Radio Communications (PIMRC’14), Washington D.C., USA, Sep. 2014.
 [31] N. Kouzayha, Z. Dawy, J. G. Andrews, and H. ElSawy, “Joint downlink/uplink RF wakeup solution for IoT over cellular networks,” IEEE Transactions on Wireless Communications, vol. 17, no. 3, pp. 1574–1588, March 2018.
 [32] S. Sesia, I. Toufik, and M. Baker, LTE: the UMTS long term evolution. Wiley Online Library, 2009.
 [33] M. Zorzi and R. R. Rao, “Capture and retransmission control in mobile radio,” IEEE Journal on Selected Areas in Communications, vol. 12, no. 8, pp. 1289–1298, Oct 1994.
 [34] A. S. Alfa, Applied DiscreteTime Queues. Springer New York, 2015.
 [35] M. Haenggi, “The meta distribution of the SIR in poisson bipolar and cellular networks,” IEEE Transactions on Wireless Communications, vol. 15, no. 4, pp. 2577–2589, April 2016.
 [36] Y. Wang, M. Haenggi, and Z. Tan, “The meta distribution of the SIR for cellular networks with power control,” IEEE Transactions on Communications, vol. PP, no. 99, pp. 1–1, 2017.
 [37] H. ElSawy and M. S. Alouini, “On the meta distribution of coverage probability in uplink cellular networks,” IEEE Communications Letters, vol. 21, no. 7, pp. 1625–1628, July 2017.
 [38] H. ElSawy and E. Hossain, “On stochastic geometry modeling of cellular uplink transmission with truncated channel inversion power control,” IEEE Transactions on Wireless Communications, vol. 13, no. 8, pp. 4454–4469, 2014.
 [39] F. J. MartinVega, G. Gomez, M. C. AguayoTorres, and M. D. Renzo, “Analytical modeling of interference aware power control for the uplink of heterogeneous cellular networks,” IEEE Transactions on Wireless Communications, vol. 15, no. 10, pp. 6742–6757, Oct 2016.
 [40] T. D. Novlan, H. S. Dhillon, and J. G. Andrews, “Analytical modeling of uplink cellular networks,” IEEE Transactions on Wireless Communications, vol. 12, no. 6, pp. 2669–2679, June 2013.
 [41] H. ElSawy and E. Hossain, “On cognitive small cells in twotier heterogeneous networks,” in Modeling Optimization in Mobile, Ad Hoc Wireless Networks (WiOpt), 2013 11th International Symposium on, May 2013, pp. 75–82.
 [42] G. Latouche and V. Ramaswami, Introduction to matrix analytic methods in stochastic modeling. Siam, 1999, vol. 5.
 [43] K. Stamatiou and M. Haenggi, “Randomaccess poisson networks: Stability and delay,” IEEE Communications Letters, vol. 14, no. 11, pp. 1035–1037, November 2010.
 [44] A. Ephremides and R.Z. Zhu, “Delay analysis of interacting queues with an approximate model,” IEEE Transactions on Communications, vol. 35, no. 2, pp. 194–201, Feb 1987.
 [45] L. Sartori and S. E. Elayoubi and B. Fourestie and Z. Nouir, “On the WiMAX and HSDPA coexistence,” in IEEE International Conference on Communications, Jun. 2007, pp. 5636–5641.
 [46] M. Haenggi, Stochastic Geometry for Wireless Networks. Cambridge University Press, 2012.
Comments
There are no comments yet.