Spatio-temporal Modeling for Massive and Sporadic Access

The vision for smart city imperiously appeals to the implementation of Internet-of-Things (IoT), some features of which, such as massive access and bursty short packet transmissions, require new methods to enable the cellular system to seamlessly support its integration. Rigorous theoretical analysis is indispensable to obtain constructive insight for the networking design of massive access. In this paper, we propose and define the notion of massive and sporadic access (MSA) to quantitatively describe the massive access of IoT devices. We evaluate the temporal correlation of interference and successful transmission events, and verify that such correlation is negligible in the scenario of MSA. In view of this, in order to resolve the difficulty in any precise spatio-temporal analysis where complex interactions persist among the queues, we propose an approximation that all nodes are moving so fast that their locations are independent at different time slots. Furthermore, we compare the original static network and the equivalent network with high mobility to demonstrate the effectiveness of the proposed approximation approach. The proposed approach is promising for providing a convenient and general solution to evaluate and design the IoT network with massive and sporadic access.


A. Motivations
Evolution of smart terminals (smartphones, smartwatch, intelligent glasses, etc.) has spawned a rich diversity of new applications, such as Virtual Reality, Augmented Reality, Industry 4.0 and so on, which post new challenges to wireless networks hosting these applications (see Figure 1).A major branch of these applications is the massive machine type communication (mMTC), also known as one of the three main application scenarios of the fifth generation mobile communications system (5G) [1], [2].The most promising applications of the mMTC include the long-term environmental monitoring with limited energy consumption, the smart city scenarios with millions of sensors, the low-delay and high-reliability scenarios in wireless factory control and so on.
For a typical mMTC application, the number of wireless devices may reach 300,000 in each cell [3].In addition to the huge number of devices, another distinctive feature of the mMTC is that the access requests from these massive number Yi Zhong and Xiaohu Ge are with School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan, P. R.China (e-mail: {yzhong, xhge}@hust.edu.cn).Guoqiang Mao is with the School of Electrical and Data Engineering, University of Technology Sydney (e-mail: Guoqiang.Mao@uts.edu.au).
This research was supported by the National Natural Science Foundation of China (NSFC) grant No. 61701183. of devices are sporadic.The characteristic of sporadic access mainly manifests in the following aspects.
• The users access the wireless network sporadically.For instance, the data generated by the mMTC devices may be periodic and driven by some regularly occurring events.Typical application is the use of intelligent water meters, which report the measured data periodically.• The size of each packet is small, and the data rate is low.
The size of the packets for a typical mMTC applications may go down to a few bytes.Meanwhile, the data rate for each user may be around 10 kb/s.
In view of this, we propose the notion of massive and sporadic access (MSA), the precise mathematical description of which will be given in the following sections.The massive and sporadic properties for such kind of accesses bring new challenges for the design of an efficient air interface.For the traditional wireless networks, a large overhead within all layers in the protocol stack is required to implement various network functions, such as access control, reliable transmission, authentication, security, and so on.However, for the massive and sporadic access, the amount of information in each attempt of transmission may be only few bytes.The large amount of redundant information generated by the protocol stack in the traditional wireless networks may greatly degrade the efficiency of MSA.Moreover, the interference as well as other networking features of MSA may also be very different from the traditional wireless networks.Therefore, the scenarios of MSA should be modeled and analyzed qualitatively so that the constructive insight for the design of wireless networks with MSA could be obtained.
Though the point process theory has been widely used to evaluate the instantaneous performance metrics for a wireless network such as the coverage probability and the achievable rate [4], [5], it is still not convenient for the characterization of plenty of significant long-term performance metrics such as the delay when the queueing process is considered.In particular, when it is necessary to characterize the coupling between the traffic with spatio-temporal variation and the performance of a wireless network, rigorous analysis based on the point process theory becomes inapplicable [6].The main difficulty of introducing the queueing analysis to the performance evaluation of a large wireless network, which is non-negligible for a practical system, lies in the complicated interaction among the queues [7]- [10], i.e., the serving rates of all queues in a wireless network are highly coupled with the statuses of the queues (i.e., empty or not).
However, for the scenario of MSA, it is very unlikely that a node will be continuously active over consecutive time slots.Thus, the set of active nodes that cause interference changes dramatically over the time.For the case of "extreme sporadic", the sets of active nodes at different moments may not intersect since the probability that a node is active at two different moments is very small, in which case the interference might be considered as independent at different time instants.For the case of "extreme massive", an active node can appear anywhere in the plane.Intuitively, the analysis of the scenario of MSA can be approximately equivalent to the analysis of a network whose nodes move extremely fast that there is no coupling between the queues, thereby greatly reducing the analytical difficulties.Therefore, in this paper, we propose to evaluate the scenario of MSA by the equivalence of high mobility [11], [12], where the nodes move extremely fast that the location of a node in the subsequent time can be considered as totally independent from that in the current time.

B. Related Works
In order to characterize the massive and sporadic properties of the MSA, we use the combination of the point process theory and the queueing theory to model both the spatial distribution of the massive number of devices and the sporadic arrival of packets at each device.The stochastic geometry tools, especially the point process theory, are widely used to model the spatial topology of wireless networks in recent years [4], [13].For example, the Poisson point process (PPP) is used to analyze the coverage probability, the achievable rate and other performance metrics in cellular networks [5], [14], [15].Related works using the point process theory to evaluate the performance of mMTC applications include [16], where a two single-hop relaying schemes exclusively designed for the MTC is proposed and analyzed.By characterizing the received signal and interference powers using the point process theory, the authors in [16] derived the outage probability and the maximum density of MTC devices that can be supported under an outage constraint.In [17], a framework to evaluate the endto-end outage probability and the uplink data transmission rate in a single-hop relay network for MTC is proposed and evaluated using the stochastic geometry.The authors in [18] propose an analytical framework based on stochastic geometry to investigate the system performance in terms of the average success probability and the average number of simultaneously served MTC devices.The authors in [19] also present a tractable analytical framework to investigate the signal-to-interference ratio (SIR), thereby deriving the success probability, the average number of successful MTC devices and the probability of successful channel utilization for the cellular-based mMTC network.The authors in [20], [21] evaluate the performance of massive non-orthogonal multiple access (NOMA) system.The above-mentioned works focus on the performance metrics by considering a snapshot of the network.As for the analysis of the long-term metrics such as the interference correlation and the delay, which requires the description of temporal variation, these approaches proposed in existing works are unable to provide an effective analysis.Moreover, the analysis based on a snapshot of the network cannot completely capture the sporadic property of MSA.
The works related to the interference correlation in static Poisson networks include [22]- [24], where the interference correlation, as well as the ways to reduce its impact, is explored.The analysis of temporal variation without considering the queueing process includes the local delay, which is defined as the number of time slots required for a packet to be successfully transmitted assuming that the networks are backlogged [25]- [27].However, since the queueing process is ignored in the evaluation of local delay, the obtained results are not of great practical significance especially for those cases where random arrival of traffic has a great impact on the network performance.
Related works considering both the spatial distribution of nodes and the temporal variation of traffic could be found in [28]- [30], in which the wireless traffic is modeled based on the granularity of total traffic in each cell.In [31], a trafficaware spatio-temporal model is proposed for the Internet of Things (IoT) supported by the uplink of a cellular network.The stability for three different transmission strategies are evaluated.In [32], the random access mechanism in the cellular-based massive IoT networks is evaluated based on a spatio-temporal model for the wireless traffic, where the spatial topology is modeled by using tools from the stochastic geometry and the evolutions of queues are assessed based on the stochastic process.In [33], a user-centric mobility management mechanism is proposed to cope with the spatial movements of users and the temporally correlation of wireless channels in ultra-dense networks.In order to model the flow at individual user, our previous work [6] combines the point process theory and the queueing theory, and bounds the statistical distribution of the signal-to-interference ratio (SIR) and the delay in heterogeneous cellular networks.Along this line of thought, continuous works such as [34] and [35] explored the delay and security performance in wireless networks.

C. Contributions
In this paper, we quantitatively model the spatio-temporal properties for the scenario of MSA.Afterwards, we explore the correlation of interference and the correlation of successful transmission events at different time instants.Then, we verify that these correlations are indeed negligible for the scenario of MSA.In view of this, we propose to evaluate the scenario of MSA by the analysis of a equivalent network in which all nodes are moving extremely fast.In order to demonstrate the accuracy of the proposed approach, we further compare the non-empty probability and the mean delay for the original static network and the equivalent network with high mobility.The main contributions are summarized as follows.
• We propose the notion of massive and sporadic access to quantitatively characterize the mMTC, and discuss the interference-limited regime and the noise-limited regime for various configurations, which plays an important role in practical MSA system design.• We evaluate the correlations of interference and successful transmission events, and verify that these correlations are ignorable for the scenario of MSA.Consequently, we propose the equivalent analysis of MSA by introducing the high mobility assumption, which significantly reduces the analytical complexity and provides a universal solution to evaluate MSA.• The non-empty probability, the success probability, the mean delay and the average queue length are derived by using the equivalence of high mobility.The numerical and simulation results demonstrate the accuracy of the proposed equivalent analysis method.The remaining paper is organized as follows.Section II describes the spatial distribution model and the arrival process.Section III gives the mathematical definition and evaluates the effect of parameters for the scenario of MSA.Section IV discusses the temporal correlation of interference and successful transmission events by considering a backlogged network.Section V assesses the accuracy of using the high mobility equivalence to analyze the scenario of MSA.Finally, Section VI concludes the paper.

II. SYSTEM MODEL
Without loss of generality, we consider the commonly used Poisson bipolar model (see [4,Definition 5.8] and [36]) to characterize the spatial distribution of the devices.In particular, we model the spatial distributions of the transmitters as a homogeneous PPP Φ = {x i } ∈ R 2 with intensity λ (see Figure 2).We assume that each transmitter is associated with a receiver at a fixed distance r 0 and a random orientation.We consider a typical link with the receiver located at the origin and the transmitter located at x 0 .Then, |x 0 | = r 0 is the distance between the typical transmitter at x 0 and the typical receiver at the origin.
As for the temporal model, we consider a discrete-time queueing system, where the time is assumed to be divided into discrete time slots with equal duration.The transmission of each packet occupies exactly one time slot.Each transmitter is equipped with an infinite queue to store the incoming packets.In each time slot, if a queue is non-empty, it attempts to transmit its head-of-line packet with probability p.If the transmission attempt is successful, the packet will be removed from the queue.Otherwise, the packet will be put back to the head-of-line of the queue and wait to be retransmitted in the next time slot.In the scenario of MSA, the packets arrive at the transmitters as stochastic processes with very small arrival rates.In this paper, we assume that the packets arrival process at each transmitter is a Bernoulli process with arrival rate ξ (0 ≤ ξ ≤ 1), which is widely used in modeling the discrete-time system.According to the definition of the Bernoulli process, ξ is also the probability that a packet arrives at a transmitter in each time slot.The arrival processes at different transmitters are assumed to be independent from each other.To be rigorous, we assume that the early arrival model is used where a potential packet departure occurs at the moment immediately before the time slot boundaries, and a potential arrival occurs at the moment immediately after the time slot boundaries (see Figure 3).If the time axis is marked by 0, 1, ..., t, ..., a potential departure occurs in the interval (t − , t), while a potential arrival occurs in the interval (t, t + ).
We assume that the network is static, i.e., the locations of all nodes are generated first as a realization of the PPP and then remain unchanged in all the following time slots.This assumption is very realistic since the locations of access points (or base stations) in most of the wireless networks are fixed after they are deployed.On the contrary, the nodes may move according to certain mobility model [37].In the extreme situation, the position of each node is assumed to be changed according to a high mobility random walk model [38,Ch. 1.3] that the location of a node in the next time slot can be considered as totally independent from that in the current time slot.We name this scenario with fast node movement as the high mobility scenario.The interference is greatly affected by the type of mobility for the nodes.For example, when the network is static, the nodes that generate interference in all time slots come from the same set of locations, which are determined when the nodes are first deployed.However, as for the case of high mobility, the locations of the nodes that generate interference change over the time.
The transmit power of all nodes is assumed to be the same, which is normalized to be one.The propagation loss of the electromagnetic wave is assumed to consist of two parts, i.e., the path loss and the fading.The standard path loss model is used, i.e., the path loss between a transmitter and a receiver with distance r apart is denoted by l(r) = r −α , where α is the path loss exponent with α > 2. The fading model is assumed to be the commonly used Rayleigh block fading with the shadowing being ignored.The power fading coefficients keep unchanged during each time slot and are spatially and temporally independent with the exponential distribution of unit mean between different time slots.
The normalized power of thermal noise is assumed to be W .Note that whether the scenario of MSA belongs to the interference-limited regime or the noise-limited regime cannot be determined since both the node intensity λ and the arrival rate ξ may influence the relationship between the interference and the noise, which will be quantitatively evaluated in the following sections.
In each time slot, a transmitter is active only when its queue is non-empty and it is allowed to transmit (with probability p).Let k ∈ N + be the index of the time slots and Φ k ∈ Φ be the set of all active transmitters in the time slot k.Note that the set Φ k varies with the time since both the statues of queues and the scheduled results of random access are different in different time slots.Then, the interference at the typical receiver located at the origin o in time slot k is where h k,x is the fading coefficient between the interfering transmitter x and the typical receiver at the origin.Since the network is static, the original set of transmitters Φ is independent of the index of time slot k.
When the typical transmitter at x 0 attempts to deliver a packet to the typical receiver at the origin, the signal to interference plus noise ratio (SINR) at the typical receiver when it is active in time slot k is The SINR threshold for successfully delivering a packet is θ, i.e., a transmission attempt of a link is successful only when the SINR of such link is above the threshold θ.Then, the success probability for the typical link when it is active in time slot k is Note that the interval between two adjacent packet arrivals is a geometrically distributed random variable due to the Bernoulli arrival.Therefore, the queueing process at the typical link is a discrete-time Geo/G/1 queueing system [39].In particular, when P k is the same for all k ∈ N + , i.e., the success probability for the typical link in all time slots is the same, the service time (in number of time slots) of each packet follows a geometric distribution with the parameter µ pP k , ∀k ∈ N + .The parameter µ is also the mean service rate for the queueing system.In this case, since the service time is also a geometric distributed random variable, the queueing system can be denoted by Geo/Geo/1.We present the following lemma which gives the mean delay for a Geo/Geo/1 queueing system.Lemma 1.For a discrete-time Geo/Geo/1 queueing system with packet arrival rate ξ and service rate µ (ξ < µ), the mean delay is Proof.From [40, Corollary 2], the mean delay (including the queueing delay and the service delay) for a Geo/G/1 queueing system with arrival rate ξ is where ξ = 1−ξ, β 1 and β 2 are the first and the second factorial moments of the service time, A(•) is the generating function of the successive interretrial times.
Since we consider a standard Geo/G/1 queueing system without retrial, i.e., the packet at the head of the queue immediately commences its service whenever the server is idle, the successive interretrial time is always zero, i.e., A(x) = 1, ∀x.Thus, the mean delay is reduced to Note that the service time is a geometric distributed random variable with mean 1/µ for the Geo/Geo/1 queueing system.The first and second raw moments of the service time are 1/µ and 2/µ 2 − 1/µ respectively.Then, the first and the second factorial moments of the service time are β 1 = 1/µ and β 2 = 2/µ 2 −2/µ.Plugging these results into (6), we get the lemma.

III. MATHEMATICAL DESCRIPTION OF MSA
To explore the performance of MSA, we first give the mathematical definition for the scenario of MSA.The two main parameters related to the MSA are the density of nodes λ and the arrival rate ξ.To characterize the property of "massive", we propose the following definition.
Remark 1.The reason for using (7) to distinguish whether the number of devices is massive or not is that the propagation features (such as the path loss exponent) may also make sense in deciding whether a scenario belongs to "massive" or not.
The assumption with ξ → 1 and p = 1 is the worst case that the interference is the largest, i.e., all transmitters are active and cause interference to the typical link.Note that the expression for the success probability in a Poisson network is well-known, which is [4,Equation (5.14)] Letting P Poisson ≤ ε, we get the inequality in Definition 1.
Note that when ignoring the interference and only considering the effect of the thermal noise, the success probability for the typical link in any time slot is e −θW r α 0 .In order to make the limitation on the success probability meaningful, the threshold ε for success probability should satisfy the following inequality 0 < ε < e −θW r α 0 .
In the scenario of MSA, the packets arrive at each transmitter sporadically.To characterize the property of "sporadic", we propose the following definition.

Definition 2. The arrival process of packets is called "sporadic" if and only if the arrival rate at each queue
where ξ 0 denotes the maximum arrival rate that guarantees the mean delay of the typical link being smaller than a predefined threshold β (β > e θW r α 0 ) when λ → 0 and p = 1, i.e., the case without interference and random access.

Remark 2.
The assumption λ → 0 corresponds to the best case where the interference is ignored.When the interference is ignored, the success probability for the typical link in any time slot is the same, which is e −θW r α 0 .Then, when p = 1, the queueing process at the typical link is a Geo/Geo/1 queueing system with the service rate e −θW r α 0 packet per time slot.If the arrival rate ξ ≥ e −θW r α 0 , the queue becomes unstable, and the mean delay D will be infinite (i.e., D = ∞).If the arrival rate ξ satisfies ξ < e −θW r α 0 , according to the equation ( 4) in Lemma 1, the mean delay for each packet is Letting the mean delay be smaller than the predefined threshold β, i.e., D ≤ β, we get the inequality in Definition 2.
Note that when the arrival rate ξ approaches 0, the mean delay given by ( 11) approaches e θW r α 0 , which is the smallest mean delay that could be achieved when the interference is ignored and p = 1.Therefore, in order to make the limitation on the mean delay meaningful, the threshold β for the mean delay should satisfy the following inequality With the above definition for "massive" and "sporadic", we define the scenario of MSA as follows.Definition 3. A scenario is named as "massive and sporadic access" if and only if the density of devices satisfies λ ≥ λ 0 , and the arrival rate of packets satisfies ξ ≤ ξ 0 , where λ 0 and ξ 0 are the critical values given by ( 7) and (10), respectively.
In particular, we define the MSA region as follows.
Definition 4. The MSA region is defined as the range of the two-tuple (λ, β) within which the corresponding scenario will be MSA (see Figure 4).

A. Effect of Parameters on MSA Region
The shape of the MSA region is affected by different network parameters, such as the path loss exponent and the thermal noise.In order to explore the effect of these parameters, we plot the processes of change for the shape of the MSA region when increasing the path loss exponent α and the normalized thermal noise W in Figure 5.We observe that the critical point (λ 0 , ξ 0 ) moves from the upper left to the lower right as α increases, indicating that the MSA region is  narrower for larger α.This is because the path loss is enlarged due to the increment of α, in which case the network should be deployed more densely to achieve the condition for "massive".Meanwhile, the arrival rate of packets should be smaller in order to achieve the condition for "sporadic".In particular, the critical density λ 0 goes to zero when α approaches to 2, which could also be observed through the equation ( 7), indicating that any density of the transmitters could be considered as "massive" in the free space propagation model with α = 2.
Figure 5 also shows that as the normalized noise W increases, the critical point (λ 0 , ξ 0 ) moves from the upper right to the lower left.Moreover, it reveals that the change of W has a great influence on the critical arrival rate ξ 0 but less effect on the critical density λ 0 .When W approaches to zero, the critical arrival rate ξ 0 goes to 1, illustrating that any arrival rate will be considered as "sporadic" for the case where the thermal noise is ignored.

B. Interference-limited and Noise-limited
Due to the spectral scarcity, most wireless networks are designed to be interference-limited, i.e. the interference rather than the thermal noise dominates the network performance.However, in the case of MSA, even if the potential transmitters are ultra dense, a wireless network may still be noise-limited since the arrival rate of packets is extremely small resulting in minor interference.In this subsection, we discuss the classification of the scenario of MSA to identify whether it belongs to interference-limited or noise-limited.
1) Interference-limited: In order to quantitatively define the interference-limited regime, we consider a simplified system in which a packet failed for transmitting will be discarded instead of being retransmitted.Therefore, the interference in the simplified system will always be a lower bound for that in the original system.We define a scenario as interferencelimited if the ratio between the success probability ignoring the noise and the success probability ignoring the interference is less than a small threshold η in the corresponding simplified system.Note that in the simplified system, the probability that a link being active in a time slot equals to the product of the probability that there is a packet arriving in the said time slot ξ and the transmit probability p, which is pξ according to the definition of the Bernoulli arrival process.The success probability for an active link when ignoring the noise and considering only the effect of the interference is The success probability for an active link when ignoring the interference and considering only the effect of the noise is Combining ( 13) and ( 14) with the condition for interferencelimitation P inter /P noise ≤ η, we get The above inequality gives the limitations on ξ and λ that makes a scenario interference-limited.
2) Noise-limited: For the noise-limited regime, the success probability for delivering a packet at each time slot is approximated the same, which is p exp(−θW r α 0 ).Then, the queueing process at each link can be considered as a Geo/Geo/1 queueing system with arrival rate ξ and service rate p exp(−θW r α 0 ).The probability of a queue being non-empty equals to the utilization of the queueing system ξ exp(θW r α 0 )/p.The active transmitters constitute an independent thinning version of the original PPP Φ with thinning probability ξ exp(θW r α 0 ).Therefore, the success probability when ignoring the noise and considering only the effect of interference is The success probability when ignoring the interference and considering only the effect of thermal noise is Fig. 6.Illustration of interference-limited regime and noise-limited regime for the scenario of massive and sporadic access.The distance between each transmitter and the associated receiver is r 0 = 5, the SINR threshold is θ = 10dB, and the transmit probability is p = 1.The threshold for success probability when ξ → 1 is ε = 0.1, and the threshold for mean delay when λ → 0 is β = 50.The path loss exponent is α = 3.5, the normalized noise is W = 10 −3.4 , and the threshold to distinguish different regimes is η = 0.5.
Similar to the case of interference-limitation, we define a scenario as noise-limited if the ratio between the success probability ignoring the interference and the success probability ignoring the noise is less than a small threshold η.Combining ( 16) and ( 17) with the condition for noiselimitation P noise /P inter ≤ η, we get ξλ ≤ θW r α 0 + ln η C 0 exp(θW r α 0 ) .
The above inequality gives the limitations on ξ and λ that makes a scenario noise-limited.Figure 6 shows the value of (λ, ξ) that belongs to the interference-limited regime or the noise-limited regime in the scenario of MSA.From Figure 6, we observe that a wireless network can still be noise-limited as long as the arrival rate is small even if the deployed nodes are highly dense.Figure 6 also reveals that there exists a minimum value for the density λ to make a network interference-limited in the scenario of MSA.In other words, if the density of the deployed nodes is less than certain value in the scenario of MSA, the network will never be interference-limited for all arrival rate.
In general, the product ξλ determines whether a wireless network works in the interference-limited regime or the noiselimited regime.Therefore, we define the product of the arrival rate ξ and the density of transmitters λ as the traffic factor.Intuitively, the traffic factor describes the intensity of the traffic in a wireless network.In Figure 7, we plot the boundaries of the traffic factor ξλ as functions of the pathloss exponent α for the interference-limited regime given by the inequality (15) and for the noise-limited regime given by the inequality (18) in the scenario of MSA.Since the Y axis is logarithmic, we observe that the boundary of the traffic factor ξλ for the interference-limited regime increases almost exponentially as the pathloss exponent α grows.We also observe that as α increases, the curve for the boundary of the noise-limited regime first grows and then goes down.This can be intuitively interpreted as that when α starts to increase, the interference decreases rapidly, and small traffic factor ξλ can make the network work in the noise-limited regime.However, if α continues to grow, the success probability of the desired link decreases due to the deterioration of the desired signal, resulting in more retransmissions, which could also be inferred from the equation ( 16).In this case, the traffic in the network becomes heavier due to the retransmitted packets, leading to increased interference and more stringent conditions on the traffic fact ξλ to make the network noise-limited.

IV. EVALUATION OF CORRELATIONS
Since the queueing process at each individual transmitter in a wireless network depends the statuses (empty or nonempty) of queues at all transmitters, the queueing processes at different transmitters are highly coupled with each other, leading to the interacting queues problem which is rather difficult to cope with.Existing works have only derived the sufficient conditions and necessary conditions for the stability [41], or the upper and lower bounds for certain performance metrics [6].However, in the case of the MSA, the set of active transmitters that cause interference to the network changes dramatically over the time slots.Moreover, in the scenario of "extreme massive" and "extreme sporadic", the sets of active transmitters in different time slots may not intersect.Then, the interference in different times slots will be independent, resulting in the decoupling of the queueing processes for different queues in the wireless networks.
Intuitively, the analysis of a wireless network with large number of coupled queues for the scenario of MSA could be approximately equivalent to the analysis of the high mobility case where the interacting queues are decoupled, thereby greatly reducing the analytical complexity.However, the accuracy of such intuition requires to be quantitatively analyzed in order to use this equivalence to simplify the analysis of the MSA.Therefore, in this section, we explore the difference between the original scenario of MSA and the high mobility case from the point of view of interference correlation and successful transmission correlation.Note that in the high mobility network, both the interference and the successful transmission are independent among different time slots, i.e. both the interference correlation and the successful transmission correlation are zero.Thus, we only need to evaluate the correlations for the scenario of MSA.
To facilitate the analysis, we consider a backlogged version of the original network, in which the queues at all transmitters are backlogged and will never be empty.In the backlogged network, each transmitter will be active independently with the same probability p.Note that the interference is temporally correlated since a subset from the same set of potential transmitters are active in different time slots.The locations of these potential transmitters are randomly deployed first as a realization of the PPP and then keep unchanged in all the following time slots, which could be considered as the "common randomness".

A. Interference Correlation
The temporal (Pearson's) correlation coefficient of the interference in the i-th time slot I i and that in the j-th time slot I j is defined as where cov( ] is the covariance between I i and I j , and is the standard deviation of I i and I j .The work in [22] has already obtained the temporal correlation coefficient in remarkably simple form as where h k,x is the power fading coefficient between an interfering transmitter x and the typical receiver at the origin in any time slot k.Due to the Rayleigh fading assumption, the power fading coefficient h k,x is exponentially distributed with unit mean.Then, the temporal correlation coefficient becomes The above equation indicates that the correlation coefficient of the interference between two different time slots grows linearly with the active probability p.Note that the correlation coefficient is not related to the density of the nodes λ.This observation indicates that in the scenario of MSA, the degree of temporally linear correlation of the interference will not be affected by increasing the density of links (i.e., more massive), while it decreases linearly as the arrival rate decreases (i.e., more sporadic).

B. Successful Transmission Correlation
In practical system, we care more about the outcome of the transmission attempt rather than the interference level.Note that the success probability relies on the SINR, the denominator of which is determined by the interference and the thermal noise.Therefore, the temporal correlation of the interference induces the temporal correlation of the successful transmissions.Let S k = 1(SINR k > θ) be the indicator that the SINR at the typical receiver in time slot k is above the threshold θ, i.e., S k is the indicator for successful transmission in time slot k.The temporal correlation coefficient between S i and S j , i = j is Note that Due to the backlogged assumption and the independent thinning, the success probability P i is given by the standard form for a Poisson network similar to the equation ( 8) as follows Using the formula (2) and the exponential distribution property of h k,x , we get the joint success probability as Plugging the above joint success probability into the equation ( 22), we get the temporal correlation coefficient between S i and S j as The temporal correlation coefficient of the successful transmissions in different time slots depends on the product of the transmit probability p and the density of transmitters λ.From the equation ( 26), we observe that when pλ → 0 or pλ → +∞, the temporal correlation coefficient ρ(S i , S j ) → 0, indicating that the successful transmissions in different time slots tend to be linearly independent when pλ is either very large or very small.This can be interpreted as that when pλ → 0, the interference could be ignored, and the thermal noise becomes the dominant factor that affects the successful transmission events.Due to the independence of the noise at different receivers, the successful transmission events at different time slots tend to be completely independent.Meanwhile, when pλ → +∞, the successful transmission events are affected by a large number of independent channel fading coefficients, leading to the independence of the successful transmission events in different time slots.In particular, we get the following lemma.
Lemma 2. The temporal correlation coefficient of the successful transmissions ρ(S i , S j ) is maximized if and only if where a = 2 − 2 δ and b = exp(−θW r α 0 ).Proof.Letting a = 2 − 2 δ , b = exp(−θW r α 0 ), and It can be verified that the numerator g(t) = (a − 1)t a − abt a−1 + 1 is monotone decreasing when t > 1.Thus, f ′ (t) is also monotone decreasing when t > 1, demonstrating that f (t) is a concave function when t > 1.Since the equality f (1) = f (+∞) = 0 holds, there is one and only one maximum value of f (t) for t ∈ (1, +∞).Let t 0 = arg max and t 0 should satisfy the following equation Due to the equation ρ(S i , S j ) = bf (exp(C 0 pλ)), we get the result in Lemma 2.
In Figure 8, we plot the temporal correlation coefficient of the successful transmissions ρ(S i , S j ) as functions of pλ for different path loss exponents.Figure 8 verifies Lemma Optimal p that maximizes correlation coefficient Fig. 9.The optimal pλ that maximizes temporal correlation coefficient ρ(S i , S j ) as functions of the normalized thermal noise W for different path loss exponents.The distance between each transmitter and the associated receiver is r 0 = 5, and the SINR threshold is θ = 10dB.
2 that there is one and only one stationary point that maximizes the temporal correlation coefficient.We also observe that as the path loss exponent α increases, the maximum temporal correlation coefficient first grows then decreases.This is because the attenuation of the interference is small for small α, in which case the interference comes from plentiful sources with non-negligible interfering signals.Since the interference levels from different sources are independent due to the independent fading, the successful transmission events in different time slots become less correlated.When α is large, the attenuation of the interference is remarkable, and the thermal noise which is independent at different receivers becomes dominant, leading to the reduction of the temporal correlation.
In Figure 9, we plot the optimal pλ that maximizes temporal correlation coefficient ρ(S i , S j ) as functions of the normalized thermal noise W for different path loss exponents.It is observed that the optimal pλ that maximizes temporal correlation increases as the path loss exponent α or the normalized thermal noise W grows. Intuitively, the thermal noise becomes dominant other than the interference when α or W increases, and pλ should be enlarged to maintain the same level of correlation.Moreover, Figure 9 implies that for large thermal noise W , the optimal pλ that maximizes the temporal correlation tends to stabilize at the same value.In particular, if W → +∞, the equation ( 27) becomes (a − 1)t a 0 + 1 = 0.By solving this equation, we get the optimal pλ that maximizes temporal correlation of successful transmission events when W → +∞ as Through the expression for the temporal correlation coefficient of the interference given by (21), the transmit probability p should be very small so that the interference correlation among different time slots can be ignored.On the other hand, through the expression for the temporal correlation coefficient of the successful transmission events given by (26), pλ should either be very small or very large so that the successful transmission correlation among different time slots can be ignored.Therefore, both the interference correlation and the successful transmission correlation can be ignored either when both p and λ are very small (corresponding to small p and small pλ) or when p is very small and λ is very large (corresponding to small p and large pλ).The former case where both p and λ are very small implies the noise-limited regime that the effect of the interference could be ignored, while the latter case where p is very small and λ is very large implies the aforementioned MSA scenario.In both cases, the wireless network can be equivalently analyzed by the high mobility assumption.Noting that the noise-limited regime is relatively much easier to be quantitatively evaluated due to the approximation of ignoring the interference, we mainly focus our research on the MSA scenario throughout this paper.
In this section, we have discussed the temporal correlation coefficients of the interference and the successful transmission events by considering a backlogged network in which the queues at all transmitters will never be empty.The transmission of each link is controlled by the transmit probability p.For the original network with complex queue evolutions, these temporal correlation coefficients are extremely complicated to quantify.However, the results obtained in the backlogged network illustrate heuristically the temporal correlation in the original system.

V. EQUIVALENCE BY HIGH MOBILITY
In the practical wireless network with random packet arrival and mutual interference, the long-term network performance metrics, such as the delay and the stability, are extremely difficult to quantify due to the interacting queues problem which is notoriously hard to coupe with [6]- [8].In the previous discussions, we have illustrated that a static wireless network in the scenario of MSA could be approximately equivalent to the high mobility case, which greatly reduce the complexity of the quantitative analysis and design.In order to assess the accuracy of using the high mobility equivalence, we evaluate the empty probability and the mean delay at the stationary regime for the typical link in the scenario of MSA.Furthermore, we compare these two metrics for the original static network and the equivalent network with high mobility.
In the high mobility case, the locations of all transmitters in current time slot can be considered as independent with that in the next time slot.Therefore, the spatial distributions of the transmitters in different time slots are independent PPP with the same intensity λ.Unlike in the static network that two queues at two nearby transmitters may affect the packet delivery of each other for long periods of time, the mutual effect between the same batch of queues in the high mobility case appears only in current time slot.Thus, the interacting queues can be decoupled and considered as independent from each other in the high mobility case.Moreover, in the static network, the stationary distributions of the queue systems at different transmitters are diversified, while in the high mobility case the stationary distribution is the same for all queues.
Let ζ be the probability that a queue is non-empty at the stationary regime in the high mobility case.Then, all queues in the high mobility case are non-empty with the same probability ζ at the stationary regime.Since only the transmitters with non-empty queues are active with probability p, the locations of all active transmitters interfering with the typical link constitute a PPP with intensity pζλ in each time slot.When the typical transmitter at x 0 attempts to deliver a packet at the stationary regime, the success probability in different time slots is the same, which is In the high mobility case, the queueing process at the typical link is a Geo/Geo/1 queueing system since the arrival process is geometric due to the Bernoulli arrival, and the service process is also geometric due to the retransmission mechanism with the same transmit probability p and the same success probability P k .For the Geo/Geo/1 queueing system at the typical link, the arrival rate is ξ while the service rate is µ = pP k .Note that the condition for the queue at the typical link being stable is ξ < µ, which turns into The probability that the queue at the typical link is nonempty at the stationary regime can be obtained by evaluating the utilization of the Geo/Geo/1 queueing system, which turns into Plugging ( 31) into (33), we get the non-empty probability as P{Queue is non-empty} = ξ p exp pζλC 0 + θW r α 0 .
Since the typical link is arbitrarily selected from a wireless network, the non-empty probability of the queue at the typical link equals to the value ζ assumed above.Therefore, we can get a fixed-point equation as follows.
Letting ω = −pζλC 0 , the above fixed-point equation ( 35) transforms into Let W(z) be the Lambert W function which solves the equation W(z)e W(z) = z.The Lambert W function has two branches, i.e., the principal branch W 0 (z) and the branch W −1 (z).The result obtained from the branch W −1 (z) is rejected since it leads to a system with a success probability which increases with arrival rate.Using the principal branch of the Lambert W function W 0 (z) to solve the above fixed-point equation (36), we get the solution ω 0 as follows Therefore, we get the following lemma  Lemma 3. In a network with high mobility, the non-empty probability for all queues at the stationary regime is the same, which is Plugging ( 38) into (32), we get the following theorem.
Theorem 1.In a network with high mobility, the stable condition for each queue is Plugging ( 38) into (31), we get the success probability in the following theorem.
Theorem 2. In a network with high mobility, the success probability for all active transmissions at the stationary regime is the same, which is The service rate of a Geo/Geo/1 queueing system at the typical link equals to the success probability.By Lemma 1 and Theorem 2, we get the mean delay in the following corollary.
Corollary 1.In a network with high mobility, the mean delay for all queues at the stationary regime is the same, which is According to the Little's Law, the average number of packets in a queue at the stationary regime L 0 equals to the product of the arrival rate and the mean delay, i.e., L 0 = ξD 0 .Thus, we get the average queue length as In Figure 10 and Figure 11, we compare the simulation results for the scenario of MSA and the numerical results obtained by the high mobility equivalence to evaluate the   accuracy of the proposed equivalent analysis method.The simulation for the scenario of MSA is conducted in an area of size 240 × 240, with the statistics within the margins less than 20 are ignored to eliminate the edge effect (i.e., the links at the edge experience less interference due to the finite edge).The network topology is regenerated for 200 times according to the point process, and for each realization of the point process, the duration of the simulation is 1000 time slots.Figure 10 and Figure 11 reveal that the proposed approximating approach by the equivalence of high mobility provides a good characterization for both the non-empty probability and the mean delay in the stable state of the scenario of MSA.
In Figure 12, we plot the success probability in the stable state as functions of the pathloss exponent α for different arrival rate ξ with the high mobility equivalence.We observe that the success probability is limited by large interference for small α and by small desired signal power for large α.We also observe that the range of α which makes the network stable become smaller when ξ increases.In Figure 13, we plot the mean delay in the stable state as functions of the arrival rate ξ for different transmit probability p with the high mobility equivalence.Figure 13 reveals that the mean delay increases rapidly when the arrival rate ξ approaches certain critical value, beyond which the network will be unstable.

VI. CONCLUSIONS
Evaluation of the long-term performance metrics of a wireless network, such as the delay and the stability, by using the stochastic geometry tools has long been a difficult problem.In this paper, we define the scenario of MSA and derive the correlation coefficients for interference and successful transmission events at different time slots.Having verified that these correlations are negligible in the scenario of MSA, we propose a convenient approach by the equivalence of high mobility to evaluate the performance for the scenario of MSA.
We further derived the non-empty probability, the success probability, the average queue length, and the mean delay in a network with the high mobility equivalence.Moreover, we compare the non-empty probability and the mean delay for the original static network and the equivalent network with high mobility to demonstrate the accuracy of the proposed approach.The proposed method is promising for providing a convenient and universal solution for the design of the mMTC with massive and sporadic access.

Fig. 1 .
Fig. 1.The practical scenario of a rich diversity of wireless applications.

Fig. 2 .Fig. 3 .
Fig.2.The Poisson bipolar model for massive and sporadic access.The red solid links denote the active links whose queues are non-empty and also selected to transmit by random access, while the blue dotted links represents the silent links whose queues are empty or silenced by random access.The arrival processes at different transmitters are independent Bernoulli processes.

Fig. 4 .
Fig. 4. Illustration of the region for massive and sporadic access.

Fig. 5 .
Fig. 5. Illustration of the region for massive and sporadic access.The distance between each transmitter and the associated receiver is r 0 = 5, and the SINR threshold is θ = 10dB.The threshold for success probability when ξ → 1 is ε = 0.1, and the threshold for mean delay when λ → 0 is β = 50.The path loss exponent α increases from 2 to 4 when fixing the normalized noise W = 10 −4 , while W increases from 10 −6 to 10 −3.3 when fixing α = 3.

Fig. 7 .
Fig.7.Boundaries of the traffic factor ξλ for the interference-limited regime and the noise-limited regime in the scenario of massive and sporadic access.The distance between each transmitter and the associated receiver is r 0 = 5, the SINR threshold is θ = 10dB, and the transmit probability is p = 1.The normalized noise is W = 10 −3.2 , and the threshold to distinguish different regimes is η = 0.5.

Fig. 8 .
Fig.8.Temporal correlation coefficient of the successful transmissions ρ(S i , S j ) as functions of pλ for different path loss exponents.The distance between each transmitter and the associated receiver is r 0 = 5, and the SINR threshold is θ = 10dB.The normalized noise is W = 10 −4 .

Fig. 10 .
Fig.10.Non-empty probability in the stable state as functions of the pathloss exponent α.The distance between each transmitter and the associated receiver is r 0 = 5, and the SINR threshold is θ = 10dB.The normalized noise is W = 10 −3.3 .The error bars show the standard deviation.

5 Fig. 11 .
Fig. 11.Mean delay in stable state as functions of the pathloss exponent α.The distance between each transmitter and the associated receiver is r 0 = 5, and the SINR threshold is θ = 10dB.The normalized noise is W = 10 −3.3 .The error bars show the standard deviation.

Fig. 12 .
Fig.12.Success probability in the stable state as functions of the pathloss exponent α.The distance between each transmitter and the associated receiver is r 0 = 5, the SINR threshold is θ = 10dB, and the transmit probability is p = 0.5.The normalized noise is W = 10 −3.3 .

Fig. 13 .
Fig.13.Mean delay in stable state as functions of the arrival rate ξ.The distance between each transmitter and the associated receiver is r 0 = 5, the SINR threshold is θ = 10dB, and the pathloss exponent is α = 3.The normalized noise is W = 10 −3.3 .