I Introduction
Future generations of cellular and satellite networks will include new services with vastly different performance requirements. In recent 3GPP releases [3gpp], a distinction is made among UltraReliable and LowLatency Communications (URLLC), with stringent delays and packet success rate requirements; enhanced Mobile Broadband (eMBB) for high throughput; and massive Machine Type Communications (mMTC) for sporadic transmissions with large spatial densities of devices [5g_tutorial, 5Goverview, opportunistic_coexistence]. In this paper, we focus on Internet of Things (IoT) scenarios, which are typically assumed to fall into the mMTC service category [nbiot_mmtc]. We take a further step as compared to the mentioned 3GPP classification by considering a beyond5G scenario characterized by the coexistence of heterogeneous IoT devices having critical or noncritical service requirements. Devices with critical service (CS) requirements must be provided more stringent throughput and packet success rate performance guarantees than noncritical service (NCS) devices. The model under study also applies to the case in which each device may require alternatively CS or NCS, as envisioned for the massive URLLC (mURLLC) service class in recent proposals [saad2019vision].
In the presence of a large number of IoT devices [mmtcsaad] requiring the transmission of small amounts of data, conventional grantbased radio access protocols can cause a significant overhead on the access network due to the large number of handshakes to be established. A potentially more efficient solution is given by grantfree radio access protocols. Under grantfree access, devices transmit whenever they have a packet to deliver without any prior handshake [grant_free_popovski, rahif_grant_free, grant_free_cavdar]. This is typically done via some variants of the classical ALOHA random access scheme [abramson1970aloha]. Grantfree access protocols are used by many commercial solutions in the terrestrial domain, e.g., by Sigfox [sigfox] and LoRaWAN [lora]; as well as in the satellite domain, using constellations of lowearth orbit satellites, e.g. Orbcomm [orbcomm] and Myriota [myriota].
In classical cellular IoT scenarios, orthogonal interservice resource allocation schemes are typically used [3gpp_nbiot]
. However, due to their static nature, orthogonal schemes may cause an inefficient use of resources when traffic patterns are hard to predict as in grantfree IoT systems. To obviate this problem, dynamic spectrum access schemes have been proposed whereby devices can use idle resources allocated to other devices. Reinforcement learning or online optimization solutions can be used to derive transmission strategies that maximize the throughput
[RL_DSA_suracruse, RL_DSA_negev, destounis2019learn2mac]. Though promising, these solutions can fall short when devices are equipped with very limited computational capabilities and battery capacity, low memory, and when they are placed in highly dynamic environments.Nonorthogonal resource allocation, which allows the allocation of multiple devices to the same timefrequency resource, presents a promising alternative solution [noma_saito, noma_performance, noma_challenges_potential, ding2014performance]. Recent work has proposed to apply nonorthogonal resource allocation to heterogeneous services [rahif_access_2018, popovski2018slicing, rahifuplink]. In order to mitigate interference in nonorthogonal schemes, one can leverage successive interference cancellation (SIC) [aloha_noma], time diversity [coded_slotted_aloha], and/or space diversity [munari_multiple_aloha][vladimir_cooperative_ALOHA].
As illustrated in Fig. 1, space diversity is provided by multiple Access Points (APs) that play the role of relays between the devices and the Base Station (BS). For the terrestrial networks, this topology reflects important deployments such as CloudRAN (CRAN), ultradense networks as well as the use of Unmanned Aerial Vehicles (UAV) as flying base stations [UAV_tutorial, flying_BS, flying_BS_2]. But, the space diversity model is also relevant for nonterrestrial scenarios. In particular, with the renewed interest for the deployment of LowEarth Orbit (LEO), megaconstellations such as, Amazon Kuiper [amazon_kuiper], SpaceX Starlink [spacex_starlink] and OneWeb [oneweb] can provide lowlatency and highspeed broadband to unserved and underserved locations. With thousands of LEO satellites, these constellations will offer connectivity to each earth location with multiple satellites at a time. These satellites can act as relays on a larger scale than in terrestrial networks.
Main Contributions: In this work, we study grantfree access for both CS and NCS in space diversitybased models for both nonterrestrial and terrestrial applications. We analytically derive throughput and packet success rate measures for both CS and NCS as a function of key parameters such as the number of APs, traffic load and frame size. The analysis accounts for orthogonal and nonorthogonal interservice access schemes, as well as for binary erasure channels modeling nonterrestrial applications (Fig. 1(a)) and for fading channels modeling terrestrial application (Fig. 1(b)). Finally, two receiver models are considered, namely, a collision model, where packets transmitted by the same device are assumed to undergo destructive collision, and a superposition model, where packets transmitted from the same device are superposed at the receiver. The analysis sheds lights on the advantages of each access scheme with respect to each type of service in addition to giving insights on different regimes as function of the number of time and space resources available.
This work was partly presented in the conference papers [frederico2019modern] and [rahif_space]. In [frederico2019modern], a simplified system with a single service was considered with nonorthogonal access, erasure channels model and a simplified collision model. In [rahif_space], the coexistence CS and NCS was considered under the erasure channels model and a simpler collision model whereby CS transmissions are assumed to be unaffected by NCS regardless of the NCS load. In addition, in contrast to this paper, the analysis performed in [rahif_space] does not allow the derivation of closedform expressions which are of practical interest. Finally, these papers do not consider the terrestrial scenario.
The rest of the paper is organized as follows. In Sec. II we describe the system model used and the performance metrics. In Sec. III we study the system in the presence of a single service while Sec. IV and V tackle the heterogeneous services case under the general collision and superposition model respectively. Finally, the heterogeneous service case is evaluated under the fading channel model in Sec. VI, conclusions and extensions are discussed in Sec. VII.
Notation: Throughout, we denote as
a Binomial random variable (RV) with
trials and probability of success ; as a Poisson RV with parameter . We also write for two independent RVs and with respective probability density functions and .Ii System model and Performance metrics
Iia System Model
We first consider the system illustrated in Fig. 1, in which APs, e.g., LEO satellites, provide connectivity to IoT devices. The APs are in turn connected to a BS, e.g., a ground station, through a shared wireless backhaul channel. We assume that time over both access and backhaul channels is divided into frames and each frame contains time slots. At the beginning of each frame, a random number of IoT devices are active. The number of active IoT devices that generate CS and NCS
messages at the beginning of the frame follow independent Poisson distributions with average loads
and , respectively, for some parameter and total load . Users select a timeslot uniformly at random among the timeslots in the frame and independently from each other. By the Poisson thinning property [billingsley2008probability], the random number of CS messages transmitted in a timeslot follows a Poisson distribution with average , while the random number of NCS messages transmitted in slot follows a Poisson distribution with average .Radio Access Model: As in, e.g., [frederico2019modern, azimi2017content, calderbank_erasure], we model the access links between any device and an AP as an independent interfering erasure channel with erasure probability . In nonterrestrial applications, as represented in Fig. 0(a), this captures the presence or absence of a lineofsight link between the transmitter and the receiver. A packet sent by a user is independently erased at each receiver with probability , causing no interference, or is received with full power with probability . The erasure channels are independent and identically distributed (i.i.d.) across all slots and frames. Interference from messages of the same type received at an AP is assumed to cause a destructive collision. Furthermore, CS messages are assumed to be transmitted with a higher power than NCS messages so as to improve their packet success rate, hence creating significant interference on NCS messages. As a result, in each timeslot, an AP can be in three possible states:

a CS message is retrieved successfully if the AP receives only one (nonerased) CS message and no more than a number of (nonerased) NCS messages. This implies that, due to their lower transmission power, NCS messages generate a tolerable level of interference on CS messages as long as their number does not exceed the threshold ;

no message is retrieved otherwise.
Backhaul model: The APs share a wireless outofband backhaul that operates in a fullduplex mode and in an uncoordinated fashion as in [frederico2019modern]. The lack of coordination among APs can be considered as a worstcase scenario in dense lowcost terrestrial cellular deployments [het_networks_no_coordination] [vladimir_cooperative_ALOHA] and as the standard solution for constellations of LEO satellites that act as relays between ground terminals and a central ground station. In fact, satellite coordination, although feasible through the use of intersatellite links [inter_satellite_links], may be costly in terms of onboard resources. In each timeslot , an AP sends a message retrieved on the radio access channel in the corresponding slot over the backhaul channel to the BS. APs with no message retrieved in slot remain silent in the corresponding backhaul slot . The link between each AP and the BS is modeled as an erasure channel with erasure probability , and destructive collisions occur at the BS if two or more messages of the same type are received. As for the radio access case, erasure channels are i.i.d. across APs, slots and frames.
In order to model interference between APs, we consider two scenarios. The first, referred to as collision model, assumes that multiple messages from the same device cause destructive collision. Under this model, in each timeslot, the BS’s receiver can be in three possible states:

no message is retrieved at the BS otherwise.
In the second model, referred to as superposition model, the BS is able to decode from the superposition of multiple instances of the same packet that are relayed by different APs on the same backhaul slot, assuming no collisions from other transmissions. In practice, this can be accomplished by ensuring that the time asynchronism between APs is no larger that the cyclic prefix in a multicarrier modulation implementation. Synchronization can be ensured, for example, by having a central master clock at the BS against which the local time bases of APs are synchronized [timesynchro_patent_AP]. Overall, the BS’s receiver can be in three possible states:

no message is retrieved at the BS otherwise.
Interservice TDMA: In addition to nonorthogonal resource allocation whereby devices from both services share the entire frame of timeslots, we also consider orthogonal resource allocation, namely interservice time division multiple access (TDMA), whereby a fraction of the frame’s timeslots are reserved to CS devices and the remaining for NCS devices. Interservice contention in each allocated fraction follows a slotted ALOHA (SA) protocol as discussed above. In the following, we derive the performance metrics under the more general nonorthogonal scheme described above. The performance metrics under TDMA for each service can be directly obtained by replacing with the corresponding fraction of resources in the performance metrics equations and setting the interference from the other service to zero.
IiB Performance Metrics
We are interested in computing the throughputs and and the packet success rate and for CS and NCS respectively. The throughput is defined as the average number of packets received correctly in any given timeslot at the BS for each type of service. The packet success rate is defined as the average probability of successful transmission of a given user given that the user is active, i.e., that it transmits a packet in a given frame.
Iii Single Service Under Collision Model
Iiia Performance Analysis
We start by considering the baseline case of a single service under the collision model. While this can account for either CS or NCS, we consider here without loss of generality only the CS by setting . An AP successfully retrieves a packet when only one of the transmitted packets arrives unerased, i.e. with probability
(1) 
where the term is the probability that the remaining packets are erased. Removing the conditioning on , one can obtain the average radio access throughput as
(2) 
which corresponds to the throughput of a SA link with erasures.
The overall throughput depends also on the backhaul channel. In particular, for a successful packet transmission, an AP must successfully decode one packet, which should then reach the BS unerased over the backhaul channel. This occurs with probability
(3) 
In addition, the packet should not collide with other packets. By virtue of the independence of erasure events, the number of incoming packets on the backhaul during a slot follows the binomial distribution
. Recalling that collisions are regarded as destructive under the collision model, a packet is retrieved only when a single packet reaches the BS, i.e. with probability . The CS throughput can then be derived as(4) 
This can be computed in closed form as stated in the following proposition.
Proposition 1: Under the collision model, assuming (single service), the throughput is given as function of the number of APs , channel erasure probabilities and , and CS packet load as
(5) 
where the auxiliary function is defined recursively as
(6) 
Proof: Denoting , and recalling the definitions of probabilities and in equations (1) and (3), the throughput (4) can be written as
(7) 
where follows by applying Newton’s binomial expansion and after some simple yet tedious rearrangements. Let us now introduce the auxiliary function
(8) 
From the definition of Taylor’s series for the exponential function, we have . Moreover, for , we have
(9)  
(10)  
(11) 
where equality applies the change of variable and equality results from applying once more Newton’s binomial expansion to . Plugging this result into the innermost summation within (7) leads to the closed form expression of the CS throughput reported in (5). ∎
We now turn to the packet success rate. Define the RV to count the number of transmitted messages given that at least one message is transmitted. This RV has the distribution
(12) 
which corresponds to a normalized Poisson distribution over the interval . For a given value , the probability that the packet of a given user reaches an AP given that is active is given by . Furthermore, the probability that the user’s packet reaches the BS is given as
(13) 
where and are defined in (1) and (3). In (13), term is the probability that the user’s packet is received at the BS from any of the APs, while denotes the probability that the BS does not receive any CS message from the remaining APs. The packet success rate , can be obtained by averaging (13) over as . This can be obtained in closed form as stated in the following proposition.
Proposition 2: Under the collision model, assuming (single service), the packet success rate is given as function of the number of APs , channel erasure probabilities and , and CS packet load as
(14) 
where the function is defined in (6) and we have .
Proof: The proof follows using the same steps as for Proposition 1. ∎
IiiB Examples
Using expressions derived in Proposition 1 and Proposition 2, in Fig. 2 we plot the throughput and packet success rate for a single service as function of the number of timeslots . Increasing is seen to improve the packet success rate: an active user has a larger chance of successful transmission when more timeslots are available for random access. In contrast, there exist an optimal value of for the throughput, as the analysis of the standard ALOHA protocol. Increasing beyond this optimal value reduces the throughput owing to the larger number of idle timeslots. The asymptotic behaviors of packet success rate and throughput can be easily verified theoretically using the expressions in Proposition 1 and Proposition 2 by taking their limit when tends to zero.
Iv Heterogeneous Services Under Collision Model
In this section, we extend the analysis in the previous section to derive the throughput and packet success rate of both CS and NCS under the collision model described in Sec. II.
Iva Heterogeneous Services with Ideal NCStoCS Interference Tolerance
We start by considering the case in which CS messages are not affected by NCS messages regardless of their number, i.e., we set . Under this assumption, the CS throughput and packet success rate expressions equals the expressions in Propositions 1 and 2. We hence focus here on the performance of NCS, as summarized in the following proposition.
Proposition 3: Under the collision model with ideal NCStoCS interference tolerance, i.e. , the NCS throughput and packet success rate can be respectively written as function of the number of APs , channel erasure probabilities and , and CS and NCS packet loads and as
(15) 
(16)  
where
(17a)  
(17b) 
Proof: The proof is detailed in Appendix A. ∎
IvB Examples
In order to study the performance tradeoffs between the two services, we start by investigating the impact of by plotting the CS and NCS throughputs versus with , , , and APs. For CS, there is an optimal value of that ensures an optimized CS load as in the standard analysis of the ALOHA protocol, discussed also in the context of Fig. 2. In contrast, the NCS throughput decreases as function of due to the increasing interference from CS transmissions. The NCS throughput is also seen to increase as a function of the channel erasure when is not too large. This is because a larger , can reduce the interference from CS transmissions.
IvC Heterogeneous Services with Limited NCStoCS Interference Tolerance
We now alleviate the assumption that CS transmissions can withstand any level of NCS interference by assuming that interference from at most NCS transmissions can be tolerated without causing a collision from CS traffic. We derive both CS and NCS performance metrics. We note that, perhaps counterintuitively, both CS and NCS performance metrics are affected by the CS interference tolerance parameter . In fact, with a lower value of , a smaller number of CS packets tends to reach the BS, reducing interference to NCS transmissions. We start by detailing the NCS performance metrics.
Proposition 4: Under the collision model with limited NCStoCS interference tolerance, i.e. finite , the NCS throughput and packet success rate are given as a function of the number of APs , channel erasure probabilities and , and CS and NCS packet loads and as
(18) 
(19) 
where ,
(20a)  
(20b) 
and the expectation in (19) is taken with respect to independent RVs and , with the latter distributed as in (12) with the index in lieu of .
Proof: The proof is detailed in Appendix D. ∎
We now address the CS analysis. With finite , a CS message is correctly received at any AP if it is the only nonerased CS message and no more than NCS messages are received erasurefree at the AP. Conditioned on the number of messages and , the probability of the first event is given by defined in (1), while the probability of the second event is given by with
(21) 
Removing the conditioning on , the probability of the second event can be written as
(22) 
where the first term in (22) is the regularized gamma function and represents the second term. For the CS performance metrics we distinguish the following two cases depending on the number of APs.
IvC1 Small number of APs ()
In this case, the effect of finite interference tolerance affects only the radio access transmission phase. In fact, in the backhaul transmission phase, if , the number of interfering NCS transmissions on a CS packet at the BS cannot exceed .
Proposition 5: Under the collision model, the CS throughput and packet success rate given as a function of the number of APs , channel erasure probabilities and and CS and NCS packet loads and as
(23) 
(24) 
where and
is the probability of receiving any CS packet at the BS and the expectation in (24) is taken with respect to independent RVs and , with the latter distributed as in (12).
Proof: The proof is provided in Appendix E. ∎
Comparing the CS throughput in (23) with the expression (5) for we observe that the effect of a finite interference tolerance is measured by the multiplicative term . It can be shown that this term is always smaller than one, which is in line with the fact that a lower CS throughput is expected when is finite.
IvC2 Large Number of APs ()
In this case, a successful CS transmission occurs in all events where a single CS packet and only up to NCS packets reach the BS. The total probability of these events given and can be computed as
(25) 
where is the probability that a CS packet reaches the BS and is the probability that a NCS packet reaches the BS (may also be not correctly received due to a collision). Removing the conditioning on and , we get the CS throughput as
(26) 
where the expectation is taken with respect to independent RVs and .
Moving to the CS packet success rate, conditioned on and , the probability of receiving a packet at an AP from a given user is given as in . The probability of receiving successfully a CS packet at the BS is then given as
(27) 
where and . The main difference between (27) and the probability inside the expectation in (24) is the multiplication by the term in (27) which corresponds to the probability that a number of NCS packets lower or equal to should be received in order to be able to recover a CS packet. Removing the conditioning on and , we obtain the packet success rate as
(28) 
where the expectation is taken with respect to independent RVs and with the latter distributed as in (12).
IvD Examples
In order to capture the effect of the number of NCS messages on the CS, in Fig. 4 we plot the throughput region for and , with the latter case corresponding to the analysis in Sec. IVA. The region includes all throughput pairs that are achievable for some value of the fraction of CS messages, as well as all throughput pairs that are dominated by an achievable throughput pair (i.e., for which both CS and NCS throughputs are smaller than for an achievable pair). For reference, we also plot the throughput region for a conventional interservice TDMA protocol, whereby a fraction for of the timeslots is allocated for CS messages and the remaining timeslots to NCS messages. For TDMA, the throughput region includes all throughput pairs that are achievable for some value of , as well as of .
A first observation from the figure is that nonorthogonal resource allocation can accommodate a significant NCS throughput without affecting the CS throughput, while TDMA causes a reduction in the CS throughput for any increase in the NCS throughput. This is due to the need in TDMA to allocate orthogonal time resources to NCS messages in order to increase the corresponding throughput. However, with nonorthogonal resource allocation, the maximum NCS throughput is generally penalized by the interference caused by the collisions from CS messages, while this is not the case for TDMA. In summary, TDMA is preferable when one wishes to guarantee a large NCS throughput and the CS throughput requirements are loose; otherwise, nonorthogonal resource allocation outperforms TDMA in terms of throughput. Furthermore, the throughput region is generally decreased by lower value of . Experiments concerning packet success rate and performance as function of the number of APs will be presented in the superposition model in the following section.
V Heterogeneous Services Under Superposition Model
In this section, we consider the superposition model described in Sec. II.
Va Performance Analysis
Unlike the collision model, in order to analyze the throughput and packet success rate under the superposition model, one needs to keep track of the index of the messages decoded by the APs. This is necessary to detect when multiple versions of the same message (i.e., sent by the same device) are received at the BS. Accordingly, we start by defining the RVs to denote the index of the message received at AP and RV for the BS at any timeslot. Accordingly, for given values and of transmitted messages, RVs can take values
(29) 
Note that we have indexed CS messages from to and NCS messages from to . As for the RV at the BS, it is defined as
(30) 
Furthermore, we define as the RVs denoting the number of APs that have message of index
. The joint distribution of RVs
given and is multinomial and can be written as follows(31) 
where we used the the probabilities in (1) and (46) that one of the CS or NCS message is received at an AP respectively in a given timeslot. The probability of retrieving a CS message in a given timeslot at the BS conditioned on , and can be then written as
(32)  
where the first sum is over all possible CS messages, the second sum is over all combinations of APs that have the CS message , and the third sum at the exponent is over all APs that have a CS message . The CS throughput can be computed by averaging (32) over all conditioning variables as
(33) 
In a similar manner, the conditional probability of receiving a NCS message at the BS can be written as
(34)  
where the first sum is over all possible NCS messages ; the second sum is over all possible combinations of APs that have message ; and the third and fourth sums at the exponent are over all APs that have a different NCS message and a CS message respectively. The NCS throughput can be then obtained by averaging over the conditioning RVs as
(35) 
VB Examples
In Fig. 4, we plot the throughput region for nonorthogonal resource allocation and interservice TDMA under the superposition model. Comparing the regions of the collision model and the superposition model, it is clear that the latter provides a larger throughput region being able to leverage transmissions of the same packets from multiple APs as compared to the collision model. This can also be seen as function of in Fig. 4.
In Fig. 5, we explore the effect of the number of APs on the CS and NCS throughputs. To capture separately the effects of the radio access and the backhaul channel erasures, we consider different values for the channel erasure probabilities and . We highlight two different regimes: the first is when is large and is small, and hence larger erasures occur on the access channel; while the second covers the complementary case where is small and is large. In the first regime, increasing the number of APs is initially beneficial to both CS and NCS messages in order to provide additional spatial diversity for the radio access, given the large value of ; but larger values of eventually increase the probability of collisions at the BS on the backhaul due to the low value of . In the second regime, when and much lower throughputs are generally obtained due to the significant losses on the backhaul channel. This can be mitigated by increasing the number of APs, which increases the probability of receiving a packet at the BS.
Finally, we consider the interplay between the throughputs and packet success rate levels for both nonorthogonal resource allocation and TDMA as function of the number of timeslots . These are plotted in Fig. 6 for , APs, and . For both services, following the discussion around Fig. 2 we observe that the packet success rate level under both allocation schemes increases as function of . This is because larger value of decrease chances of packet collisions. However, this not the case for the throughput, since large values of may cause some timeslots to be left unused, which penalizes the throughput. For the CS in Fig. 5(a), it is seen that nonorthogonal resource allocation outperforms TDMA in both throughput and packet success rate level due to the larger number of available resources. In contrast, Fig. 5(b) shows that TDMA provides better NCS throughput and packet success rate level than nonorthogonal resource allocation. The main reason for this is that the lower number of resources in TDMA is compensated by the absence of interservice interference for NCS messages.
Vi Throughput and packet success rate Analysis Under Fading Channels
The binary erasure channel model discussed in the previous sections offers a tractable setup that facilitates the analysis of the throughput and packet success rate, enabling the derivation of closedform expressions in various cases of interest. It is also of practical interest as a simplified model for mmwave channels [mmwave_erasurechannels] and nonterrestrial communications scenarios represented in Fig. 1. In this section, we briefly study a more common scenario that accounts for fading channels in both radio and backhaul channels. This typically represents terrestrial scenarios as shown in Fig. 0(b). More complex models that include both fading and erasures [fading_and_erasure] can also be analyzed following the same steps presented below (see Section VII for some details). We first detail the channel and signal models, and then we derive the throughput and packet success rate metrics.
Via Channel and Signal Models
At any timeslot , the channels between each user and AP and between each AP and the BS are assumed to follow the standard Rayleigh fading model, and are denoted as and , respectively. Ensuring consistency with the erasure model, we assume that all channels are independent and that the average channel gains and are fixed. Furthermore, as detailed below, we assume that each AP and the BS decode at most one packet in each slot. Finally, we denote the transmission rates of CS and NCS messages as and , respectively. Assuming that both access and backhaul channels are allocated the same amount of radio resources, the transmission rates are the same for both channels.
Given the numbers
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