I Introduction
During the past few years, there has been a growing consensus that 5G wireless systems will support three generic services, which, according ITUR, are classified as enhanced mobile broadband (eMBB), massive machinetype communications (mMTC), and ultrareliable and lowlatency communications (URLLC) (also referred to as missioncritical communications)
[1, 2]. A succinct characterization of these services can be put forward as follows: (a) eMBB supports stable connections with very high peak data rates, as well as moderate rates for celledge users; (b) mMTC supports a massive number of Internet of Things (IoT) devices, which are only sporadically active and send small data payloads; (c) URLLC supports lowlatency transmissions of small payloads with very high reliability from a limited set of terminals, which are active according to patterns typically specified by outside events, such as alarms. This paper studies the problem of enabling the coexistence of the three heterogeneous services within the same Radio Access Network (RAN) architecture. We describe below in more details the requirements of the three services.eMBB traffic can be considered to be a direct extension of the 4G broadband service. It is characterized by large payloads and by a device activation pattern that remains stable over an extended time interval. This allows the network to schedule wireless resources to the eMBB devices such that no two eMBB devices access the same resource simultaneously. The objective of the eMBB service is to maximize the data rate, while guaranteeing a moderate reliability, with packet error rate (PER) on the order of .
In contrast, an mMTC device is active intermittently and uses a fixed, typically low, transmission rate in the uplink. A huge number of mMTC devices may be connected to a given base station (BS), but at a given time only an unknown (random) subset of them becomes active and attempt to send their data. The large number of potentially active mMTC devices makes it infeasible to allocate a priori
resources to individual mMTC devices. Instead, it is necessary to provide resources that can be shared through random access. The size of the active subset of mMTC devices is a random variable, whose average value measures the mMTC traffic arrival rate. The objective in the design of mMTC is to maximize the arrival rate that can be supported in a given radio resource. The targeted PER of an individual mMTC transmission is typically low, e.g., on the order of
.Finally, URLLC transmissions are also intermittent, but the set of potential URLLC transmitters is much smaller than for mMTC. Supporting intermittent URLLC transmissions requires a combination of scheduling, so as to ensure a certain amount of predictability in the available resources and thus support high reliability; as well as random access, in order to avoid that too many resources being idle due to the intermittent traffic. Due to the low latency requirements, a URLLC transmission should be localized in time. Diversity, which is critical to achieve high reliability [3], can hence be achieved only using multiple frequency or spatial resources. The rate of a URLLC transmission is relatively low, and the main requirement is ensuring a high reliability level, with a PER typically lower than , despite the small blocklengths.
In 5G, heterogeneous services are allowed to coexist within the same network architecture by means of network slicing [4]. Network slicing allocates the network computing, storage, and communication resources among the active services with the aim of guaranteeing their isolation and given performance levels. In this paper, we are interested in the “slicing” of RAN communication resources for wireless access. The conventional approach to slice the RAN is to allocate orthogonal radio resources to eMBB, mMTC, and URLLC devices in time and/or frequency domains, consistently with the orthogonal allocation of wired communication resources. However, wireless resources are essentially different due to their shared nature. Using communicationtheoretic analysis, this work demonstrates that a nonorthogonal allocation that is informed by the heterogeneous requirements of the three services can outperform the standard orthogonal approach. Importantly, the considered nonorthogonal approach multiplexes heterogeneous services, and is hence markedly distinct from the conventional NonOrthogonal Multiple Access (NOMA) methods that share radio resources only among devices of the same type (see, e.g., [5]). This is further discussed next.
Ia Network Slicing of Wireless Resources: HOMA and HNOMA
Consider an uplink scenario in which a set of eMBB, mMTC and URLLC devices is connected to a common BS, as shown in Fig. 1. We note that the designing uplink access is more complex than the corresponding problem for the downlink due to the lack of coordination among users. Orthogonal and nonorthogonal slicing of the RAN among the three services are illustrated in Fig. 2(a) and (b), respectively.
The conventional orthogonal allocation depicted in Fig. 2(a) operates in the frequency domain and allots different frequency channels to eMBB, mMTC, or URLLC devices. eMBB and mMTC transmissions are allowed to span multiple time resources. In contrast, in order to guarantee the latency requirements discussed above, URLLC transmissions are localized in time and are spread over multiple frequency channels to gain diversity. Furthermore, since the URLLC traffic is bursty, the resources allocated to URLLC users may be largely unused. This is because the channels reserved for URLLC are idle in the absence of URLLC transmission.
Importantly, orthogonal slicing does not preclude the sharing of wireless resources among devices of the same type. For example, multiple eMBB users may transmit on the same allotted frequency channels by using NOMA [5]. Therefore, in order to distinguish orthogonality among signals originating from devices of the same type, as in conventional Orthogonal Multiple Access (OMA), from the orthogonality among different services, we refer to the approach in Fig. 2(a) as Heterogeneous Orthogonal Multiple Access (HOMA).
As mentioned, in this work, we investigate the potential advantages of a nonorthogonal allocation of RAN resources among multiple services, which we refer to as Heterogeneous NonOrthogonal Multiple Access (HNOMA). Fig. 2(b) depicts an instance of HNOMA. By comparison with the HOMA solution in Fig. 2(a), under HNOMA, the frequency resources that were allocated only to mMTC or URLLC traffic can also be granted to the eMBB users. In this way, HNOMA may allow for a more efficient use of radio resources as compared to HOMA by avoiding unused resources due to URLLC or mMTC inactivity. This may yield a higher spectral efficiency for the eMBB users that can benefit from the intermittent nature of mMTC and URLLC traffic. However, the mutual interference between eMBB and mMTC or URLLC transmissions may significantly degrade the performance for all the involved services. Ensuring desired performance levels is hence more challenging with HNOMA.
In this paper, we tackle this problem by developing a communicationtheoretic model that aims at capturing the essential performance tradeoffs and design insights for HOMA and HNOMA. More specifically, the main goals of this work can be illustrated using Fig. 3, as discussed next.
To start, Fig. 3(a) depicts the type of results that are of interest when studying conventional OMA and NOMA within a given service type, as done in a growing line of work [6, 5]. These results rely on the classical analysis of the multiple access channel, in which all users have identical reliability requirements and block lengths and the goal is to characterize the region of achievable rates [7] as the block length grows large.
In contrast, Fig. 3(b,c,d) exemplify the type of results that are of interest when evaluating the performance tradeoffs between heterogeneous services that are allowed by HOMA and HNOMA. As a first example to be further elaborated on in the paper, Fig. 3
(b) shows the tradeoff between the URLLC activity, i.e., the probability of URLLC devices being active, and the eMBB transmission rate or spectral efficiency. The figure illustrates the fact that the eMBB rate is not affected by the URLLC packet arrival rate under HOMA, while the resulting interference impairs the performance of HNOMA. As an alternative performance evaluation, Fig.
3(c) shows the tradeoff between the reliability of URLLC transmissions and the eMBB rate. The example highlights the fact that a nontrivial tradeoff exists for both HOMA and HNOMA. In fact, URLLC reliability can be improved by taking away frequency resources from eMBB and allocating them to URLLC. As a final illustration, Fig. 3(d) depicts the tradeoff between the arrival rate of mMTC devices and the eMBB rate. In a manner similar to Fig. 3(c), this figure suggests that, even under HOMA, the spectral efficiency of the eMBB user that shares the resources with mMTC devices can be traded off for the mMTC arrival rate by a proper allocation of radio resources.IB Further Related Works
In addition to the mentioned literature on conventional NOMA, here we briefly review works that directly tackle the coexistence of heterogeneous services. A logical architecture for network slicing in 5G in the presence of orthogonal slicing has been presented in [4] and [8]. The downlink multiplexing of URLLC and eMBB is studied in [9] and [10]. These works investigates the dynamic scheduling of URLLC traffic over ongoing eMBB transmissions by abstracting the operation at the physical layer. In [11], the authors treat the problem of resource allocation for mMTC and URLLC in a new radio (NR) setting by focusing on the role of feedback. Orthogonal resource allocation for mMTC and eMBB users is studied in [12] by accounting for intercell interference. In [13], grantfree uplink transmissions are considered for the three services by considering concrete transmission/modulation/spreading methods for supporting the three services.
IC Main Contributions
The main contributions are as follows.

We propose a communicationtheoretic model that is tractable and yet captures the key features and requirements of the three services. Unlike [13], in which the authors focus on grantfree access for all services, the proposed model takes into account the difference in arrival processes and traffic dynamics that are inherent to each individual service. The proposed model can be seen as an extension of the classical multiple access channel model that underlies the analysis of conventional NOMA in the sense that it accounts for the coexistence of heterogeneous services.

We first analyze the performance of orthogonal slicing, or HOMA, for all three services. We focus on achievable transmission rates for eMBB and URLLC, under the respective target reliability, and on the throughput for mMTC.

We then consider the performance of HNOMA. Although the modeling approach allows to study an arbitrary combination of services, in this paper we have focused on the analysis of two specific cases as illustrated in Fig. 2, namely: (i) slicing for URLLC and eMBB, and (ii) slicing for mMTC and eMBB. In the case of URLLCeMBB slicing, among other schemes, we consider the technique of puncturing, which is currently under consideration in 3GPP [10]. It is noted that, while of interest, HNOMA between URLCC and mMTC may be problematic due to the need to ensure reliability guarantees for URLLC devices in the presence of the random interference patterns caused by mMTC transmissions.

Among the main conclusions, our study demonstates that nonorthogonal slicing, or HNOMA, can achieve service isolation in the sense of ensuring performance levels for all services by leveraging their heterogeneous reliability requirements. We refer to this design principle as reliability diversity. As it will be discussed, the heterogeneity leveraged by reliability diversity is not only in terms of the numerical values of the reliability levels, but also in terms of very definition of reliability across the three services. For example, the reliability metric typically considered for mMTC is the fraction of detected devices among the massive set of active users, whereas for eMBB and URLLC services one typically adopts the classical frame error rate. Our results show that, if reliability diversity is properly exploited, nonorthogonal slicing can lead, in some regimes, to important gains in terms of performance tradeoffs among the three generic services.
The paper is organized as follows. The next section presents the system model and provides a performance analysis of each of the three services when considered in isolation. Section III treats the slicing of resources to support eMBB and URLLC, while Section IV is dedicated to the slicing of resources for eMBB and mMTC. Both sections provide a description of the proposed theoretical framework as well as numerical results illustrating the tradeoff between the services for both HOMA and HNOMA schemes. The conclusions are given in Section VA, while Section VB contains discussion on possible generalizations of the model considered in this paper. Two appendices, containing the technical details of some of the derivations, conclude the paper.
Ii System Model
We are interested in understanding how the three service described in Section I, i.e., eMBB, URLLC, and mMTC, should efficiently share the same radio resources in the uplink when communicating to a common BS. We consider radio resources, where each resource occupies a single frequency channel and a single time slot. A radio resource, which is indexed by , contains symbols. The symbols are further divided into minislots, where each minislot consists of symbols. Fig. 4 shows an example of a timefrequency grid.
We assume that the transmission of an eMBB user occupies a single radio resource at a given frequency . In contrast, due to latency constraints, a URLLC user transmits within a single minislot across a subset of frequency channels. An URLLC device may be active in an allocated minislot with probability . Finally, the set of mMTC users is allowed to access the channel only at a specified radio resource ; for the example on Fig. 4 we have . The number of active mMTC devices in such a resource is distributed as , where is the mean value, referred to as mMTC arrival rate.
Some comments regarding the modelling choices made above are in order. First, for eMBB traffic, we focus on the standard scheduled transmission phase, hence assuming that radio access and competition among eMBB devices have been resolved prior to the considered time slot. Second, we do not model collisions among URLLC devices. We assume instead that a single URLLC device is allocated a number of minislots in the given slot, over which it is active with some probability. On the contrary, we do model the random access phase for mMTC traffic, since this is the key transmission phase for this type of traffic, due to the massive population of devices. Extensions of our model will be discussed in Section VB.
Each radio resource is assumed to be within the time and frequencycoherence interval of the wireless channel, so that the wireless channel coefficients are constant within each radio resource. Furthermore, we assume that the channel coefficients fade independently across the radio resources. The channel coefficients of the eMBB, URLLC, and the mMTC devices, which we denote by , , and , ,^{1}^{1}1Throughout, we use the convention that the subscripts , , and indicate a quantity referring to eMBB, URLLC, and mMTC, respectively. are independent and Rayleigh distributed, i.e., , , and for across all radio resources . The channel gains for the three services in a radio resource are denoted by , , and for .
The average transmission power of all devices is normalized to one. The differences in the actual transmission power across various users and in the path loss are accounted for through the average channel gains , , and . Furthermore, the power of the noise at the BS is also normalized to one, so that the received power equals the signaltonoise ratio (SNR) for each device. The number of symbols in a minislot is assumed sufficiently large to justify an asymptotic informationtheoretic analysis. Extensions of our analysis to capture finiteblocklength effects [14] will be considered in future works. Due to latency and protocol constraints to be detailed later, no channelstate information (CSI) is assumed at the URLLC and at the mMTC devices. In contrast, the eMBB devices are assumed to have perfect CSI. Finally, the BS is assumed to have perfect CSI.
The error probabilities of the eMBB, URLLC, and mMTC devices are denoted as and , respectively. These probabilities must satisfy the reliability requirements , and , where
(1) 
The large differences in reliability levels among the services, as well as their different definitions, which we will introduce shortly, motivate the introduction of the concept of reliability diversity. Reliability diversity refers to system design choices that leverage the differences among the supported services in terms of reliability requirements and definitions. For example, as we will see, strict perpacket reliability guarantees are typically enforced for eMBB and URLLC devices, whereas the notion of reliability for mMTC devices is less stringent and typically involves the computation of averages over a large group of active devices.
Iia Signal Model
To summarize the main assumptions discussed so far and to fix the notation, we assume that each eMBB user is scheduled on a single frequency channel within the considered frequency resources; each URLLC device occupies frequencies resources, numbered without loss of generality as , in a given minislot; and a set of mMTC devices is available for transmission in a channel frequency .
Let
denote the received vector corresponding to the minislot
and the frequency channel . Based on the given assumptions, the received signal can be written as(2)  
where is the signal transmitted by an eMBB user scheduled in the frequency resource ; is the signal transmitted by a URLLC device transmitting in minislot and frequency ; is the signal transmitted by one of the active mMTC devices in frequency ; and
represents the noise vector, whose entries are i.i.d. Gaussian with zero mean and unit variance. The notation
for the mMTC devices, which indicates ordering, will be formally introduced in Section IID.We emphasize that the transmitted eMBB signal in (2) is zero if no eMBB user is scheduled in frequency channel ; similarly, the URLLC signal is zero if no URLLC device transmits in minislot and frequency , e.g., if ; and the mMTC signals are similarly all equal to zero if the channel is not allocated to mMTC traffic, i.e., if .
As discussed, with HOMA, resources are allocated exclusively to one of the three services, while, with HNOMA, resources can be shared. In the remainder of this section, we study the performance of the three traffic types in an HOMA setting, that is, in the absence of mutual interference. We also introduce the metrics that will be used to evaluate the performance of the three services.
IiB eMBB
Consider a radio resource allocated exclusively to an eMBB user. As mentioned, the eMBB is aware of the CSI and can use it in order to select its transmission power . The objective is to transmit at the largest rate that is compatible with the outage probability requirement under a longterm average power constraint. This can be formulated as the optimization problem
(3)  
The optimal solution to this problem is given by truncated power inversion [15]. Accordingly, the eMBB device chooses a transmission power that is inversely proportional to the channel gain if the latter is above a given threshold , while it refrains from transmitting otherwise.
Beside being theoretically justified by the mentioned ratemaximization problem, the thresholdbased transmission strategy discussed above also captures the fact that eMBB devices only transmit if the current SNR is sufficient to satisfy minimal rate requirements. This is the case in most communication standards, such as LTE, in which the transmission mode is selected from a set of allowed modulation and coding schemes with given SNR constraints. As we will discuss below, the scheme has the additional analytical advantage of relating directly outage probability and probability of activation for an eMBB user. We remark that the analysis could be extended to other design criteria such as the maximization of the average transmission rate.
Based on the discussion above, the probability that the eMBB user transmits is given by
(4) 
Furthermore, in the absence of interference from other services, the only source of outage for an eMBB transmission is precisely the event that an eMBB does not transmit because of an insufficient SNR level. Hence, the probability of error equals
(5) 
Imposing the reliability condition
(6) 
we obtain the value of the threshold SNR
(7) 
Note that, in the absence of interference and under the given assumptions, the threshold SNR does not depend on the frequency channel . This dependence is kept here in view to the extension to HNOMA in the next sections.
Based on the powerinversion scheme, the instantaneous power is chosen as a function of the instantaneous channel gain as
(8) 
where is the target SNR, which is obtained from the threshold by imposing the average power constraint as
(9) 
with being the lower incomplete gamma function. This implies that the target SNR is
(10) 
IiC Urllc
The URLLC device transmits data in the allocated frequency channels of a minislot, with activation probability . Hence, the number of URLLC transmissions during the time slot is a random variable . We assume that each URLLC transmission carries a different message, and that, due to the low latency requirement, each message must be decoded as soon as the relevant minislot is received. This implies that the URLLC device cannot code across multiple minislots.
Unlike eMBB users, the URLLC device is not aware of the CSI for the allocated frequency resources. This assumption is justified by the fact that CSI at the URLLC device would require signaling exchange before transmission, which entails extra latency as well as a potential loss in terms of reliability. In fact, the high reliability constraint would enforce an even higher reliability requirement on the auxiliary procedure of CSI signaling. As a result of the lack of CSI, no power or rate adaptation is possible for URLLC devices.
We choose the rate as the performance metric of choice. In the absence of interference from other services, outage occurs with probability
(12) 
Imposing the reliability condition allows us to obtain the maximum allowed rate . We will refer to this quantity as for reference. Note that increasing enhances the frequency diversity and, hence, makes it possible to satisfy the reliability target at a larger rate .
IiD mMTC
The key property of the mMTC traffic is that the set of mMTC devices that transmit in a given radio resource is random and unknown. An mMTC transmission has a fixed rate and consumes one radio resource of channel uses. Given the rate and the reliability constraint , we focus on the maximum arrival rate that can be supported by the system as the performance criterion of interest. As detailed below, the probability of error measures the fraction of incorrectly decoded devices among the active ones.
SIC at the BS is a useful strategy to improve the performance of mMTC traffic. As discussed next, a SIC decoder can leverage power imbalances and other mechanisms not reviewed here (see, e.g., [16]), in order to sequentially improve the reliability of simultaneous mMTC transmissions.
To characterize the performance achievable with SIC, we let denote the index of the mMTC device with the th largest channel gain for the allocated frequency . In the rest of this section, we drop the dependence on for simplicity of notation. By definition, we then have the inequalities . In the absence of interference from eMBB and URLLC traffic, the SINR available when decoding the signal of the th mMTC device, under the additional assumption that the devices with indices are correctly decoded, depends only on its channel gain and on the channels gains of the other active mMTC devices as
(13) 
The th mMTC device is correctly decoded if the inequality holds; and, if decoding is successful, the signal from the device is subtracted from the received signal. We let be the random number of mMTC devices in outage, i.e., is the largest integer in satisfying, for all , the inequality
(14) 
The error rate of the mMTC devices is then quantified as the ratio
(15) 
between the average number of users in outage, namely , and the average number of active users. The maximum rate that can be supported under the reliability condition is defined for reference as
(16) 
This quantity can be computed by means of Monte Carlo numerical methods.
Iii Slicing for eMBB and URLLC
In this section, we consider the coexistence of eMBB and URLLC devices, while assuming that there is no mMTC traffic, i.e., that the mMTC arrival rate is . We first briefly recall, using the results in the previous section, how the performance of the two services can be evaluated for the case of HOMA, and then analyze the more complex scenario of HNOMA.
Iiia Orthogonal Slicing: HOMA for eMBB and URLLC
In the case of orthogonal slicing, i.e., under HOMA, we assume that out of the frequency radio resources for all minislots in the given radio resource are allocated to the URLLC transmissions, while the remaining radio resources are each allocated to one eMBB user. Note that, in each minislot, the probability that the frequency channels allocated to URLLC traffic are unused is the complement of the activation probability, i.e., .
The performance of the system is specified in terms of the the pair of eMBB sumrate and URLLC rate achievable at the given reliability levels . The eMBB sumrate is obtained as
(17) 
where is obtained as explained in Section IIB. The URLLC rate is computed from (12) by imposing the equality as detailed in Section IIC.
IiiB Nonorthogonal Slicing: HNOMA for eMBB and URLLC
We now consider nonorthogonal slicing, or HNOMA, whereby all frequency channels are used for both eMBB and URLLC transmissions. Hence, . With nonorthogonal slicing, eMBB and URLLC transmissions interfere, and, hence, the rate pair cannot be directly obtained from the analysis in Section II. We next describe different decoding architectures, and derive corresponding achievable pairs and for nonorthogonal slicing.
Decoding Architectures: A key observation in the design of decoding schemes is that, due to latency constraints, the decoding of a URLLC transmission cannot wait for, and hence depend upon, the decoding of eMBB traffic. In fact, decoding of a URLLC transmission can only rely on the signal received in the given minislot. This constraint prevents SIC decoders whereby eMBB transmissions are decoded first and canceled from the received signal prior to decoding of the URLLC messages. Note also that, because of the heterogeneity of reliability requirements, decoding eMBB first and then URLLC in a SIC fashion would require decoding the eMBB traffic at the same level of reliability needed for the URLLC traffic. As a result of these considerations, in HNOMA with SIC the URLLC transmissions should be decoded while treating eMBB signals as an additional noise.
In contrast to URLLC traffic, eMBB requirements are less demanding in terms of latency, and hence eMBB decoding can wait for URLLC transmissions to be decoded first. This enables a SIC mechanism whereby URLLC messages are decoded and then canceled from the received signal prior to decoding of the eMBB signal. Since the reliability of URLLC is two or more orders of magnitude higher than eMBB, the performance of eMBB under the described SIC decoder is expected to be close to the ideal orthogonal case in which no interference from URLLC traffic is present. This design choice is an instance of reliability diversity.
That said, the SIC decoder may be ruled out by considerations such as complexity. In such circumstances, one could adopt another decoding approach that is, in a sense, diametrically opposite to SIC in its treatment of URLLC interference. Such a decoder treats any minislot that contains a URLLC transmission as erased or punctured. This option of HNOMA with puncturing is currently being considered within the 5G community [10]. Note that this approach requires the decoder at the BS to be able to detect the presence of URLLC transmissions, e.g., via energy detection.
Encoding: If HNOMA with SIC is used, we set the eMBB rate to
(18) 
where represents the target SNR for eMBB transmission, which is to be determined.
In contrast, in HNOMA with puncturing and erasure decoding, the eMBB device applies an outer erasure code with rate , which is concatenated to the codebook used in the physicallayer transmission of the eMBB encoder. Thanks to the erasure code, the decoder is able to correct erased minislots, while, if the number of URLLC transmissions is larger than , the decoding process fails. The parameter needs to be designed so as to satisfy the target error rate for eMBB users. The resulting data rate for eMBB transmission in frequency channel is
(19) 
Regarding the selection of the target SNR , we recall that in the orthogonal case, as shown in (10), the variable is uniquely determined by the error probability target via the threshold SNR defined in (7). In contrast, with nonorthogonal slicing, it may be beneficial to choose a smaller target SNR than the one given by (10), so as to reduce the interference caused to URLLC transmissions. This yields the inequality
(20) 
Summarizing the discussion so far, decoding of URLLC traffic cannot leverage SIC and treats eMBB transmissions as noise. In contrast, the eMBB decoder at the BS can either leverage SIC by decoding URLLC traffic first, or rather treat any minislot occupied by URLLC traffic as erased. These are two extreme points among all possible eMBB decoders.
Rate Region: The objective of the analysis is to determine the rate region for which the target error probabilities of the two services are satisfied. To this end, we fix the URLLC rate , and compute the maximum attainable eMBB rate
. We recall that the available degrees of freedom in the design are the target SNR
and the minimum channel gain at which an eMBB device is active (or equivalently the activation probability (4)), as well as the erasure code parameter if a puncturing approach is adopted for eMBB decoding. We also emphasize that, unlike the orthogonal case, the target SNR and the minimum SNR are separate degrees of freedom, which are related by the inequality (20).We start by imposing the reliability constraint for the URLLC user, which yields the following condition for both SIC and erasure decoder:
(21) 
Here, are independent Bernoulli random variables with parameter given in (4). Recall that is a function of . The term represents the interference power caused by an eMBB transmission on frequency channel to the URLLC traffic. The inequality (21) imposes a joint constraint on both and . Next, we impose the reliability constraint for eMBB traffic by considering separately SIC and erasure decoders.
IiiB1 SIC decoder
Under HNOMA the decoding of an eMBB message is generally affected by the interference from the URLLC users. However, this interference is not present if: (i) there are no URLLC transmissions, i.e., ; or (ii) if URLLC transmissions are present, i.e., , but the corresponding signals are decoded successfully and canceled by the SIC decoder. As for the latter event, since the interference from eMBB users and the fading gains are constant across the minislots, either all URLLC transmissions are decoded incorrectly (event ) or they are all correctly decoded (event ).
Based on the discussion above, we can bound the eMBB error probability by distinguishing the case in which the eMBB transmission is subject to interference from URLLC signals, and the case in which is not, using the law of total probability, as follows:
(23)  
(24) 
Here, equality (23) holds because the only source of outage for eMBB in absence of URLLC interference is the instantaneous SNR being below the minimum SNR, which implies ; moreover, (23) follows by using that and that canceling URLLC interference results in the same performance achievable in the absence of URLLC transmissions, so that we have the equality . Imposing the reliability condition and using (24) we obtain the inequality
(25) 
As already pointed out, this equivalently imposes a constraint on through (4).
From (25), we see that, unlike the orthogonal case, with nonorthogonal slicing the eMBB activation probability is larger than . This is becasue URLLC interference may cause an eMBB decoding error even when the eMBB’s SNR is above the threshold. However, the impact of URLLC interference is typically minimal. Indeed, the high reliability requirements for URLLC, which are reflected by the very small value of , imply that is close to .
To summarize, for a given feasible URLLC rate , the maximum eMBB rate is obtained by maximizing subject on the constraints on and implied by (20), (21), and (25). This maximization requires the use of a twodimensional numerical search. Note that the activation probability is typically very close to , and hence, when solving this problem, one can conservatively assume that the eMBB interference is always present in (21), i.e., . In contrast, the dependence of the righthand side of (21) on causes a nontrivial interdependence between and .
IiiB2 Puncturing and erasure decoder
Turning now to the erasure decoder, we can write the probability of error for an eMBB user by means of the law of total probability as
(26) 
where we have distinguished the case in which the erasure code is able to correct the erasures caused by URLLC transmissions, i.e., , and the case in which an error is instead declared, i.e., . When the latter event occurs, a decoding error occurs, and hence we have . In contrast, when , the only source of outage is the instantaneous SNR being below threshold, which results in . Overall, the resulting eMBB reliability requirement is
(27) 
IiiC Numerical Illustration
Here we present simulation results for the rate region for HOMA as well as HNOMA, with both SIC and puncturing decoders. In addition to the results obtained from the previous analysis, we also show curves obtained from the expressions derived in Appendix A, which are easier to evaluate and are shown to provide a performance lower bound (“LB”).
In Figs. 5 and 6, we plot the rate regions for . Fig. 5 considers the case with dB and dB, while Fig. 6 focuses on the complementary setup with when dB and dB. For both figures, HNOMA with puncturing uses the optimal puncturing parameter .
For both setups considered in the figure, HMONA with puncturing is outperformed by both HOMA and HNOMA with SIC. Furthermore, when , we see from Fig. 5 that the SIC region dominates the region achievable by orthogonal slicing. This is thanks to the capability of the BS to decode and cancel URLLC transmissions by leveraging reliability diversity.
In contrast, when , Fig. 6 shows that orthogonal slicing can attain pairs that are not attainable by HNOMA with SIC. In particular, HOMA is preferable if one wishes to obtain large values of the URLLC rate. This is due to the difficulty of ensuring high reliability in the presence of eMBB transmissions when . We recall that this is a consequence of the impossibility to decode and cancel eMBB transmissions prior to URLLC decoding owing to the URLLC latency constraint. In contrast, if one is interested in guaranteeing large eMBB sumrates, HNOMA offers significant performance gains. This is because nonorthogonal transmission allows eMBB users to operate over a larger number of spectral resources while not being significantly affected by URLLC interference.
We see that the lower bound is able to capture the shape of the region obtained through more accurate and timeconsuming MonteCarlo simulations.
Iv Slicing for eMBB and mMTC
In this section, we treat the slicing of wireless resources to jointly support eMBB and mMTC services, while assuming that the URLLC traffic, if present, has been allocated orthogonal resources. Analogously to the case of eMBBURLLC coexistence, we consider separately orthogonal slicing (HOMA) and nonorthogonal slicing (HNOMA). We shall focus without loss of generality on the case , since the mMTC users are assumed to be active on a single frequency channel. The extension to the case , in which the mMTC devices are allowed to randomly access all channels, is rather straightforward, as further elaborated in Section VB. Since a single channel is considered, in this section we omit all frequency indices .
Iva Orthogonal Slicing: HOMA for eMBB and mMTC
For the case of orthogonal slicing, we assume that the eMBB and the mMTC devices use the frequency radio resource in a timesharing manner. Let and be fraction of time in which the resources are allocated to the eMBB device and the mMTC devices, respectively. We aim at characterizing the region of pairs of eMBB rate and mMTC arrival rate that can be supported by orthogonal slicing for a given mMTC transmission rate requirement and probability of error .
For a given timesharing factor , the achievable pair of eMBB rate and mMTC arrival rate can be written in terms of the quantities derived in Sections IIB and IID as
(28)  
(29) 
respectively, where is obtained as explained in Section IIB and is defined in (16). In fact, with orthogonal slicing, both the achievable eMBB rate and the achievable mMTC transmission rate (specified on the righthandside of (14)) are scaled according to the fraction of time resources allocated to the service.
IvB Nonorthogonal Slicing: HNOMA for eMBB and mMTC
In HNOMA, the eMBB device is allowed to use the radio resource at the same time as the mMTC devices.
Decoding Achitecture As argued in Section IID, a SIC decoder may enhance the reliability of mMTC decoding. Furthermore, when radio resources are allocated exclusively to mMTC devices, optimal decoding follows the order of descending channel gains. The situation is more complicated in the presence of an interfering eMBB transmission.
In light of the higher reliability requirements of eMBB transmissions as compared to mMTC traffic, i.e., , one may be tempted to consider decoding the eMBB traffic before attempting to decode any mMTC traffic. This appears to be in line with the discussion in the previous section concerning SIC for eMBB and URLLC coexistence. However, this approach is suboptimal, since it neglects to account for the different definition of reliability of mMTC traffic. In fact, the probability of error (15) measures the fraction of incorrectly detected active users and not a perdevice decoding probability. As such, some of the active mMTC devices may well have very high channel gains, hence, causing large interference, making it beneficial to decode and cancel them prior to decoding the eMBB signal. Selecting a SIC decoder that accounts for this important feature of mMTC traffic is another example of a design choice that utilizes reliability diversity.
Based on this discussion, we assume that, at each decoding step, the BS decodes either the eMBB device, provided that it has not been decoded yet, or the next available mMTC device in order of decreasing channel gains. Note that this implies that the decoding step at which the eMBB device is decoded is random, as it depends on the realization of the channel gains. The process ends when no more transmissions can be reliably decoded.
As in nonorthogonal slicing of eMBB and URLLC, the eMBB rate is set to
(30) 
where is the target SNR for the eMBB transmission, which is to be determined. Similar to the eMMBURLLC coexistence case (see Section IIIB), this quantity needs to satisfy
(31) 
Again, as in Section IIIB, we allow the eMBB device not to use the maximal power since it may be beneficial to use a value of lower than the righthand side of (31) in order to control the impact of eMBB interference on the overall SIC procedure.
Next, we formalize the SIC decoding procedure. When the eMBB is inactive because of an insufficient SNR, i.e., , the SIC decoding procedure is equivalent to the procedure described in Section IID, namely, the mMTC devices are decoded in the order of decreasing channel gains. When the eMBB is active, the SIC procedure runs as follows. Starting from , the receiver computes the SINR for the th mMTC device as
(32) 
If , the th mMTC is decoded, canceled, is incremented by one, and the procedure starts over. Otherwise, the receiver attempts to decode the eMBB user. To this end, it computes the SINR of the eMBB transmission as
(33) 
and decodes and cancels the eMBB if the condition is satisfied. If the eMBB is decoded successfully, the decoding procedure continues as in Section IID. If , the procedure terminates.
Let and be the random variables denoting the number of decoded mMTC and eMBB devices. With this notation, the probabilities of error for mMTC and eMBB users are given as and , respectively.
In order to characterize the achievable pairs (, ), we evaluate the maximum supported mMTC arrival rate as a function the eMBB rate as
(34)  
We remark that, the probability distributions of
and depend on the parameters , , , , and . The computation of (34) requires Monte Carlo simulations.IvC Numerical Illustration
We present numerical simulation results illustrating the tradeoffs between the eMBB rate and the mMTC arrival rate for orthogonal and nonorthogonal slicing. In addition to numerical results obtained by solving (34) through Monte Carlo methods, we also report results obtained upper and lower bounds on in (34), which are easier to evaluate and are derived in Appendix B. Throughout this section, we set , and .
In Fig. 7, we plot the maximum mMTC arrival rate for both orthogonal and nonorthogonal slicing as a function of when dB, dB, and . When orthogonal slicing (HOMA) is used, the supported mMTC arrival rate is seen to decrease in an approximately linear fashion with the eMBB rate. As for nonorthogonal slicing (HNOMA), we observe three fundamentally different regimes as changes from zero towards is maximal value .
The first regime consists of very small values of the eMMB rate , for which the supported arrival rate is almost constant. At such values of , the eMBB device can be reliably decoded before the mMTC devices. Therefore, interference from eMBB user can be cancelled, and the performance of mMTC traffic is unaffected by small increases in . The second regime spans intermediate values of . In this case, the eMBB signal can only be decoded after some of the strongest mMTC signals are decoded and canceled. Hence, the mMTC performance is reduced by the interference from eMBB transmissions. Also, the SIC decoder tends to stop the decoding process after detecting the eMBB user, and decoding typically fails while detecting an mMTC device because of the interference from the other, yet undecoded, mMTC devices. In the third regime, eMBB decoding fails with a probability comparable to that of the weaker mMTC devices due to the mutual interference between the two services. As a result, in this regime, the supported mMTC arrival rate decays to zero as the eMBB rate increases.
The first and the third regime identified in Fig. 7 can also be understood with the help of the lower and upper bounds derived in Appendix B. In particular, when is very low, as mentioned, it is almost always possible to decode the eMBB transmission before decoding any mMTC device. This is the premise of the lower bound, which, as shown in Fig. 7, agrees with the simulation results in the first regime. On the contrary, the upper bound is computed by first identifying the subset of mMTC devices whose channels are so weak that the additive noise and the eMBB interference alone make their decoding impossible. The upper bound is seen to agree the simulation results in the third regime, i.e., when is large.
In Figs. 8 and 9, we plot the supported mMTC arrival rate as a function of for different values of the eMBB SNR and , and for different eMBB reliability levels and dB, respectively. This figures allows us to assess the impact of the average eMBB gain and of the eMBB reliability on the operation of the system in the three regimes identified above and on relative performance of orthogonal and nonorthogonal slicing, as discussed next.
As it pertains to three regimes, we observe from Fig. 8 that the rate at which the transition from the second to the third regime occurs does not change as the eMBB average channel gain is increased from to . On the contrary, the value corresponding to the transition from the first to the second regime becomes larger with this increase in . This increase, in fact, allows the eMBB transmission to be decoded earlier in the SIC process for a larger set of values of . From Fig. 9, we observe that the eMBB error probability constraint significantly affects the supported mMTC arrival rate in the second and third regimes. In fact, in these case, the higher eMBB transmission power required to ensure a higher reliability impairs the decoding of mMTC users via interference.
We now elaborate on the comparison between orthogonal and nonorthogonal slicing. The presented figures emphasize the fact that there are points in the rate region (,) that can be attained by nonorthogonal slicing and not by orthogonal slicing, and vice versa. Specifically, nonorthogonal slicing is seen to be beneficial when is across the second and the third regimes, especially for not too large reliability levels (see Fig. 8). For such values, the eMBB rate is large, and yet low enough not to hamper the decoding of the mMTC users. Once again, reliability diversity is crucial to ensure the effectiveness of nonorthogonal slicing. In contrast, for large values of the rate , when nonorthogonal slicing is deeply in the third operating regime, orthogonal slicing is always superior. This is because in this regime the performance is limited by the interference caused by eMBB users.
V Discussion, Conclusions and Future Work
Va Discussion and Conclusions
In this work, we have presented a communicationtheoretic model that enables the investigation of the fundamental tradeoffs associated with the sharing of the wireless resources among the three 5G traffic types, namely eMBB, mMTC and URLLC. Albeit simple, the model accounts for the differences among the services in reliability, latency, and number of supported devices. Specifically, we have considered the slicing of resources among the services in the uplink over a shared multiple access resource. We have utilized the term “slicing” in order to emphasize the heterogeneous performance requirements that need to be satisfied for each service as well as the performance isolation among services. Two slicing paradigms have been investigated, orthogonal and nonorthogonal and the respective transmission schemes HOMA and HNOMA, where the latter is inherently possible only in shared wireless channels.
We have applied the model to the study of the slicing for two services in two different cases: (i) eMBB and URLLC and (ii) eMBB and mMTC. In both cases, we have shown that, in order to be effective, the design of nonorthogonal slicing solutions must be guided by reliability diversity. For the case of eMBBURLLC coexistence, reliability diversity dictates that, in HNOMA with SIC, the URLLC device should be decoded first, as its decoding cannot depend on the decoding of eMBB, whose reliability and latency requirements are much looser compared to URLLC. The implications of reliability diversity are more subtle in the case of nonorthogonal slicing between eMBB and mMTC. In this case, considering the fact that the number of active mMTC devices is large with high probability, it is natural to introduce a reliability metric that accounts for the fraction of correctly decoded transmissions. The analysis demonstrated that there are regimes in which the decoding of eMBB should be performed after the decoding of one or multiple mMTC devices in order to benefit from nonorthogonal slicing.
Our numerical results show that there are regimes in which HNOMA is advantageous over HOMA and vice versa. In the case of eMBBURLLC, HNOMA with SIC is always beneficial when the eMBB rate is very large. In the case of eMBBmMTC, nonorthogonal slicing is beneficial when the eMBB rate takes values that are small enough not to hamper the decoding of the mMTC devices.
VB Generalizations and Future Work
The analysis presented in this paper is based on some simplifying assumptions. However, the basic model and the methodology developed here can be extended to more general models and other operation regimes, as briefly discussed here.
Starting with eMBBURLLC coexistence, one could devise another HNOMA scheme, where eMBB and URLLC users are allowed to access partially nonorthogonal resources, so that only a subset of frequency channels potentially occupied by URLLC traffic may be interfered by eMBB transmissions. Another direct generalization is to assume that the minislots are preallocated to different URLLC devices. Recall that, when all transmissions are made by the same URLLC device, the block fading model dictates that either all or none of the transmissions in a minislot are decoded correctly. If each URLLC transmission is carried out by a different device, then then error decoding events are independent across the minislots. As a more involved extensions of the model, one may consider the impact of frequency diversity also for eMBB traffic, and the performance under alternative decoding strategies, such as treating interference as noise.
As for the coexistence of mMTC and eMBB services, an interesting extension is to allow multiple channels for mMTC traffic. In particular, mMTC devices may be allowed to use frequency hopping, and the number of allocated frequency channels may depend on the reliability requirements. Another aspect that deserves study is the impact of the arrival process, which here has been assumed to be Poisson. Namely, the higher burstiness of the arrival process can potentially improve the gain that one can obtain with nonorthogonal slicing. Finally, following the approach in NBIoT systems, the transmission of a single mMTC device may consist of replicas of the same packet in multiple time slots. This makes the nonorthogonal slicing of mMTC and eMBB even more relevant, as it is not feasible to reserve resources exclusively for replicas of packets generated by sporadically active mMTC devices.
Appendix A Lower Bound on the URLLC rate
By setting , we can upper bound (21) as
(35)  
(36)  
(37)  
(38) 
We obtained (35) by multiplying both terms in the inequality by and then by exponentiating them; (36) follows from Markov inequality; and (37) holds because and , are i.i.d., and hence we can set in (38) without loss of generality. The inequality in (38) can be rewritten as
(39) 
where the strict lower bound follows by assuming that the eMBB interference is always present, i.e., . The expectation in (39) can be calculated by a Monte Carlo simulation and the value of in (39) is chosen such as to maximize the lower bound.
Appendix B Bounds for NonOrthogonal Slicing for eMBB and mMTC
B1 A first upper bound
The idea behind the bound is as follows: if the eMBB decoding fails, then the decoding of all mMTC devices for which must also fail. We obtain next a lower bound on the eMBB error probability, which will give us the desired upper bound on , by assuming that all mMTC devices that do not satisfy are decoded correctly and cancelled before eMBB decoding, and that the remaining ones, which are not decoded correctly, cause interference to the eMBB. This yields
(40)  
(41) 
Here, is the indicator function of the event . The random variable in (41
) is the sum of a Poissondistributed number of truncated, exponentialdistributed random variables, which allows for an efficient numerical evaluation of the probability term in (
41). Next, we set the righthand side of (41) equal to , and find the values of and that result in the largest . This value is precisely the desired upper bound on .B2 A lower bound and an alternative upper bound
We upperbound the eMBB error probability by considering the following suboptimal decoding scheme. We force the decoder to always decode the eMBB first and subsequently decode the mMTC devices. The maximal supported arrival rate with this modified decoder is clearly a lower bound on . While deriving this bound, we shall derive as byproduct also an alternative upper bound on .
As already mentioned, decoding the eMBB as the first device results in an upper bound on the eMBB error probability. Mathematically,
(42)  
(43) 
Here, the random variable follows an Erlang distribution. It will turn out convenient to denote by the righthand side of (43). Let now be the mMTC error probability as a function of mMTC arrival rate in the absence of the eMBB device. By the law of total probability, the mMTC error probability when the eMBB is present can be upper and lowerbounded as
(44) 
Set now
(45)  
(46) 
In words, these are the largest arrival rates for which the righthand side and the lefthand side of (44) are smaller than , respectively. Furthermore, let be given by
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