## I Introduction

With the advent of the fifth generation (5G) of cellular systems, wireless cellular systems are undergoing an evolution in terms of services and of network architecture. enhanced Mobile BroadBand (eMBB) services, which are aimed at consumers, will in fact share radio and network resources with Ultra-Reliable Low-Latency Communications (URLLC) and machine-type traffic, which cater to vertical industries [1]. Furthermore, the cellular network architecture is evolving from a base station-centric deployment to a fog-like set-up [2][3], referred to here as Fog-Radio Access Network (F-RAN). In the latter, network functionalities are distributed among edge and cloud – defined as distributed and central units by 3GPP [2] – depending on their latency and reliability requirements [3][4]. An extreme instance of this type of network architectures is Cloud RAN (C-RAN), whereby all processing, apart from radio-frequency components, is carried out in the cloud [5]. Motivated by the discussed evolution of services and architectures, this work considers the problem of engineering the coexistence of URLLC and enhanced MBB (eMBB) services in an F-RAN system for both uplink (UL) and downlink (DL).

To elaborate, consider the multi-cell scenario shown in Fig. 1, in which each cell contains an Edge Node (EN), along with its connected computing platform, and multiple eMBB and URLLC users. URLLC traffic is handled at the ENs in order to meet the low-latency requirements. For example, URLLC devices may be vehicles in vehicle-to-cellular use cases[6], or they may be devices serving automation chains in Industry 4.0 scenarios [7][8]. In contrast, eMBB devices can be served by a centralized cloud processor as in a C-RAN architecture in order to benefit from the cloud’s interference management capabilities [9]. The proposed hybrid edge-cloud solution fully leverages the unique features of F-RAN systems in order to cater to the heterogeneous requirements of URLLC and eMBB systems.

Beside the architectural considerations discussed above, a key design question for the system in Fig. 1 is how to share the radio resources between URLLC and eMBB services. A conventional approach would be to allocate orthogonal spectral resources to the two traffic types in order to ensure isolation, and to then carry out a separate design that is tailored to the heterogeneous performance metrics and guarantees for the two services.

As an alternative to orthogonal resource allocations, Non-Orthogonal Multiple Access (NOMA) is grounded in classical results in information theory that prove the capacity-achieving property of non-orthogonal transmissions in multiple access channels and of superposition coding in broadcast channels, modeling single-cell UL and DL scenarios, respectively [10]. Motivated by these information-theoretic results, NOMA has been proposed as a key component of 5G systems as a means to share radio resources among transmissions belonging to the same traffic type, see, e.g., [11], [12] for eMBB and [13] for URLLC.

In the presence of multiple coexisting services, radio resources can be shared not only among services of the same type, but also across different services. Following [14], we refer to this approach as Heterogeneous NOMA (H-NOMA), in order to highlight the key distinction with respect to conventional NOMA of accommodating services with heterogeneous performance requirements. The idea was first proposed in [14], in which a communication-theoretic model was put forth for the study of the performance of H-NOMA for a single-cell scenario in the UL, demonstrating the potential advantages of H-NOMA.

In contrast to prior work, the goal of the present contribution is to analyze and compare the performance of conventional Heterogeneous-Orthogonal Multiple Access (H-OMA) techniques with H-NOMA in the multi-cell F-RAN system of Fig. 1 for both UL and DL. As illustrated in Fig. 2, the communication model assumes random activation of URLLC users, which implies possible collisions in the UL and blockages in the DL, and scheduled access for eMBB. The analysis also accounts for capacity limitations on the fronthaul links connecting cloud and ENs, as well as for different H-NOMA strategies including puncturing, treating URLLC interference as noise and superposition coding (see Fig. 5 for the UL and Fig. 6 for the DL).

A preliminary study was published by some of the authors in [15], which considered only the UL and a simplified Wyner-type channel model with no fading and inter-cell interference limited to neighbouring cells. We also refer to [16] that considers the same setup as [15] but with an analog fronthaul link. Another related work for the UL Wyner model is [17], in which higher-latency messages are decoded by means of cooperation between adjacent cells, while lower-latency messages are decoded without cooperation.

The rest of the paper is organized as follows. In the next section, we detail the system model, while Sec. III describes the signal model and the performance metrics. In Sec. IV and V, the performance of H-OMA is evaluated for UL and DL respectively, while in Sec. VI and VII, the performance of H-NOMA is analyzed for UL and DL respectively. Finally, numerical results are presented in Sec. VIII, and conclusions are drawn in Sec. IX.

Notation:

Bold upper-case characters denote matrices and bold lower-case characters denote vectors.

represents the expectation of the argument with respect to the distribution of the random variable

. denotes the Hermitian transpose of matrix .indicates a Bernoulli distribution with parameter

. indicate a Binomial random variable distribution with parameters and . denotes the mutual information between random variables and for the given constant value of random variable , i.e., . is the determinant of matrix .## Ii System Model

As illustrated in Fig. 1, we study UL and DL communications in a cellular network with an F-RAN architecture that encompasses both eMBB and URLLC users. Each one of the cells contains an Edge Node (EN) and multiple eMBB and URLLC users. All ENs are connected to a Baseband Unit (BBU) in the cloud by mean of orthogonal fronthaul links of capacity bit/s/Hz, or equivalently bits for each symbol of the wireless channel. The RAN uses Frequency Division Duplex (FDD) in order to facilitate grant-free URLLC transmissions, as detailed below.

F-RAN topology and operation: As illustrated in Fig. 1, we assume that the URLLC users are located close to the ENs, and hence URLLC communications take place with non-negligible power only with the EN in the same cell. As a result, URLLC users do not cause interference to ENs in other cells while transmitting in the UL, and they only receive transmissions from the same-cell EN in the DL. This condition can be ensured by allowing URLLC transmissions only from users with large average channel gain to the target EN, so that the high reliability requirement of URLLC traffic can be satisfied. As an example, as seen in Fig. 1, the EN may serve a nearby vehicle for transmission of time-sensitive control information in vehicle-to-cellular user cases [6]. Alternatively, in mission-critical or Industry 4.0 scenarios, ENs can be deployed in locations that contain URLLC devices. The eMBB users, instead, need not guarantee this condition, and are assumed to be in arbitrary positions with potentially non-negligible channel gains to all ENs for both UL and DL.

As illustrated in Fig. 1, for the UL, due to latency constraints, the URLLC signals are decoded locally at the EN, while the eMBB traffic is decoded centrally at the BBU as in a C-RAN architecture[18] [19]. In a similar manner, in the DL, the eMBB traffic is assumed to be generated at the cloud, e.g., as a result of web searches or broadband streaming, and C-RAN precoding and quantization are applied [20] [21]. In contrast, URLLC traffic is generated at the edge, with each EN serving same-cell URLLC users, in line with the use cases mentioned above. Note that these assumptions imply that the higher layers of eMBB and URLLC services are implemented separately at cloud and edge, respectively.

Frame structure: As illustrated in Fig. 2, we consider a radio interface that operates in frames of mini-slots and frequency channels. The time frequency plane is organized into Resource Units (RUs), and each RU spans one mini-slot of index and one frequency channel of index , and it contains symbols in time domain and subcarriers^{1}^{1}1As an example, in 3GPP release 15 [22], the RU consists of =12 subcarriers and =14 symbols.. We index as the RU located at frequency channel and mini-slot .

While eMBB users are scheduled, URLLC transmissions are assumed to be grant-free, and packets are randomly generated in each mini-slot for URLLC users. As illustrated in Fig. 2 and Fig. 3, we assume that at most one URLLC packet per cell and per mini-slot may be generated at an URLLC user in the UL or at the EN in the DL. Each packet is generated at, or intended for, a different URLLC user. The probability of generation of such packet is

and packet generation is independent across mini-slots and cells. We note that, in practice, there may be a chance of multiple URLLC packets being generated within a cell in a mini-slot.Due to latency constraints, each URLLC transmission can span only a single mini-slot, and hence the blocklength of an URLLC transmission is equal to symbols. Unlike URLLC transmissions, eMBB codewords span all mini-slots in a frame and a set of frequency channels.

Channel and Channel State Information (CSI) model: For both UL and DL, we consider Rayleigh fading channels that are constant over time in the considered frame, but vary independently from one frequency channel to another. Accordingly, the complex channel gain between the -th EN and an eMBB user in the -th cell at RU is modeled as , where accounts for path loss and large scale fading. The channel gains are i.i.d. over the frequency index ; independent for different pairs ; and constant during the mini-slots of the considered frame. In a similar manner, the channel gains between an URLLC user in cell and the -th EN are modeled as , where reflects the path loss and large-scale fading. Note that the dependence on mini-slot index is kept for URLLC transmissions in order to highlight that, under the given assumptions, different URLLC users transmit, or are served in each mini-slot .

Following a standard path-loss model, we write and , where is the distance between -th EN and the -th eMBB user and is the distance between the -th URLLC user and the EN in the -th cell; is the path loss exponent and constants and are used to set the signal-to-noise ratio () levels at the reference distances and .

Due to latency constraints, CSI is assumed to be unavailable at the transmitter side in the communications between a URLLC user and an EN, while receiver CSI is available. This assumption reflects the fact that, in an FDD system, transmitter CSI would require feedback from the receiver. This would limit reliability since it would add another potential cause of error, and it would increase latency. In contrast, CSI is conventionally assumed to be available at both the transmitter and the receiver for the eMBB traffic.

### Ii-a Heterogeneous Orthogonal and Non-Orthogonal Multiple Access

In this work, we consider two access schemes, namely H-OMA and H-NOMA. As discussed in Sec. I, these schemes allow the sharing of time and frequency resources between eMBB and URLLC services. As seen in Fig. 2(a), under H-OMA, URLLC packets can only occupy preallocated mini-slots over which eMBB transmissions are not allowed. In particular, a mini-slot is allocated for transmission of URLLC traffic every mini-slots. Parameter hence represents the access latency, i.e., the maximum number of mini-slots a URLLC packet has to wait before transmission. We note that, for the DL, it would also be possible to consider a dynamic schedule of eMBB and URLLC transmissions (see, e.g., [23]).

In the UL, as illustrated in Fig. 3(a), if multiple URLLC users in a cell generate a packet within the mini-slots between two allocated mini-slots, then a collision occurs in the allocated mini-slot. In this case, all packets are discarded due to latency constraints. In the DL, instead, as illustrated in Fig. 3(b), when multiple URLLC packets are generated at an EN, the EN can select one such packet uniformly at random and discard, hence blocking from access, all the other packets. Collisions in the UL and blockages in the DL contribute to the overall error rate for URLLC.

In contrast, H-NOMA enables eMBB and URLLC traffic to share the same radio resources. More precisely, as shown in Fig. 2(b), URLLC packets are transmitted in the same mini-slot in which they get generated. It follows that H-NOMA has the minimal access latency of at the price of possible interference between eMBB and URLLC signals. Furthermore, under the assumed model, no collisions or blockages occur with H-NOMA.

## Iii Signal Model and Performance Metrics

In this section, we detail signal models for UL and DL, as well as the performance metrics of interest. Throughout the analysis, we focus our attention on the analysis of eMBB and URLLC traffic that occupies the frame shown in Fig. 2. We first concentrate on detailing the signal transmitted and received in any of the symbols of an RU and then describe the performance metrics of interest. Throughout, we avoid introducing an explicit notation for the indices pointing to one of the symbols in each RU, and hence refer generically with the index to any symbol within RU .

### Iii-a Uplink Signal Model

As discussed in the previous section, with H-OMA, one mini-slot is exclusively allocated to URLLC users in the UL every mini-slots (see Fig. 2(a)). We denote as the signal received by each at any symbol within the RU . This can be written as

(1) |

for all different from and where denotes any symbol transmitted in RU by the eMBB user that is active in cell over the given frequency channel ;

is complex Gaussian noise with zero mean and unit variance, which is i.i.d. across the indices

, and , and across the symbols in an RU. Furthermore, for all mini-slots allocated to URLLC users, the received signal for mini-slot when there is no collision is(2) |

where denotes the signal sent by an URLLC user in the -th cell. We recall that, in contrast to the eMBB channel, the URLLC channel coefficients depend on the time index due to the assumption that URLLC packets are generated at different URLLC users.

Unlike H-OMA, under H-NOMA, URLLC users transmit immediately in the mini-slot in which a packet is generated. Accordingly, the signal received at each at RU can be written as

(3) |

where is the indicator variable that equals to one if an URLLC packet is generated in cell at mini-slot .

The power constraint for eMBB and URLLC users are defined respectively as

(4) |

where the average in (4) is taken over all uniformly selected information messages.

It will be convenient to write the signal models in matrix form. To this end, the channel matrix for eMBB users at RU is denoted by with -th row given by the vector . The channel matrix for URLLC users is diagonal due to the discussed lack of inter-cell interference and is denoted as . Consequently, we can write the signals (3) received at RU across all ENs under H-NOMA as

(5) |

where , , and . Models (1)-(2) can be written in matrix form in an analogous way.

Following the general assumptions introduced in Sec. II, the BBU and the ENs are assumed to have available the channel matrices and for both eMBB and URLLC users. Note that providing CSI to the BBU causes a fronthaul overhead that can be considered negligible as the coherence interval size increases (see, e.g., [24]). eMBB users are informed by the BBU about the transmission rate at which to operate, while URLLC users have no CSI. As a result, URLLC transmitters adapt their rates only to the distribution of the channel, while the eMBB transmitters adjust their rates to the current channel realization.

### Iii-B Downlink Signal Model

In the DL, for both H-OMA and H-NOMA, the signal received at an eMBB user in cell at RU can be written as

(6) |

where denotes the symbol transmitted by the -th EN; and is Gaussian noise received at the eMBB users, with , which is i.i.d. across the indices and and across all symbols in an RU. We also write (6) in vector form as

(7) |

where .

In contrast, the signal received by the -th URLLC user at RU is given as

(8) |

where represents Gaussian noise. We recall that, for the same reason as in the UL, the URLLC users’ CSI depend on the mini-slot index . As we will detail in Sec. V, under H-OMA, the signal is either intended for an URLLC user or an eMBB, while for H-NOMA the signal may carry the superposition of URLLC and eMBB signals. In all cases, the power constraint for each EN in the DL is defined as

(9) |

Finally, following the general channels assumptions described in Sec. II, all eMBB and URLLC users are assumed to have available the local channels and signal-to-interference-plus-noise ratio (). The BBU is informed about the eMBB channel matrices . Finally, as in the UL scenario the URLLC rate is adjusted to the statistics of the channels, while the eMBB rate is adjusted to the current channel realization.

In the remainder of the paper, we will drop the dependence on when no confusion may arise.

### Iii-C Performance Metrics

We are interested in the following performance metrics. For eMBB users, we study the average per-cell sum-rate , where the average is taken over the fading distribution. The transmission rates are adapted to the fading realizations thanks to the availability of transmitter CSI. The per-cell sum-rate measures the sum-rate, or, sum-spectral efficiency, in bit/s/Hz across all eMBB users in the system normalized by the number of cells.

As for URLLC users, we define the access latency as the maximum number of mini-slots an URLLC user has to wait before receiving a generated packet. By construction, for H-NOMA, we have which corresponds to the minimum access latency when a packet is transmitted in the same mini-slot in which it is generated. Furthermore, following 3GPP[25, Sec. 7.9], we define URLLC reliability as the probability to successfully transmit a packet within a given time constraint, here . Accordingly, we explicitly define a constraint on the URLLC error probability as

(10) |

for some desired error level . The error event accounts for two possible types of error, namely collision or blockage and decoding failure. As illustrated in Fig. 3, a collision or a blockage, which only applies to H-OMA, happens in UL and DL respectively, when two or more packets are generated in the mini-slots between two transmission opportunities. In contrast, decoding failure occurs when an URLLC packet is transmitted, and hence it is not subject to collision or blockage, but decoding fails at the receiver. For a given outage probability, due to the absence of CSI at the transmitter, open-loop transmission with no rate adaptation is assumed, and we adopt the maximum transmission rate under an outage probability constraint, or outage capacity, as the rate metric of interest [26].

## Iv Uplink Orthogonal Multiple Access

In this section, we consider the UL system performance in terms of eMBB rate and URLLC rate for a fixed URLLC access latency and URLLC probability of error requirement when assuming the conventional H-OMA. The operation of the ENs and of the BBU are illustrated in Fig. 4(a).

### Iv-a URLLC Performance

As discussed in Sec. II, due to latency constraints, URLLC packets are decoded at the local EN upon reception in the transmission mini-slot (see Fig. 1). For a given decoding error probability , the outage capacity of the -th URLLC user is given as

(11) |

where denotes the outage probability

(12) |

and is the signal-to-noise ratio (SNR) at the EN. The per-cell sum-outage capacity is obtained as .

Following Sec. III-C, the probability of error of an URLLC packet can be written as

(13) |

where is the distribution of random variable . The binomial random variable represents the number of additional URLLC packets generated by the URLLC users during the remaining mini-slots between two transmission opportunities. The first term in (13) is the probability that a packet is lost due to collisions, which occurs if additional packets are generated. The second term in (13) is the probability of a decoding error at the receiver.

### Iv-B eMBB Rate

Unlike delay-constrained URLLC traffic, eMBB messages are decoded jointly at the cloud in order to leverage the centralized interference management capabilities of the BBU. To this end, following the standard C-RAN operation, each EN quantizes and compresses the received signal for the mini-slots by using point-to-point compression (see [18] [19][27]), and forwards the resulting signal to the cloud over the fronthaul links (see Fig. 4(a)). Using (1), for each frequency channel , the quantized signal received at the BBU from can be written as

(14) |

where represents the quantization noise with power . From classical results in rate-distortion theory, we have the following relationship between the quantization noise power and the fronthaul capacity[28][27]:

(15a) | ||||

(15b) |

In (15), the factor captures the fact that a fraction of all mini-slots of the wireless channel are occupied by eMBB transmissions. The value of can be obtained by solving (15b) via numerical methods.

Considering the signals received by the BBU from all ENs, the eMBB per-cell sum-rate for a given channel realizations can be finally written as

(16a) | ||||

(16b) |

where . The average per-cell sum-rate is obtained by averaging (16) over the channel realizations .

## V Downlink Orthogonal Multiple Access

In this section, we consider the performance under H-OMA for the DL. Recalling that, as seen in Fig. 2(a), one every mini-slots is allocated to URLLC users, the signal sent by the -th EN for any symbol of the RU can be written as

(17) |

where and are the signals intended for URLLC and eMBB users, respectively, over the given RU. Note that we have if no URLLC packet was generated in mini-slots . As a result of the power constraint (9), we have the conditions and .

### V-a URLLC Performance

The rate analysis of the performance of URLLC traffic under H-OMA in the DL yields the same results (11)-(12) as for the UL with the caveat that should be replaced by the EN power constraint . This is because in both cases, under H-OMA, URLLC links are interference free. Furthermore, the probability of error (13) should be modified as

(18) |

since, in the DL, in case multiple URLLC packets are generated at an EN in the mini-slots per transmission opportunity, one packet can be selected at random and delivered to the corresponding user by the EN. In (18), the first term is the probability that more than one additional packets are generated and the packet of interest is blocked from access (see Fig. 3). The second term accounts instead for the decoding error of the transmitted packet.

### V-B eMBB Rate

Conventional C-RAN transmission based on linear precoding at the BBU and fronthaul quantization is carried out to communicate with eMBB users (see, e.g., [21][20]). To elaborate, we define as the independent encoded symbols intended for the eMBB user active in cell over frequency channel at a given mini-slot. The assumption reflects the use of standard Gaussian random codebooks. As illustrated in Fig. 4(b) the BBU carries out linear precoding separately for each frequency channel , producing the vector

(19) |

where we have defined the vectors , and is the channel precoding matrix for all eMBB users active on frequency channel . We write , where and are the -th column and -th row of matrix , respectively.

Assuming the standard C-RAN operation where the BBU compresses and forward the eMBB signal to each EN, the signal received at all ENs from the BBU over each frequency channel can be written as

(20) |

where we have defined and represents the quantization noise with power . The quantization noise is independent across the EN index and frequency channel . Consequently, the received signal (7) at the -th eMBB user can be written as

(21) |

In order to obtain the quantization noise’s power , in a manner similar to (15), we impose the conventional rate distortion condition [28]

(22) | ||||

for all ENs , which follows from the fronthaul capacity constraint of each EN .

Based on the derivations above, the eMBB achievable per-cell sum-rate for all eMBB users in cell for given channel realizations can be written as

(23a) |

where the effective noise accounts both for the disturbance due to fronthaul quantization and for eMBB inter-cell interference.

Based on the available CSI, precoding can be optimized at the BBU by maximizing the eMBB per-cell sum-rates as

maximize | (24) | |||||

subject to | ||||||

where the maximization is over the variables . The second constraint represents the power constraint at each EN, while the third constraint results from the fronthaul capacity limitations. The problem is non-convex, but it can be tackled using standard methods based on Semidefinite Relaxation (SDR)[29] and Concave-Convex Procedure (CCP) [30]. Accordingly, by performing the change of variables , adding the constraint and relaxing the constraint , the problem falls in the class of difference of convex problems (DC) [30] and thus CCP can be used to solve it as in, e.g., [31, Sec. IV].

## Vi Uplink Non-Orthogonal Multiple Access

In this section we consider the UL performance under H-NOMA. As discussed in Section II, with H-NOMA, eMBB and URLLC users may interfere with each other. For eMBB users, interference is dealt with at the BBU when jointly decoding the eMBB signals. To this end, as illustrated in Fig. 5, three decoding strategies are studied, namely Treating URLLC Interference as Noise (TIN), puncturing, and Successive Interference Cancellation (SIC). With TIN, ENs quantize and forward the received signals to the BBU on the fronthaul links, and the BBU decodes the eMBB transmissions while treating URLLC signal as noise. Under puncturing, whenever an URLLC user is active in a mini-slot, the receiving EN discards the corresponding eMBB signal received in the same mini-slot prior to quantizing the received signal and forwarding it to the BBU over the fronthaul links. Consequently, the BBU decodes using only interference-free eMBB mini-slots. Finally, with the more advanced SIC decoder, the ENs decode and cancel the URLLC transmission before fronthaul quantization. In contrast, for URLLC transmissions, due to reliability and latency constraints, the ENs cannot wait for the entire eMBB frame to be received, and hence URLLC decoding cannot benefit from interference cancellation of the eMBB signal. Therefore, the only affordable decoding strategy for URLLC transmissions is treating eMBB transmissions as noise.

### Vi-a URLLC Rate

With H-NOMA, as illustrated in Fig. 2(b), URLLC users transmit in any mini-slot in which a packet is generated with no additional access latency. We hence have the minimal access latency of . As for the probability of error, an error can only occur when decoding fails, since no collisions may occur under the given assumptions (see Sec. III). Hence, the probability of error coincides with the decoding error probability and the reliability constraint (10) imposes the inequality . For a given reliability level , the URLLC outage capacity is thus given as in (11) and (12) with , and with the caveat that the signal-to-noise-plus-interference ratio at the -th EN can be written as

(25) |

where accounts for interference from eMBB users in all cells active over frequency channel .

### Vi-B eMBB Rate under Treating URLLC Interference as Noise

Turning to the eMBB performance, we first study the standard C-RAN solution whereby the EN quantizes and forwards all the received signals to the BBU. The BBU decodes the eMBB messages while treating URLLC signals as noise. Under these assumptions, the signal received at the cloud from all ENs at RU can be written in matrix form using (5), as

(26) |

where and . In (26), the URLLC activation matrix contains i.i.d. variables. In order to obtain the quantization noises’ powers , in a similar manner as in (15), we impose the fronthaul capacity constraint for as

(27) | ||||

We note that equation (27) assumes that the BBU is able to detect the presence of URLLC transmissions. This is reflected in the expectation over the URLLC users’ activations. Finally, the eMBB per-user rate for given channel realizations is given by (28) where we recall the notation . The expectation in (28) can in practice be computed exactly by summing over the possible values for matrix as long as is not too large. Otherwise, stochastic approximation methods can be used.

(28) |

### Vi-C eMBB Rate under Puncturing

With puncturing, as seen in Fig. 5, whenever an URLLC user is active in a given cell , and RU , the signal received at the EN is discarded and not forwarded to the BBU. Note that, with the assumed grant-free URLLC transmissions, this requires the detection of URLLC user’s activity prior to fronthaul quantization, e.g., based on the detection of URLLC references sequences. A similar approach is under consideration within 3GPP [32].

To elaborate, we assume that each EN detects correctly that there are transmissions of URLLC devices. The assumption is well justified by the high reliability of URLLC transmissions. The EN compresses and forwards only the signals received during mini-slots free of interference from URLLC transmissions. Under this assumption, the signal received at the cloud from over RU can be written as

(29) |

According to (29), the received signal carries no information, i.e., only noise, when an URLLC user is active () in the corresponding mini-slot. Otherwise, when , the signal contains the contributions of the eMBB users and of the quantization noise . In matrix form, the signal in (29) received across all ENs over RU can be equivalently written as

(30) |

In order to enable decoding, the BBU at the cloud needs to be informed not only of the signals (29) for all the mini-slots with for all ENs , but also of the location of such mini-slots. To this end, each collects the i.i.d. binary vector containing the Bernoulli variables . The number of bits needed to be communicated from to the BBU in order to ensure the lossless reconstruction of this sequence is given by where is the binary entropy function [28]. Based on the discussion above, imposing fronthaul capacity constraint yields the condition

(31) | ||||

where and we recall that is the total number of bits per frame available for transmission on each fronthaul link, and the mutual information term accounts for the compression of the received signals over the symbols unaffected by URLLC interference (i.e., with ). The quantization noise’s powers can be obtained by solving (31) using numerical means for all . Following (28), the eMBB per user rate for given channel realization is finally given by

(32) |

### Vi-D eMBB Rate under Successive Interference Cancellation of URLLC Users

We finally study a more complex receiver architecture whereby SIC of URLLC packets is carried out at the ENs prior to fronthaul quantization. More specifically, as seen in Fig. 5, if an URLLC user is active and its message is decoded correctly at the receiving EN, the URLLC message is canceled by the EN. If decoding is unsuccessful, the signal received in the corresponding mini-slots is treated as in puncturing. Accordingly, with this scheme, each EN quantizes the received signals only for the minislots that are either free of URLLC transmissions or that contain URLLC messages that were successfully decoded and canceled at the EN. As a result, the received signal at the BBU from -th EN can be written as (29) but with an erasure probability of instead of . This is because a mini-slot is dropped if an URLLC user in the cell is active, which happens with probability , and if its transmission is decoded incorrectly, which happens with probability . As a result, the eMBB rate can be evaluated as (32) with the caveat that the random variables for are i.i.d. .

## Vii Downlink Non-Orthogonal Multiple Access

In this section we consider H-NOMA for the DL scenario. We follow an approach similar to the UL in Sec. V by allowing for different interference management strategies between URLLC and eMBB. A key new aspect in the DL is that interference arises from the URLLC transmissions originating at the ENs, which are a priori unknown to the BBU. Accordingly, as illustrated in Fig. 5, we consider two interference management strategies at the ENs, namely puncturing and superposition coding. Under puncturing, in any mini-slot in which an URLLC packet is generated, the EN transmits only the URLLC packet, dropping any eMBB information. Instead, with superposition coding, the EN transmits a superposition of both eMBB and URLLC signals to both users.

### Vii-a Puncturing

For both puncturing and superposition coding, the BBU precodes the eMBB signals and forwards them to the ENs over the fronthaul links. Under puncturing, whenever an URLLC packet is generated at an EN in a given mini-slot, the EN discards the eMBB signal received for the same mini-slot from the BBU. Note that this does not affect the scheduling decision made at the BBU for eMBB traffic, whose packet still spans the frame, with the exclusion of the punctured mini-slots. Consequently, the transmitted signal by can be written as

(33) |

where we recall that is the binary random variable denoting the generation of a URLLC packet at the EN in mini-slot .

URLLC Rate: URLLC users’ outage capacity is the same as in the H-OMA case discussed in Sec. V due the absence of inter-cell interference at URLLC users. However, with H-NOMA, the probability of error is equal to the decoding error probability due to the absence of collisions between URLLC packets. As a result, the URLLC rate is given by (11) with .

eMBB Rate: By assuming linear precoding at the BBU with precoding matrix as in Sec. V, the signal received by the -th eMBB user in a given mini-slot can be written in a manner similar to (21) as

(34) |

with definitions given in Sec. III. According to (34), an eMBB user receives useful information only from the ENs in cells for which no URLLC packet is generated i.e., . The per-user eMBB rate at the -th user and for given channel realizations can be written as (35) where the term accounts for URLLC interference and accounts for eMBB interference. We remark that achievability of (35) requires the capability of eMBB users to detect URLLC transmissions, e.g., using a specific preamble in the URLLC mini-slots. Furthermore, we note that, with puncturing, eMBB and URLLC transmissions are orthogonal within each cell, but inter-service interference still arises due to the asynchronous URLLC packet generation across cells.

(35) |

In a manner similar to (23), optimal linear precoding can be carried out by maximizing the sum-rates at the BBU. An interesting aspect of this problem is that while the channel matrix is known at the BBU, the effective channels are not known to the BBU due to the presence of the random URLLC activation matrix . The sum-rate maximization problem can be formulated and tackled in a manner similar to (24).

### Vii-B Superposition Coding

Under this strategy, each transmits a superposition of the signal intended for URLLC users and the signal for eMBB users. We fix the power of the signal intended to the URLLC user to . When designing the beamforming matrices , the BBU assumes an available power of since it is not aware of the URLLC activations for . We hence have the constraint as for puncturing. Accordingly, the signal transmitted by the -th EN over RU can be written as

(36) |

with the scaling factor . The signal (36) is such that, when , only the eMBB signal is transmitted; and, when , the transmitted signal is given by the superposition

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