In fifth generation (5G), it is expected to support real-time and mission-critical applications (e.g., autonomous vehicles, remote surgery, and so on) by ultra-reliable low-latency communications (URLLC)   . In URLLC, a target packet error rate is as low as or less, and low packet delays (e.g., msec) are expected . For (ultra) reliable transmissions in wireless communications, channel coding can be used for (short) coded packets  . Due to time-varying fading, scheduling might be essential to limit latency as studied in  for downlink and in   for uplink transmissions.
Non-orthogonal multiple access (NOMA) has been extensively studied for cellular systems    , because it can provide a higher spectral efficiency than orthogonal multiple access (OMA). In particular, in order to support multiple users with the same radio resource block for downlink with beamforming in cellular systems, power-domain NOMA with successive interference cancellation (SIC) is considered  . It is also possible to employ the notion of NOMA for reliable transmissions in ad-hoc networks as in .
In , it is suggested to consider NOMA for URLLC. For example, in , the impact of NOMA on delay performance is considered based on the notion of effective capacity  . In , for URLLC in downlink, NOMA with multiple-input multiple-output (MIMO) is considered.
In this paper, we study the application of NOMA to URLLC in uplink with certain limitations. Due to the nature of URLLC over time-varying fading channels, we do not assume that the channel state information (CSI) of users is available at a base station (BS) to perform full resource allocation for NOMA including the uplink power allocation. Note that the BS can estimate the instantaneous CSI for the current packet from a user to perform the coherent detection when a packet includes a (short) pilot sequence to allow the channel estimation. However, for the resource allocation, the BS needs to know the CSI of users in advance, which might be difficult unless fading channels are quasi-static. Consequently, throughout the paper, we assume that the BS only allocates the radio resource blocks to users, while users perform the power control with known CSI.
For URLLC, we assume that each user has a finite number of packets to be transmitted within a certain time and one packet can be transmitted within a slot in this paper. Thus, if there is no decoding failure at the BS, a user needs to have slots to deliver packets (usually, may not be too large in mission-critical applications, e.g., remote surgery, with URLLC). However, due to deep fading, with a limited power budget, a user may not be able to successfully transmit some packets. Therefore, in order to complete the delivery of packets, in general, we need more than slots, say slots, where . Thus, for URLLC, it is expected that is sufficiently small with a high probability that all packets can be successfully transmitted within slots.
In this paper, we show that for OMA, cannot be close to (under Rayleigh fading) with a high probability of successful transmissions of packets. However, using opportunistic NOMA mode, which is proposed in this paper to transmit more than one packet per slot using others’ channels based on power-domain NOMA, can be close with a high probability of successful transmissions of packets. For example, under Rayleigh fading, the probability of successful transmissions of packets becomes about with slots if a proposed NOMA scheme is used. On the other hand, the probability of successful transmissions of packets becomes more than 0.5 if OMA is used.
In summary, the main contributions of the paper are two-fold: i) opportunistic NOMA schemes are proposed for low-latency uplink transmissions; ii) a closed-form expression for an upper-bound on the outage probability is derived to see the impact of NOMA on the outage probability under independent Rayleigh fading (as well as a closed-form for the NOMA factor).
I-a Remainder of the Paper
In Section II, we present the system model for low-latency uplink transmissions based on the notion of (power-domain) NOMA. The power allocation is studied in Section III with a limited power budget. The probabilities of multi-packet transmissions by different NOMA schemes are considered and their closed-form expressions are derived under independent Rayleigh fading in Section IV. With a scenario for low-latency uplink transmissions, an outage probability is defined and an upper-bound on the outage probability is derived in Section V. Simulation results are presented in Section VI and the paper is concluded with some remarks in Section VII.
Matrices and vectors are denoted by upper- and lower-case boldface letters, respectively. The superscriptsand denote the transpose and complex conjugate, respectively. The Kronecker delta is denoted by , which is 1 if and 0 otherwise. and
denote the statistical expectation and variance, respectively.represents the distribution of circularly symmetric complex Gaussian (CSCG) random vectors with mean vector and covariance matrix .
Ii System Models
Suppose that there are radio resource blocks or (multiple access) channels that are orthogonal (in the frequency or code domain). We assume that users are assigned to radio resource blocks and each user transmits packets to a BS for uplink transmissions.
Ii-a OMA System
In OMA, we have so that each user can have one (orthogonal) radio resource block for uplink transmissions. Let represent the received signal at the BS from user through the th radio resource block or channel at time slot . In OMA, we have
where denotes the channel coefficient from user through the th radio resource block at time slot , is the packet transmitted by user , and is the background noise in the th channel at the BS. Throughout the paper, we assume block fading channels , where remains unchanged over the duration of a slot and randomly varies from a time slot to another. If a user’s packets are not successfully transmitted (e.g., due to decoding errors at the BS under deep fading), there should be re-transmissions through the same radio resource block using automatic repeat request (ARQ) or hybrid ARQ (HARQ) protocols for reliable transmissions , which results in delay.
If short packets are considered, the overhead of feedback signals for HARQ to each user might be high. To avoid a high feedback overhead, we can consider the power control at users with known channel coefficients. That is, each user can decide the transmit power of packets to meet the required signal-to-noise ratio (SNR) or signal-to-interference-plus-noise ratio (SINR) for successful decoding. To this end, throughout the paper, we assume time-division duplexing (TDD) mode. The BS transmits a beacon signal prior to each slot so that all the users can estimate their channel coefficients to the BS thanks to the channel reciprocity and perform power control. In this case, a user cannot transmit a packet when the channel gain is not sufficiently high (to meet the required SINR with a limited power budget), which leads to delay.
In order to avoid a long delay, the notion of NOMA can be used in an opportunistic manner to transmit more than one packet per slot. There are two different systems to apply opportunistic NOMA mode to uplink, which are discussed below.
Ii-B Symmetric NOMA System
Since there are channels, a user can transmit packets simultaneously if necessary. Thus, for example, if user has additional packets to transmit, the received signals at the BS from user are given by
where represents the index of channel (or radio resource block) that is chosen by user to transmit the th packet at slot , denoted by , to be transmitted at power level111In this paper, we assume power-domain NOMA  , where multiple signals in a radio resource block are characterized by their (different) power levels. for (power-domain) NOMA mode. In Section III, we discuss the power allocation and power levels for opportunistic NOMA mode in detail.
Throughout the paper, for convenience, we assume that and the primary channel of user is channel , . Furthermore, we assume that
where . Here, “” represents the modulo operation. Since there are channels, each user can transmit up to packets simultaneously. However, to avoid the high transmit power in NOMA mode, we assume that the maximum number of packets to be simultaneously transmitted is limited to , which is called the depth. It can be easily shown that as long as , there is at most one packet at each level in every radio resource block or channel. Note that the depth is the number of levels in power-domain NOMA. In addition, we have OMA if , i.e., opportunistic NOMA mode with becomes OMA.
In Fig. 1 (a), we show the structure of the channels with NOMA mode when and , where each pattern is associated to a user’s (NOMA) channels. Suppose that each user can have a different number of packets to transmit. For example, if users 1, 2, and 3 have 1, 3, and 2 packets to transmit, respectively, the channels to be used are as shown in Fig. 1 (b).
Denote by the number of packets to be transmitted from user at time , which depends on the CSI and the power budget at user . Then, is given by
At the BS, the received signal through channel is given by
At time slot , user can transmit packets using opportunistic NOMA mode. On the other hand, in OMA, user can transmit up to one packet per slot. Therefore, if each user has a finite number of packets to transmit within a certain number of slots due to a delay constraint, opportunistic NOMA mode becomes an attractive approach as it can transmit more than one packet per slot.
Ii-C Asymmetric NOMA System
In this subsection, we consider a system that supports users differently depending on their distances from the BS.
Suppose that among users, one user, say user 1, is close to the BS and the other users are far away from the BS. In this case, user 1 can exploit opportunistic NOMA mode to transmit multiple packets through the others’ primary channels. With depth , to exploit the selection diversity gain if , it is possible that user 1 can choose one channel from channel 2 to channel that has the highest channel gain. For example, as shown in Fig. 2, user 1 is able to transmit another packet through either channel 2 or 3 in level 2 using opportunistic NOMA mode. On the other hand, users 2 and 3 can only transmit their packets through their primary channels.
Clearly, there is only one packet transmitted by user 1 in channel 1. On the other hand, there can be two packets in the received signal through channel as follows:
if channel is chosen by user 1 to transmit an additional packet. In the asymmetric system, user 1 (i.e., a near user) can take advantage of a high channel gain for opportunistic NOMA. To see this, we can consider the SINR as follows:
where is the transmit power of user ’s packet in level at time . Since user is a far user, we expect that
Thus, the resulting SINR can be high without having a high transmit power of user 1’s packet in level 2. In other words, user 1 can employ opportunistic NOMA mode without a high transmit power thanks to the difference propagation loss between near and far users. In addition, if , user 1 can choose the channel that has the highest gain among others’ channels, i.e., , which provides a (selection) diversity gain. The resulting case is referred to as the selection diversity based opportunistic NOMA (SDO-NOMA) mode.
Alternatively, it is possible to transmit up to additional packets in level 2 in the asymmetric system. In this case, additional packets from user 1 can be received at slots , , as follows:
for all . The resulting case is referred to as fully opportunistic NOMA (FO-NOMA) mode.
It can be seen that SDO-NOMA exploits the others’ channels to transmit one additional packet using the selection diversity gain, while FO-NOMA can transmit up to additional packets through the others’ channels. Thus, at the BS, one SIC is required in SDO-NOMA, but multiple SICs are required for FO-NOMA. This implies that FO-NOMA can transmit more packets than SDO-NOMA at the cost of a higher receiver complexity.
Iii Power Allocation for Opportunistic NOMA Mode
In this section, we discuss the power allocation when opportunistic NOMA mode is used with a limited power budget.
To allow SIC at the BS, we assume that each level has a target or required SINR and the power allocation is performed to meet the target SINR. Let represent the received signal power at level . Then, provided that SIC is successful to decode the signals in levels , the SINR of the packet in level becomes
For simplicity, we assume that all the packets are encoded at the same rate. Thus, the required SINR for successful decoding, denoted by , becomes the same for all levels, i.e., , . Thus, from (7), can be recursively decided as follows:
Provided , we can see that increases with .
In Fig. 3, the required received power levels (i.e., ) are shown for different power levels in NOMA mode from (8). It is shown that grows exponentially with , which implies that a large is not practical due to a limited power budget.
Iii-a Symmetric System
Consider user 1 at time slot . For convenience, we omit the time index and user index . In this case, represents , i.e., . In the symmetric system, when user 1 transmits a packet through channel , its power level is also according to (1). Thus, the transmit power of the packet of user 1 to be transmitted through channel (and level ), denoted by , is decided to satisfy
Since each user has a limited power budget to transmit packets in each slot, denoted by , the maximum number of the packets per slot time interval in the symmetric system becomes
Iii-B Asymmetric System
In the asymmetric system, only user 1 can employ opportunistic NOMA mode to allow to transmit more than one packet per time slot. Thus, for user , it follows
and for user 1, the power allocation is performed to hold
in SDO-NOMA mode. As a result, user 1 can transmit two packets if
In FO-NOMA mode, user 1 can transmit (at least) packets if
where denotes the th largest order statistic of .
Iii-C Some Issues
In this paper, we assume that each user has perfect CSI so that the power allocation can be performed to hold (9). However, due to the background noise, a user only has an estimate of CSI, which may lead to imprecise power allocation and the resulting SINR can be different from the target SINR, . Due to the different SINR from the target SINR, decoding failure and erroneous SIC become inevitable . Thus, in order to avoid them due to imprecise power allocation, a margin can be given to the target SINR, .
It is also assumed that if the SINR is greater than or equal to , the BS is able to decode packets   . If the length of packet is sufficiently long and capacity-achieving codes are used, can be decided to satisfy
where is the transmission or code rate  . However, as shown in  , for short packets, it is necessary to take into account the channel dispersion, which makes the required SINR higher than . It is noteworthy that even if the target SINR is higher than , there is a non-zero probability of decoding error with short packets. Thus, needs to be decided such that the resulting decoding error is sufficiently low and negligible compared to the target outage probability, which will be explained later.
Iv Probability of Multi-Packet Transmissions When
In this section, we find the probability of multi-packet transmissions using opportunistic NOMA mode. For tractable analysis, we only consider the case of .
Iv-a Symmetric System
In the symmetric system, each user can equally employ opportunistic NOMA mode. Thus, it might be reasonable to assume that the statistical channel conditions of the users are similar to each other (i.e., all the users are either far or near users). For example, might be independent and identically distributed (iid). In this subsection, we find the probability of multi-packet transmissions for iid channels.
According to (10), the probability that each user, say user 1, can transmit at least packets is given by
Denote by the probability that user 1 can transmit packets. For OMA (i.e., ), it follows
while for all .
In opportunistic NOMA mode with , we only need to consider and to find , . Clearly, , , and .
Note that the ’s are independent of the depth , which is also true for , (with ). However, , , depends on as shown above (while for ). Thus, in order to emphasize it, with a finite , the probability of that user 1 can transmit packets is denoted by instead of .
Suppose that is
iid and has
an exponential distribution, i.e.,
is iid and has an exponential distribution, i.e.,. That is, independent Rayleigh fading channels are assumed. Then, we have
where is the modified Bessel function of the second kind which is given by .
It is sufficient to consider user 1 as is iid. It can be shown that
Iv-B Asymmetric System
Unlike the symmetric system, when the asymmetric system is considered, it is expected that the long-term channel gain of user 1 is greater than those of the others. Thus, we assume the following:
where . In general, the long-term channel coefficient of a user is decided by the distance between the BS and the user . Then, letting denote the distance between the BS and user , we have , , where represents the path loss exponent.
Assume that the channel coefficients are given as in (27). Suppose that far users (e.g., users ) have the power budget, . Then, the probability of transmission through primary channel from user , denoted by , is given by
In SDO-NOMA mode, the probability that user 1 can transmit at least packets with power budget , denoted by , is given by
From (13), we have
where and . Under (27
), since the cumulative distribution function (cdf) of the order statistic is given by , it can be shown that
As in (25), we can show that
which is identical to (31). This completes the proof.
Iv-C The Case of
In this section, as mentioned earlier, we mainly focus on the case of depth 2, i.e., . In general, if , it is difficult to obtain closed-form expressions for the ’s (or ’s). However, we can show that the case of provides the worst performance of opportunistic NOMA mode with . In particular, with the average number of transmitted packets per slot, we have the following result.
With in opportunistic NOMA mode, let be the average number of transmitted packets per slot. Then, increases in , i.e.,
With , let . Consider the case of , where have , , and . Note that when , we have and . It can be shown that
From this, it follows that , because is a probability. Similarly, we can also show that , which completes the proof.
Consequently, throughout the paper, for opportunistic NOMA mode, we only consider the case of for analysis, which can be used as performance bounds for the case of .
V Outage Analysis
In this section, it is assumed that each user has a set of packets, which is called a stream, to be delivered to the BS within a finite time. If a user can transmit one packet during every slot, the transmission of a stream can be completed within slots. However, due to deep fading, some packets are to be re-transmitted, which requires additional time slots. As a result, we may need to have slots for the transmission of a stream. For convenience, is called the length of session222It is assumed that one session is required to transmit packets or a stream, which is to be completed within a time duration of slots.. Clearly, it is expected to design a system for low-latency uplink transmissions that can complete the delivery of a stream within a session time (corresponding to the time period of slots) with a high probability. In this section, we discuss the outage probability, which is the probability that a stream cannot be delivered within a session time, in terms of and .
Note that after a session, the BS needs to send a feedback signal if an outage happens. In addition, the BS can send the indices of unsuccessfully decoded packets among packets so that a user can re-transmit them. Since an outage event results in a long delay, it is necessary to keep it low for low-delay transmissions.
V-a An Upper-bound on Outage Probability
Let denote the difference between the slot time, , and the accumulated number of successfully transmitted packets at time . If a user can transmit one packet in every slot, , , has to be 0. However, due to deep fading and opportunistic NOMA mode, can vary. Letting denote the number of successfully transmitted packets at time slot , it can be shown that
where . Thus,
becomes a Markov chain with the following transition probability:
where represents the indicator function of event .
The outage event happens if there are packets that are not yet transmitted after uses of channel. From (42), it can be shown that . Thus, the outage probability is given by
In order to have a low outage probability, it is necessary to hold or
For convenience, define the relative delay as . In OMA, it is expected to have . Thus, cannot be close to , which means a long relative delay. On the other hand, if opportunistic NOMA mode is used, we can have (as more than one packet can be transmitted within a slot). In this case, can be close 1 (or even greater than 1). Thus, a short relative delay can be achieved (with a low outage probability). This clearly demonstrates the advantage of opportunistic NOMA mode over OMA for low-latency uplink transmissions.
In general, the outage probability decreases as increases or decreases. However, a small or a large is not desirable for low-latency uplink transmissions. Therefore, it is necessary to decide a minimum with a certain target outage probability. To this end, we need to have a closed-form expression for the outage probability in terms of key parameters including and . However, since it is not easy to find an exact expression, we resort to an upper bound using the Chernoff bound .
For an upper-bound on the outage probability, we consider the following inequality:
The Chernoff bound is given by
Here, that minimizes is given by
V-B The Case of OMA
In OMA (i.e., with ), the Chernoff bound on the outage probability is given by
Here, it is necessary that for the condition (46) since .
With , since , we have
Taking the differentiation with respect to and setting it to zero (since the upper-bound is convex in ), it can be shown that
Note that if (for the necessary condition in (46)), we can show that .
Note that using the weighted arithmetic mean (AM) and geometric mean (GM) inequality, we can show that
which implies that the upper-bound in (51) cannot be greater than 1.
According to (46), the minimum achievable relative delay, , becomes in OMA, which can be achieved as . However, in this case, the outage probability can be high since as for a finite . Thus, we need for a low outage probability, which implies a long relative delay. In other words, OMA is not suitable for low-latency uplink transmissions.
V-C The Case of Opportunistic NOMA
Using the Chernoff bound in (49), we can find an upper-bound on the outage probability when opportunistic NOMA mode is employed as follows.
In opportunistic NOMA mode with , if , the Chernoff bound is given by
Since the proof is similar to that of Lemma 4, we omit it.
Although we can obtain the outage probabilities of OMA and opportunistic NOMA mode from (51) and (54) using upper-bounds, respectively, it is difficult to directly see the gain by using opportunistic NOMA mode for low-latency uplink transmissions. Thus, based on the upper-bound in (48), we consider the NOMA factor that is given by
It can be seen that for a given length of session , becomes the minimum possible ratio of the outage probability of opportunistic NOMA mode to that of OMA. Note that it is desirable that the NOMA factor, , is being independent of the values of and so that can demonstrate the pure gain of opportunistic NOMA.
The NOMA factor is given by
where . Clearly, (59) is a fractional program , where the numerator is convex and the denominator is concave in . In particular, it is a convex-concave fractional program, which can be reduced to a convex program . Thus, the (unique) solution can be found by taking the differentiation with respect to and setting it to zero. Then, the optimal , which is denoted by , needs to satisfy the following equation:
After some manipulations, we have
From (57), we can see that the NOMA factor, , decreases with and increases with . In addition, as long as , becomes less than 1. From this, with , it is expected that the outage probability will be dramatically lowered by opportunistic NOMA mode compared to OMA for a reasonably long session length, . For example, assuming that the upper-bound on the outage probability of OMA is 1, if (for ), the outage probability of opportunistic NOMA mode becomes (note that it might be a lower-bound as is the minimum possible ratio of outage probabilities). In particular, if , the outage probability of opportunistic NOMA mode can be as low as .
Vi Simulation Results
In this section, we present simulation results for user 1’s performance under the assumption that the channels of radio resource blocks experience independent Rayleigh fading, i.e., or , . For convenience, we also assume that .
Vi-a Results of Symmetric NOMA
In symmetric NOMA, the depth, , becomes the maximum number of packets that a user can transmit in a slot (under the assumption that ). Thus, as increases, it is expected that the average number of transmitted packets increases according to Lemma 3. Fig. 4 shows the average number of transmitted packets, , in a session time (i.e., slots) for different values of depth, , when , , (packets), and (slots). It is shown that although increases, becomes saturated due to a high value of , (e.g., see Fig. 3). Thus, in most cases, (i.e., two power levels) becomes a reasonable choice unless the required SINR, , is sufficient low or the power budget, , is extremely high, which is however impractical.
Fig. 5 shows the outage probabilities and their ratio/NOMA factor as functions of power budget, , when , , and . It is shown that the improvement of the outage probability of OMA is slow as increases. However, the outage probability significantly decreases with if opportunistic NOMA mode is employed, which clearly demonstrates that opportunistic NOMA mode is an attractive scheme for low-latency uplink transmissions over fading channels. In Fig. 5 (a), it is shown that the bound in (54) can successfully predict the decreases of the outage probability when opportunistic NOMA mode is used. In addition, we also see that the performance with is almost the same as that with , which means that the depth is sufficient to take advantage of opportunistic NOMA mode. In Fig. 5 (b), the NOMA factor, , in (57) is shown with the outage probability ratio from simulation results (which is represented by the dashed line). It is confirmed that (57) can reasonably well predict the behavior of the outage probability ratio.
Fig. 6 shows the outage probabilities and their ratio/NOMA factor as functions of session length, , when , , and . It is clearly shown that the increase of decreases the outage probability at the cost of increasing delay.
Fig. 7 shows the outage probabilities and their ratio/NOMA factor as functions of target SINR, , when , , and . As the target SINR decreases, the outage probabilities decrease. However, since the code or transmission rate decreases with the target SINR, the target SINR cannot be low. With , we can see that the outage probability of OMA is almost 1, while that with opportunistic NOMA mode becomes sufficiently low (i.e., less than ). This again shows that opportunistic NOMA mode can play a key role in low-latency uplink transmissions as it can make the outage probability sufficiently low with a reasonably delay constraint.
Vi-B Results of Asymmetric NOMA
In this subsection, we present simulation results of asymmetric NOMA with SDO-NOMA and FO-NOMA.
Fig. 8 shows the outage probabilities of SDO-NOMA and FO-NOMA as functions of power budget, , when , , , and . From simulation results (with the two dashed lines), we can see that there is no significant performance difference between SDO-NOMA and FO-NOMA, which means that with a reasonable power budget, it is unlikely to transmit more than two packets using FO-NOMA mode. Thus, SDO-NOMA becomes preferable to FO-NOMA as it only uses one additional resource block with and one SIC at the BS.
The outage probability versus is illustrated in Fig. 9 when , , , and . It is noteworthy that even if , SDO-NOMA and FO-NOMA can provide a low outage probability, which is about , while the outage probability of OMA is 1. With , the outage probability of SDO-NOMA or FO-NOMA can approach .
Fig. 10 shows the outage probabilities of SDO-NOMA and FO-NOMA as functions of the number of radio resource blocks, , when , , , and . We can see that the outage probability decreases with in SDO-NOMA due to the increase of the selection diversity gain and in FO-NOMA due to the increase of radio resource blocks to transmit more packets per slot. Note that although FO-NOMA can transmit more packets than SDO-NOMA as increases (FO-NOMA can transmit up to packets per slot, while SDO-NOMA can transmit up to two packets per slot), there is no significant performance difference between SDO-NOMA and FO-NOMA in terms of the outage probability. This implies that the impact of the selection diversity gain in SDO-NOMA on the outage probability is similar to that of up to transmissions per slot.
Vii Concluding Remarks
In this paper, we studied opportunistic NOMA mode for low-latency uplink transmissions. It was assumed that each user has a set of packets that are to be transmitted within slots for low-latency uplink transmissions. With OMA, it was shown that has to be larger than for a low outage probability, which results in a long delay. On the other hand, it was shown that opportunistic NOMA mode can dramatically lower the outage probability compared with OMA. In particular, under independent Rayleigh fading, with , it was shown that the outage probability of opportunistic NOMA can approach , while that of OMA is 0.5. It was also confirmed by the derived NOMA factor that shows the outage probability of opportunistic NOMA can be significantly lower than that of OMA although is not significantly larger than .
There are issues to be investigated in the future. Although an upper-bound on the outage probability was found as a closed-form expression to see the behavior of the outage probability, it was noted that there is a gap between the upper-bound and simulation results. Thus, it is desirable to find a tighter bound in the future. In addition, it is necessary to study the impact of imperfect CSI on SIC, which results in degraded performance (in the paper, we assumed no errors in SIC thanks to perfect CSI).
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