Short-Packet Communication Over a Two-User Rayleigh Fading Z-Interference Channel: From Stability Region to the Age of Information

One of the fundamental challenges in 5G and beyond technologies is to support short packet transmissions while ensuring ultra-reliable communication. Due to the distributed nature of the networks, such as machine-to-machine (M2M) communications, interference is unavoidable. The impact of interference on the system’s performance must be better understood when users are constrained to transmit short packets. In addition, users’ traffic is bursty. Thus, they may not always have data to send. This work considers a two-user Z-interference channel (Z-IC) under Rayleigh fading. The work characterizes the stability region corresponding to prominent interference mitigation schemes such as treating interference as noise, successive interference cancellation, and joint decoding schemes using the finite block-length information theory framework. The developed results consider the packet length, rate, and underlying channel model. Evaluating stability region involves determining the probability of successful decoding for the various interference mitigation techniques. The different probabilities of successful decoding are characterized for various interference mitigation techniques. These results are not explored in the existing literature in the context of Z-IC. The developed results also help to explore the impact of interference on average delay and the average age of information for various interference mitigation techniques.


I. INTRODUCTION
S HORT packet communication is considered to be a key enabler in supporting two essential application scenarios of 5G, namely: (a) massive machine type communication (mMTC) and (b) ultra-reliable low-latency communication (uRLLC).Many existing performance metrics, such as capacity and outage probability, cannot be used for short packet communication, as these metrics have the underlying assumption of long packet lengths.Ensuring reliable communication and latency requirements for interference-limited scenarios is challenging when devices are constrained to use short packets for communication.In addition to the short size of the packet, data arrival at the users is random in Internet of Things (IoT) or Machine-to-Machine (M2M) communication, which is in contrast to the infinitely backlogged users assumption considered in classical information theory.Thus, capturing the bursty nature of the sources is essential, which can help explore the impact of interference on latency.To address these problems, this work explores the impact of packet length and random data arrival at the transmitter on the system's performance in an interference-limited environment.
There is a need to develop a theory for short-packet communication that considers reliability and latency.Information theory has provided models to capture the uncertainty associated with the underlying channel model, such as noise, fading, and interference.Generally, these works do not capture random data arrival at the users.On the other hand, network theory provides mathematical tools or models to capture the random arrival of data and latency aspects of communication.However, it does not capture the characteristics of the underlying physical channel.Unifying these two theories can provide a more accurate model where it is required to consider the aspects mentioned above jointly [2].

A. RELATED WORKS
To capture the effect of the random arrival of data on the performance, the notion of stable throughput has been used in the existing literature [3], [4], [5], [6].For multi-user scenarios, stability region becomes a relevant metric.The stability region is analogous to the capacity region in information theory, and it has been explored for many important communication models such as the broadcast channel and interference channel [6], [7].However, the characterization of stability region in multi-user scenarios is a challenging problem due to interaction among the queues [3], [4].The technique of stochastic dominance has been used in the existing literature to overcome the difficulty in analyzing interacting queues [3], [6].The characterization of stability region in systems involving more than two users remains a challenging task.
Latency is another critical aspect for many applications in 5G and beyond communication systems, where the data rate is not the most appropriate key performance index (KPI).Delay and age of Information (AoI) are considered to be relevant metrics to capture latency and timing aspects of information [8].Delay takes account of transmission and queuing delays in communication.AoI measures the freshness of information [8], [9], [10] and is a relevant metric in status updating systems apparent in IoT and industrial automation scenarios [11].
Characterizing the stability region involves determining the probability of successful decoding at the receiver.In the existing literature, most of the works on stability region either assume that the probability of successful decoding is known or determined using signal to noise ratio (SNR)/ signal to interference plus noise ratio (SINR) based metrics.However, these metrics do not capture the packet length, an essential parameter for reliability and latency requirements in short packet communication.The framework of finite block-length information theory allows to capture the impact of packet length, rate, and power budget at the transmitter and underlying channel model on the error performance [12], [13], [14], [15], [16], [17].The result in [13] provided the mathematical framework to explore the trade-off between rate, error, and block length for a point-to-point memoryless channel.Progress has also been made in developing the results for multi-user scenarios in the finite block length information theory framework.The second-order achievable rate region for multiple access channels is obtained in [14].The achievable results for other important models, such as the broadcast and interference channels, can be found in [16], [17].More details on point-to-point channel and multi-user scenarios under finite block-length regime can be found in [18].
The results from finite-block length information theory have also been used to analyze the performance in case of fading scenario for various communication models [19], [20], [21], [22], [23].The work in [19] characterizes the average throughput and explores the trade-off between energy efficiency and spectral efficiency for the point-to-point Rayleigh fading channel when no channel state information is available at the transmitter.In general, developing results for multi-user scenarios in the context of short packet communication is a non-trivial problem.In recent years, there has been an increased interest in exploring the role of NOMA in improving the performance metrics such as spectral efficiency in short packet communication [20], [21], [22], [23].The works in [20], [24] show the benefits of NOMA for downlink scenarios in contrast to the orthogonal counterpart, provided both the schemes use the same block length.The work in [25] proposes opportunistic NOMA for uplink transmission where more than one packet can be sent per slot.The work in [26] explored the throughput of NOMA users in the presence of an external eavesdropper when the users are constrained to use short packets.In [27], stable secure throughput of a Rayleigh faded wiretap channel with a friendly jammer has been characterized for a finite block length coding regime.
When multiple users share a common channel, interference is unavoidable due to the distributed nature of devices.In the existing literature, various interference mitigation techniques, such as treating interference as noise (TIN) and successive interference cancellation (SIC), have been studied extensively for various models under the asymptotic regime [28].Characterization of second-order coding rates of the Gaussian interference channel (GIC) in the strong interference regime has been considered in [17], and a case where the interference does not affect the channel dispersion has been studied.However, developing a second-order coding rate for the two-user Gaussian IC for general channel conditions still needs to be explored.

B. CONTRIBUTIONS
In short packet communication, various interference mitigation approaches need to take account of the reliability and latency aspects of communication.This paper aims to address the following two related problems: 1) Performance of various interference mitigation techniques, such as TIN, SIC, and joint decoding, are well explored in the existing literature when packets are of large length (for the asymptotic regime).However, the impact of interference on reliability is not well understood when users are constrained to communicate short packets.
2) Performance of various interference mitigation techniques by taking account of random arrival of data at the users, packet length, and underlying physical model is not explored in the existing literature.To address the above problems, this paper considers a twouser Z-interference channel (Z-IC) under Rayleigh fading, where transmitters can buffer the incoming packets.The metric named stability region is used to capture the bursty nature of the sources.Determination of the stability region requires the characterization of the probability of successful decoding at the receiver.The probabilities of successful decoding for various interference mitigation schemes are characterized for the considered model using the finite block length information theory framework, which allows capturing the impact of packet size and rate on the reliability.The developed results are used to explore the impact of interference on the delay and average age of Information (AAoI) for various interference mitigation techniques.Some works considering delay and stability region under finite block length regime can be found in [29], [30].The work in [29] explores the benefits of NOMA in reducing queuing delay.It has been shown that SIC may not be an attractive choice for low-latency communication in case of uplink communication under channel uncertainty.The work in [30] examines the stability condition of non-cooperative timedivision multiple access (TDMA) and multiple access relay channel with TDMA scheduling and bursty traffic under finite block length constraint.The stability region has been considered for the two-user interference channel in [7].However, it does not consider the size of the packet in determining the probability of successful decoding.When the arrival of data is random at the users, minimizing only the length of the codeword or packet size in short packet communication may not reduce the delay in communication, as it does not consider the queuing delay.To explore the performance of various interference mitigation schemes in the context of short packet communication, it is required to consider packet size, underlying channel condition, and random arrival rate at the user.The main contributions of the work are summarized below: 1) To capture the bursty arrival of data at the source, the notion of stability region is used, and it is characterized for the considered system model (See Section III).
The evaluation of the stability region requires the characterization of the probability of successful decoding at the receiver.To the best of the authors' knowledge, the stability region for the two-user Z-IC has not been explored in the existing literature.
2) The probability of successful decoding at both the receivers corresponding to different queue states is characterized for the two-user Rayleigh fading Z-IC, for various interference mitigation techniques (See Section IV).The results are derived using the finite block-length information theory framework, which captures the packet's length.For the SIC scheme, the derivation of probability of successful decoding takes account of the product of the error terms involved in both the stages of decoding, which is ignored in the existing works to the best of authors' knowledge [23], [29], [31], [32], [33].3) In addition to the stability region, it is essential to understand how interference affects latency in the case of different interference mitigation schemes.To address this problem, average delay and AAoI are characterized for the considered system model.The developed results help to explore the interplay between average delay and AAoI for various interference mitigation techniques, which needs to be explored in the literature.The complete list of notations and mathematical functions used in this work is given in Table 1.

II. SYSTEM MODEL
This work considers a two-user Rayleigh-faded Z-IC with random data arrival at the transmitters, as shown in Fig. 1.Each transmitter has a queue to store the incoming packets.Rx-i is intended to receive packets from Tx-i (i ∈ {1, 2}).In Z-IC, only one of the receivers (Rx-1 in this case) experiences interference.Z-IC can be considered as a special case of interference channel, and real-world scenarios can be modeled using Z-IC [34], [35].Examples of such cases are when one of the users is far from the interfering transmitter or one of the receivers is blocked by an obstacle.In such cases, one of the receivers does not experience interference, whereas the other receiver experiences interference.Time is assumed to be slotted.The packet arrival processes at Tx-1 and Tx-2 are assumed to be stationary and independent with arrival probabilities λ 1 and λ 2 (packets/slot), respectively.Both the transmitters have an infinite capacity to store the incoming packets, and Q i denotes the length of the queue corresponding to the i th transmitter, i ∈ {1, 2}.A transmitter is assumed to send a packet when its queue is non-empty.Otherwise, it remains silent.Packets that arrive at the queue are served on a first come-first serve (FCFS) basis, i.e., FCFS queuing policy is adopted.Even though the last come-first serve (LCFS) queuing policy ensures a relatively lower AoI, it can increase the average delay.This paper aims to give a unified view of the impact of interference on the stability region, average delay, and AoI when users are constrained to use short packets.In applications where the packet arrival order is of prime concern, the FCFS queuing policy is more relevant.Hence, this work adopts an FCFS queuing policy.Further, it is assumed that the acknowledgments (ACKs) are instantaneous and error-free.If a receiver fails to decode its intended packet, it will remain in its queue at the transmitter and be re-transmitted in the next time slot.Re-transmissions at the transmitter are necessary when it is required to receive all the information at the receiver.
Tx-i encodes k i information nats into a codeword of length N (in channel uses), and the rate is defined as R i k i N .The power budget at Tx-i is P i in Watts, (i ∈ {1, 2}).The channel between the nodes (h ij ) are assumed to undergo Rayleigh fading and are independent of each other.Let 1 {A} denotes indicator variable which takes a value 1 if A is true, otherwise it takes the value 0.Then, the input-output relation for the model is given below: where noise Z i is modelled as a complex additive white Gaussian noise (AWGN), i.e., Z i ∼ CN (0, 1).The system's performance is analyzed in block-fading conditions [36], [37], [38], i.e., where the channel coefficients remain constant during the transmission of the codeword, and then change according to the underlying distribution of the channel.The receivers are assumed to have perfect channel state information (CSI) [37], [38].However, the transmitter does not know instantaneous channel gain but knows the channel statistics.The probability density function of the square of the magnitude of channel gain as given below: where φ ij represents the average value of channel gain between Tx-i and Rx-j.

III. CHARACTERIZATION OF STABILITY REGION
In many emerging scenarios such as M2M communication or IoT, stability region is a relevant metric as it accounts for random data arrival at the users.Stability region is defined as the set of all arrival rates such that all the queues in the system are stable [7].In the following theorem, the stability region is characterized for the two-user Z-IC without assuming any specific encoding or decoding schemes at the transmitter or receiver, respectively.In this case, Tx-i sends a packet whenever its queue has a packet, i.e., when In the following theorem, the term D τ i denotes the event that Rx-i is able to successfully decode the packet sent from Tx-i (i ∈ {1, 2}) given that a set of transmitters denoted by τ is sending packet.For example, D {1,2} 1 denotes the event of successfully decoding the packet sent by Tx-1 at Rx-1 provided that both the transmitters (τ = {1, 2}) are sending packets.Similarly, D {1} 1 denotes the event of successfully decoding the packet sent by Tx-1 at Rx-1, when only Tx-1 is sending the packets.The term Pr(D τ i ) denotes the probability of event D τ i .The same notation is used in the rest of the paper.
Theorem 1: The stability region of the two-user Z-IC with bursty arrival of data at the transmitters is given by Proof: The service probability μ i associated with Tx-i (i ∈ {1, 2}) is given by the following expression: From ( 6), one can observe that the service rate corresponding to Rx-2 does not depend on the status of the other queue as there is no interference from Tx-1.Hence, there is no coupling between the queues at the transmitters in the case of Z-IC.From Loyne's criterion [6], it is known that queue at the Tx-2 is stable if and only if λ 2 < μ 2 .Thus, the stability condition for Tx-2 is given by: where μ 2 is as given in (6).Note that λ 2 denotes the number of packet arrivals per unit time slot at Tx-2, whereas μ 2 denotes the number of packets that are being serviced at Rx-2 per unit time slot.To determine the service rate corresponding to Rx-1, it is required to determine the probability that queue at Tx-2 is empty (Pr(Q 2 = 0)) or non-empty (Pr(Q 2 > 0)).From Little's theorem, the probability that queue at Tx-2 is non-empty is given by: Substituting ( 8) into (5), service rate for Rx-1 is given by: The queue at Tx-1 is stable if and only if λ 1 < μ 1 , and hence, the stability condition is given by The stability region R stated in the theorem is obtained using (10) and (7).Remark: The result mentioned above regarding the stability region holds in general for any interference management technique, and the general form of the stability region is depicted in Fig. 2. Hence, it is necessary to evaluate the probability of successful decoding (Pr(D {τ } i )) corresponding to different interference mitigation schemes to determine the stability region.The probability of successful decoding corresponding to different interference mitigation schemes is evaluated in Section IV.

A. STABILITY REGION ANALYSIS WITH RANDOM ACCESS
When users operate in interference-limited scenarios, random access may improve the stability region further.In this case, whenever a transmitter has a packet to send, it sends that packet with a certain probability.Since only one of the transmitters causes interference in the Z-IC, it is assumed that Tx-2 sends a packet using random access protocol, and the other user sends a packet whenever its queue is not empty.The stability region with random access is stated in the following corollary.
Corollary 1: The stability region of the two-user Z-IC with bursty arrival of data at the transmitters and with random access at Tx-2 is given by the following expression: where Tx-2 sends a packet with probability q 2 (q 2 = 0), when its queue is non-empty.
Proof: The service probability of user-1 (μ 1 ) is given by the following expression: Pr(Tx-2 is transmitting) Pr(Tx-2 is not transmitting).(12) To evaluate μ 1 , it is required to determine Pr(Tx-2 is transmitting), which is given by: Hence, the service probability of Tx-1 is simplified to the following: It can be seen that service probability of Tx-2 is given by 2 ) and from Little's theorem, probability that queue at Tx-2 is non-empty is given by: Substituting ( 15) in ( 14), the service probability for user 1 becomes: For the queue at Tx-1 to be stable, following condition needs to be satisfied: λ 1 < μ 1 .Hence, following holds: Substituting ( 16) in ( 17), we obtain: Similarly, for the Queue at Tx-2 to be stable, following condition needs to be satisfied: Thus, from ( 18) and ( 19), stability region of Z-IC with random access at Tx-2 is obtained as stated in the corollary. Remarks: 1) The second inequality in (11) depends on random access probability q 2 , whereas the first inequality does not depend on q 2 .One can observe that the stability region in ( 11) is a sub-region of the stability region without random access at Tx-2, i.e., q 2 = 1.Hence, random access at Tx-2 does not improve the stability region of the Z-IC.2) If one seeks to maximize the first user's performance under constraints on the stable throughput of the second user, random access at the second transmitter can still provide benefits, as it regulates the transmission.However, this is out of this work's scope and left as a future extension.3) When q 2 = 0, it can be observed that μ 2 = q 2 Pr(D {2} 2 ) = 0.As Tx-2 does not send any packet irrespective of any new arrival of packets, Pr(Q 2 > 0) = 1.Hence, the second queue in the considered system model is unstable for any non-zero arrival rate λ 2 .Both the queues in the system need to be stable for the stability region.

IV. PROBABILITY OF SUCCESSFUL DECODING FOR VARIOUS INTERFERENCE MITIGATION TECHNIQUES
In this section, the probability of successful decoding based on the queue states at the transmitters is characterized for the following interference mitigation schemes: (a) TIN, (b) SIC, and (c) Joint decoding (JD).This, in turn, enables characterizing the stability region of different interference mitigation techniques.In many future applications, such as M2M communication, packet length is short, and it is necessary to consider the packet length in determining the probability of successful decoding.The problem is non-trivial for the following reasons: 1) The probability of successful decoding depends on factors such as rate, packet length, underlying channel model, and interference mitigation techniques at Rx-1.
To capture these, this work uses the finite blocklength information theory framework to determine the probability of successful decoding for various interference mitigation schemes.
2) The probability of successful decoding requires the determination of error expression for various decoding schemes, and these results are not explored in the existing literature for the two-user Z-IC under the framework of finite block-length information theory.This work obtains instantaneous error expression for the corresponding decoding scheme to determine the expression for the average block error rate (BLER).Moreover, the distribution of the signal at the transmitter (or the codeword corresponding to the message) is chosen carefully for mathematical tractability.To the best of the authors' knowledge, the average error for various interference management schemes has yet to be characterized in the literature for the considered model under finite blocklength coding regime.

A. TREATING INTERFERENCE AS NOISE
TIN is one of the conventional schemes to mitigate interference.In addition to its low complexity in decoding, TIN is robust to channel uncertainty.Hence, it is essential to understand the performance of the TIN scheme in different interference regimes under finite block length coding.To characterize the probability of successful decoding at the Rx-1, it is necessary to determine the instantaneous BLER when Rx-1 treats interference as noise.In this case, the channel between Tx-1 and Rx-1 can be modeled as a point-to-point channel with a modified noise variance.The achievable result in [13] uses a non-Gaussian signaling scheme.When both the transmitters use such a non-Gaussian signaling scheme, the effective noise (h 21 X 2 + Z 1 ) at Rx-1 is no longer Gaussian, and the model reduces to a non-Gaussian Z-IC, which is hard to analyze.Similar observations have been made regarding multiple access channels in [29].To overcome this problem, signaling schemes at the transmitters are carefully chosen, and more details in this regard can be found in the proof of the following theorem.
Theorem 2: In a two-user Rayleigh faded Z-IC channel, when Rx-1 treats interference as noise, the probability of successful decoding at the receivers, depending on the status of the queues, are given as follows: 1) When Q 1 = 0 and Q 2 = 0: and 2) When Q 1 = 0 and Q 2 = 0: 3) When Q 1 = 0 and Q 2 = 0: where Proof: The probability of successful decoding based on the status of the queues is evaluated as follows.
a) When Q 1 = 0 and Q 2 = 0: In this case, as both the queues have packets to send, Rx-1 is subjected to interference.When Rx-1 treats interference as noise, the noise floor at the receiver increases, and the effective noise is given by h 21 X 2 + Z 1 .The achievable result for the pointto-point memoryless channel in [13] uses non-Gaussian signaling.When such signaling is used at both the transmitters, the model reduces to a non-Gaussian Z-IC, as the effective noise (h 21 X 2 + Z 1 ) is no longer Gaussian.Analysing non-Gaussian Z-IC is a complex problem.Similar observations have been made in the case of MAC channel [29].To overcome this problem, Tx-2 uses Gaussian signaling so that the effective noise at Rx-1 (h 21 X 2 + Z 1 ) is Gaussian.As the effective noise follows Gaussian distribution, it is possible to use the achievable result in [13] for Rx-1.As non-Gaussian signaling results in smaller dispersion than Gaussian signaling in the finite block-length regime [39], Tx-1 uses non-Gaussian signaling.The probabilities of successful decoding at the receivers are obtained as follows.
Evaluation of Pr D {1,2} 1 : Let TIN and TIN denote the instantaneous BLER and average BLER, respectively in case of TIN scheme at Rx-1.Then where TIN is given by [13]: , and (31) The term V NG (γ 1,TIN ) defined in (32) denotes the channel dispersion term corresponding to the case where Tx-1 transmits non-Gaussian codewords (γ 1,TIN is the SINR at Rx-1).The subscript NG in V NG (γ 1,TIN ) is used to emphasize that Tx-1 uses non-Gaussian signaling.V(SNR) denotes the channel dispersion, which captures the variability of the channel relative to a deterministic bit pipe with the same capacity.Channel dispersion is a function of SNR and varies according to the signaling scheme employed at the transmitter.Note that in (29) higher order terms O( log n n ) has been ignored.For more details, one can refer to [13], [19], [30].In the subsequent analysis, such higher order terms are not considered.
To evaluate the average BLER, it is required to calculate the average of the instantaneous BLER with respect to the distribution of the SINR ( 1,TIN ).As Q-function is involved in (29), it is difficult to find a closed form expression for TIN .By using the linear approximation of Q-function, TIN can be approximated with (24) as given in the statement of Theorem 2. The details of the linearization approximation of the Q-function and the derivation of average BLER ( TIN ) can be found in Appendix A. In the following, the probability of successful decoding at Rx-2 is derived.

Evaluation of Pr(D {2}
2 ): Let 22 G denotes the instantaneous BLER at Rx-2 and is given by the following equation [39]: where The term V G (γ 22 ) defined in (34) denotes the channel dispersion term corresponding to the case where Tx-2 sends Gaussian codewords.The subscript G in V G (γ 22 ) is used to emphasize that Tx-2 uses Gaussian signaling.One can notice that 22 G does not depend on status of the queue at Tx-1 as there is no interference at Rx-2.Hence, Note that the dispersion term V G (x) in ( 34) is different from the dispersion term V NG (x) in ( 32), as Tx-2 uses Gaussian signaling.Hence, the following is used as linear approximation for the Q-function in (33) instead of (104): where and To determine the average BLER at Rx-2, it is required to determine the cumulative distribution function associated with the SNR ( 22 ) at Rx-2, and it is given by It can be seen that F 22 (x) can be obtained from F 1,TIN (x) by substituting P 2 = 0 and by replacing P 1 and φ 11 with P 2 and φ 22 , respectively in (105).Hence, the average BLER at Rx-2 ( 22 G ) can be obtained from (107) by substituting P 2 = 0 and by replacing α 1 , β 1 NG , φ 11 , and P 1 with α 2 , β 2 G , φ 22 , and P 2 , respectively and after some simplification, the expression in ( 25) is obtained.b) When Q 1 = 0 and Q 2 = 0: As queue 2 is empty, Rx-1 does not experience any interference due to Tx-2.It is required to determine the probability of successful decoding for Rx-1 only.It is obtained as follows: where 11 NG denotes the instantaneous BLER at Rx-1 when only Tx-1 is active and is given by the following equation [13]: where 2 ) is the same as in the first case, i.e., (36) (where both the transmitters have packets to send).

B. SUCCESSIVE INTERFERENCE CANCELLATION SCHEME
In successive interference cancellation, decoding occurs in two stages.In the first stage, Rx-1 decodes the codeword of Tx-2 and treats its intended message as noise.In the second stage, Rx-1 decodes its message after subtracting the interference caused due to Tx-2.As the decoding happens in two stages, the average error must consider the error in both stages.In this work, the expression for the probability of successful decoding for SIC scheme is obtained by taking account of coupling between the errors in both stages, which has been ignored in earlier works [23], [29], [31], [32], [33].This provides accurate error expression for SIC, as discussed in the later part of the paper.
Theorem 3: In a two-user Rayleigh faded Z-IC channel, when Rx-1 employs the SIC scheme, the probability of successful decoding at the receivers, depending on the status of queues, are given as follows: 1) When Q 1 = 0 and Q 2 = 0: and 2) When Q 1 = 0 and Q 2 = 0: 3) When Q 1 = 0 and Q 2 = 0: where different error probabilities are defined at the bottom of next page.
Proof: The probabilities of successful decoding based on the status of the queues are evaluated as follows: a) When Q 1 = 0 and Q 2 = 0: When both the queues are active, Tx-2 creates interference at Rx-1.In this case, both the transmitters send the codewords drawn from Gaussian distribution, and this also ensures that Tx-2 does not cause non-Gaussian interference at Rx-1.The probabilities of successful decoding at both receivers are obtained as follows.
Evaluation of Pr(D ): In this case, probability of successful decoding at Rx-1 is given by: where SIC denotes instantaneous BLER associated with the SIC scheme at Rx-1.
Let E 1 represents the error event in decoding the codeword of Tx-2 at Rx-1, while the codeword of Tx-1 is treated as noise, and E 2 represents error event in decoding codeword of Tx-1 at Rx-1, then the instantaneous BLER SIC is obtained as follows: Here, 21 denotes the instantaneous BLER in the first stage of decoding, i.e., while decoding codeword of Tx-2 at Rx-1, and 11 G denotes the instantaneous BLER in decoding the codeword of Tx-1 at Rx-1 (second stage of decoding).When there is an error in the first stage of decoding in SIC, there will be an error in the second stage with high probability, and hence, Pr(E 2 |E 1 ) is set to 1 in the above equation.
The term 21 11 G was ignored in [29] while evaluating the average error, and this term is taken into account for the calculation of the average BLER in this work.
In the first stage of SIC, the codeword of Tx-2 is decoded by treating the codeword from Tx-1 as noise.In this case, both the transmitters use Gaussian signaling.In the first stage of SIC decoding, the effective noise is given by h 11 X 1 + Z 1 .For a given channel realization, h ij are modeled as constant, and hence, h 11 X 1 + Z 1 is modeled as Gaussian noise.Hence, the channel between Tx-2 and Rx-1 can be modeled as an AWGN channel with an equivalent SINR γ 21 which is defined in (56), for a given realization of h 11 and h 21 .Hence, the instantaneous BLER in decoding the codeword of Tx-2 in the first stage can be obtained by replacing γ 22 in (33) by γ 21 and the corresponding instantaneous BLER is given by the following expression [39]: where U G (γ 21 ) is as defined in (37), and  (46), shown at the bottom of the page can be obtained in the same way as TIN that was found in TIN scheme by replacing α 1 and β 1 NG with α 2 and β 2 G , respectively and the terms P 1 (and φ 11 ) need to be interchanged with P 2 (and φ 21 ) in (24).
Instantaneous BLER in the second stage of decoding, i.e., 11 G is given by where U G (γ 11 ) is as defined in (37), and Noticing the similarity between γ 11 in (58) and γ 22 in (34), the average BLER 11 G defined in (47), shown at the bottom of the previous page is evaluated in the same way as that of average BLER With a slight abuse of notation, the probability density function f X (x) is presented as f (x) in the rest of the derivation of the theorem.Similarly, f X|Y (x|y) is represented as f (x|y).Using (63), the term 21 11 G in (59) is approximated with the expression given in (48), shown at the bottom of the previous page.The details of the derivation can be found in Appendix B. In the following, the probability of successful decoding at Rx-2 is derived.

Evaluation of Pr(D {2}
2 ): Probability of successful decoding at Rx-2 is given by where 22 G is the instantaneous BLER in decoding the codeword of Tx-2 at Rx-2 as defined in (34).Hence, it is evident that 22 G defined in (49), shown at the bottom of the previous page is same as that defined in (25).b) When Q 1 = 0 and Q 2 = 0: In this case, Tx-2 does not cause any interference to Rx-1.The probability of successful decoding at Rx-1 in this scenario is given by where 11 G is the instantaneous BLER in decoding the codeword of Tx-1 at Rx-1 as in (57).Note that 11 G corresponds to (47).c) When Q 1 = 0 and Q 2 = 0: In this case, queue 1 is empty and it is required to consider decoding only at Rx-2.The probability of successful decoding at Rx-2 is given by where 22 G is same as (25), which is characterized when both the queues are active.

C. JOINT DECODING SCHEME
Joint decoding scheme requires decoding all the messages simultaneously.In the joint decoding scheme, Rx-1 needs to decode messages of Tx-1 and Tx-2 simultaneously.The joint decoding scheme needs to consider all codeword pairs for optimal decoding.To reduce joint-decoding complexity, one can consider iterative decoding [40], [41].An example of joint decoding can be found in [40].The channel between Tx-1, Tx-2, and Rx-1 is modeled as a multiple access channel (MAC), and the result derived in [14] is used to characterize the average error for the joint decoding scheme.To the best of the authors' knowledge, the performance of the joint-decoding scheme under finite block length coding for the Z-interference channel is yet to be explored in the existing literature.The probabilities of successful decoding at both the receivers, corresponding to the joint decoding scheme are stated in the following theorem.Theorem 4: In the two-user Rayleigh faded Z-IC, when Rx-1 uses a joint decoding scheme, the different probabilities of successful decoding at the receivers depending on the status of queues are given as follows: 1) When Q 1 = 0 and Q 2 = 0: and 2) When Q 1 = 0 and Q 2 = 0: 3) When Q 1 = 0 and Q 2 = 0: where and Proof: The different probabilities of successful decoding are evaluated in the following cases: a) When Q 1 = 0 and Q 2 = 0: When both the transmitters have a packet to send, one can see that Tx-1, Tx-2, and Rx-1 form a MAC channel.Rx-1 can use the joint decoding scheme to decode its message.Using the result for the MAC channel [14], [42], the rates R 1 and R 2 are achievable with instantaneous BLER J = I + II + III and the following holds: where and As it is difficult to evaluate the error probability I , II , and III exactly, each error term is set equal to the RHS in (78).The probability of successful decoding is given by where Since each error term is set equal to RHS in (78), J in (85) can be greater than the actual error probability.Thus, the probability of successful decoding obtained using J is a lower bound.A minimum of average error probability and 1 is considered for the numerical evaluation and simulation.First consider the evaluation for I .By noticing the similarity between the expressions of I stated in (78) and 11 NG stated in (41), it can be seen that I defined in (73) can be obtained by following similar approach as used for deriving 11 NG in Theorem 2. By noticing the similarity between the expressions of II stated in (78) and 11 NG stated in (41), it can be seen that II defined in (74) can be obtained by following similar approach as used for deriving 11 NG in Theorem 2.
Determining the average BLER III ) is non-trivial due to the involvement of the Q-function and the term V(γ 1 J , γ 2 J ) is defined as follows: The term V NG (γ 1 J + γ 2 J ) can also be expressed in the following manner: From the above equation, it can be seen that the approximation in (87) is accurate when the following condition is satisfied: The above condition is satisfied when at least one of the terms γ 1 J or γ 2 J is large.The second term of V NG (γ 1 J , γ 2 J ), i.e., 2 is a ratio of the product of SNR (γ 1 J ) and INR (γ 2 J ), and square of the sum of SNR and INR.Due to this, obtaining a closed-form expression for III becomes difficult.Hence, the second term in (86) is ignored in the evaluation of E[ III ] for mathematical tractability.However, this term has been considered in the simulation, and a close match has been found between the simulation and numerical results (See Section VI).
To determine III , it is required to find the cumulative distribution function of t J , where t J = 1 J + 2 J .Thus, the cumulative distribution function associated with t J is obtained as follows: Then the average BLER can be expressed as: By using the linear approximation for the Q-function as given in (104), and the cumulative distribution function of t,J derived in (91), ( 92) is expressed as follows: By substituting (91) in (93) and with some algebraic manipulation, expression in (75) is obtained.By adding the terms in (73), (74), and (75), J can be obtained which is eventually used to evaluate Pr(D ).In the following, the probability of successful decoding at Rx-2 is obtained.

Evaluation of Pr(D {2}
2 ): We know that where It can be seen that the expression of 22 NG in ( 95) is same as that of the expression of 22 G in (33), except that V G (γ 22 ) is replaced by V NG (γ 22 ), as Tx-2 uses non-Gaussian signaling.Hence, 22 NG stated in (76) can be evaluated in the same way as that of 22 G , that has been derived in the case of SIC scheme.b) When Q 1 = 0 and Q 2 = 0: The probability of successful decoding, in this case, is given by It can be seen that I corresponds to (73) and has been previously derived for the case where both the queues are active.c) When Q 1 = 0 and Q 2 = 0: The probability of successful decoding, in this case, is given by It can be seen that Pr(D {2} 2 ) corresponds to (94) as Rx-2 does not experience any interference from Tx 1. Remarks: 1) The result of Theorem 3 considers the product of error terms in both the decoding stages while evaluating the probability of successful decoding of the SIC scheme.From Fig. 3, it can be observed that at lower transmission rates, the probability of successful decoding is almost the same whether we ignore or consider the product of error terms.However, at higher transmission rates (R ≥ 0.6), one can see that the probability of successful decoding is underestimated when the product of the error terms is ignored.When R increases beyond 1, the probability of successful decoding becomes zero, as the product of error terms is ignored.However, when the product of error terms is considered, this does not happen even if R ≥ 1.Hence, it is required to consider the product of error terms for a high transmission rate.The product of error terms in the evaluation of average BLER gives accurate results compared to the work which has ignored the product of error terms [23], [29], [31], [32], [33].2) One can use the results for a very strong interference regime in the case of the two-user AWGN IC [17] to obtain the achievable results for two-user Z-IC for specific channel conditions.This is left as a future extension of this work.
3) The probability of successful decoding obtained for various interference mitigation techniques does not consider path loss.The effect of path-loss can be taken into consideration by replacing , where α ij (d ij ) corresponds to the path-loss exponent (distance) between Tx-i and Rx-j (i, j ∈ {1, 2}).

V. AVERAGE DELAY AND AVERAGE AGE OF INFORMATION ANALYSIS
The stable throughput or the stability region does not capture the delay or latency associated with the communication.It is essential to understand how interference affects the timeliness of the data when devices are constrained to use short packets for communication, as in IoT or M2M scenarios.To get more insight into this problem, the impact of various interference mitigation techniques on average delay and the AAoI are explored in this section.The AAoI metric fundamentally differs from delay and captures the freshness of information.As the queues are decoupled in the Z-IC, it is possible to characterize the AAoI and delay using the result in [43], [44], [45].In the following, results related to Rx-1 (which experiences interference) are given, and one can obtain the results for Rx-2 similarly.
The average delay ( 1 ) and AAoI (AAoI 1 ) corresponding Tx-1 and Rx-1 are given by the following expressions [43], [44]: and From ( 100) and (101), it can be seen that it is required to determine μ 1 to evaluate both delay and AAoI.The service probability corresponding to user 1 can be obtained from (16) as given below: Hence, from (102), it can be seen that μ 1 not only depends on the mean arrival probability of Tx-2 (λ 2 ) but also on the scheme that is adopted at Rx-1 to mitigate interference.The different probabilities of successful decoding at the receiver can be obtained for the various interference mitigation schemes using the results developed in Section IV.Hence, one can use the service probability given in (102) to determine the average delay and AAoI for Tx-1 and Rx-1 where (λ 1 , λ 2 ) ∈ R, i.e., both the queues are stable.Recall that R corresponds to the stability region given in (4).The AoI and delay performance of various decoding schemes depend on the probability of successful decoding of a packet at the receiver and arrival rate.The probability of successful decoding of the packet depends on the relative strength between signal and interference along with other parameters such as power budget at transmitter and blocklength.

VI. RESULTS AND DISCUSSION
This section presents results to illustrate the performance of various interference mitigation techniques when users are constrained to use short packets for communication.The average channel gain between Tx-i and Rx-i is fixed at φ ii = 1 for all the results.Two scenarios are considered by taking account of the relative strength between the interference and intended signal: (a) φ 11 > φ 21 and (b) φ 11 < φ 21 .

A. PROBABILITY OF SUCCESSFUL DECODING AND AVERAGE BLER
In Fig. 4, probability of successful decoding (Pr(D )) for various interference mitigation schemes stated in Theorems 2-4 are plotted against N, when both the queues are active.The simulation results for various schemes are obtained by averaging over the corresponding instantaneous error expressions in Theorems 2-4 over 10 6 channel realizations.For the TIN scheme, the instantaneous error in ( 29) is averaged over 10 6 channel realizations.To obtain the simulation result for the probability of successful decoding in the case of SIC scheme, the instantaneous error terms in (55), and (57) are averaged over 10 6 channel realizations.Similarly, for the joint decoding scheme, the RHS of different error terms in (78) are averaged over 10 6 various channel realizations.The results presented in the framework of finite block length coding are accurate when the block-length is in the order of 100 channel uses.
The number of information bits and power at both the transmitters are set at k 1 = k 2 = 30 nats and P 1 = P 2 = 50W, respectively.The probability of successful decoding for all the schemes increases with block-length N. From Fig. 4(a) (φ 11 = φ 22 = 1 and φ 21 = 0.5), it is interesting to note that the joint decoding scheme performs better in comparison to other schemes even when φ 21 < φ 11 .The SIC scheme achieves the lowest probability of successful decoding as the interfering user's average channel gain is weak compared to the intended user's average channel gain.Due to this, the average BLER in the first stage of the SIC scheme is higher, affecting the decoding of the intended message in the second stage.To illustrate the error performance of various decoding schemes, average BLER (1 − Pr(D )) at Rx-1 (when both the queues are active) is plotted against block length N in Fig. 5.The joint decoding scheme can achieve low average BLER even with short block-length compared to other schemes but at the cost of increased complexity at the receiver.
In Fig. 4(b), the average channel gain corresponding to the interfering link is stronger than the direct link (φ 11 = φ 22 = 1 and φ 21 = 1.5).In this case, the SIC scheme performs better than the TIN scheme for a larger Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.In Fig. 6, the average BLER for Rx-2 is plotted when different decoding schemes are used at Rx-1.Recall that for the TIN and the SIC scheme (See Theorems 2-3), Tx-2 uses Gaussian signalling, whereas for the joint decoding scheme (See Theorem 4), Tx-2 uses non-Gaussian signalling.Hence, the average BLER expression is the same for Rx-2 in the case of TIN and SIC schemes.One can notice that the average BLER in the case of Rx-2 is less compared to Rx-1 as Rx-2 does not experience any interference.It is also found that the average BLER for different cases is almost the same, although the non-Gaussian signalling scheme achieves lower dispersion compared to Gaussian signalling.
In Fig. 7, the probability of successful decoding Pr(D ) at Rx-1 is plotted versus the number of information nats (k i = k) for a given block-length N = 500 channel uses, when both the queues are active.From Fig. 7(a), it is interesting to note that the joint decoding scheme gives the best performance, even when the average channel gain of the interfering user is less than the average channel gain of the direct link for a wide range of k.However, when k increases beyond a specific value (around 1000 nats), the TIN scheme performs best among the considered schemes.This is because the error in decoding the messages increases with an increase in the rate (R i = k i N ), as Rx-1 needs to decode both the messages in joint decoding.However, in the TIN scheme, Rx-1 does not need to decode the message of the unintended user.From Fig. 7(b), one can also observe that the joint decoding scheme outperforms other schemes for a broader range of k when compared to Fig. 7(a), as φ 21 > φ 11 .As the interfering user's average channel gain is stronger than the intended user's average channel gain, Rx-1 can support a higher rate from both users with less error.One can also notice from Fig. 7(b) that even when φ 21 > φ 11 , the TIN scheme performs better in comparison to the SIC scheme for high values of k, as Rx-1 does not need to decode the interference in case of TIN scheme.

B. STABILITY REGION
In Fig. 8, the stability region of Z-IC in Theorem 1 is plotted for P 1 = P 2 = 50W, N = 500 channel uses, and k 1 = k 2 = 300 nats.It can be seen from Fig. 8(a) (φ 11 > φ 21 ) that there is no single scheme that can achieve the largest stability region.From Fig. 8(a), one can notice that when user 1 supports a maximum arrival rate of 0.97 packets/slot, it is not possible to support any packet at user 2. When the TIN scheme is used at Rx-1, Tx-2 can support an arrival rate of 0.97 packets/slot, but at the cost of a reduced arrival rate that can be supported at Tx-1 in comparison to SIC scheme.When the arrival probability at Tx-1 increases beyond 0.7 packet/slot, it is preferable to use a joint decoding scheme at Rx-1, as it allows both the users to support a higher arrival rate without violating the stability criteria in comparison to other schemes.This gain comes from decoding the interference caused by Tx-2 at Rx-1.The SIC scheme is not able to provide a larger stability region, as the decoding error in the first stage of SIC propagates to the next stage of decoding at Rx-1, as φ 21 < φ 11 .
When φ 21 > φ 11 , the joint decoding scheme provides the largest stability region, and both the users can support a maximum arrival rate close to 1 packet/slot (See Fig. 8(b)).The stability region corresponding to the SIC scheme is larger than the TIN scheme, which contrasts the result in Fig. 8(a).
In Fig. 9, the stability region is plotted for the case where both the users transmit at a lower rate (k 1 = k 2 = 30 nats) as compared to the result in Fig. 8. From Fig. 9(a), it can be observed that stability regions corresponding to TIN and joint decoding schemes are almost similar at lower transmission rates.Similar observations can be made from Fig. 9(b).

C. AVERAGE DELAY AND AAOI
In Figs. 10 (φ 11 > φ 21 ) and 12 (φ 11 < φ 21 ), average delay in (100) corresponding to different interference mitigation schemes at Tx-1 is plotted as a function of arrival probability λ 1 where (λ 1 , λ 2 ) ∈ R in (4).The impact of the arrival rate at Tx-2 (λ 2 ) and the relative strength between the intended signal and interference is also illustrated.Note that average delay or AAoI depends on μ 1 and λ 1 .The service probability μ 1 depends on the arrival rate at Tx-2, the decoding scheme used at Rx-1, and the underlying channel condition.The different probabilities of successful decoding involved in (102) are obtained for various interference management schemes as mentioned in Section IV.From Figs. 10 and 12, it can be observed that when the arrival rate at user one increases, the average delay gradually increases for all the schemes.The TIN scheme ensures lower delay compared to the SIC scheme when φ 21 < φ 11 (See Fig. 10).When φ 21 > φ 11 , the SIC scheme provides a smaller delay than the TIN scheme (See Fig. 12).However, the joint-decoding scheme provides the lowest delay among all the considered schemes.
Moreover, it can be seen from Figs. 10(a) and 10(b) that as the arrival rate at Tx-2 (λ 2 ) increases, the delay increases for both SIC and TIN schemes, as the interference increases with an increase in the arrival rate at Tx-2.However, joint decoding provides almost similar delay performance even when λ 2 increases.A similar phenomenon can be observed in Figs.12(a) and 12(b).
In Figs.11 and 13, average AoI at Tx-1 as stated in (101) is plotted as a function of λ 1 for different arrival probabilities at Tx-2 as well as for different relative strength between the intended signal and interference.When the arrival probability is low at the Tx-1, AAoI is high, as the receiver is not getting fresh updates.As the arrival probability at user-1 increases, the AAoI decreases as the receiver gets enough updates.However, after a certain arrival probability, AAoI increases with an increase in the arrival probability due to the increase in the queuing delay at the transmitter.It can also be observed that joint decoding can achieve low AAoI and low average delay simultaneously.
Moreover, it can be seen from Figs. 11(a) and 11(b) that as the arrival rate at Tx-2 (λ 2 ) increases, AAoI increases for both SIC and TIN schemes.This is due to an increase in the interference with an increase in λ 2 at Tx-2.However, it can be seen that joint decoding provides almost similar AAoI even when λ 2 increases.Similar behaviour can be observed in Figs.13(a) and 13(b).
From the results, it can be noticed that when the average channel gain of the interfering user is less than the average channel gain of the intended user, the receiver can employ the TIN scheme as it can support high stable throughput, low delay, and low AAoI in comparison to the SIC scheme.Moreover, when it is required to support a high arrival rate at both users simultaneously, it is preferred to use a joint decoding scheme at Rx-1, as it can provide low delay and low AAoI.However, these benefits come at the cost of increased complexity in joint decoding.When the average channel gain of the interfering user is less than the average channel gain of the intended user, a joint decoding scheme can be used as it can provide significant improvement in throughput and latency compared to other schemes, even though it has high implementation complexity.

D. RESULTS WITH DIFFERENT POWER CONSIDERATIONS AT THE USERS
This section presents results to investigate the performance of different interference mitigation schemes under different power budget assumptions at the transmitters.For all the results, it is assumed that P 1 = 50W, P 2 = 130W, φ 11 = φ 22 = 1, and φ 21 = 0.5.
In Fig. 14, the probability of successful decoding at Rx-1 (Pr(D )) is plotted as a function of block length N for different decoding schemes when both the queues are active.Interestingly, even in the weak interference regime, the SIC scheme performs better than the TIN scheme.As Tx-2 has a high-power budget compared to Tx-1, the error in the first stage of decoding reduces, where the message of Tx-1 is treated as noise.The reduced error in the first stage of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.decoding improves the overall error performance of the SIC scheme.Due to this, the stability region corresponding to the SIC scheme is larger than the TIN scheme (See Fig. 15).When the same power budget is used at both transmitters, the TIN scheme simultaneously supports a larger arrival rate for both users than the SIC scheme (See Fig. 9(a)).The stability region corresponding to the joint decoding scheme is the largest when compared to other schemes, as the joint decoding scheme has the highest probability of successful decoding compared to other schemes.In Fig. 16, the average delay and AAoI of user-1 are plotted as a function of λ 1 .The SIC scheme ensures lower average delay and average AoI as compared to the TIN scheme even when φ 21 < φ 11 .The joint decoding scheme provides better AAoI and delay performance and supports a higher arrival rate at Tx-1 compared to the other schemes.

VII. CONCLUSION
This work characterizes the stability region of the two-user Rayleigh fading Z-IC, which has not been explored in the  theory framework was used to obtain an approximate closedform expression for the probability of successful decoding corresponding to various decoding schemes at the receiver.The performance of various decoding schemes depends not only on the channel conditions but also on the arrival rate at the user, packet length, and rate associated with the user.The developed results further help to explore the interplay between packet length, average delay, and AAoI in the case of interference-limited scenarios.

APPENDIX A DERIVATION OF TIN IN THEOREM 2
Linearization technique [46] is used to approximate the Qfunction in (29), at point x = α as given below: + αβ NG , and .
The cumulative distribution function (CDF) associated with SINR ( 1,TIN ) at Rx-1 is given by the following expression: Using ( 31), ( 103) and (105), the average BLER ( TIN ) can be evaluated as follows Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.(107) Using the definition of exponential integral defined in (26) in (107), the average error at Rx-1 given in ( 24) is obtained.

APPENDIX B DERIVATION OF E[ 21 11 G ] IN THEOREM 3
From (55), (57), and (60), 21 11 G in (59) is further simplified as shown at the bottom of next page, where δ l 1 , δ u 1 , δ l 2 , and δ u 2 are defined in (50), shown at the bottom of the p. 9.By using the linear approximation of U G (γ 21 ) as defined in (37) and f (γ 21 |γ 11 ) obtained in (61), the terms I 1 and I 2 simplify to the following:  where ρ 2 G is defined in (50).The term I in (109), shown at the bottom of the next page is evaluated using (110) and ( 111), and after some simplifications, the following is obtained: where h(δ l 2 ) and h(δ u 2 ) are defined in (51), shown at the bottom of the p. 9 and Ei(x) is defined in (26).I 4 specified in (113) can be obtained by following the same procedure used in evaluation of I 3 and is given by the following expression: By making use of I obtained in (112), I 5 specified in (113) is evaluated as follows: (119) Using (62), I 7 and I 8 in (117) are evaluated as follows:

FIGURE 2 .
FIGURE 2. Stability region of Z-IC for the general case.

TABLE 1 . Notation List.
11 ) 2 .(42) By observing the expressions of 22 G and 11 NG stated in (33) and (41) respectively, it can be seen that 11 NG stated in (23) can be obtained from the expression of 22 G by replacing P 2 , φ 22 , α 2 , and β 2 G in (25) with P 1 , φ 11 , α 1 , and β 1 NG respectively.It is required to determine the probability of successful decoding only in the case of Rx-2.One can notice that Pr(D c) When Q 1 = 0 and Q 2 = 0: In this case, Rx-1 does not receive any packet from Tx-1.