RAN Slicing Performance Trade-offs: Timing versus

RAN Slicing Performance Trade-offs: Timingversus Throughput Requirements

Federico Chiariotti, Member, IEEE, Israel Leyva-Mayorga, Member, IEEE, Cedomir Stefanovic, SeniorMember, IEEE, Anders E. Kalør, Student Member, IEEE, and Petar Popovski, Fellow, IEEE

Abstract—The coexistence of diverse services with hetero-geneous requirements is a fundamental feature of 5G. Thisnecessitates efficient radio access network (RAN) slicing, definedas sharing of the wireless resources among diverse services whileguaranteeing their respective throughput, timing, and/or reliabil-ity requirements. In this paper, we investigate RAN slicing for anuplink scenario in the form of multiple access schemes for twouser types: (1) broadband users with throughput requirementsand (2) intermittently active users with timing requirements,expressed as either latency-reliability (LR) or Peak Age ofInformation (PAoI). Broadband users transmit data continuously,hence, are allocated non-overlapping parts of the spectrum. Weevaluate the trade-offs between the achievable throughput of abroadband user and the timing requirements of an intermit-tent user under Orthogonal Multiple Access (OMA) and Non-Orthogonal Multiple Access (NOMA), considering capture. Ouranalysis shows that NOMA, in combination with packet-levelcoding, is a superior strategy in most cases for both LR andPAoI, achieving a similar LR with only slight 2% decrease inthroughput with respect to the upper bound in performance.However, there are extreme cases where OMA achieves a slightlygreater throughput than NOMA at the expense of an increasedPAoI.

Index Terms—Age of Information (AoI), Heterogeneous ser-vices, Non-Orthogonal Multiple Access (NOMA), Reliability,Slicing.

I. INTRODUCTION

The fifth generation of mobile networks (5G) aims to sup-port three main service categories: enhanced mobile broadband(eMBB), ultra-reliable low-latency communications (URLLC),and massive machine-type communications (mMTC) [1].eMBB is the direct evolution of the 4G mobile broadbandservice with higher data rates, along with greater spectral andspatial efficiency. Even though eMBB use cases mainly occurin the downlink (e.g., video streaming or file download) noveleMBB use cases in the uplink are quickly gaining relevance.These include remote driving and live-video streaming in, forexample, tactile internet applications and sport and culturalevents. URLLC services, on the other hand, usually involveexchange of small amounts of data, but require latency in theorder of a few milliseconds and high reliability guarantees,e.g., a packet loss ratio below 10−5. Finally, mMTC alsoinvolve transmissions of small amounts of data per device,but consist of hundreds or thousands of devices in the servicearea. The main challenge in mMTC is to design accessnetworking mechanisms that maximize the success probability

The authors are with the Department of Electronic Systems, AalborgUniversity, 9220 Aalborg, Denmark (e-mail: [email protected]; [email protected];[email protected]; [email protected]; [email protected]).

while maintaining an adequate timing in data delivery andresource efficiency.

The main strategy for service co-existence adopted by3GPP [2], [3] is network slicing, which refers to the allocationof subsets of the network resources to the active services.The idea is to provide performance guarantees by limiting themutual impact among services and/or service categories [4].Arguably, in the context of the radio access network (RAN)the most natural form of slicing is frequency division multipleaccess (FDMA), which provides a high degree of isolationbetween services in different frequency bands. However, whileFDMA slicing works well for high throughput services thatare constantly demanding resources, it yields low utilizationfor intermittent services that are active only a fraction of thetime. This motivates the search for alternative slicing schemesthat are more suitable for heterogeneous services types.

In general, RAN slicing can be implemented in the formof Orthogonal Multiple Access (OMA) and Non-OrthogonalMultiple Access (NOMA). OMA schemes, such as FDMA,time division multiple access (TDMA), and code divisionmultiple access (CDMA), have been extensively studied andimplemented in commercial and cellular systems. Moreover,OMA seems to be the approach preferred by 3GPP for 5Gand beyond 5G systems, contextualized in the concept ofbandwidth part [2]. On the other hand, in NOMA the sametime-frequency resources are assigned to multiple services orusers. This allows, for example, to increase the number ofserved users with the available resources and/or the spectral ef-ficiency of the system [5]. To enable communication in sharedtime-frequency resources, NOMA is usually accompanied bymulti-user detection techniques, like separation of the users inthe code domain, or in the power domain accompanied withsuccessive interference cancellation (SIC) where the individualsignals are in turn decoded and subtracted from the receivedcomposite signal [5]–[8].

The difference between OMA and NOMA slicing is illus-trated in Fig. 1. Here, it can be seen that TDMA and NOMAachieve higher resource utilization than FDMA when theintermittent service transmits infrequently, while the differencewill be less pronounced when the intermittent service transmitsfrequently. In the latter case, NOMA may even be inefficientdue to the high amount of interference introduced to thebroadband service. Further, TDMA faces the challenge ofcarefully dimensioning the allocated resources to the intermit-tent service, which is essential to achieve an acceptably lowamount of idle resources (i.e., empty slots) and waiting timefor transmission.

1

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1

Base station

Broadbandservice

Intermittentservice

Long transmissi

on

Short and infrequenttransmissions

OrthogonalRAN slicing

Non-orthogonalRAN slicing

Freq

uenc

y

Time

FDMA

Freq

uenc

y

Time

TDMA

Empty slot

Freq

uenc

y

Time

NOMA

Signal overlap

Fig. 1: Orthogonal (i.e., FDMA and TDMA) and non-orthogonal RAN slicing (i.e., NOMA) between broadband andintermittent services.

Despite this apparent trade-off between efficiency andtimeliness of the slicing schemes, only a few studies havecompared the performance between TDMA and NOMA withheterogeneous services [9]–[11]. On the other hand, OMA andNOMA slicing has been widely studied in the presence ofmultiple users of the same service type [6], [12]–[14]. Forinstance, the trade-offs in achievable data rates for eMBBservices are characterized in an additive white Gaussian noise(AWGN) channel with OMA and power-domain NOMA [13],[14]. In our previous work, we derived the performance trade-offs with heterogeneous services with TDMA and NOMAwith packet-level coding in a simplified collision channelmodel, which provides conservative results for NOMA [11],[15]. Some results with capture, obtained by simulation, wereprovided in [11], which served as one of the main motivationsfor this study, as these illustrated the potential gains of NOMA.The aim of this work is to provide an extensive and exactanalytical treatment of OMA and NOMA slicing in an uplinkscenario with two different service types: broadband andintermittent users with throughput and timing requirements, re-spectively. Because the trade-off between TDMA and NOMAis least apparent, this combination is our primary focus andinclude FDMA as benchmark that neglects resource efficiency.Furthermore, to ensure a realistic evaluation, we considera channel model with capture, which inevitably occurs inpractice.

As mentioned, we assume that broadband users transmitdata continuously and are primarily interested in achievinga high throughput. In contrast, intermittent users transmitshort packets sporadically and are primarily interested inthe timeliness of their data, according to the underlyingapplication. To consider packet-level and flow-level timingrequirements, we have selected two different Key PerformanceIndicators (KPIs). The first KPI, which reflects flow-levelrequirements, is Peak Age of Information (PAoI), relevant

. . .Time

AoI

`1 `3 `4

Arrival Success Error

0 `1 `3 `4 Time0

1/4

1/2

3/4

1Error

probability

Lat

ency

-rel

iabi

lity

(LR

)

Fig. 2: Exemplary diagram of the Age of Information (AoI)and latency-reliability KPIs in a period with four packettransmissions. The latency ì for packets transmitted witherrors is set to ∞.

for users that send updates of an ongoing process in whichthe freshness of information is the most important objective.PAoI measures the time elapsed since the generation of thelast received update until a new update is received [16],and it is therefore determined by the transmission latency,reliability and the update generation pattern. PAoI-focusedapplications can tolerate individual packet losses, as thereare no strict reliability requirements and new updates cansupersede old ones. The second KPI, which reflects packet-level requirements, is denoted by latency-reliability (LR) andcaptures the probability of delivering individual packets withina given latency threshold [17]. For this, we use the distributionof latency where lost packets are defined to have infinitelatency. LR captures, for example, URLLC traffic with strictconstraints on the reliability of communication within a max-imum latency. Our specific focus will be on computing highpercentiles of LR and PAoI, which can be used to designsystems with probabilistic reliability guarantees.

The scenario assumed in this paper comprises a singlefrequency band (i.e., bandwidth part in 5G New Radio (NR)terminology) sliced in time to accommodate one broadbandand one intermittent user.1. We study OMA, focusing onTDMA but also including FDMA, and NOMA with SIC. Inthe latter case, SIC can be applied (1) in conjunction with thecapture effect, such that the colliding packets are immediatelyresolved, and (2) coupled with the packet-level coding, suchthat after decoding of the broadband user block, the interfer-ence is removed and past packets from the intermittent usercan be recovered. We analyze the achievable performance andthe inherent trade-offs, providing closed-form expressions forthroughput of the broadband user and timing of the intermittentuser. The derivations are contextualized for a simple fading-based channel model, however, the elaborated approach isgeneral and easily transferable to other settings.

In particular, the main contributions of this paper are thefollowing.• We provide exact expressions that allowed us to analyze

the operating regions and trade-offs with OMA andNOMA with a realistic channel model that includes thepossibility of capture. The results with this model showfundamental differences when compared to the analyses

1The scenario is inspired by the latest non-orthogonal multiplexing ap-proaches in the uplink studied by the 3GPP [18].

2

carried out in our previous work and confirm the trendsobserved through simulations [11], [15].

• We analyze the impact of the wireless conditions of theintermittent user, including distance from the base station(BS) and path loss, in the performance of NOMA.

• We provide design guidelines for selecting the multipleaccess scheme and its parameters, depending on:

1) The requirements and features of the different typesof services in the system.

2) The available bandwidth.3) The wireless conditions of the intermittent user.

We observe that, while FDMA provides the upper bound inperformance, NOMA schemes offer significant benefits w.r.t.OMA when the target KPI for the intermittent user is LR.Specifically, NOMA can achieve similar performance trade-offs as FDMA but with a much higher resource utilization.On the other hand, the potential gains of NOMA w.r.t.TDMA decrease when the target KPI is PAoI, with TDMAoutperforming NOMA in extreme cases where throughput ismaximized in exchange for a longer PAoI.

The rest of the paper is organized as follows. Sec. IIpresents the related work on AoI and slicing-based access forheterogeneous user classes. The system model and KPIs arespecified in Sec. III. We then derive the analytical distributionsof those metrics for OMA and NOMA in Sec. IV andSec. V, respectively. Sec. VII presents simulation results anddiscussion of the performance of the different access schemes.Finally, Sec. VIII concludes the paper.

II. RELATED WORK

Non-orthogonal slicing, in the form of NOMA, offers thepossibility of increasing the spectral efficiency and the numberof supported users with respect to OMA in exchange for agreater decoding complexity at the receiver to perform SIC [5],[19] or other multi-user detection techniques. Hence, NOMAhas been widely studied in the literature in systems with asingle service type [5], [6], [12]–[14]. NOMA often assumesuser separation in the power domain such that the benefits ofSIC can be fully exploited. However, different performancegains have been observed for NOMA in the uplink and inthe downlink. In particular, the effect of power control in theuplink can be eclipsed by the channel conditions of the usersin combination with imperfect channel state information [19].

A particularly interesting approach towards heterogeneousservice coexistence with NOMA is presented in [6], whichemphasizes the importance of power control in NOMA andformulates resource allocation as a non-cooperative game andas a matching problem. While not addressing them directly,this non-cooperative game may be able to handle heteroge-neous service types, as each user defines and attempts tomaximize its own utility.

One of the first studies that addresses the coexistence ofheterogeneous services in OMA and NOMA was presentedin [10], considering different combinations of 5G services inan uplink scenario. Specifically, eMBB users are allocatedorthogonal resources between them; these coexist with eitherone URLLC user or with mMTC traffic, which follows a

Poisson distribution. It was observed that NOMA may offerbenefits with respect to OMA depending on the rate of theeMBB users and on the type of coexisting traffic. Specifically,these benefits were evaluated in terms of the achievable ratesfor eMBB and URLLC traffic and the achievable eMBB ratesas a function of the arrival rate of mMTC packets. This workwas later extended to a multi-cell scenario with strict latencyguarantees for URLLC traffic [20], where it was observed thatNOMA leads to a greater spectral efficiency w.r.t. OMA. Thissame conclusion was drawn by Maatouk et al. [12] in an uplinkscenario with two users with and one service type. The aimof the latter study was to minimize the average AoI, however,it was also observed that a greater spectral efficiency does notdirectly translate in a lower average AoI.

AoI is a relatively new performance metric, but it hasbeen rapidly adopted due to its relevance in remote controltasks [21]. Most papers in the literature have examined it inthe context of queuing theory, often in ideal systems withMarkovian service [22], because of the relative simplicityof the analysis, but a few have considered the effect ofphysical layer issues and medium access schemes on it. Recentworks compute the average AoI in Carrier Sense MultipleAccess (CSMA) [23], ALOHA [24] and slotted ALOHA [25]networks, considering the impact of the different mediumaccess policies on the age.

Another important missing piece in the AoI literature isthe worst-case performance analysis: while studies on averageAoI are common, the tail of its distribution is rarely consid-ered [26], limiting the relevance of the existing body of workfor reliability-oriented applications. The analytical complexityof deriving the complete distribution of the age is a dauntingobstacle; only recently, advances have been made in this line.A recent work [27] uses the Chernoff bound to derive an upperbound of the quantile function of the AoI for two queues intandem with deterministic arrivals. Using a more analyticalapproach, the PAoI distribution was computed over a single-hop link with fading and retransmissions in [28]. We alsomention the work in [29], where different service classes aredefined and the system is modeled as an M/G/1/1 clockingqueue with hyperexponential service time. However, in thelatter, only the service rate is different among classes. Then,the classes can adapt the arrival rate to minimize the AoI.finally, for a more detailed overview of the literature on AoI,we refer the reader to [30].

III. SYSTEM MODEL

We consider an uplink scenario with a set of users Utransmitting data to a BS through an Orthogonal Frequency-Division Multiple Access (OFDMA) system whose time-frequency resources are divided into time slots and bandwidthparts as in 5G NR [31]. A bandwidth part is defined as aset of contiguous resource blocks in the frequency domain.We consider the case where the users transmit up to onepacket per time slot, occupying the whole bandwidth part.Herein, we consider the case where two heterogeneous usersmust be allocated resources. The options for the BS are 1)allocate the users in the same bandwidth part and define how

3

the resources should be shared among them or 2) allocate adifferent bandwidth part for each of the users using FDMA.

User 1 is a broadband user following the eMBB model [1]:it is a full-buffer user, that is, it always has data to transmit,and maintains an infinite transmission queue. To counteractpotential packet losses due to fading and noise, the broadbanduser implements a packet-level coding scheme, where blocksof K (source) packets are encoded to generate a frame of Ncoded packets. The coded packets are transmitted in the samebandwidth part, one after the other, and have a zero probabilityof linear dependence, which can be achieved, for example,with Maximum Distance Separable (MDS) codes. Hence, theblock of packets is decoded when any K packets from thesame frame are received without errors.

User 2 is an intermittent user that, with (a relatively low)probability α, may generate a short packet at each slot. Eachgenerated packet is transmitted just once at the next availabletime slot. User 2 maintains up to Q of the generated packetsin a transmission queue. We denote by qt ≤ Q the length ofthe queue at time slot t. If a new packet is generated whenqt = Q, user 2 discards the oldest buffered packet and addsthe newly generated one at the end of the queue.

When both users are allocated to the same bandwidth part,the BS must allocate the time slots that are available for thetransmission of each of the two users. For this, we define theresource allocation set At ⊆ U as the subset of users that canaccess the bandwidth part at slot t. We define the followingthree types of slot allocations.

1) Broadband: The slot is reserved for the broadband user.Hence, At = 1.

2) Intermittent: The slot is reserved for the intermittent userand may use it if it has one or more packets in its queue.Hence, At = 2.

3) Mixed: Both users are allowed to access the slot, imply-ing that the signals will overlap if the intermittent usertransmits. Hence, At = {1, 2}.

Building on these, we define the following different accessschemes.

1) TDMA: Both users are allocated resources in a singlebandwidth part with separate broadband and intermittentslots. Specifically, the intermittent user has a reservedintermittent slot once every Tint slots, while the rest ofthe slots are reserved for the broadband user. As such,this is a non-orthogonal slicing in the frequency domainbut orthogonal in the time domain where there is nointerference among the users.

2) NOMA: Both users are allocated resources in a singlebandwidth part with only mixed slots. Hence, the in-termittent user may transmit at any slot. The two usersinterfere any time the intermittent user transmits, butthe packets can be recovered through SIC by decodingone of the signals immediately or at a later time slot.More details on the operation of SIC are given inSection III-A.

3) FDMA: The users are allocated resources in differentand non-overlapping bandwidth parts. Hence, one of the

1 2 3 4 5 N 1 2

1 2 3 4 5 N

1 2 3 4 5 N 1 2

TDMA

NOMA

FDMA(benchmark)

· · ·

· · ·

· · ·

· · ·

Broadband

Intermittent

Mixed

Time

TintSlot

Frame

Fig. 3: Frame structure for the TDMA, NOMA, and FDMAschemes with K = 4 and N = 6.

bandwidth parts contains only broadband slots and theother bandwidth part contains only intermittent slots.

The frame structures for these access schemes are illustratedin Fig. 3. Naturally, FDMA can only take place when there aretwo bandwidth parts available. Needles to say, the bandwidthpart allocated to the intermittent user with FDMA is likely tobe under-utilized, as α is relatively small. Hence, this approachresults in a low resource efficiency, and is used here only as abenchmark scheme in which the performance of the users isfully independent of each other.

A. Channel model

We consider a block fading channel, where the receivedsignal by the BS at time slot t is given as

yt =∑

u∈{1,2}

hu,tau,txu,t + zt (1)

where hu,t ∈ C is the random fading coefficient for user uand zt is the circularly-symmetric Gaussian noise with powerσ2. The variable au,t is an activity indicator, equal to 1 if theuser is active in slot t and 0 otherwise. A user is active attime t if and only if u ∈ At and if its packet queue qu,t is notempty

au,t = I(u ∈ At)I(qu,t > 0), (2)

where I(·) is the indicator function, equal to 1 if the conditionis true and 0 otherwise. Let Pu ≤ Pmax be the selected(i.e., fixed) transmission power for user u, where Pmax is themaximum transmission power. The signal-to-noise ratio (SNR)of user u is given as

SNRu,t =|hu,t|2Puau,t

σ2=|h′u,t|2Puau,t

ùσ2, (3)

where ù is the constant large-scale fading, including path loss,and |h′u,t| is the envelope of the channel coefficient due to fastfading. The path loss is a function of the distance of user u tothe BS ru, the carrier frequency fc, and a path loss exponentη. We assume the standard path loss model

ù =(4πfc)

2rηuc2

, (4)

where c is the speed of light.The expected SNR for a transmission by user u is

SNRu =E[|hu,t|2

]Pu

σ2=

E[|h′u,t|2

]Pu

ùσ2, (5)

4

By using the standard assumption of treating the interferingsignal as AWGN noise, the signal-to-interference-plus-noiseratio (SINR) for user u in the considered scenario is

SINRu,t =|hu,t|2Puau,t

σ2 + |hv,t|2Pvav,t=

SNRu,t1 + SNRv,t

, s.t. v 6= u.

(6)

B. Reception model

Let X be the random variable (RV) of the number of packetsfrom user 1 that belong to the same block and are receivedwithout errors. The success probability of user 1, denoted asps,1, is defined as the probability of receiving K or morepackets out of the N that comprise the block. That is,

ps,1 , Pr [X ≥ K|N ] . (7)

We define γu as the threshold in the SINR to decode a packettransmitted by user u. In practice, the threshold is mainly afunction of the modulation and coding scheme and the receiversensitivity. In the following, we consider the case in which thefading envelope |hu,t| is Rayleigh distributed and define theerasure probabilities for the two users.

Erasure probability for the broadband user: The BS hascollected sufficient channel state information (CSI) about thebroadband user so that the appropriate transmission powerP1 ≤ Pmax, blocklength, and data rate (i.e., modulation andcoding) to achieve a target erasure probability ε1 are signaledback to the broadband user. Therefore, user 1 transmits withpower

P1 ≤ Pmax : Pr [SNR1,t < γ1] = ε1 (8)

Erasure probability for the intermittent user: Due to theinfrequent transmissions, the CSI of this user at the BS isinsufficient to perform a precise selection of parameters asdone for the broadband user. Instead, the user always transmitsat P2 = Pmax and its erasure probability ε2 is determined byits path loss `2 and by γ2. Hence, the erasure probability foruser 2 is calculated from (5) as

ε2 = Pr [SNR2,t < γ2] = 1− e−γ

SNR2 = 1− e−γ`2σ

2

Pu (9)

since E[|h′u,t|2

]= 1 for unitary Rayleigh fading.

Based on these probabilities, we define six different out-comes for the cases where both users transmit in the sametime slot. These outcomes are ordered pairs (o1, o2) whereou ∈ {I, E ,R} indicates the outcome of user u’s signal,described in the following.• I: Either 1) the signal of interest has sufficient SINR to be

immediately decoded or 2) the other signal has sufficientSINR to be immediately decoded, its interference isremoved through SIC, and the signal of interest hassufficient SNR and is decoded.

• E : The signal has insufficient SNR to be decoded, evenif the interference from the other signal is removed.

• R: None of the signals has sufficient SINR to be decodedimmediately, but the signal of interest has sufficient SNRto be decoded if interference from the other is removed.

The probability of each of the outcomes, denoted as πo1o2 , isderived in the following based on the operation of SIC andunder Rayleigh fading.• (I, I): The signal with the highest SINR is decoded

and its interference is immediately removed through SIC.Then, the second signal is decoded. This occurs withprobability

πII = Pr

[SNR1,t

1 + SNR2,t≥ γ1 ∧ SNR2,t ≥ γ2

]+ Pr

[SNR2,t

1 + SNR1,t≥ γ2 ∧ SNR1,t ≥ γ1

]=

SNR1

SNR1 + γ1SNR2

e−γ1SNR1 e

−γ2(γ1

SNR1+ 1

SNR2

)

+SNR2

γ2SNR1 + SNR2

e−γ2SNR2 e

−γ1(

1SNR1

+γ2

SNR2

).

• (I, E) and (E , I): The signal with the higher SINR isdecoded and its interference is immediately removedthrough SIC. However, the second signal cannot bedecoded due to the impact of noise, i.e., a low SNR.These outcomes occur with probabilities

πIE = Pr

[SNR1,t

1 + SNR2,t≥ γ1 ∧ SNR2,t < γ2

]=

SNR1e−γ1SNR1

SNR1 + γ1SNR2

(1− e−γ2

(γ1

SNR1+ 1

SNR2

)). (10)

πEI = Pr

[SNR2,t

1 + SNR1,t≥ γ2 ∧ SNR1,t < γ1

]=

SNR2e−γ2SNR2

γ2SNR1 + SNR2

(1− e−γ1

(1

SNR1+

γ2SNR2

)). (11)

• (E , E): The SNR of both signals is insufficient and, thus,neither can be decoded even if the interference from theother user were removed. This occurs with probability

πEE = Pr [SNR2,t < γ2 ∧ SNR1,t < γ1] (12)

=

(1− e

−γ1SNR1

)(1− e

−γ2SNR2

). (13)

• (R, E): The signal from user 2 has insufficient SNR,while the signal from user 2 has a sufficient SNR butinsufficient SINR. Since the system cannot remove theinterference from user 2 without decoding it first, bothpackets remain undecoded. This outcome occurs withprobability

πRE = Pr [γ1 ≤ SNR1,t < γ1 (1 + SNR2,t)

∧ SNR2,t < γ2]

=

(SNR1

SNR1 + γ1SNR2

(e−γ2

(γ1

SNR1+ 1

SNR2

)− 1

)+

(1− e

−γ2SNR2

))e−γ1SNR1

• (·,R): In this case, none of the signals can be im-mediately recovered but the signal from user 2 could

5

be decoded if the interference from user 1 is removedvia SIC after decoding the block of user 1. Therefore,this outcome includes the cases (E ,R) and (R,R), andoccurs with probability

π·R = 1− πII − πIE − πEI − πEE − πRE . (14)

Note that the cases (I,R) and (R, I) are not feasible, asoutcome I indicates that a signal is immediately decoded andthat its interference to the other signal is removed.

C. Key Performance IndicatorsThe broadband user (user 1) is interested on maximizing its

throughput S under the constraint that the desired reliabilityps,1 must be greater than 1 − ε1. Note that increasing thereliability of the broadband user entails a reduction in thecoding rate K/N .

The intermittent user (user 2) is interested on the timelinessof its data, i.e., either LR or PAoI, where we have selectedtheir 90th percentile as the main KPIs. Let T and ∆ be theRVs of LR and PAoI, respectively. Then, the 90th percentileof LR is defined as

T90 := minn{n : Pr [T ≤ n] > 0.9} (15)

and the 90th percentile of PAoI ∆90 is defined analogously.Note that the latter allows us to evaluate the tail distributionof the PAoI in a general scenario and can be used to comparethe performance with different values of α [32].

Since S and the timeliness of the intermittent user areinterlinked, we evaluate their trade-offs for a specific activationprobability α and erasure probabilities εu, via the Paretofrontier defined in the following.

Definition 1. Let C be the set of feasible configurations for aspecific access method and f : C → R2. Next, let

Y = {(S, τ) : (S, τ) = f(c), c ∈ C},where S is the throughput of user 1 and τ is the timeliness ofuser 2; τ ∈ {T90,∆90}. The Pareto frontier is the set

P(Y ) = {(S, τ) ∈ Y : ∀(S′, τ ′) ∈ Y : S > S′ ∨ τ < τ ′}.Besides obtaining the Pareto frontiers, we evaluate the

schemes by setting a minimum requirement for S, the through-put of user 2. Then, the optimal configuration of an accessmethod is defined as the combination of parameters thatminimizes the timing, either LR or PAoI while maintainingS above the minimum required.

Table I summarizes the relevant notation introduced in thissection. To simplify the analytical expressions in the rest ofthe paper, we define the binomial function Bin(K;N, p) as

Bin(K;N, p) =

(N

K

)pK(1− p)N−K (16)

and the multinomial function Mult(K;N,p) as

Mult(K;N,p) =N !∏|p|i=1 p

Kii (1−

∑|p|i=1 pi)

N−∑|p|i=1Ki

(N −∑|p|i=1Ki)!

∏|p|i=1Ki!

,

(17)where |p| is the length of vector p. Finally, we denote δ(x)as the delta function, which is equal to 1 if x = 0 and 0otherwise, and [x]+ = max(x, 0).

IV. PERFORMANCE WITH TDMA

Here we derive the KPIs for the TDMA system, for a LR-or PAoI-oriented intermittent user. For LR, the length of theintermittent user’s queue is assumed to be fixed to some Q ≥1. On the other hand, for PAoI, the length of the intermittentuser’s queue is set to Q = 1. This is because transmitting thenewest packet is the optimal strategy to minimize PAoI butpacket retransmissions are not allowed.

In the assumed TDMA system, the broadband user hasframes of N slots, each of which contains K data packetsand N − K redundancy packets, while the intermittent userhas one reserved slot every Tint. The success probability foruser 1 is easy to compute

ps,1 =

N∑m=K

Bin(m;N, 1− ε1). (18)

The expected throughput of user 1 is

S = ps,1(Tint − 1)K

TintN. (19)

That is, the throughput measures the rate of innovative (i.e.,non-redundant) packets received at the BS from user 1 pertime slot. As the broadband user can only use Tint−1 slots foreach Tint, setting up more frequent transmission opportunitiesfor the intermittent user reduces the throughput.

A. Latency-reliability (LR)

In order to derive the probability mass function (pmf) of LRfor the intermittent user, without loss of generality, we takethe origin of time to be a slot in which a transmission occurs.We define a Markov chain representing the state of the queueqt for the intermittent user, i.e., the number of packets in thequeue at time t. The transition matrix of the chain is P(0),whose elements P (0)

ij represent the probability of transitioningfrom state i to state j in the queue of the intermittent user atthe end of such slot [33]. The elements P (0)

ij are obtained as

P(0)ij =

0 if j < i− 1;

Bin(j − i+ 1;Tint, α) if i− 1 ≤ j < Q;∑Tintm=Q−i+1 Bin(m;Tint, α) if j = Q.

(20)Let ϕ(0) =

[ϕ

(0)0 , ϕ

(0)1 , . . . , ϕ

(0)Q

]be the steady-state distri-

bution vector of the queue immediately after a transmission.From the transition matrix computed in (20), we can easilyderive ϕ(0) as the left-eigenvector of P (0) with eigenvalue 1,normalized to sum to 1 to be a valid probability metric

ϕ(0)(I−P(0)) = 0 ∧Q∑q=0

ϕ(0)q = 1. (21)

It is easy to derive the steady-state distribution of the queueqn (i.e., n slots after a transmission) as

ϕ(n)q =

{∑qs=0 ϕ

(0)s Bin(q − s;nα) if q < Q;∑Q

s=0

∑nm=Q−s ϕ

(0)s Bin(m;n, α) if q = Q,

(22)

6

TABLE I: Notation summary

Symbol Description Symbol Description

U = {1, 2} Set of users; u = 1 is the broadband user Pu Transmission power of user uand u = 2 is the intermittent user SNRu Expected SNR for user u

K Size of the source block for user 1 SNRu,t SNR of user u at slot tN Size of the coded block for user 1 SINRu,t SINR of user u at slot tQ Maximum queue length for user 2 εu Erasure probability of user uTint Period between slots allocated to user 2 in TDMA ou ∈ {I,R, E} Outcome for user u when signals overlapt ∈ Z Time slot index (o1, o2) Outcome when signals overlapqt Length of the queue for user 2 at t πo1o2 Probability of outcome (o1, o2)

At ⊆ U Allocation of time slot t ps,u Success probability of user uau,t Activity indicator for user u at t; 1 if active S Throughput of user 1hu,t Fading envelope for user u at t T RV of LR for user 2σ2 Noise power ∆ RV of AoI for user 2ù Path loss of user u T90, ∆90 90th percentile of LR and AoIr Distance between user 2 and the BS δ(x) Delta function, equal to 1 if x = 0 and 0 otherwise

where ϕ(0)q is the q-th element of vector ϕ(0). If a packet is

queued behind q others, it will be transmitted at the (q+1)-thopportunity, unless new arrivals make the system drop someof the packets ahead of it in the queue: we remind the readerthat, if the queue is full, the oldest packet (i.e., the first in thequeue) is dropped. Let gi ∈ {0, 1, . . . , Tint} for i ≥ 1 be thenumber of packets generated by user 2 between the i-th and(i + 1)-th intermittent slots after the current one. Further, letg0 be the number of packets generated between the currenttime slot and the next intermittent slot. We define

G(n)` =

{[g0 ∈ {0, 1, . . . , Tint − n}, g1, . . . , g`]

}(23)

to be the set of possible vectors for the number of packetsgenerated by user 1 given that there are Tint−n slots until thenext intermittent slot.

The probability of occurrence of each element g ∈ G(n)` is

pgen(g; `, n) = Bin(g0;Tint − n, α)∏i=1

Bin(gi;Tint, α). (24)

At each intermittent slot, up to one packet is transmitted and,hence, removed from the queue. Other packets are removedif the number of generated packets exceeds the number ofremaining spaces in the queue. For a given vector g ∈ G(n)

` ,the considered packet is transmitted at the `-th transmissionopportunity, where ` is the first index that satisfies conditionψ

(g,q)k if the packet has q others ahead of it in the queue

ψ(g,q)k = δ

k∑i=1

q + 1−Q+

i∑j=1

gj

+

+ k − (q + 1)

.

(25)We now define the set S(n,q)

` , which contains the elementsg ∈ G(n)

` for which the considered packet is transmitted at the`-th opportunity

S(n,q)` =

{g ∈ G(n)

` : ψ(g,q)` −

`−1∑k=1

ψ(g,q)k = 1

}. (26)

Since the packet is either transmitted within q+1 transmissionattempts or discarded, the conditioned success probabilityps,2(n, q;Tint) for the intermittent user is given by

ps,2(n, q) =

q+1∑`=1

∑g∈S(n,q)

`

pgen(g; `, n)(1− ε2). (27)

We can now compute the latency pmf pT (t), considering thefact that it takes 1 slot to transmit the packet

pT (t) =

Tint∑n=1

Q∑q=0

∑g∈S(n,min(q,Q−1))

`

pgen

(g; t+n−1

Tint, n)

Tintps,2(n,min(q,Q− 1))

×ϕ(n−1)q δ

(t+ n− 1

Tint−⌊t+ n− 1

Tint

⌋). (28)

The success probability of the intermittent user is

ps,2 =

Tint∑n=1

Q∑q=0

ϕ(n−1)q ps,2(n, q)

Tint. (29)

B. Peak age

In the PAoI-oriented case, the pmf is given by the sum ofthe waiting time W and the inter-update interval Z [21].

Since Q = 1, the generated packets are always sent at thefirst available transmission opportunity. The pmf of the waitingtime W for a successful transmission is given by:

pW (w) =α(1− α)w−1

1− (1− α)Tint, w ∈ {1, . . . , Tint}. (30)

We now compute the pmf of Z. Since exactly one slot everyTint is reserved for the intermittent user, Z is Tint timesthe number of reserved slots between consecutive successfultransmissions. This is a geometric random variable, whoseprobability of successful transmission is given by:

ξ = (1− (1− α)Tint)(1− ε2). (31)

The pmf of Z is then

pZ(z) = (1− ξ)zTint−1ξδ (mod(z, Tint)) . (32)

7

The pmf of the PAoI is

p∆(t+ 1) = pZ (t−mod(t, Tint)) pW (1 + mod(t, Tint)) .(33)

V. PERFORMANCE WITH NOMA

We now derive the distributions of the KPIs in the NOMAcase, in which the broadband user has frames of N slots, allof which are mixed, i.e., allocated both to the intermittent andbroadband user.

First, we define p1 as the probability that a packet from thebroadband user is received in a given slot, which is given by

p1 = ((1− α)(1− ε1) + α(πII + πIE)) . (34)

The probability that the block from the broadband user isdecoded in the d-th slot of the frame, denoted as pD(d), is

pD(d) =p1Bin(K − 1; d− 1, p1), (35)

The Cumulative Distribution Function (CDF) of the decodinginstant D, PD(d), is

PD(d) =

d∑m=K

Bin(m; d, p1). (36)

We then simply have ps,1 = PD(N). The average throughputfor the broadband user is

S =KPD(N)

N. (37)

A. Latency-reliability (LR)

We now analyze the latency distribution for the intermittentuser. All the intermittent packets transmitted after decodingslot d – once the block from the broadband user has beendecoded – can be either decoded immediately or lost. Specif-ically, these packets are lost with probability ε2 = πIE +πEE + πRE . On the other hand, if the intermittent user packetis sent before the decoding slot d, it is decoded instantly withprobability πII+πEI , while it can be decoded after SIC withprobability π·R. In order to compute ps,2, we need to computethe conditioned probability of having ab collisions between thetwo users before the broadband user block is decoded in slotd, ib of which result in an immediate decoding, while vb aredecoded after SIC, for the intermittent user. This is denotedas pAb,Ib,Vb|D(ab, ib, vb|d), and given by:

pAb,Ib,Vb|D(ab, ib, vb|d) =

min(ab,K−1)∑`=K−d+ab

Bin(ab; d− 1, α)

× p1

pD(d)

min(ib,`)∑m=0

Bin(K − 1− `; d− 1− ab, 1− ε1)

×Mult ([m, ib −m, vb, `−m]; ab, [πII , πEI , π·R, πIE ]) .(38)

We can then simply take the four cases for packets from theintermittent user (transmitted before slot d, in slot d, afterslot d, or in lost frames), and compute ps,2. We can computepAd,Id(ad, id), the probability that a packet from user 2 is sent

and correctly decoded in the same slot as the broadband userblock decoding:

pAd,Id(ad, id) =

απIIp1

, ad = 1, id = 1;απIEp1

, ad = 1, id = 0;(1−α)(1−ε1)

p1, ad = 0, id = 0;

0, otherwise.

(39)

We then give the probability of having Aa packets after thedecoding of the broadband user block in slot d, Ia of whichare correctly received:

pAa,Ia|D(aa, ia|d) =Bin(ia; aa, πII + πEI + π·R)

pD(d)

× Bin(aa;N − d, α).

(40)

Finally, we can consider the case in which the broadband userframe is not decoded: in this case, the only intermittent packetsthat are decoded are immediate captures. We can then computethe probability pAz,Iz|D(az, iz):

pAz,Iz|D(az, iz) =

min(K−1,N−a)∑c=0

Bin(c;N − a, 1−)

min(az,K−1−c)∑`=0

Bin(az;N,α)

×min(`,iz)∑m=0

Mult([m, iz −m, `−m]; az, [πII , πEI , πIE ])

1− ps,1.

(41)We now know that all packets transmitted by the intermittentuser at or after the decoding of the broadband block, or inframes for which the broadband block is not decoded, areeither lost or decoded immediately. To compute the latencydistribution, we then only need to distinguish the case in whicha packet transmitted before d is decoded instantly or after SIC.The probability of a packet from the intermittent user beingdecoded instantly is then pT (1):

pT (1) =(1− ps,1)

N∑az=1

az∑iz=0

izpAz,Iz|D(az, iz)

(1− Bin(0;N,α))az

+N∑d=K

d−1∑ab=0

1∑ad=0

N−d∑aa=0

a∑ib=0

1∑id=0

m∑ia=0

ab−ib∑vb=0

pD(d)

×(ib + id + ia)pAd,Id(ad, id)pAa,Ia|D(aa, ia|d)

(1− Bin(0;N,α))(ab + ad + aa)

× pAb,Ib,Vb|D(ab, ib, vb|d).(42)

As the delay from any packet decoded after SIC is distributeduniformly between 2 and d+1, we can easily compute pT (t):

pT (t) =

N∑d=min(K,t−1)

pD(d)

d−1∑ab=0

1∑ad=0

N−d∑aa=0

ab∑ib=0

ab−ib∑vb=0

vbd

×1∑

id=0

m∑ia=0

pAd,Id(ad, id)pAa,Ia|D(aa, ia|d)

(ab + ad + aa)(1− Bin(0;N,α))

× pAb,Ib,Vb|D(ab, ib, vb|d), t ∈ {2, . . . , N}.

(43)

The combination of (42) and (43) is the latency-reliability pmffor the intermittent user. We then have:

ps,2 =

N∑t=1

pT (t). (44)

8

B. Peak age

In order to derive the pmf of the PAoI, we first need tocompute some auxiliary values. First, we derive the probabilitythat the first decoded packet from the intermittent user in aframe is decoded in slot f , denoted as pF (f) and given in (45).

It is then easy to get pe, the probability of decoding no newintermittent packets in a frame:

pe = 1−N∑f=1

pF (f). (46)

The pmf of the number of slots Y from the beginning of agiven frame until the first decoded packet from the intermittentuser is

pY (y) = pb yN ce pF (mod(y,N)), (47)

where mod(m,n) is the integer modulo function.We now consider the probability pU (u) of receiving an

update from the intermittent user, i.e., a packet with newerinformation than the one already available. We have thefollowing pmf, conditioned on the decoding slot d of thebroadband block. First, we consider the case in which d < u

pU |D(u|d) =

d−1∑ab=1

ab∑ib=1

ab−ib∑vb=0

ibpAb,Ib,Vb|D(ab, ib, vb|d)

d− 1, d < u.

(48)Next, for d > u

pU |D(u|d) =

N−d∑aa=1

aa∑ia=1

iaN − d

pAa,Ia|D(aa, ia|d), d > u.

(49)Finally, for d = u

pU |D(d|d) =

d−1∑ab=1

ab−1∑ib=0

ab−ib∑vb=1

d−1∑m=ib+vb

1∑ad=0

vbpAd,Id(ad, 0)

d− 1

×Hm−1,d−1(ib + vb − 1, ib + vb − 1)

× pAb,Ib,Vb(ab,ib,vb|d)|D + pAd,Id(1, 1),(50)

where HM,N (m,n) is the hypergeometric distribution, whosepmf is given by

HM,N (m,n) =

(Mm

)(N−Mn−m

)(Nn

) . (51)

We also consider the probability pU |D(u), i.e., the probabilityof receiving an update in slot u if the broadband user blockis not decoded

pU |D(u) =

N∑az=1

az∑iz=1

izpAz,Iz|D(az, iz)

N. (52)

By applying the law of total probability, we obtain pU (u)

pU (u) =

N∑d=K

pD(d)pU |D(u|d) + (1− ps,1)pU |D(u). (53)

We now compute the probability that a given update is thelast in the frame, given that the decoding happens in slot

d, denoted as pL|D(`|d). Again, we distinguish three cases,starting from ` < d:

pL|D(`|d) =

d−1∑ab=1

∑ib=1

`−ib∑vb=0

1∑ad=0

N−d∑aa=0

ibpAd,Id(ad, 0)

d− 1

× pAb,Ib,Vb|D(ab, ib, vb|d)pAa,Ia|D(aa, 0|d)

× H`−1,d−2(vb + ib − 1, vb + ib − 1)

pU |D(`|d), ` < d.

(54)If ` = d, we have:

pL|D(d|d) =

N−d∑aa=0

pAa,Ia|D(aa, 0|d). (55)

Finally, if ` > d the probability is

pL|D(`|d) =

`−d∑aa=1

aa∑ia=1

iapAa,Ia|D(aa, ia|d)

(N − d)pU |D(`|d)

×H`−d−1,N−d−1(ia − 1, ia − 1), ` > d.

(56)

The probability that an update in slot ` is the last in the frame,given that the broadband user frame is lost, pL|D(`), is

pL|D(`) =∑az=1

az∑iz=1

pAz,Iz (az, iz)izHN−`,N−1(0, iz − 1)

NpU |D(`).

(57)Combining the expressions derived above, we get

pL(`) =

N∑d=K

pD(d)pL|D(`|d) + (1− ps,1)pL|D(`). (58)

If the update is not the last in the frame, we can compute theconditioned pmf pZ|U,D,L(z|u, d) of the inter-update intervalZ. We first consider the case in which z + u < d

pZ|U,D,L(z|u, d) =

d−1∑ab=2

ab∑ib=2

ib(ib − 1)Hz−1,d−3(0, ib − 2)

pU |D(u|d)

×ab−ib∑vb=0

pAb,Ib,Vb|D(ab, ib, vb|d)

(d− 1)(d− 2)(1− pL|D(u|d)), u + z < d.

(59)In this case, the only possibility to have another update after zis to have two immediate captures in slots u and u+z, withoutany immediate captures in between. Further, for u > d

pZ|U,D,L(z|u, d) =

N−d−z+1∑aa=2

aa∑ia=2

pAa,Ia|D(aa, ia|d)

(N − d)(N − d− 1)

× ia(ia − 1)Hz−1,N−d−2(0, ia − 2)

(1− pL|D(u|d))pU |D(u|d), u > d ∧ u + z ≤ N.

(60)Next, for u = d:

pZ|U,D,L(z|d, d) =

N−d−z+1∑aa=1

aa∑ia=1

iapAa,Ia|D(aa, ia|d)

(N − d)(1− pL|D(d|d))

×Hz−1,N−d−1(0, ia − 1), d+ z ≤ N.(61)

9

pF (f) =

f−1∑a=0

αεC(·, I)

min(K−1,a)∑`=0

Bin(a; f − 1, α)

min(K−1−`,f−a−1)∑m=0

Mult([0, 0, `]; a, [πII , πEI , πIE ])× Bin(m; f − a− 1, 1− ε1)

+ pD(f)

1∑ad=0

pAb,Ib,Vb|D(ab, 0, vb|f)pAd,Id(ad, 0) +

f−1∑d=K

pD(d)×f∑

ab=0

pAb,Ib,Vb|D(ab, 0, 0|d)

×1∑

ad=0

pAd,Id(ad, 0)Bin(0; f − d− 1, απ·Rαπ·R.

(45)

We then consider the case that u + z = d

pZ|U,D,L(d− u|u, d) =

d−1∑ab=1

ab∑ib=1

ab−ib∑vb=0

pAb,Ib,Vb|D(ab, ib, vb|d)

× Hu−1,d−2(ib − 1, ib − 1)

(d− 1)pU |D(u|d)(1− pL|D(u|d))

(pAd,ID (1, 1)

+1(vb − 1)

1∑ad=0

pAd,Id(ad, 0)(1−Hu−ib,d−ib−1(vb, vb))

),

(62)where 1(x) is the step function, equal to 1 if x ≥ 0 and0 otherwise. Finally, we can derive pZ|U,D,L(z|u), if thebroadband user frame is not decoded:

pZ|U,D,L(z|u) =

N∑az=2

az∑iz=2

Hz−1,N−2(0, iz − 2)

N(N − 1)(1− pL|D(u)pU |D(u)

× pAz,Iz (az, iz), u + z ≤ N.(63)

We now compute the pmf of the inter-update interval Z ifthe next packet is in the same frame

pZ(z|u) =

N∑d=K

pZ|U,D,L(z|u, d)pD(d) + (1− ps,1)

× pZ|U,D,L(z|u), z ≤ N − u.

(64)

On the other hand, if z > N − u, we have

pZ(z|u) = pL(u)pb z−(N−u)

N ce pF (mod(z − (N − u), N)),

∀z > N − u.(65)

The decoding delay W component of PAoI applies only ifthe update transmitted before the decoding slot d and decodedwith SIC only after the decoding of the broadband user block.We then give pW |U,D(w|u, d), the pmf of W for an update inthe same slot d which the broadband user block is decoded in

pW |U,D(w|d, d) =1

pU |D(u|d)

(pAd,Id(1, 1)δ(w − 1)+

×min(ab,d−w+1)∑

vb=1

Hw−2,d−2(0, vb − 1)

min(ab,d−w+1)−vb∑ib=0

Hw−2,d−vb−1(0, ib)

×pAb,Ib,Vb|D(ab, ib, vb|d)

(d− 1)

1∑ad=0

pAd,Id(ad, 0)

), u = d.

(66)

In all other cases, the packet is captured instantaneously, andwe simply have

pW |U,D(w|u, d) = δ(w − 1), u 6= d. (67)

(E, I) (I, E) (R, E) or (E, E)

(I, I)

(·,R)

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

Distance between user 2 and the BS r (m)

Prob

abili

tyπo1o2

(a) η = 2.6

(E, I) (R, E) or (E, E)

(I, I)

(I, E)(·,R)

0 100 200 300 4000

0.2

0.4

0.6

0.8

1

Distance between user 2 and the BS r (m)

Prob

abili

tyπo1o2

(b) η = 3

Fig. 4: Area plot for the probabilities of the different outcomes(o1, o2) when the signals of both users collide for (a) η = 2.6and (b) η = 3. The dashed line indicates the value of ε2.

If the broadband user frame is not decoded, the decoding delayis always 1, as the only updates are due to immediate capture

pW |U,D(w|u) = δ(w − 1). (68)

By applying the law of total probability, we get

pW |U (w|u) =

N∑d=K

pD(d)pW |U,D(w|u, d)

+ (1− ps,2)pW |U,D(w|u).

(69)

Finally, we get the PAoI as the convolution of Z and W andremoving the condition on U

p∆(t) =

N∑u=1

pU (u)

min(u,t−1)∑w=1

pW |U (w|u)pZ|U (t− w|u). (70)

10

TABLE II: Parameter settings

Parameter Symbol Setting Parameter symbol Setting

Coded block length for user 1 N {2, 3, . . . , 32} Source block length for user 1 K < N

Erasure probability of user 1 ε1 0.1 Transmission power of user 2 P2 23 dBmActivation probability for user 2 α {0.01, 0.05, 0.1} Period between intermittent slots in TDMA Tint {1, 2, . . . , 40}SINR threshold to decode a packet γ1 = γ2 3 dB Noise power σ2 −127.216 dBmDistance from user 2 to the BS r {50, 100, . . . , 400}m Carrier frequency fc 2 GHzPath loss exponent η {2.6, 3} Queue length in TDMA Q 4 packets

VI. BENCHMARK: PERFORMANCE WITH FDMA

In case of FDMA, the two users are occupying a dedicatedbandwidth part each, and their KPIs are independent. Thesuccess probability for user 1 is equal to that in TDMA, givenby (18). The throughput of user 1 can be computed by settingTint →∞ in (19), which gives

S =Kps,1N

. (71)

For user 2, the latency for all successfully decoded packetsis 1. Further, ps,2 = 1 − ε2 and the pmf of LR is simplypL(t) = δ(t− 1).

The PAoI for user 2 can be obtained as the latency T = 1plus the inter-decoding time Z when setting Tint = 1 in (31)and (32). Hence, it is simply a function of the inter-arrivaltime and ε2. Namely,

p∆(t) = (1− α (1− ε2))t−2

α (1− ε2) , t ≥ 2. (72)

VII. EVALUATION

We assume that user 1 (the broadband user) selects itstransmission power to achieve ε1 = 0.1. On the other hand,user 2 (the intermittent user) transmits infrequently, and thuscannot get up-to-date information on the channel state. Thebest possible strategy for it is then to always transmit atmaximum power; in this case, ε2 depends on its distance fromthe BS r and the erasure probability ε2 is minimized.

For performance evaluation, we set parameters that rep-resent a typical 5G urban scenario [1]. Namely, the carrierfrequency is 2 GHz, the path loss exponent is η ∈ {2.6, 3} dB,the noise power σ2 is determined by the noise temperatureand the subcarrier spacing, set to a typical ∆f = 15 kHz,plus a noise figure of 5 dB. The resulting noise power andother relevant parameter settings are listed in Table II. Forsimplicity’s sake, the SINR thresholds for decoding both usersare set to the same value γ1 = γ2 = 3 dB. As a reference, theSINR threshold when calculating the maximum coverage in5G is 0 dB [1]. Fig. 4 show the area plots for the probabilityof the outcomes when both users transmit in the same slotfor η ∈ {2.6, 3}. The figure shows that a high reliability forthe intermittent user is only achievable when it is close tothe base station, particularly when η = 3. On the other hand,recovering packets after decoding the broadband user block iscrucial, as case (·,R) occurs with a relatively high probabilityfor both values of η and is critical to achieve high reliabilityfor the intermittent user.

An essential aspect of our analysis is to identify the valuesof K and N that maximize the throughput S of user 1.

These can be selected independently of user 2’s parametersfor TDMA and FDMA and, hence, represent the optimalconfiguration for user 1 with these schemes.

Note that implementing a longer coded block size N wouldgrant a greater throughput, bounded by 1−ε1 for N →∞, butwould also lead to a longer decoding latency and complexity.Hence, we limit the value of N ≤ 32 to achieve an adequatebalance between S and decoding latency and complexity. Byrestricting N ≤ 32, the optimal configuration for user 1 forboth TDMA and FDMA is K = 26 and N = 32, which leadsto ps,1 = 0.964. With this configuration, FDMA achieves athroughput of S = 0.7833 for all cases, as user 1 operates ina separate channel from user 2 and, hence, there is no trade-off between S and the KPI of user 2. On the other hand, theoptimal configuration for TDMA and NOMA depends on thedesired performance trade-off and, hence, these are given atthe end of this section.

A. Pareto analysis

We first present the Pareto frontier for throughput of user1 S and timing of user 2, for LR T90 or PAoI ∆90, whichdescribes the best achievable trade-offs between these KPIs.We consider three different distances (50, 150, and 250 m) forthe intermittent user, with three different activation probabili-ties. It is easy to see in Fig. 5 that NOMA easily outperformsTDMA in terms of LR and throughput in all scenarios.

Furthermore, Fig. 5a-c show that T90 = 1 can be achievedwith NOMA if the distance and path loss allow to immediatelydecode more than 90% of the packets from user 2 due tocapture and the use of SIC in the same slot. In these cases,the throughput with NOMA is only up to 2% lower than withFDMA. Therefore, NOMA is the most efficient access schemein these cases, as it achieves a similar performance to FDMAbut with half the resources: one bandwidth part instead of two.

On the other hand, there is a strict trade-off between LR andthroughput for all cases with TDMA, as the only way to reducethe latency is to decrease the period between intermittent slotTint, which decreases the resources assigned to the broadbanduser. The same trade-off appears with NOMA for the caseswhere πI,I + πE,I < 0.9 due to an increase in path loss,shown in Fig. 5d-e. In these, reducing the latency also requiresreducing the efficiency of the code. However, the Paretofrontier for NOMA is always above and to the left of thecurve for the equivalent scenario with TDMA, showing thatNOMA is clearly the best choice in this scenario. The frontiersfor r = 250 m and path loss exponent η = 3 are not shown, as

11

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)

S(p

acke

ts/s

lot)

NOMA, α = 0.01TDMA, α = 0.01NOMA, α = 0.05TDMA, α = 0.05NOMA, α = 0.1TDMA, α = 0.1

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)

S(p

acke

ts/s

lot)


(a) r = 50 m, η = 2.6.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)

S(p

acke

ts/s

lot)


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)

S(p

acke

ts/s

lot)


(b) r = 50 m, η = 3.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)

S(p

acke

ts/s

lot)


0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)

S(p

acke

ts/s

lot)


(c) r = 150 m, η = 2.6.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)S

(pac

kets

/slo

t)


(d) r = 150 m, η = 3.

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

FDMA

T90 (slots)

S(p

acke

ts/s

lot)


(e) r = 250 m, η = 2.6.

Fig. 5: Pareto frontiers for latency-reliability versus throughput with TDMA and NOMA, with different values of α. The crossmarks indicate the performance with FDMA (benchmark).

in this case it is impossible for the intermittent user to deliver90% of packets at any finite latency (i.e., T90 =∞).

NOMA also achieves a lower PAoI with high throughput forall cases in Fig. 6a-c. In these, the Pareto frontier increasesabruptly and reaches its maximum S ≈ 0.78, which is closeto the one achieved with FDMA of 0.7833. This occurs atexactly or only a few time slots later than the minimum ∆90.Thus, the resource efficiency of NOMA is much greater thanthat of FDMA and achieves similar trade-offs.

On the other hand, for r ≥ 150 m, TDMA achieves a higherthroughput than NOMA at the expense of an increase in PAoI.This is expected, as greater values of Tint increase S but also∆. Specifically, as described in Section VI, the throughputwith TDMA for Tint →∞ is equal to that with FDMA.

Note, however, that the activation rate α has the greatestimpact on the PAoI. This is because the interval time between

consecutive packets with low values of α can be so long thatreducing the latency for each individual packet has only aminor effect on the PAoI.

B. Distance analysis

We now investigate the performance of the schemes as afunction of the distance r between user 2 and the base station.In this case, we also consider the case for NOMA with fullydestructive interference and, hence, no capture, which wasinvestigated in our previous work [11]. In this later case, wehave π·,R = 1 − ε2 and πE,E = ε2 for any slot in which thetwo users collide, eliminating the possibility of instantaneousSIC. This scenario is naturally a lower bound for NOMA’sperformance, as removing the possibility of capture makes thescheme perform significantly worse.

12

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1 α = 0.1α = 0.05 α = 0.01

FDMA

∆90 (slots)

S(p

acke

ts/s

lot)

NOMA, α = 0.01 TDMA, α = 0.01NOMA, α = 0.05 TDMA, α = 0.05NOMA, α = 0.1 TDMA, α = 0.1

(a) r = 50 m, η = 2.6.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1 α = 0.1α = 0.05 α = 0.01

FDMA

∆90 (slots)

S(p

acke

ts/s

lot)


(b) r = 50 m, η = 3.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1 α = 0.1α = 0.05 α = 0.01

FDMA

∆90 (slots)

S(p

acke

ts/s

lot)


(c) r = 150 m, η = 2.6.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1 α = 0.1α = 0.05 α = 0.01

FDMA

∆90 (slots)

S(p

acke

ts/s

lot)


(d) r = 150 m, η = 3.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1 α = 0.1α = 0.05 α = 0.01

FDMA

∆90 (slots)

S(p

acke

ts/s

lot)


(e) r = 250 m, η = 2.6.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1 α = 0.1α = 0.05 α = 0.01

FDMA

∆90 (slots)

S(p

acke

ts/s

lot)


(f) r = 250 m, η = 3.

Fig. 6: Pareto frontiers for PAoI versus throughput with TDMA and NOMA, with different values of α. The cross marksindicate the performance with FDMA (benchmark).

Fig. 7 shows the performance of the schemes in termsof the minimum LR T90 that can be achieved while fulfill-ing a relatively high throughput requirement S ≥ 0.7 forα = 0.01, 0.05. In general, NOMA can outperform TDMAin most cases, but it is interesting to observe the behaviorof the schemes when α is high. In these cases, we notice aperformance drop for both TDMA, which has to allocate moreslots to the intermittent user, and NOMA without capture,which has to increase the robustness of the packet-level code toprotect the transmission from the additional intermittent userpackets. On the other hand, capture allows NOMA to be morerobust to the increased activation probability, maintaining aperformance that is close to FDMA. In fact, while not shownin the figures, NOMA and, naturally, FDMA are the only

schemes that can achieve S ≥ 0.7 with α = 0.1, while theother schemes do not achieve the required throughput for anyconfiguration.

On the other hand, we can confirm the trend that weobserved in Fig. 6 for PAoI at different distances, as Fig. 8shows that NOMA achieves a slightly lower ∆90 than TDMA.However, capture is essential for the NOMA scheme withhigher values of α: without it, it performs slightly worsethan TDMA for α = 0.05, and it never reaches the requiredthroughput for α = 0.1. Finally, it can be seen that NOMAachieves similar values of ∆90 than FDMA for (1) most valuesof r with η = 2.6 and (2) short distances r ≤ 150 with η = 3.This demonstrates that, in the cases where the system canbenefit from capture and SIC, NOMA is nearly equivalent to

13

50 100 150 200 250 300 350 4000

20

40

60

80

100

r (m)

T90

(slo

ts)

NOMA NOMA (no capture)TDMA FDMA

(a) α = 0.01, η = 2.6.

50 100 150 200 250 300 350 4000

20

40

60

80

100

r (m)

T90

(slo

ts)


(b) α = 0.01, η = 3.

50 100 150 200 250 300 350 4000

20

40

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80

100

r (m)

T90

(slo

ts)


(c) α = 0.05, η = 2.6.

50 100 150 200 250 300 350 4000

20

40

60

80

100

r (m)

T90

(slo

ts)


(d) α = 0.05, η = 3.

Fig. 7: Minimum LR with S ≥ 0.7 as a function of the distance between user 2 and the BS for different values of α.

50 100 150 200 250 300 350 4000

50

100

150

200

r (m)

∆90

(slo

ts)


(a) α = 0.05, η = 2.6.

50 100 150 200 250 300 350 4000

25

50

75

100

125

150

r (m)

∆90

(slo

ts)


(b) α = 0.05, η = 3.

50 100 150 200 250 300 350 4000

25

50

75

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125

150

r (m)

∆90

(slo

ts)

NOMA TDMAFDMA

(c) α = 0.1, η = 2.6.

50 100 150 200 250 300 350 4000

25

50

75

100

125

150

r (m)

∆90

(slo

ts)

NOMA TDMAFDMA

(d) α = 0.1, η = 3.

Fig. 8: Minimum PAoI with S ≥ 0.7 as a function of the distance between user 2 and the BS for different values of α.

14

50 100 150 200 250 300 350 4000

10

20

30

40

r (m)

Val

ueK (NOMA) K (NOMA, no cap.)N (NOMA) N (NOMA, no cap.)Tint (TDMA)

(a) LR-oriented system, α = 0.01.

50 100 150 200 250 300 350 4000

10

20

30

40

r (m)

Val

ue

K (NOMA) K (NOMA, no cap.)N (NOMA) N (NOMA, no cap.)Tint (TDMA)

(b) LR-oriented system, α = 0.05.

50 100 150 200 250 300 350 4000

10

20

30

40

r (m)

Val

ue


(c) PAoI-oriented system, α = 0.01.

50 100 150 200 250 300 350 4000

25

50

75

100

125

150

r (m)V

alue


(d) PAoI-oriented system, α = 0.05.

Fig. 9: Optimal settings for the three schemes with S ≥ 0.7 as a function of the distance between user 2, with η = 2.6.

FDMA in terms of performance, even when the latter utilizestwice the amount of resources. These cases occur, for example,when pairing the broadband user with an intermittent userlocated near the BS and, hence, that achieves a high meanSNR.

C. Parameter analysis

We conclude by investigating the optimal configurations forthe schemes as a function of the distance of user 2, under theconstraint S ≥ 0.7. Fig. 9 shows the optimal values of Kand N for NOMA and Tint for TDMA, for both LR-orientedand PAoI-oriented systems with η = 2.6. Fig. 9a-b, whichare related to LR-oriented systems, show that the value of Tintis always 14, independently from the distance. On the otherhand, LR-oriented NOMA systems tend to slightly increaseboth K and N as the distance increases. This occurs becausethe capture probability decreases as the distance from user2 to the BS increases. Increasing N and K then increasesthe robustness of the codes to errors in the transmission.This also implies that, when the capture probability is high,the NOMA system can significantly reduce the frame size,which simplifies the decoding and reduces the latency, evenfor intermittent user packets that need to wait for SIC.

On the other hand, if PAoI is the main objective, Fig. 9c-dshow a different picture: the value of K and N for NOMAwithout capture is almost constant, as is the value of Tintfor TDMA, while the best possible values of K and N forNOMA are higher at some distances and lower for others.

This phenomenon is likely due to the interplay between thedifferent outcome probabilities and their effects on the PAoI.

VIII. CONCLUSIONS

In this paper, we evaluated orthogonal and non-orthogonalslicing for heterogeneous services, namely, broadband andintermittent, in the RAN. Our model considered power controland packet-level coding for the broadband user and the useof SIC at the BS. Our analyses and results highlighted thedifferent characteristics and achievable performance of TDMAand NOMA access schemes when compared to FDMA, whichutilizes double the amount of resources – two bandwidthparts instead of one for TDMA and NOMA. In addition, weobserved stark differences in terms of achievable trade-offs,impact of the inter-arrival times, and optimal configuration ofthe access schemes between the cases where the intermittentuser aims to minimize either LR and PAoI. Hence, our resultshighlight the importance of the considered performance indi-cator for the intermittent user and of its wireless conditions,which must be taken into account for an efficient user pairingin NOMA.

In particular, our results showed that, with the consideredschemes, NOMA achieves a closely similar performance asFDMA if the intermittent user has a sufficiently high meanSINR as a result of a relatively low path loss. Furthermore,NOMA achieved better trade-offs between throughput andLR than TDMA in every studied scenario. Since NOMAutilized half of the resources of FDMA (the upper bound inperformance), it achieved the best balance between resource

15

efficiency and performance when the intermittent user aimsto minimize LR. Furthermore, even NOMA without captureachieved a better performance than TDMA in terms of LR.

Moreover, NOMA achieved better throughput and PAoItrade-offs in most cases. In particular, TDMA only showed asuperior performance when aiming for the highest throughputpossible in exchange for a longer PAoI. However, the differ-ences in PAoI were considerably smaller than those for theLR cases, especially for short distances from user 2 to theBS. Hence, TDMA may be preferred in the cases where theintermittent user is close to the BS due to its simplicity. Finally,NOMA without capture achieved the worst performance forPAoI.

Finally, is it important to note that, since the slicing isperformed independently for each bandwidth part, our modeland analyses can be easily extended to the case with multipleusers and multiple bandwidth parts. This is the case withmultiple broadband users, each with its own bandwidth partthat can be shared with up to one of the intermittent users.Further, the FDMA scheme could be used to allocate multipleintermittent users in the same bandwidth part. However, theanalysis of this scenario is out of the scope of this paper asFDMA is used as a benchmark for TDMA and NOMA and itrequires to define an access scheme for the intermittent users.

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