ALOHA Receivers: a Network Calculus Approachfor Analyzing Coded Multiple Access with SIC

Tzu-Hsuan Liu, Che-Hao Yu, Yi-Jheng Lin, Cheng-Shang Chang, Fellow, IEEE,and Duan-Shin Lee, Senior Member, IEEE

Abstract—Motivated by the need to hide the complexity of thephysical layer from performance analysis in a layer 2 protocol, aclass of abstract receivers, called Poisson receivers, was recentlyproposed in [1] as a probabilistic framework for providingdifferentiated services in uplink transmissions in 5G networks.In this paper, we further propose a deterministic framework ofALOHA receivers that can be incorporated into the probabilisticframework of Poisson receivers for analyzing coded multipleaccess with successive interference cancellation. An ALOHAreceiver is characterized by a success function of the number ofpackets that can be successfully received. Inspired by the theoryof network calculus, we derive various algebraic properties forseveral operations on success functions and use them to provevarious closure properties of ALOHA receivers, including (i)ALOHA receivers in tandem, (ii) cooperative ALOHA receivers,(iii) ALOHA receivers with traffic multiplexing, and (iv) ALOHAreceivers with packet coding. By conducting extensive simula-tions, we show that our theoretical results match extremely wellwith the simulation results.

Keywords: multiple access, network calculus, successiveinterference cancellation, ultra-reliable low-latency communi-cations.

I. INTRODUCTION

The fifth-generation networks (5G) and beyond aim to coverthree generic connectivity types: (i) enhanced mobile broad-band (eMBB), (ii) ultra-reliable low-latency communications(URLLC), and (iii) massive machine-type communications(mMTC) (see, e.g., [2]–[4] and references therein). The relia-bility defined in 3GPP for supporting URLLC services, suchas autonomous driving, drones, and augmented/virtual reality,requires the 1 − 10−5 success probability of transmitting alayer 2 protocol data unit of 32 bytes within 1ms. On the otherhand, the number of devices is in general much larger thanthe number of orthogonal resources for mMTC. Motivated bythese emerging needs in 5G, many multiple access schemeshave been proposed in the literature, see, e.g., ContentionResolution Diversity Slotted ALOHA (CRDSA) [5], Irreg-ular Repetition Slotted ALOHA (IRSA) [6], coded slottedALOHA (CSA) [7]–[10], Low-Density Signature (LDS) basedspreading [11], Sparse Code Multiple Access (SCMA) [12],Multi-User Sharing Access (MUSA) [13], Pattern Division

T.-H. Liu, C.-H. Yu, Y.-J. Lin, C.-S. Chang, and D.-S. Lee are with theInstitute of Communications Engineering, National Tsing Hua University,Hsinchu 30013, Taiwan, R.O.C. Email: carina000314@gmail.com;chehaoyu@gapp.nthu.edu.tw; s107064901@m107.nthu.edu.tw;cschang@ee.nthu.edu.tw; lds@cs.nthu.edu.tw. This work was supportedin part by the Ministry of Science and Technology, Taiwan, under Grant109-2221-E-007-091-MY2, and in part by Qualcomm Technologies underGrant SOW NAT-435533.

Multiple Access (PDMA) [14], T -fold ALOHA [15], Asyn-chronous Multichannel Transmission Schedules (AMTS) [16],Grant-free Hybrid Automatic Repeat reQuest (GF-HARQ)[17], and Polar Code Based TIN-SIC [18]. Though manyof the above multiple access schemes use elegant messagepassing algorithms (or density evolution methods) for de-coding as in the low-density parity-check codes [19], thedecoding results depend heavily on the performance of theunderlining physical channel. As such, it is very difficult tocarry out further analysis in a layer 2 protocol, where thereare multiple classes of input traffic and interconnected basestations/receivers/relays/antennas.

To hide the complexity from the physical layer, one needsan abstract model at the Medium Access Control (MAC)layer. Motivated by this, we proposed in our previous work[1] an abstract receiver, called Poisson receiver, that modelsthe input-output relations of multiple classes of input trafficsubject to an (independent) Poisson offered load. It was shownin [1] that various CSA systems in [6]–[10] can be modelledby Poisson receivers. In this paper, we take one step furtherto show that the block fading channel with capture in [10],[20], [21] can also be modelled by Poisson receivers. Poissonreceivers have two elegant closure properties: (i) Poissonreceivers with packet routing are still Poisson receivers, and(ii) Poisson receivers with packet coding are still Poissonreceivers. Thus, one can use smaller Poisson receivers asbuilding blocks for analyzing a larger Poisson receiver. In-tuitively, Poisson receivers can be viewed as a probabilisticframework for analyzing coded random access.

In this paper, we propose a new class of abstract receivers,called ALOHA receivers, that can be viewed as a determin-istic framework for analyzing coded multiple access withsuccessive interference cancellation (SIC). Our approach isinspired by the theory of network calculus (see, e.g., theseminal works by Rene Cruz [22], [23], the books and thesurvey papers [24]–[28] and references therein). The theoryof network calculus is a queueing theory that analyzes the net-work performance by viewing each queue as a building blockwith a certain arrival-departure property. By exploiting variouselegant algebraic properties of the min-plus algebra [29], end-to-end delay bounds can be derived for various schedulingpolicies, including Generalized Processor Sharing (GPS) [30]and Service Curve Earliest Deadline first scheduling (SCED)[31].

Like a Poisson receiver, an ALOHA receiver is also anabstract receiver that views the underlining physical layer as

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a network element with a deterministic input-output func-tion. Specifically, when there are nk class k packets, k =1, 2, . . . ,K, arriving at a φ-ALOHA receiver, the number ofclass k packets that are successfully received (or successfullydecoded) is exactly φk(n), k = 1, 2, . . . ,K. The functionφ(n) = (φ1(n), φ2(n), . . . , φK(n)) is called the successfunction of the φ-ALOHA receiver when it is subject to adeterministic load n = (n1, n2, . . . , nK). On the other hand,the failure function, denoted by φc with φc(n) = n − φ(n)represents the number of packets remained to be decoded.

One nice feature of coded multiple access with SIC is thata user can send multiple copies of a packet to a receiver. Aslong as one of the copies is successfully received, then thepacket is successfully received, and the other copies of thatpacket can be removed from the receiver. Such a feature isknown as the perfect SIC assumption in the literature (see,e.g., [10], [18]). As such, one can repeatedly apply the SICoperation to a receiver until there are no more packets thatcan be successfully decoded. Such an operation is even moreinteresting and powerful when a set of cooperative receiverscan exchange information of successfully received packets.

The SIC operation induces several operations on functions,including minimum, composition, closure, and complement.Like the theory of network calculus, we develop variousalgebraic properties for these operations that can be usedfor proving various closure properties of ALOHA receivers,including

(i) two ALOHA receivers in tandem is also an ALOHAreceiver in Section III-C,

(ii) two cooperative receivers is an ALOHA receiver inSection III-D,

(iii) ALOHA receivers with traffic multiplexing is anALOHA receiver in Section III-E,

(iv) ALOHA receivers with packet coding is an ALOHAreceiver in Section III-F, and

(v) multiple cooperative D-fold ALOHA receivers is anALOHA receiver in Section III-G.

An ALOHA receiver can be easily converted into a Poissonreceiver. However, it is not the other way around. As such,the deterministic framework of ALOHA receivers can beincorporated into the probabilistic framework of Poisson re-ceivers. To illustrate this, we provide a numerical example thatuses two cooperative D-fold ALOHA receivers with packetcoding to provide differentiated services between URLLCtraffic and eMBB traffic. For D = 1, such a system isreduced to the CSA system with two cooperative receivers in[1]. The theoretical results in this example match extremelywell with the simulation results (except for those data pointswith extremely small error probabilities). Moreover, there isa significant performance gain by using the 2-fold ALOHAreceivers over the 1-fold ALOHA receivers.

To provide a general overview of the probabilistic frame-work of Poisson receivers and the deterministic frameworkof ALOHA receivers, we depict in Figure 1 the basic build-ing blocks and the associated calculus for analyzing codedmultiple access with SIC.

Fig. 1. An overview of the probabilistic framework of Poisson receivers andthe deterministic framework of ALOHA receivers.

The rest of the paper is organized as follows. In SectionII, we briefly review the framework of Poisson receiversin our previous work [1]. There we also show how tomodel the Rayleigh block fading channel with capture as aPoisson receiver. We develop the deterministic framework ofALOHA receivers in Section III by proving various algebraicproperties for several operations on functions. Using thesealgebraic properties, we then prove various closure propertiesof ALOHA receivers. In Section IV, we provide numerical ex-amples to show how ALOHA receivers and Poisson receiverscan be used for providing differentiated services betweenURLLC traffic and eMBB traffic. The paper is then concludedin Section V.

II. POISSON RECEIVERS

A. Review of the framework of Poisson receivers

In this section, we briefly review the framework of Poissonreceivers in our previous work [1]. We say a system with Kclasses of input traffic is subject to a Poisson offered loadρ = (ρ1, ρ2, . . . , ρK) if these K classes of input traffic areindependent, and the number of class k packets arriving atthe system follows a Poisson distribution with mean ρk, fork = 1, 2, . . . ,K.

Definition 1: (Poisson receiver with multiple classesof input traffic [1]) An abstract receiver is called a(Psuc,1(ρ), Psuc,2(ρ), . . . , Psuc,K(ρ))-Poisson receiver with Kclasses of input traffic if the receiver is subject to a Poissonoffered load ρ = (ρ1, ρ2, . . . , ρK), a tagged (randomly se-lected) class k packet is successfully received with probabilityPsuc,k(ρ), for k = 1, 2, . . . ,K.

The throughput of class k packets (defined as the expectednumber of class k packets that are successfully received) for a(Psuc,1(ρ), Psuc,2(ρ), . . . , Psuc,K(ρ))-Poisson receiver subjectto a Poisson offered load ρ is thus

Sk = ρk · Psuc,k(ρ), (1)

k = 1, 2, . . . ,K.It was shown in [1] that many systems can be modelled

by Poisson receivers, including SA, SA with multiple non-cooperative receivers, and SA with multiple cooperative re-ceivers. Moreover, there are two elegant closure properties ofPoisson receivers in [1].

(i) Poisson receivers with packet routing are still Pois-son receivers.

(ii) Poisson receivers with packet coding are still Poissonreceivers.

These two closure properties allow us to use smaller Poissonreceivers as building blocks for analyzing a larger Poissonreceiver.

Theorem 2: (Poisson receivers with packet rout-ing [1]) Consider a Poisson receiver with K2 classesof input traffic and the success probability functionsPsuc,1(ρ), Psuc,2(ρ), . . . , Psuc,K2

(ρ). There are K1 classes ofexternal input traffic to the Poisson receiver. With the routingprobability rk1,k2 , a class k1 external packet transmittedto the Poisson receiver becomes a class k2 packet at thePoisson receiver. Let G = (G1, G2, . . . , GK1) and ρ =(ρ1, ρ2, . . . , ρK2

) with

ρk2 =

K1∑k1=1

Gk1rk1,k2 , (2)

k2 = 1, 2, . . . ,K2. Then the system is a(Psuc,1(G), Psuc,2(G), . . . , Psuc,K1

(G))-Poisson receiver,

Psuc,k1(G) =

K2∑k2=1

rk1,k2Psuc,k2(ρ), (3)

k1 = 1, 2, . . . ,K1.Using packet coding to increase system throughput has

been widely addressed in the literature, see, e.g., [5]–[8],[10], [32], [33]. The basic idea is to send a packet multipletimes. If any one of them is successfully received, then theother copies of that packet can be removed from the system.Such an assumption is known as the perfect SuccessiveInterference Cancellation (SIC) assumption. The decodingprocess is known as a peeling decoder that repeatedly re-moves decoded packets from the system. To analyze sucha decoding process, the tree evaluation method [34]–[36]tracks the evolution of the decodability probability after eachSIC iteration. Such an analysis leads to the exact asymptoticsystem throughput under the tree assumption. The frameworkin [1] added the reduced Poisson offered load step in the treeevaluation method that leads to the following closure propertyfor Poisson receivers with packet coding.

Theorem 3: (Poisson receivers with packet coding [1])Consider a system with GkT class k active users, k =1, 2, . . . ,K, and T independent Poisson receivers with Kclasses of input traffic and the success probability func-tions Psuc,1(ρ), Psuc,2(ρ), . . . , Psuc,K(ρ). Each class k usertransmits its packet for Lk times (copies) uniformly andindependently to one of the T Poisson receivers. Let Λk,`be the probability that a class k packet is transmitted ` times,i.e.,

P (Lk = `) = Λk,`, ` = 1, 2, . . . (4)

Define the function

Λk(x) =

∞∑`=0

Λk,` · x`, (5)

and the function

λk(x) =Λ′(x)

Λ′(1). (6)

Denote by � the element-wise multiplication of two vec-tors, i.e., for two vectors u = (u1, u2, . . . , uK) and v =(v1, v2, . . . , vK),

u� v = (u1v1, u2v2, . . . , uKvK). (7)

Then under the perfect SIC assumption and the tree assump-tion for a very large T , the system of coded Poisson receiversafter the ith SIC iteration is a Poisson receiver with the successprobability function for a tagged class k packet

P(i)suc,k(G) = 1− Λk

(1− Psuc,k(q(i−1) �G� Λ′(1))

), (8)

k = 1, 2, . . . ,K, where q(i) = (q(i)1 , q

(i)2 , . . . , q

(i)K ) can be

computed recursively from the following equation:

q(i+1)k = λk

(1− Psuc,k(q(i) �G� Λ′(1))

), (9)

with q(0) = (1, 1, . . . , 1).

B. Rayleigh block fading channel with capture

In this section, we consider the block fading channel withcapture in [10], [20], [21]. We show that such a channel modelcan also be modelled by a Poisson receiver. In a wirelesschannel with N active users and one receiver, the signal at thereceiver is commonly represented by the sum of transmittedsignals and the added white Gaussian noise, i.e.,

N∑n=1

hnsn +N , (10)

where sn is the signal transmitted by the nth active user, Nis the added white Gaussian noise, and hn is the channel gainbetween the nth active user and the receiver. Suppose that(i) the transmitted signals are orthogonal, (ii) the transmittedpower at each active user is P , and (iii) the noise power isPnoise. Then the total received power at the receiver is

N∑n=1

||hn||2P + Pnoise. (11)

In the threshold-based decoding model, the signal si can besuccessfully decoded if the signal-to-interference-and-noiseratio (SINR) is higher then a predefined threshold b?, i.e.,

||hi||2P∑n 6=i ||hn||2P + Pnoise

≥ b?. (12)

Let Xn = ||hn||2, γ = P/Pnoise, and b = b?/P . Then (12)can be written as follows:

Xi∑n 6=iXn + 1

γ

≥ b. (13)

As in [10], [20], [21], we assume the independent Rayleighfading model for each active user, i.e., Xn’s are independent

and exponentially distributed with mean 1. Then the proba-bility that the signal si can be successfully decoded is

P(Xi∑

n 6=iXn + 1γ

≥ b) =e−b/γ

(1 + b)N−1. (14)

Now we consider the capture effect in the threshold-basedmodel. Like the SIC decoding algorithm described in theprevious section, the threshold-based model with the captureeffect first decodes the signal with the largest power. If theSINR of the signal with the largest power is not smaller thanthe threshold, then that signal is successfully decoded, and it isremoved from the received signal. The process is then repeateduntil no signal can be successfully decoded. The probabilitythat the r signals si, i = 1, 2, . . . , r, are successfully decodedin the order 1, 2, . . . , r, is

P( X1∑N

n=2Xn + 1γ

≥ b, X2∑Nn=3Xn + 1

γ

≥ b,

. . . ,Xr∑N

n=r+1Xn + 1γ

≥ b)

=e−

1γ ((1+b)

r−1)

(1 + b)r(N−1−r−12 )

. (15)

The detailed derivation of (15) is shown in Appendix A. Assuch, the probability that there are at least r successfullydecoded signals is

N !

(N − r)!e−

1γ ((1+b)

r−1)

(1 + b)r(N−1−r−12 )

. (16)

Since E[X] =∑∞r=1 P(X ≥ r) for any nonnegative discrete

random variable X , we know that the average number ofsuccessfully decoded signals is

N∑r=1

N !

(N − r)!e−

1γ ((1+b)

r−1)

(1 + b)r(N−1−r−12 )

. (17)

For a Poisson offered load ρ, the number of active usersN follows a Poisson distribution with mean ρ. From (17), thethroughput for the Rayleigh block fading channel with capture(subject to a Poisson offered load ρ) is

S =

∞∑N=0

e−ρρN

N !

N∑r=1

N !

(N − r)!e−

1γ ((1+b)

r−1)

(1 + b)r(N−1−r−12 )

=

∞∑N=0

N∑r=1

e−ρρN

(N − r)!e−

1γ ((1+b)

r−1)

(1 + b)r(N−1−r−12 )

. (18)

Thus, the Rayleigh block fading channel with capture is aPsuc(ρ)-Poisson receiver with

Psuc(ρ) =

∞∑N=0

N∑r=1

e−ρρN−1

(N − r)!e−

1γ ((1+b)

r−1)

(1 + b)r(N−1−r−12 )

=

∞∑N=0

N−1∑τ=0

e−ρρN−1

(N − (τ + 1))!

e−1γ ((1+b)

τ+1−1)

(1 + b)(τ+1)(N−1− τ2 )

=

∞∑t=0

t∑τ=0

e−ρρt

(t− τ)!

e−1γ ((1+b)

τ+1−1)

(1 + b)(τ+1)(t− τ2 ). (19)

III. ALOHA RECEIVERS

The framework of Poisson receivers in [1] is a probabilisticframework for analyzing coded random access. Such a frame-work relies on the tree assumption to keep the independence ofvarious classes of the input traffic. For the tree assumption tohold, the number of Poisson receivers T in Theorem 3 needsto be very large. To deal with the scenario where input trafficindependence is difficult to justify, we propose a deterministicframework, called ALOHA receivers.

A. Definitions and examples of ALOHA receivers

Denote by Z+ the set of nonnegative integers. We saya system with K classes of input traffic is subject to adeterministic load n = (n1, n2, . . . , nK) ∈ Z+K if thenumber of class k packets arriving at the system is nk. Fortwo vectors n′ and n′′ in Z+K , we say n′ ≤ n′′ if n′k ≤ n′′kfor all k = 1, 2, . . . ,K.

Definition 4: (ALOHA receiver with multiple classes ofinput traffic) Consider a deterministic function

φ : Z+K → Z+K

that maps a K-vector n = (n1, n2, . . . , nK) to the K-vector(φ1(n), φ2(n), . . . , φK(n)). An abstract receiver is called φ-ALOHA receiver (with K classes of input traffic) if thenumber of class k packets that are successfully received isexactly φk(n), k = 1, 2, . . . ,K, when the receiver is subjectto a deterministic load n = (n1, n2, . . . , nK) (see Figure 2).The function φ is called the success function of the ALOHAreceiver. Define the failure function φc by

φc(n) = n− φ(n). (20)

A φ-ALOHA receiver is called monotone if the failure func-tion φc is increasing in the deterministic load n, i.e., for anyn′ ≤ n′′,

φc(n′) ≤ φc(n′′). (21)

Fig. 2. A φ-ALOHA receiver with K classes of input traffic.

The naming of ALOHA receivers is from the SlottedALOHA (SA) system [37]. In the classical collision channelmodel, if there is more than one packet transmitted to areceiver, then there is a collision, and collided packets are

assumed to be lost. On the other hand, if there is exactly onepacket transmitted to a receiver, then that packet is assumed tobe successfully received. Thus, the SA system is a φ-ALOHAreceiver with a single class of input traffic, where

φ(n) =

{1 if n = 10 otherwise . (22)

The D-fold ALOHA system proposed in [15] is a general-ization of the SA system. If there are less than or equal toD packets transmitted in a time slot, then all these packetscan be successfully decoded. On the other hand, if there aremore than D packets transmitted in a time slot, then all thesepackets are lost. Clearly, the D-fold ALOHA system in a timeslot is a monotone φ-ALOHA receiver with a single class ofinput traffic, where

φ(n) =

{n if n ≤ D0 otherwise . (23)

In this paper, we simply call the D-fold ALOHA system in atime slot a D-fold ALOHA receiver.

In addition to the SA system with a single class of inputtraffic, one can use the φ-ALOHA receiver to model the near-far SIC decoding in the following example.

Example 1: (Near-far SIC decoding) Suppose that thereare two classes of input traffic to a receiver. The power ofa class 1 packet at the receiver is much stronger than thatof a class 2 packet at the receiver. One may view class 1(resp. 2) packets as the users who are near (resp. far from)the receiver. Suppose that there are two packets arriving atthe receiver: one is a class 1 packet, and the other is a class2 packet. The SIC decoding algorithm first decodes the class1 packet and then removes it to reduce the interference to theclass 2 packet (under the perfect SIC assumption). By doingso, the class 2 packet can also be decoded. For such a near-far SIC decoding, we can model it as a monotone φ-ALOHAreceiver with

φ(n1, n2) =

{(n1, n2) if (n1, n2) ≤ (1, 1)(0, 0) otheriwse .

Note that a φ-ALOHA receiver with K1 classes of inputtraffic and a ψ-ALOHA receiver with K2 classes of inputtraffic in parallel can be viewed as a (φ, ψ)-ALOHA receiverwith K1 + K2 classes of input traffic. The near-far SICdecoding model in Example 1 can be viewed as two SAsystems in parallel, where the first SA system is for class1 traffic, and the second SA system is for class 2 traffic.

In the following theorem, we show that a φ-ALOHAreceiver is also a Poisson receiver. Such a Poisson receiveris called the induced Poisson receiver from the φ-ALOHAreceiver. This allows us to incorporate our deterministic frame-work of ALOHA receivers into the probabilistic frameworkof Poisson receivers.

Theorem 5: A φ-ALOHA receiver with K classes ofinput traffic is a (Psuc,1(ρ), Psuc,2(ρ), . . . , Psuc,K(ρ))-Poissonreceiver, where

Psuc,k(ρ) =1

ρk

∑n

φk(n)

K∏`=1

e−ρ`ρ`n`

n`!, (24)

k = 1, 2, . . . ,K.Proof. Note that the throughput of class k packets in aφ-ALOHA receiver subject to a Poisson offered load ρ =(ρ1, . . . , ρK) is

Sk =∑n

φk(n)

K∏`=1

e−ρ`ρ`n`

n`!. (25)

Using (1) yields (24).

B. Operations on functions

For the ease of our presentation, we introduce several op-erations on functions and their algebraic properties. Considerthe class of functions

F = {f : Z+K → Z+K with f(n) ≤ n}. (26)

Define two binary operations on this class of functions: theminimum operation ∧ and the composition operation ◦.

(f ∧ g)(n) = min[f(n), g(n)], (27)(f ◦ g)(n) = f(g(n)). (28)

The minimum operation in (27) is a component-wise opera-tion. We say f = (resp. ≤)g if f(n) = (resp. ≤)g(n) forall n ∈ Z+K .

It is easy to see that these two operations have the followingproperties:

f ∧ g = g ∧ f, (commutativity) (29)f ∧ (g ∧ h) = (f ∧ g) ∧ h, (associativity) (30)f ◦ (g ◦ h) = (f ◦ g) ◦ h. (associativity) (31)

Let ε be the identity mapping, i.e., ε(n) = n. Clearly, ε is theidentity element for these two operations, i.e.,

f ∧ ε = ε ∧ f = f, (32)f ◦ ε = ε ◦ f = f. (33)

A function is said to be increasing if

f(n′) ≤ f(n′′), (34)

for all n′ ≤ n′′. Define the class of increasing functions

F↑ = {f ∈ F : f is increasing.}. (35)

In the following proposition, we state the closure propertyand the monotone property for the two operations in F↑. Theproofs are rather straightforward and thus omitted.

Proposition 6:

(i) (Closure property) If f, g are in F↑, then f ∧ g andf ◦ g are also in F↑.

(ii) (Monotone property) For f1, f2, g1, g2 in F↑, if f1 ≤f2, g1 ≤ g2, then

f1 ∧ g1 ≤ f2 ∧ g2, (36)f1 ◦ g1 ≤ f2 ◦ g2. (37)

Let f (0) = ε and f (i+1) = f (i) ◦ f . For any f ∈ F↑, wehave

f (i+1) = f (i) ◦ f ≤ f (i) ◦ ε = f (i). (38)

As such, {f (i)(n), i = 1, 2, . . .} is a decreasing sequence andthus converges to a vector n∗. In view of this, we can define aunary operation ∗, called the closure operation, on a function fas the limit of the decreasing sequence of functions {f (i), i =1, 2, . . .}, i.e.,

f∗ = limi→∞

f (i). (39)

Example 2: (Star-shaped functions) For K = 1, a functionf is called a star-shaped function if f(n)/n is increasing inn for all n ≥ 1. As f ∈ F↑, f(n)/n ≤ 1. Let

n1 = inf{n : f(n)/n = 1}.

Then for n ≥ n1, we have f(n)/n ≥ f(n1)/n1 = 1 and thusf(n) = n for n ≥ n1. On the other hand, for n < n1, wehave f(n)/n < 1. If f (i)(n) ≥ 1, then for n < n1,

f (i+1)(n)

f (i)(n)=f(f (i)(n))

f (i)(n)≤ f(n)

n< 1.

This implies that f (i+1)(n) < f (i)(n). Thus, {f (i)(n), i =1, 2, . . .} decreases to 0 for n < n1. As such, we have

f∗(n) =

{0 if n < n1n otherwise . (40)

In the following lemma, we show several properties of theclosure operation.

Lemma 7: For any f, g in F↑, we have(i) (Monotone property) If f ≤ g, then f∗ ≤ g∗.(ii) f ◦ f∗ = f∗ ◦ f = f∗.(iii) f∗ ◦ f∗ = f∗.(iv) (f∗)∗ = f∗.(v) (f ◦ g)∗ = (g ◦ f)∗

(vi) (f ◦ g)∗ = (f∗ ◦ g∗)∗.(vii) (f ◦ g)∗ = (f ∧ g)∗.

Proof. (i) From the monotone property in Proposition 6, wehave f (i) ≤ g(i) for all i. Letting i → ∞ completes theargument.

(ii) That (ii) holds follows trivially from the definition ofthe closure operation as the limit of the decreasing sequenceof functions {f (i), i = 0, 1, 2, . . .}.

(iii) From (ii), it follows that f∗◦f (i) = f∗ for all i. Lettingi→∞ completes the argument.

(iv) From (iii), we have (f∗)(i) = f∗ for all i. Lettingi→∞ completes the argument.

(v) First, note from the associative property of ◦ that

(f ◦ g)(i) ◦ f = f ◦ (g ◦ f)(i). (41)

Letting i→∞ yields

(f ◦ g)∗ ◦ f = f ◦ (g ◦ f)∗. (42)

Since f, g ≤ ε, we then have the monotone property inProposition 6 that

(f ◦ g)∗ = (f ◦ g)∗ ◦ ε ≥ (f ◦ g)∗ ◦ f= f ◦ (g ◦ f)∗ = (ε ◦ f) ◦ (g ◦ f)∗

≥ (g ◦ f) ◦ (g ◦ f)∗ = (g ◦ f)∗. (43)

Interchanging f and g in (43) yields

(g ◦ f)∗ ≥ (f ◦ g)∗. (44)

Thus, we have from (43) and (44) that

(g ◦ f)∗ = (f ◦ g)∗. (45)

(vi) Since f ≥ f∗ and g ≥ g∗, we have from the monotoneproperty in (i) of this lemma that

(f ◦ g)∗ ≥ (f∗ ◦ g∗)∗.

On the other hand, note that f ≥ (f ◦ g) and g ≥ (f ◦ g). Itthen follows from the monotone property that f∗ ≥ (f ◦ g)∗

and g∗ ≥ (f ◦ g)∗. Thus, from Lemma 7(iii),

(f ◦ g)∗ = (f ◦ g)∗ ◦ (f ◦ g)∗ ≤ f∗ ◦ g∗.

Using (iv) of this lemma and the monotone property in (i) ofthis lemma yields

(f ◦ g)∗ = ((f ◦ g)∗)∗ ≤ (f∗ ◦ g∗)∗.

(vii) Since f ≥ f ◦ g and g ≥ f ◦ g, we have fromProposition 6(ii) that

f ∧ g ≥ (f ◦ g) ∧ (f ◦ g) = f ◦ g.

It then follows from the monotone property in (i) of thislemma that

(f ∧ g)∗ ≥ (f ◦ g)∗.

On the other hand, we also have from f∧g ≤ f and f∧g ≤ gthat

(f ∧ g)(2) = (f ∧ g) ◦ (f ∧ g) ≤ f ◦ g.

This then leads to

(f ∧ g)∗ ≤ (f ◦ g)∗.

In addition to the closure operation, we also need anotherunary operation c, call the complement operation of a functionf ∈ F . Specifically, we denote by f c the complement functionof f with

(f c)(n) = n− f(n). (46)

Clearly, f c is in F and (f c)c = f . In our definition ofALOHA receivers, the success function φ and the failure func-tion φc are complement functions of each other. Moreover, ifa φ-ALOHA receiver is monotone, then the failure functionφc is in F↑. As such, we can use the closure operation on thefailure function φc.

C. Two ALOHA receivers in tandem

Consider a system with two receivers and K classes ofinput traffic. The first receiver is a φ-ALOHA receiver, and thesecond receiver is a ψ-ALOHA receiver. These two receiversare subject to the same K classes of input traffic. The tworeceivers are arranged in tandem so that only packets thatare successfully received by receiver 1 can be forwarded toreceiver 2 for SIC decoding, but not the other way around. Fora deterministic load n = (n1, n2, . . . , nK), there are φk(n)class k packets that are successfully received by receiver 1,and thus the number of class k packets arriving at receiver 2 iseffectively reduced from nk to nk − φk(n), k = 1, 2, . . . ,K.As such, receiver 2 is subject to a deterministic load n −φ(n). Since receiver 2 is a ψ-ALOHA receiver, the numberof class k packets that are successfully received by receiver2 is ψk(n− φ(n)). Thus, the total number of class k packetsthat are successfully received by these two receivers is

φk(n) + ψk

(n− φ(n)

).

This shows that the system is a ζ-ALOHA receiver, where

ζk(n) = φk(n) + ψk

(n− φ(n)

), (47)

k = 1, 2, . . . ,K.Another way to see this is to count the numbers of packets

that are not successfully received. After the receiver 1, thedeterministic load is effectively reduced from n to φc(n).Then after receiver 2, the deterministic load is further reducedfrom φc(n) to ψc(φc(n)) = (ψc ◦ φc)(n). Thus, the numbersof packets that are successfully received are (ψc ◦ φc)c(n).This is stated in the following theorem.

Fig. 3. A system with a φ-ALOHA receiver and a ψ-ALOHA receiver intandem.

Theorem 8: For the system with a φ-ALOHA receiver and aψ-ALOHA receiver in tandem (see Figure 3), it is a ζ-ALOHAreceiver, where

ζ = (ψc ◦ φc)c. (48)

D. Two cooperative ALOHA receivers

Consider the same setting as in Section III-C. Now weassume that these two receivers are cooperative and packetsthat are successfully received by receiver 2 can also beforwarded to receiver 1 for SIC decoding. As such, we canrepeat the SIC decoding between these two receivers untilno packets can be decoded. To illustrate this, let n(i) =

(n(i)1 , n

(i)2 , . . . , n

(i)K ), where n

(i)k is the number of class k

packets remained to be decoded in the system after the ith

SIC iteration, k = 1, 2, . . . ,K, and i = 0, 1, . . .. Clearly,n(0) = n. Note that the first SIC iteration corresponds tothe two receivers in tandem in Section III-C. After the firstSIC iteration, the number of class k packets that are remainedto be decoded is

n(1)k = n

(0)k − ζk(n(0)) (49)

where ζk is defined in (47). In general, we have

n(i+1) = n(i) − ζ(n(i))

= ζc(n(i)) = (ζc)(i)(n). (50)

As this sequence of vectors {n(i), i ≥ 0} is nonnegative andmonotonically decreasing, it converges to the vector

n(∞) = (ζc)∗(n) = (ψc ◦ φc)∗(n). (51)

This leads to the following theorem.

Fig. 4. Two cooperative receivers: a φ-ALOHA receiver and a ψ-ALOHAreceiver.

Theorem 9: Consider a system with a φ-ALOHA receiverand a ψ-ALOHA receiver (see Figure 4). If these two receiversare cooperative, then it is a η-ALOHA receiver, where

η = ((ψc ◦ φc)∗)c. (52)

One interesting question is whether the system converges tothe same η-ALOHA receiver in (52) if the SIC decoding orderof the two receivers is interchanged. In the following theorem,we show this is true if the two receivers are monotone.

Theorem 10: Suppose that both the φ-ALOHA receiver andthe ψ-ALOHA receivers are monotone. Then the system withtwo cooperative receivers converges to the same η-ALOHAreceiver in (52) if the SIC decoding order of the two receiversis interchanged, i.e., the order of the SIC decoding processdoes not affect the decoding result. Moreover, the system withtwo cooperative receivers is also monotone.Proof. Note that if we interchange the decoding order ofthe two receivers, then it is a ((φc ◦ ψc)∗)c-ALOHA receiver(from Theorem 9). Since both the φ-ALOHA receiver and theψ-ALOHA receiver are monotone, both φc and ψc are in F↑.As a result of Lemma 7 (v), we have

(ψc ◦ φc)∗ = (φc ◦ ψc)∗.

Thus, the system converges to the same η-ALOHA receiverin (52) if the SIC decoding order of the two receivers isinterchanged.

Since (ψc ◦ φc)∗ is still in F↑ from the closure propertyin Proposition 6(i), the η-ALOHA receiver in (52) is alsomonotone.

By using an inductive argument, we note that the result inTheorem 10 still holds for the scenario with more than twomonotone cooperative receivers.

E. ALOHA receivers with traffic multiplexing

In this section, we consider ALOHA receivers with trafficmultiplexing. To provide insight into our analysis, we firstconsider a system with K classes of (external) input trafficthat are multiplexed into a D-fold ALOHA receiver. Recallthat a D-fold ALOHA receiver is a φ-ALOHA receiver withφ in (23). Let n = (n1, n2, . . . , nK) be the deterministicload to the system. As the K classes of (external) inputtraffic are multiplexed into the D-fold ALOHA receiver, thedeterministic load to the D-fold ALOHA receiver is

∑Ki=1 ni.

Thus, the number of class k packets that are successfullyreceived is nk if

∑Ki=1 ni ≤ D. On the other hand, if∑K

i=1 ni > D, then none of the arriving packets can besuccessfully received. This shows that such a system is a ψ-ALOHA receiver with the success function

ψ(n) =

{(n1, n2, . . . , nK) if

∑Ki=1 ni ≤ D

(0, 0, . . . , 0) otherwise. (53)

One important insight of the above D-fold ALOHA receiveris that the arriving packets are either all successfully received

or all failed to be decoded. This motivates us to introduce thenotion of all-or-nothing ALOHA receivers.

Definition 11: A φ-ALOHA receiver with K classes ofinput traffic is said to be an all-or-nothing receiver if itsatisfies the following two properties:

(i) (All-or-nothing property) For all n =(n1, n2, . . . , nK) and k = 1, 2, . . . ,K, eitherφk(n) = nk or φk(n) = 0.

(ii) (On-off property) If n′ ≥ n′′ and φk(n′) = n′k forsome k, then φk(n′′) = n′′k . On the other hand, ifn′ ≤ n′′ and φk(n′) = 0 for some k, then φk(n′′) =0.

The on-off property in Definition 11 implies that an all-or-nothing φ-ALOHA receiver is monotone as φck(n′) ≤ φck(n′′)k = 1, 2, . . . ,K, for all n′ ≤ n′′.

Clearly, a D-fold ALOHA receiver with a single class ofinput traffic is an all-or-nothing φ-ALOHA receiver, wherethe success function φ is specified in (23). When it is subjectto K classes of external input traffic, it is an all-or-nothingψ-ALOHA receiver with the success function ψ in (53).

Traffic multiplexing that maps K classes of external inputtraffic into T classes of internal input traffic can be representedby a K × T bipartite graph, where the K classes of externalinput traffic are the K nodes on the left (the external nodes)and the T classes of internal input traffic are the T nodeson the right (the internal nodes). The constraint for trafficmultiplexing is that each external node has at most one edge.

Let H = (Hk,t) be the K × T bi-adjacency matrix for thebipartite graph, where Hk,t = 1 if there is an edge betweenthe external node k and the internal node t, and 0 otherwise.Denote by the set Ct the set of nonzero elements in the tth

column of H . As each external node has at most one edge,each row of H has at most one nonzero element, and thusthe sets Ct, t = 1, 2, . . . , T , are disjoint. Such a bi-adjacencymatrix is called a traffic multiplexing matrix in this paper.Specifically, a traffic multiplexing matrix H is a binary matrix,where each row of H has at most one nonzero element.

Fig. 5. ALOHA receivers with traffic multiplexing: a system with K classesof external input traffic to an all-or-nothing φ-ALOHA receiver with T classesof internal input traffic through a K × T traffic multiplexing matrix H .

Theorem 12: Consider a system with K classes of externalinput traffic to an all-or-nothing φ-ALOHA receiver withT classes of internal input traffic through a K × T traffic

multiplexing matrix H (see Figure 5). Then the system is anall-or-nothing ψ-ALOHA receiver, where

ψk(n) =

{nk if

∑Tt=1Hk,t · φt(nH) > 0

0 otherwise. (54)

k = 1, 2, . . . ,K.Proof. Consider a deterministic load n = (n1, n2, . . . , nK)for the external input traffic. Through traffic multiplexing, theload to the φ-ALOHA receiver is nH . As such, the numberof internal class t packets that are successfully received isφt(nH), t = 1, 2, . . . , T . Since the φ-ALOHA receiver isassumed to be an all-or-nothing receiver, the number ofexternal class k packets that are successfully received is nk ifthere is one internal class t such that Hk,t = 1 and the internalclass t packets are all successfully received, i.e., φt(nH) > 0.Note that for each external class k, there is at most oneinternal class t such that Hk,t = 1. These two conditionscan be simplified as the condition

∑Tt=1Hk,tφt(nH) > 0.

This shows that the all-or-nothing property.Note that if an external class k is not connected to any

internal class, i.e., Hk,t = 0 for all t = 1, 2, . . . , T , then noneof class k packets can be successfully received. To show theon-off property, it thus suffices to consider an external classk that is connected to an internal class t(k). In this case, wehave from (54) that

ψck(n) =

{0 if Hk,t(k) · φct(k)(nH) = 0

nk otherwise. (55)

Since an all-or-nothing receiver is monotone, we haveφct(k)(n

′H) ≤ φct(k)(n′′H), k = 1, 2, . . . ,K, for all n′ ≤ n′′.

Using this in (55) yields ψck(n′) ≤ ψck(n′′), k = 1, 2, . . . ,K.It is easy to see from the all-or-nothing property and themonotone property that the on-off property is also satisfied.

F. ALOHA receivers with packet codingAnalogous to the iterative decoding algorithm for Poisson

receivers with packet coding, we develop an iterative decodingalgorithm for ALOHA receivers with packet coding. As intraffic multiplexing in Section III-E, we consider a systemwith K classes of external input traffic. Inside the system,there is an all-or-nothing φ-ALOHA receiver with T classesof internal input traffic. A packet coding scheme for the Kclasses of external input traffic is to multicast an external classk packet to a set of internal classes, Bk, of the φ-ALOHAreceiver so that the sets Bk, k = 1, 2, . . . ,K, are disjoint. Astraffic multiplexing in Section III-E, a packet coding schemealso corresponds to a K × T bipartite graph with a K × Tbi-adjacency matrix H . The constraint for packet coding isthat each internal node has at most one edge. Thus, a packetcoding matrix H is a binary matrix, where each column of Hhas at most one nonzero element. Note that the set Bk is theset of nonzero elements in the kth row of H .

Consider a deterministic load n = (n1, n2, . . . , nK) forthe external input traffic. As in Section III-D, let n(i) =

(n(i)1 , n

(i)2 , . . . , n

(i)K ), where n

(i)k is the number of class k

packets remained to be decoded in the system after the ith SICiteration, k = 1, 2, . . . ,K, and i = 0, 1, . . .. Clearly, n(0) = n.For the first SIC iteration, the load to the all-or-nothing φ-ALOHA receiver is the T -vector n(0)H . As such, the numberof internal class t packets that are successfully received isφt(n

(0)H), t = 1, 2, . . . , T . Since an external packet is suc-cessfully received if one of its multicast copies is successfullyreceived, the number of external class k packets that aresuccessfully received is maxt∈Bk φt(n

(0)H), k = 1, 2, . . . ,K(as the φ-ALOHA receiver is an all-or-nothing receiver). Afterthe first iteration, the number of external class k packets thatare remained to be decoded is

n(1)k = n

(0)k −max

t∈Bkφt(n

(0)H), (56)

k = 1, 2, . . . ,K. As the φ-ALOHA receiver is an all-or-nothing receiver, we know that n(1)k is either 0 or n

(0)k .

Such an all-or-nothing property is preserved through everySIC iteration. Define the function θ(n) = (θ1(n), . . . , θK(n))with

θk(n) = maxt∈Bk

φt(nH). (57)

Then we have from the argument for (56) that

n(i+1) = n(i) − θ(n(i)). (58)

As this sequence of vectors {n(i), i ≥ 0} is nonnegative andmonotonically decreasing, it converges (in a finite number ofiterations) to the vector

n(∞) = (θc)∗(n). (59)

This leads to the following theorem.

Fig. 6. The θ-ALOHA receiver in (57) for analyzing a system of ALOHAreceivers with packet coding. In such a system, there are K external classesof input traffic to an all-or-nothing φ-ALOHA receiver with T internal classesof input traffic through a K × T packet coding matrix H .

Theorem 13: Consider a system with K external classes ofinput traffic to an all-or-nothing φ-ALOHA receiver with Tinternal classes of input traffic through a K×T packet codingmatrix H (see Figure 6). Then the system is an all-or-nothing((θc)∗)c-ALOHA receiver, where the function θ is defined in(57).

Interestingly, one can view the system of two cooperativeALOHA receivers in Section III-D as an ALOHA receiverwith packet coding. First, an all-or-nothing φ-ALOHA re-ceiver with K classes of input traffic and an all-or-nothing ψ-ALOHA receiver with K classes of input traffic can be viewed

as an all-or-nothing (φ, ψ)-ALOHA receiver with 2K classesof input traffic. As the two cooperative ALOHA receivers aresubject to the same K classes of external input, the K × 2Kcoding matrix H is simply a concatenation of two K × Kidentity matrices, i.e., both the first K columns of H and thelast K columns of H are K ×K identity matrices. In viewof (57), we have

θk(n) = max[φt(n), ψt(n)], (60)

and thusθc = φc ∧ ψc. (61)

It then follows from Lemma 7(vii) that

(θc)∗ = (φc ∧ ψc)∗ = (φc ◦ ψc)∗. (62)

The result in Theorem 13 then recovers the result in Theorem9.

We note that if the two receivers are not all-or-nothingALOHA-receivers, then we have to identify the indices ofthe packets that are successfully received in order to carryout SIC decoding. As such, the identity in (56) is no longervalid as the class k packets that are successfully received inφt(n

(0)H) for t ∈ Bk might not be the same. One interestingexample is to consider the two cooperative ALOHA receiversin Section III-D with a single class of input traffic and

φ(n) = ψ(n) =

{1 if n = 1 or 30 otherwise . (63)

When the system is subject to the deterministic load n = 3,the decoding result from Theorem 9 is

(φc ◦ ψc)∗(3) = 2.

On the other hand, if we use the decoding method for packetcoding and the packet that is successfully received by receiver1 is different from that by receiver 2, then after the first SICiteration, the number of packets remained to be decoded isreduced from 3 to 1. After the second iteration, it will befurther reduced to 0. This example shows that the decodingresult by the method for packet coding might be different fromthat from Theorem 9 if the two receivers are not all-or-nothingALOHA-receivers.

For the general setting with joint traffic multiplexing andpacket coding, it can be represented by a general K × Tbipartite graph with a bi-adjacency matrix H . For such abipartite graph, one can always add an intermediate stageof T nodes to form a tripartite graph such that (i) the firstpart is a K × T bipartite graph with a packet coding matrixH1, (ii) the second part is a T × T bipartite graph with atraffic multiplexing matrix H2, and (iii) H = H1H2. Onesimple way of doing this is to represent each edge in thegeneral bipartite graph by a node in the intermediate stage.By using the results for traffic multiplexing in Theorem 12and packet coding in Theorem 13, we can then analyze thesetting with joint traffic multiplexing and packet coding. Wewill further illustrate this by considering multiple cooperativeD-fold ALOHA receivers in the next section.

G. Multiple cooperative D-fold ALOHA receivers

In this section, we show that a system of multiple cooper-ative D-fold ALOHA receivers with joint traffic multiplexingand packet coding is a φ-ALOHA receiver. Moreover, thesuccess function φ for such a system can be computed bya max-sum message passing algorithm.

Consider a system with T cooperative D-fold ALOHAreceivers and K classes of input traffic through a K × T bi-adjacency matrix H . The bi-adjacency matrix H is a binarymatrix that needs not to satisfy the constraints imposed on atraffic multiplexing matrix or a packet coding matrix. Let Bk,k = 1, 2, . . . ,K, be the set of receivers associated with class kinput traffic (the set of nonzero elements in the kth row of H),and Ct, t = 1, 2, . . . , T , be the set of input traffic multiplexedinto the tth receiver (the set of nonzero elements in the tth

column of H). To compute the success function for such asystem subject to the deterministic load n = (n1, n2, . . . , nK),we let n(i) = (n

(i)1 , n

(i)2 , . . . , n

(i)K ), where n(i)k is the number

of class k packets remained to be decoded in the systemafter the ith SIC iteration, k = 1, 2, . . . ,K, and i = 0, 1, . . ..Clearly, n(0) = n. Also, let n(i)k,t be the number of class k thatare successfully received by the tth D-fold ALOHA receiverafter the ith iteration. For the first SIC iteration, we knowfrom (53) that n(1)k,t = n

(0)k if k ∈ Ct and

∑`∈Ct n

(0)` ≤ D,

and n(1)k,t = 0 otherwise. From (56), we also know the numberof class k packets that are remained to be decoded after thefirst iteration is

n(1)k = n

(0)k −max

t∈Bkn(1)k,t , (64)

k = 1, 2, . . . ,K. In general, for the ith SIC iteration, we have

n(i)k,t =

{n(i−1)k if k ∈ Ct,

∑`∈Ct n

(i−1)` ≤ D

0 otherwise, (65)

andn(i)k = n

(i−1)k −max

t∈Bkn(i)k,t, (66)

k = 1, 2, . . . ,K. These two recursive equation leads to themax-sum message algorithm in Algorithm 1. Note that Steps1 and 2 correspond to the computation of n(i)k,t in (65), and Step3 corresponds to the computation of n(i)k in (66) (as the D-foldALOHA receivers are all-or-nothing ALOHA receivers).

We note there is only a finite number of deterministic loadsthat need to be computed. To see this, note that if nk ≥D + 1 for some k, then there are at least D + 1 packetstransmitted to the D-fold ALOHA receivers in Bk. As such,no packets can be successfully received by the receivers inBk, and we can remove all the edges connected to the receivenodes in Bk. Thus, the decoding results are the same for all thedeterministic loads with nk ≥ D+1 for some k. This impliesthat all the deterministic loads with nk > D+ 1 is equivalentto nk = D + 1. As such, there are (D + 2)K equivalenceclasses (as we only need to consider nk = 0, 1, . . . , D+1 foreach k).

Certainly, one can use (25) to compute the throughputof class k users subject to a Poisson offered load ρ =

ALGORITHM 1: The max-sum message passing algo-rithm

Input A K × T bi-adjacency H and a deterministic loadn = (n1, n2, . . . , nK).

Output The number of class k packets that aresuccessfully received, φk(n), k = 1, 2, . . . ,K.

0: Initially, set n(0) = n and i = 1.1: For each class k node, send a message n(i−1)k to each

receiver node in Bk.2: For each receiver t node, compute the sum of theincoming messages. If the sum is not larger than D,return the original message to the sender. Otherwise,return a message 0 to all the senders in Ct.

3: For each class k node, compute the maximum of thereturning messages. If the maximum is 0, setn(i)k = n

(i−1)k . Otherwise, set n(i)k = 0.

4: If for every k = 1, 2, . . . ,K, the maximum of thereturning messages is 0, then no more packets can besuccessfully received. Set φk(n) = nk − n(i)k ,k = 1, 2, . . . ,K. Otherwise, increase i by 1 and repeatfrom Step 1.

(ρ1, ρ2, . . . , ρK) for such a φ-ALOHA receiver. However, itis more convenient to compute the throughput of class k usersfrom the probabilities of the (D + 2)K equivalence classes.Specifically, let p(n) is the probability of the deterministicload (equivalence class) n subject to a Poisson offered loadρ. For d = 0, 1, . . . , D, let

hd(ρk) =e−ρkρdkd!

,

and

hD+1(ρk) = 1−D∑d=0

e−ρkρdkd!

.

For the Poisson distribution with mean ρk, the probabilitiesfor nk are hd(ρk), d = 0, 1, . . . , D + 1. Since the Poissonoffered loads from the K classes are independent, we have

p(n) =

K∏k=1

hnk(ρk). (67)

Using (1) yields

Psuc,k(ρ) =Skρk

=1

ρk

∑n

φk(n)

K∏k=1

hnk(ρk). (68)

Example 3: (Two cooperative 2-fold ALOHA receivers)Consider the system with two cooperative 2-fold ALOHAreceivers and three classes of users. Suppose that class 1(resp. 2) packets are sent to receiver 1 (resp. 2), and class

3 packets are sent to both receivers. The bi-adjacency matrixof the association graph is

H =

1 00 11 1

. (69)

As discussed in this section, it is a φ-ALOHA receiver withthe success function φ = (φ1, φ2, φ3) being specified in TableI.

IV. NUMERICAL RESULTS

A. Two cooperative D-fold ALOHA receivers with URLLCtraffic and eMBB traffic

Fig. 7. An illustration for two cooperative D-fold ALOHA receivers withtwo classes of external input traffic: URLLC packets are multicast to bothreceivers, while eMBB packets can only be routed to one of the two receivers.

In this section, we demonstrate how the framework ofPoisson receivers and the framework of ALOHA receivers canbe used for providing differentiated services between URLLCtraffic and eMBB traffic in uplink transmissions.

1) Wireless channel model: We consider a wireless channelthat is modelled by two cooperative D-fold ALOHA receiversin Example 3. There are three classes of input traffic. Class1 (resp. 2) packets are sent to the first (resp. 2) receiver, andclass 3 packets are sent to both receivers (see Figure 7 foran illustration). For such a channel model, it is a φ-ALOHAreceiver with the success function φ specified in Table I. Byusing (68), we can compute the success probability functionsPsuc,k(ρ1, ρ2, ρ3), k = 1, 2, 3, of the induced Poisson receiver.The first two classes are eMBB traffic and the third class isURLLC traffic. Each eMBB packet can be routed to a class1 packet or a class 2 packet with an equal probability. Asa result of the inverse multiplexer in Example 3 of [1], onecan use Theorem 2 to model such a channel as a Poissonreceiver with two classes of external input traffic, URLLCtraffic (external class 1) and eMBB traffic (external class 2).The success probability functions of these two external classesare

Psuc,1(ρ1, ρ2) = Psuc,3(ρ2/2, ρ2/2, ρ1)

Psuc,2(ρ1, ρ2) =1

2Psuc,1(ρ2/2, ρ2/2, ρ1)

+1

2Psuc,2(ρ2/2, ρ2/2, ρ1). (70)

TABLE ITHE φ-ALOHA RECEIVER FOR A SYSTEM WITH TWO COOPERATIVE

2-FOLD ALOHA RECEIVERS IN EXAMPLE 3.

n1 n2 n3 φ1(n) φ2(n) φ3(n)0 0 0 0 0 00 0 1 0 0 10 0 2 0 0 20 0 3 0 0 00 1 0 0 1 00 1 1 0 1 10 1 2 0 1 20 1 3 0 0 00 2 0 0 2 00 2 1 0 2 10 2 2 0 2 20 2 3 0 0 00 3 0 0 0 00 3 1 0 0 10 3 2 0 0 20 3 3 0 0 01 0 0 1 0 01 0 1 1 0 11 0 2 1 0 21 0 3 0 0 01 1 0 1 1 01 1 1 1 1 11 1 2 0 0 01 1 3 0 0 01 2 0 1 2 01 2 1 1 2 11 2 2 0 0 01 2 3 0 0 01 3 0 1 0 01 3 1 1 0 11 3 2 0 0 01 3 3 0 0 02 0 0 2 0 02 0 1 2 0 12 0 2 2 0 22 0 3 0 0 02 1 0 2 1 02 1 1 2 1 12 1 2 0 0 02 1 3 0 0 02 2 0 2 2 02 2 1 0 0 02 2 2 0 0 02 2 3 0 0 02 3 0 2 0 02 3 1 0 0 02 3 2 0 0 02 3 3 0 0 03 0 0 0 0 03 0 1 0 0 13 0 2 0 0 23 0 3 0 0 03 1 0 0 1 03 1 1 0 1 13 1 2 0 0 03 1 3 0 0 03 2 0 0 2 03 2 1 0 0 03 2 2 0 0 03 2 3 0 0 03 3 0 0 0 03 3 1 0 0 03 3 2 0 0 03 3 3 0 0 0

2) Coded random access for a use case: To providedifferentiated services between URLLC traffic and eMBBtraffic over the wireless channel model in Section IV-A1, weconsider the particular use case in Section IV.C of [1] forsupporting precise cooperative robotic motion control definedin use case 1 of mobile robots in [38]. For this use case, themessage (packet) size of URLLC traffic is 40 bytes, and thetransmission time interval (TTI) is 1 ms. According to Table4.1A-2 in [39], the maximum number of bits of an uplinkshared channel (UL-SCH) transport block transmitted withina TTI is 105,528 bits (i.e., 13,191 bytes). As a conservativedesign, the number of packet transmissions (minislots) withinone TTI is set to be 256. As there can be as many as D suc-cessful packet transmissions in a D-fold ALOHA receiver, weset T = 256/D. Thus, for each TTI, there are T independentPoisson receivers for the wireless channel model in SectionIV-A1. The number of active URLLC users N1 = G1T is setto be 50, and each URLLC user uses a coded random accessscheme that transmits its packet for L = 5 times uniformlyand independently to the T Poisson receivers. On the otherhand, each eMBB user is scheduled to transmit exactly onepacket in a (randomly assigned) time slot in each TTI. Thepackets transmitted by URLLC users are superposed with thescheduled eMBB traffic [40] over the wireless channel inSection IV-A1. By viewing each minislot as a Poisson receiverwith two classes of input traffic in Section IV-A1, the codedrandom access scheme corresponds to a system of Poissonreceivers with packet coding in Theorem 3, where the degreedistribution of URLLC traffic is Λ1(x) = xL with L = 5 (see(5)), and the degree distribution of eMBB traffic is Λ2(x) = x.We note that URLLC transmissions in such a system aredecoded first by using the SIC technique. eMBB transmissionsthat overlap with undecodable URLLC transmissions can betreated as erased or punctured [41] and can be protected byusing another layer of error correction codes.

We are interested in addressing the question of how manyeMBB users N2 = G2T can be admitted to the system sothat the packet error probability of URLLC traffic is less than10−5.

In Figure 8, we show the theoretical results and the sim-ulation results for D = 1 and D = 2. Each data pointfor the estimated error probability is obtained by averagingover 100,000 independent runs. Also, we set the numberof iterations for SIC to be 100 (i = 100). As shown inFigure 8, our theoretical results match extremely well with thesimulation results except for those data points with extremelysmall error probabilities. For D = 1, the 1-fold ALOHAreceiver is the simply the SA system considered in SectionIV.C of [1]. One can see from this figure that the performanceof the 2-fold ALOHA receivers is significantly better thanthat of the SA system. In particular, the number of eMBBusers can be admitted to the system is increased from 194 forthe system with two 1-fold ALOHA receivers to 292 for thesystem with two 2-fold ALOHA receivers (while keeping theerror probability of URLLC packets lower than 10−5).

50 100 150 200 250 300 350 400Number of eMBB users

10-10

10-8

10-6

10-4

10-2

100

1-P

suc

error prob of eMBB (1-fold sim) error prob of URLLC (1-fold sim)error prob of eMBB (1-fold theory) error prob of URLLC (1-fold theory)error prob of eMBB (2-fold sim) error prob of URLLC (2-fold sim)error prob of eMBB (2-fold theory) error prob of URLLC (2-fold theory)

Fig. 8. The effect of the number of eMBB users on the error probability ofURLLC users in a system of two cooperative D-fold ALOHA receivers forD = 1 and D = 2.

B. Rayleigh block fading channel with URLLC traffic andeMBB traffic

In this section, we consider the Rayleigh block fadingchannel with capture in Section II-B. Such a wireless channelcan be modelled by a Poisson receiver with a single class ofinput in (19). By using Theorem 2 for Poisson receivers withpacket routing, such a wireless channel can be extended to aPoisson receiver with two classes of input traffic. The successprobability functions for these two classes of input traffic areas follows:

Psuc,1(ρ1, ρ2) = Psuc,2(ρ1, ρ2)

=

∞∑t=0

t∑τ=0

e−(ρ1+ρ2)(ρ1 + ρ2)t

(t− τ)!

e−1γ ((1+b)

τ+1−1)

(1 + b)(τ+1)(t− τ2 ).

(71)

To provide differentiated services between URLLC traffic andeMBB traffic, we use the coded random access scheme withT minislots as described in Section IV-A2. We assume thatthe channel gains for these two classes in T minislots areindependent, and thus such a system corresponds to a systemof Poisson receivers with packet coding in Theorem 3. In ourexperiments, we use the same parameters as those in SectionIV-A2, i.e., the number of minislots T = 256, the number ofURLLC users N1 = 50, the number of SIC iterations i =100, the degree distribution of URLLC traffic Λ1(x) = x5,and the degree distribution of eMBB traffic Λ2(x) = x. Forthe Rayleigh block fading channel, we set γ = 20dB andb = 3dB. As in Section IV-A2, each data point is obtainedby averaging over 100,000 independent runs. In Figure 9, weshow the theoretical results and the simulation results for theeffect of the number of eMBB users on the error probabilityof URLLC users in the Rayleigh block fading channel withcapture. Once again, our theoretical results match extremelywell with the simulation results. The number of eMBB userscan be admitted to the system is roughly 58 while keepingthe error probability of URLLC packets lower than 10−5.

0 50 100 150 200 250 300 350 400Number of eMBB users

10-10

10-8

10-6

10-4

10-2

100

1-P

suc

error prob of eMBB (sim)error prob of URLLC (sim)error prob of eMBB (theory)error prob of URLLC (theory)

Fig. 9. The effect of the number of eMBB users on the error probability ofURLLC users in the Rayleigh block fading channel with capture.

V. CONCLUSION

In this paper, we developed a deterministic frameworkof φ-ALOHA receivers that can be incorporated into theprobabilistic framework of Poisson receivers for analyzingcoded multiple access with SIC. Like the theory of networkcalculus, there are various algebraic properties for severaloperations on functions, including minimum ∧, composition◦, closure ∗, and complement c. As such, small ALOHAreceivers can be used as building blocks for constructinga large ALOHA receiver. In particular, we showed variousclosure properties of ALOHA receivers, including (i) ALOHAreceivers in tandem, (ii) cooperative receivers, (iii) ALOHAreceivers with traffic multiplexing, and (iv) ALOHA receiverswith packet coding. As an illustrating example, we com-puted/simulated the numerical results of a system that usestwo cooperative D-fold ALOHA receivers with packet codingto provide differentiated services between URLLC traffic andeMBB traffic. The theoretical results in this example matchextremely well with the simulation results (except for thosedata points with extremely small error probabilities).

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[38] 3GPP, “Service requirements for cyber-physical control applicationsin vertical domains,” 3rd Generation Partnership Project(3GPP), Technical Specification (TS) 22.104, 03 2019, version16.1.0. [Online]. Available: https://portal.3gpp.org/desktopmodules/Specifications/SpecificationDetails.aspx?specificationId=3528

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[40] A. Anand, G. De Veciana, and S. Shakkottai, “Joint scheduling of urllcand embb traffic in 5g wireless networks,” IEEE/ACM Transactions onNetworking, vol. 28, no. 2, pp. 477–490, 2020.

[41] P. Popovski, K. F. Trillingsgaard, O. Simeone, and G. Durisi, “5gwireless network slicing for embb, urllc, and mmtc: A communication-theoretic view,” IEEE Access, vol. 6, pp. 55 765–55 779, 2018.

Tzu-Hsuan Liu received the B.S. degree in com-munication engineering from National Central Uni-versity, Taoyuan, Taiwan (R.O.C.), in 2018. She iscurrently pursuing the M.S. degree in the Institute ofCommunications Engineering, National Tsing HuaUniversity, Hsinchu, Taiwan (R.O.C.). Her researchinterest is in 5G wireless communication.

Che-Hao Yu received his B.S. degree in mathemat-ics from National Tsing-Hua University, Hsinchu,Taiwan (R.O.C.), in 2018, and the M.S. degree incommunications engineering from National TsingHua University, Hsinchu, Taiwan (R.O.C.), in 2020.His research interest is in 5G wireless communica-tion.

Yi-Jheng Lin received his B.S. degree in electricalengineering from National Tsing Hua University,Hsinchu, Taiwan, in 2018. He is currently pursuingthe Ph.D. degree in the Institute of Communica-tions Engineering, National Tsing Hua University,Hsinchu, Taiwan. His research interests includewireless communication and cognitive radio net-works.

Cheng-Shang Chang (S’85-M’86-M’89-SM’93-F’04) received the B.S. degree from National Tai-wan University, Taipei, Taiwan, in 1983, and theM.S. and Ph.D. degrees from Columbia University,New York, NY, USA, in 1986 and 1989, respec-tively, all in electrical engineering.

From 1989 to 1993, he was employed as aResearch Staff Member with the IBM Thomas J.Watson Research Center, Yorktown Heights, NY,USA. Since 1993, he has been with the Departmentof Electrical Engineering, National Tsing Hua Uni-

versity, Taiwan, where he is a Tsing Hua Distinguished Chair Professor. He isthe author of the book Performance Guarantees in Communication Networks(Springer, 2000) and the coauthor of the book Principles, Architectures andMathematical Theory of High Performance Packet Switches (Ministry ofEducation, R.O.C., 2006). His current research interests are concerned withnetwork science, big data analytics, mathematical modeling of the Internet,and high-speed switching.

Dr. Chang served as an Editor for Operations Research from 1992 to1999, an Editor for the IEEE/ACM TRANSACTIONS ON NETWORKINGfrom 2007 to 2009, and an Editor for the IEEE TRANSACTIONS ONNETWORK SCIENCE AND ENGINEERING from 2014 to 2017. He iscurrently serving as an Editor-at-Large for the IEEE/ACM TRANSACTIONSON NETWORKING. He is a member of IFIP Working Group 7.3. He receivedan IBM Outstanding Innovation Award in 1992, an IBM Faculty PartnershipAward in 2001, and Outstanding Research Awards from the National ScienceCouncil, Taiwan, in 1998, 2000, and 2002, respectively. He also receivedOutstanding Teaching Awards from both the College of EECS and theuniversity itself in 2003. He was appointed as the first Y. Z. Hsu ScientificChair Professor in 2002. He received the Merit NSC Research Fellow Awardfrom the National Science Council, R.O.C. in 2011. He also received theAcademic Award in 2011 and the National Chair Professorship in 2017 fromthe Ministry of Education, R.O.C. He is the recipient of the 2017 IEEEINFOCOM Achievement Award.

Duan-Shin Lee (S’89-M’90-SM’98) received theB.S. degree from National Tsing Hua University,Taiwan, in 1983, and the MS and Ph.D. degreesfrom Columbia University, New York, in 1987 and1990, all in electrical engineering. He worked as aresearch staff member at the C&C Research Labo-ratory of NEC USA, Inc. in Princeton, New Jerseyfrom 1990 to 1998. He joined the Department ofComputer Science of National Tsing Hua Universityin Hsinchu, Taiwan, in 1998. Since August 2003,he has been a professor. He received a best paper

award from the Y.Z. Hsu Foundation in 2006. He served as an editor for theJournal of Information Science and Engineering between 2013 and 2015. Heis currently an editor for Performance Evaluation. Dr. Lee’s current researchinterests are network science, game theory, machine learning and high-speednetworks. He is a senior IEEE member.

APPENDIX A

It this section, we show the derivation of (15).

Pr

{X1∑N

j=2Xj + 1γ

≥ b, X2∑Nj=3Xj + 1

γ

≥ b, · · · Xr∑Nj=r+1Xj + 1

γ

≥ b

}

=

∫ ∞0

dxN · · ·∫ ∞0

dxr+1 ×∫ ∞b(∑Nj=r+1 xj+

1γ )

dxr · · ·∫ ∞b(∑Nj=2 xj+

1γ )

dx1 × e−xN · · · e−x1

=

∫ ∞0

dxN · · ·∫ ∞0

dxr+1 ×∫ ∞b(∑Nj=r+1 xj+

1γ )

dxr · · ·∫ ∞b(∑Nj=2 xj+

1γ )

e−x1dx1 × e−∑Nj=2 xj

=

∫ ∞0

dxN · · ·∫ ∞0

dxr+1 ×∫ ∞b(∑Nj=r+1 xj+

1γ )

dxr · · ·∫ ∞b(∑Nj=3 xj+

1γ )

dx2 × e−b∑Nj=2 xj−b

1γ e−

∑Nj=2 xj

=e−b1γ ×

∫ ∞0

dxN · · ·∫ ∞0

dxr+1 ×∫ ∞b(∑Nj=r+1 xj+

1γ )

dxr · · ·∫ ∞b(∑Nj=3 xj+

1γ )

e−(1+b)x2dx2 × e−(1+b)∑Nj=3 xj

=e−b1γe−b(1+b)

1γ

(1 + b)×∫ ∞0

dxN · · ·∫ ∞0

dxr+1×∫ ∞b(∑Nj=r+1 xj+

1γ )

dxr · · ·∫ ∞b(∑Nj=4 xj+

1γ )

dx3 × e−(1+b)∑Nj=3 xje−(1+b)b

∑Nj=3 xj

=e−b1γe−b(1+b)

1γ

(1 + b)×∫ ∞0

dxN · · ·∫ ∞0

dxr+1×∫ ∞b(∑Nj=r+1 xj+

1γ )

dxr · · ·∫ ∞b(∑Nj=4 xj+

1γ )

e−(1+b)2x3dx3 × e−(1+b)

2 ∑Nj=4 xj

=e−b1γe−b(1+b)

1γ

(1 + b)

e−b(1+b)2 1γ

(1 + b)2×∫ ∞0

dxN · · ·∫ ∞0

dxr+1×∫ ∞b(∑Nj=r+1 xj+

1γ )

dxr · · ·∫ ∞b(∑Nj=5 xj+

1γ )

e−(1+b)3x4dx4 × e−(1+b)

3 ∑Nj=5 xj

...

=

r−2∏k=0

(e−b

1γ (1+b)

k

(1 + b)k

)×∫ ∞0

dxN · · ·∫ ∞0

dxr+1 ×∫ ∞b(∑Nj=r+1 xj+

1γ )

e−(1+b)rxrdxr × e−(1+b)

r ∑Nj=r+1 xj

=

r−1∏k=0

(e−b

1γ (1+b)

k

(1 + b)k

)×∫ ∞0

dxN · · ·∫ ∞0

dxr+1 × e−(1+b)r ∑N

j=r+1 xj

=

r−1∏k=0

(e−b

1γ (1+b)

k

(1 + b)k

)×

N∏k=r+1

(∫ ∞0

e−(1+b)rxkdxk

)=

r−1∏k=0

(e−b

1γ (1+b)

k

(1 + b)k

)×(

1

(1 + b)r

)N−r

=

(e−b

1γ

∑r−1k=0(1+b)

k

(1 + b)∑r−1k=0 k

)×(

1

(1 + b)r

)N−r=

(e−

1γ ((1+b)

r−1)

(1 + b)r(r−1)

2

)×(

1

(1 + b)r

)N−r=

e−1γ ((1+b)

r−1)

(1 + b)r(r−1)

2 +r(N−r)=

e−1γ ((1+b)

r−1)

(1 + b)r(N−1−r−12 )