Optimal Slotted ALOHA under Delivery Deadline Constraint for Multiple-Packet Reception
aa r X i v : . [ c s . I T ] J u l Optimal Slotted ALOHA under Delivery DeadlineConstraint for Multiple-Packet Reception
Yijin Zhang, Yuan-Hsun Lo, Feng Shu, and Jun Li
Abstract —This paper considers the slotted ALOHA protocol in a communication channel shared by N users. It is assumed that thechannel has the multiple-packet reception (MPR) capability that allows the correct reception of up to M ( ≤ M < N ) time-overlappingpackets. To evaluate the reliability in the scenario that a packet needs to be transmitted within a strict delivery deadline D ( D ≥ ) (inunit of slot) since its arrival at the head of queue, we consider the successful delivery probability (SDP) of a packet as performancemetric of interest. We derive the optimal transmission probability that maximizes the SDP for any ≤ M < N and any D ≥ , andshow it can be computed by a fixed-point iteration. In particular, the case for D = 1 (i.e., throughput maximization) is first completelyaddressed in this paper. Based on these theoretical results, for real-life scenarios where N may be unknown and changing, we developa distributed algorithm that enables each user to tune its transmission probability at runtime according to the estimate of N . Simulationresults show that the proposed algorithm is effective in dynamic scenarios, with near-optimal performance. Index Terms —Slotted ALOHA, multiple-packet reception, optimal transmission probability, successful delivery probability ✦ NTRODUCTION
Since Abramson’s seminal work [1] in 1970, ALOHA-typeprotocols have been widely used for initial terminal accessor short packet transmissions in a variety of distributedwireless communication systems due to their simplicity.There were extensive studies on the slotted ALOHA underthe traditional model of a single-packet reception (SPR) chan-nel: a packet can be correctly received if there is no otherpacket transmission during its transmission. However, theSPR has become restrictive due to the advent of multiple-packet reception (MPR) techniques that allow the correctreception of time-overlapping packets. Hence, there is anatural interest in gaining a clear insight into the fundamen-tal impact of MPR on the behavior of the slotted ALOHAprotocol.Differently from previous studies that dealt with thestability, throughput or delay issue of slotted ALOHA underMPR [2]–[10], we in this paper concentrate on achievingmaximum reliability in the scenario that a packet needsto be transmitted within a strict delivery deadline D (inunit of slot) since its arrival at the head of queue. Such ascenario can be safety message dissemination in vehicularnetworks [11] or machine to-machine communications inInternet of Things [12]. Some recent work on a similar issuecan be found in [13]–[17] for an SPR channel, and [18]–[20]for a multichannel system. • Y. Zhang, F. Shu and J. Li are with the School of Electronic and Optical En-gineering, Nanjing University of Science and Technology, Nanjing, China,and also with National Mobile Communications Research Laboratory,Southeast University, Nanjing, China. E-mails: [email protected]; { shufeng, jun.li } @njust.edu.cn. • Y.-H. Lo is with the School of Mathematical Sciences, Xiamen University,Xiamen, China. E-mail: [email protected].
Along the lines of [6]–[10], this paper considers a specificMPR channel, namely the M -user MPR channel, in whichup to M time-overlapping packets can be received correctly.Our key contributions are summarized as follows: • We derive the optimal transmission probability forthe reliability maximization when N users contendfor the channel access. Our work can be seen as ageneralization of the work in [13] that only focusedon the SPR channel. Moreover, we show that theoptimal transmission probability can be computedby a fixed-point iteration. • As explained in Section 3, the saturation through-put maximization of finite-user slotted ALOHA [8],[10] can be studied as a special case D = 1 ofreliability maximization that we investigate here. Itshould be pointed out that Bae et al. [10] obtainedthe optimal transmission probability for saturationthroughput maximization necessarily relying on anunproved technical condition dd τ E [ X ] E [ X ] < N − ( X is the number of users involved in each successfultransmission and τ is the transmission probability).In this paper, we present analysis to prove that thistechnical condition always holds for an arbitrary ≤ M < N . Hence, the issue of saturation through-put maximization under an M -user MPR channel isfirst completely addressed in this paper. • Clearly, deriving the optimal transmission proba-bility requires priori knowledge of N . To achievemaximum reliability in real-life scenarios where N isunknown and changing over time, built on the the-oretical results derived above, we propose a tuningalgorithm that allows each user to estimate N with-out requiring any access parameter input, and adjustits transmission probability accordingly at runtime.Through an extensive performance study we show that the tuning algorithm is effective in a variety ofdynamic network configurations considered in thepaper, with near-optimal performance. The first attempt to study slotted ALOHA under MPRwas made by Ghez et al. [2], [3], in which they proposedthe symmetric MPR channel model and analyzed stabilityproperties under an infinite-user assumption. Naware etal. [4] extended the stability study to finite-user systemswithout posing any limitation on the MPR model, and inaddition investigated the average delay in capture channels.Luo et al. [5] further established the throughput and stabilityregions for finite population over a standard MPR channelin which simultaneous packet transmissions are not helpfulfor the reception of any particular group of packets.After the aforementioned studies for various general-ized MPR channels, the throughput performance of slot-ted ALOHA over an M -user MPR channel has receivedmuch attention recently. Gau [6], [7] derived the satura-tion and non-saturation throughput for finite-user cases.To demonstrate the capacity-enhancement, Zhang et al.in [8] proved that the maximum achievable throughputincreases superlinearly with M for both finite-user casewith saturation traffic and infinite-user case with randomtraffic, and in [9] further showed that superlinear scalingalso holds under bounded delay-moment requirements. Fol-lowing [8], to fully utilize the M -user MPR channel, Bae etal. [10] derived the optimal transmission probability thatmaximizes the saturation throughput in the finite-user caseunder some unproved technical conditions. To the best ofour knowledge, little work has been done to investigate thereliability issue over an M -user MPR channel. Our paperhere is an attempt along this direction.Under unknown and time-varying operating conditions,developing an estimation algorithm to acquire knowledgeof the number of users N is of significant importance. Manyapproaches have been proposed for the SPR channel. Themethod in [21] estimates N based on the channel state in theprevious slot, and the schemes in [22], [23] estimate N withstatistics of consecutive idle and collision slots. However,all of them require that N follows a Poisson distribution.To remove the assumption on the distribution of N , thealgorithm in [24] applies an ARMA filter to estimate N relying on the measured collision probability, the algorithmin [25] estimates N by the number of idle slots between twosuccessful transmissions, and the algorithm in [26] estimates N from the knowledge of number of consecutive idle slots.The extension of the estimation algorithm to the MPR caseis rarely reported. We are aware of only one previouslyproposed algorithm in [10] that estimates N according tothe collected information on the number of users involvedin each successful transmission. However, we find it isineffective in some dynamic scenarios, which will be shownin Section 5.The remainder of this paper is organized as follows.In Section 2, we describe the considered system model. InSection 3, we derive the optimal transmission probabilitythat maximizes the reliability for any ≤ M < N andany D ≥ , and show it can be computed by a fixed-point iteration. In section 4, we propose a tuning algorithm by which each user can achieve a reliability level close tothe theoretical limit at runtime. In Section 5, simulationresults verify the accuracy of our analytical results andthe effectiveness of the proposed tuning algorithm. Finally,Section 6 concludes this paper. YSTEM MODEL
As adopted in [8], [10], [13], we develop our analyticalmodel based on the following assumptions:(i) There are N ( N ≥ ) users with saturated traffic in thenetwork, and all of them are within the transmissionrange of each other.(ii) The system is limited by user interference and the chan-nel has an M -MPR capability, which means a packetcan be correctly decoded by the receiver if at most M − other packet transmissions overlap with it at any time,and is unrecoverable otherwise. To avoid some trivialcases, we assume ≤ M < N . Specially, M = 1 corresponds to the SPR channel.(iii) The channel time is divided into time slots of an equallength, and every packet exactly occupies the durationof one time slot.(iv) Each user knows the slot boundaries, and attempts totransmit a packet with a common transmission proba-bility τ at the beginning of a time slot, ≤ τ ≤ .(v) Every packet is neither acknowledged nor retransmit-ted, since that an acknowledgement mechanism wouldincur extra overhead, waiting time and energy cost forshort packets, and meanwhile, for some periodic traffic,the content can simply be replaced, and hence there isno need to retransmit an outdated packet.(vi) Every packet should be delivered within a strict de-livery deadline D ( D ≥ ) (in unit of slot), whichis defined as the duration from the moment of itsarrival at the head of the queue to the completion ofits transmission. PTIMAL T RANSMISSION P ROBABILITY
Given any real number τ ∈ [0 , and integer D ≥ ,let P D ( τ ) , called successful delivery probability (SDP), be theprobability that a packet will be successfully received withinthe delivery deadline D under the common transmissionprobability τ . Consider a tagged user. Let Y denote the num-ber of packets transmitted by the other N − users in a timeslot. It is easy to see Y follows a binomial distribution withparameters N − and τ , and then for i = 0 , , . . . , N − ,we have P ( Y = i ) = N − i ! τ i (1 − τ ) N − − i . (1) Furthermore, the value P D ( τ ) can be obtained as: P D ( τ ) = D X k =1 τ (1 − τ ) k − P ( Y ≤ M − D X k =1 τ (1 − τ ) k − M − X i =0 N − i ! τ i (1 − τ ) N − − i = (cid:0) − (1 − τ ) D (cid:1) M − X i =0 N − i ! τ i (1 − τ ) N − − i . (2)In this section, we aim to obtain the optimal transmissionprobability for maximizing P D ( τ ) .For a given integer D ≥ , let P maxD denote the maxi-mum SDP going through all possible τ ∈ [0 , , that is, P maxD := max τ ∈ [0 , P D ( τ ) . Then, define the optimal transmission probability , denoted by τ optD , to be the transmission probability such that the SDPachieves P maxD , i.e., τ optD := arg max τ ∈ [0 , P D ( τ ) . Note that τ optD may not be unique by definition. Remark 1: As P ( τ ) refers to the individual saturationthroughput defined as the time average of the number ofpackets successfully transmitted by a user provided that allusers have saturated traffic, τ opt is indeed the optimal trans-mission probability maximizing the saturation throughputunder MPR, which has been investigated in [10]. Remark 2:
When M = 1 , τ optD is the optimal transmissionprobability maximizing the SDP within the delivery dead-line D under SPR, which has been derived in [13].It is easy to see from (2) that, when D is fixed, P D ( τ ) is a continuous function of τ on the closed interval [0 , .Hence, by The Extreme Value Theorem, τ optD exists. In theremainder of this subsection, we shall show the uniquenessof τ optD , and present how to obtain it.Define the following semi open interval I := h − ( N − N − D ) D , (cid:17) . We first provide some properties of τ optD . Lemma 1.
For any integers D ≥ and ≤ M < N ,(i) τ optD is a solution of ddτ P D ( τ ) = 0 , and(ii) τ optD must lie in I .Proof. We prove these two statements by investigating themonotonicity of P D ( τ ) . Define f ( τ ) := M − X i =0 N − i ! τ i (1 − τ ) N − − i (3)and f ( τ ) := M − X i =0 i N − i ! τ i (1 − τ ) N − − i . (4)Obviously, f ( τ ) > and f ( τ ) ≥ for < τ < , ≤ M < N and D ≥ . By adopting the notation f and f , the derivative of P D ( τ ) with respect to τ can be written asdd τ P D ( τ ) = D (1 − τ ) D − f ( τ )+ (cid:16) f ( τ ) τ (1 − τ ) − ( N − f ( τ )1 − τ (cid:17)(cid:0) − (1 − τ ) D (cid:1) = (cid:16) ( N + D − − τ ) D − − N − − τ (cid:17) f ( τ )+ (cid:16) − (1 − τ ) D τ (1 − τ ) (cid:17) f ( τ ) (5) = 1 τ (1 − τ ) (cid:16)(cid:0) − (1 − τ ) D (cid:1) f ( τ ) − (cid:0) N − − ( N + D − − τ ) D (cid:1) τ f ( τ ) (cid:17) . (6)It is easy to see that dd τ P D ( τ ) is continuous on the interval (0 , .Since that P D ( τ ) > P D (0) = P D (1) = 0 if τ ∈ (0 , , weknow the continuous function P D ( τ ) has a local maximumat τ optD , which lies in (0 , . As dd τ P D ( τ ) always exists onthe interval τ ∈ (0 , , by the Fermat’s Theorem, τ optD is asolution of dd τ P D ( τ ) = 0 .Furthermore, we have from (5) that dd τ P D ( τ ) > for τ ∈ (0 , \ I , and from (6) that dd τ P D ( τ ) < as τ → − . By TheIntermediate Value Theorem, the solutions of dd τ P D ( τ ) = 0 must be in I , i.e., τ optD must lie in I .Let H ( τ ) := f ( τ ) f ( τ ) and H ( τ ) := τ (cid:16) N + D − − D − (1 − τ ) D (cid:17) . Following the proof of Lemma 1, in (6), τ ∗ is a solution ofequation dd τ P D ( τ ) = 0 if and only if it is a solution of thefollowing equation: H ( τ ) − H ( τ ) = 0 . (7)In what follows, we will show that the equation (7) has aunique solution in the interval τ ∈ (0 , by investigatingthe monotonicity of H ( τ ) and H ( τ ) separately. Lemma 2.
For τ ∈ (0 , , we have dd τ H ( τ ) = 0 if M = 1 , andotherwise < dd τ H ( τ ) < N − . Proof.
The case for M = 1 obviously holds as H ( τ ) = 0 . Inthe following, we only consider ≤ M < N .We first show that dd τ H ( τ ) > by a known resultin [10]. Let T ( τ ) := P Mi =1 i (cid:0) Ni (cid:1) τ i (1 − τ ) N − i P Mi =1 i (cid:0) Ni (cid:1) τ i (1 − τ ) N − i . (8) By letting j = i − , after some algebraic manipulations, wehave T ( τ ) − P Mi =1 ( i − i ) (cid:0) Ni (cid:1) τ i (1 − τ ) N − i P Mi =1 i (cid:0) Ni (cid:1) τ i (1 − τ ) N − i = P M − j =0 j ( j + 1) (cid:0) Nj +1 (cid:1) τ j +1 (1 − τ ) N − − j P M − j =0 ( j + 1) (cid:0) Nj +1 (cid:1) τ j +1 (1 − τ ) N − − j = τ N P M − j =0 j (cid:0) N − j (cid:1) τ j (1 − τ ) N − − j τ N P M − j =0 (cid:0) N − j (cid:1) τ j (1 − τ ) N − − j = H ( τ ) . (9)Since it has been proven in [10] that dd τ T ( τ ) ≥ for τ ∈ (0 , and M > , we have dd τ H ( τ ) = dd τ T ( τ ) > by (9).Now, we will show that dd τ H ( τ ) < N − . By thebinomial theorem, H ( τ ) can be rewritten as H ( τ ) = ( N − τ − P N − i = M i (cid:0) N − i (cid:1) τ i (1 − τ ) N − − i − P N − i = M (cid:0) N − i (cid:1) τ i (1 − τ ) N − − i = ( N − τ − P N − i = M (cid:0) i − ( N − τ (cid:1)(cid:0) N − i (cid:1) τ i (1 − τ ) N − − i − P N − i = M (cid:0) N − i (cid:1) τ i (1 − τ ) N − − i ( ∗ ) = ( N − τ − ( N − M ) (cid:0) N − M − (cid:1) τ M (1 − τ ) N − M P M − i =0 (cid:0) N − i (cid:1) τ i (1 − τ ) N − − i , = ( N − τ − ( N − M ) N − M − ! R ( τ ) , (10)where R ( τ ) := τ M (1 − τ ) N − M P M − i =0 (cid:0) N − i (cid:1) τ i (1 − τ ) N − − i . The proof of ( ∗ ) is as follows.For m = 2 , , . . . , N − , we have N − m − ! ( N − m ) τ m (1 − τ ) N − m + (cid:0) m − − ( N − τ (cid:1) N − m − ! τ m − (1 − τ ) N − m = N − m − ! ( N − m ) τ m (1 − τ ) N − m − ( N − m ) N − m − ! τ m − (1 − τ ) N − m + ( N − N − m − ! τ m − (1 − τ ) N − m +1 = − N − m − ! ( N − m ) τ m − (1 − τ ) N − m +1 + ( N − N − m − ! τ m − (1 − τ ) N − m +1 =( m − N − m − ! τ m − (1 − τ ) N − m +1 = N − m − ! ( N − m + 1) τ m − (1 − τ ) N − m +1 . Then by recursively using the above equation, we have N − X i = M (cid:0) i − ( N − τ (cid:1) N − i ! τ i (1 − τ ) N − − i = N − N − ! τ N − (1 − τ )+ N − X i = M (cid:0) i − ( N − τ (cid:1) N − i ! τ i (1 − τ ) N − − i = N − M − ! ( N − M ) τ M (1 − τ ) N − M . To prove dd τ H ( τ ) < N − , by (10), it suffices to showthat dd τ R ( τ ) > . Let Q ( x ) := x M − P M − i =0 (cid:0) N − i (cid:1) x i . By plugging x = τ − τ , we have R ( τ ) = τ Q ( x ) . (11)Note that as τ increases from to , x increases from 0 to ∞ , and hence Q ( x ) > . By taking the derivative of Q ( x ) with respect to x , we have, for x > ,dd x Q ( x ) = P M − i =0 ( M − − i ) (cid:0) N − i (cid:1) x M + i − (cid:16) P M − i =0 (cid:0) N − i (cid:1) x i (cid:17) > . (12)Then,dd τ R ( τ ) = dd τ (cid:16) τ Q ( x ) (cid:17) = Q ( x ) + τ (1 − τ ) · dd x Q ( x ) > . (13)Hence the result follows. Lemma 3.
For τ ∈ (0 , , we havedd τ H ( τ ) > N − . Proof.
Taking the derivative of H ( τ ) with respect to τ derives thatdd τ H ( τ ) = N + D − − D − (1 − τ ) D − Dτ (1 − τ ) D − (cid:0) − (1 − τ ) D (cid:1) = N − D (cid:16) − − (1 − τ ) D − Dτ (1 − τ ) D − (cid:0) − (1 − τ ) D (cid:1) (cid:17) So we havedd τ H ( τ ) > N − ⇔ (cid:0) − (1 − τ ) D (cid:1) > − (1 − τ ) D − Dτ (1 − τ ) D − ⇔ (cid:0) − (1 − τ ) D (cid:1) + (1 − τ ) D + Dτ (1 − τ ) D − − > ⇔ (1 − τ ) D − (1 − τ ) D + Dτ (1 − τ ) D − > ⇔ (1 − τ ) D +1 + ( D + 1) τ − > (14)Let G ( τ ) := (1 − τ ) D +1 + ( D + 1) τ − . We havedd τ G ( τ ) = − ( D + 1)(1 − τ ) D + D + 1= ( D + 1) (cid:0) − (1 − τ ) D (cid:1) , τ M =1 M =2 M =3 M =8 M =9 D =1 D =5 D =50 H ( τ ) H ( τ ) Fig. 1. H ( τ ) for the varying M and τ , H ( τ ) for the varying D and τ when N = 10 . which is larger than for τ ∈ (0 , . Therefore, G ( τ ) isstrictly increasing for τ ∈ (0 , , which implies that G ( τ ) > lim τ → + G ( τ ) = 0 . Hence we complete the proof by (14).To illustrate that ≤ dd τ H ( τ ) < N − < dd τ H ( τ ) onthe interval τ ∈ (0 , for any D ≥ and ≤ M < N obtained by Lemma 1 and Lemma 2, a numerical exampleis presented in Fig. 1, which plots H ( τ ) and H ( τ ) for thevarying τ , M and D when N = 10 .Now, we are ready to derive the uniqueness of τ optD . Theorem 4.
For any integers D ≥ and ≤ M < N , theequation (7) has a unique solution on τ in the interval < τ < ,denoted by τ ∗ , and τ optD = τ ∗ .Proof. Suppose there are two distinct solutions to the equa-tion (7) in (0 , . By The Mean Value Theorem, there existsa solution of dd τ H ( τ ) = dd τ H ( τ ) . However, by Lemma 2and Lemma 3, we havedd τ H ( τ ) < N − < dd τ H ( τ ) , ∀ τ ∈ (0 , . This implies a contradiction to dd τ H ( τ ) = dd τ H ( τ ) . Hencewe conclude that the equation (7) has a unique solution on τ in the interval < τ < , which by Lemma 1 promises theuniqueness of τ optD , and yields τ optD = τ ∗ . Remark 3:
In the context of saturation throughput max-imization, Bae et al. in [10] derived the τ opt under theassumption ddτ T ( τ ) < N in the interval τ ∈ (0 , . Theyclaimed that they proved dd τ T ( τ ) < N for M = 1 , , ,but could not prove it for any arbitrary M due to ex-tremely complex algebraic manipulations. Here, the proofin Lemma 2 has addressed this unsolved question, as dd τ T ( τ ) = dd τ H ( τ ) < N − . In other words, the issueof saturation throughput maximization under an M -userMPR channel is first completely addressed in this paper.Moreover, we in Theorem 4 obtained the existence anduniqueness of τ optD for any D ≥ and any ≤ M < N without any assumption. For the case M = 1 , which implies H ( τ ) = 0 , it is easyto see from (7) that τ optD = 1 − (cid:0) N − N − D (cid:1) D . For the case < M < N , as H ( τ ) > H ( τ ) = 0 when τ = 1 − ( N − N − D ) D , we by Lemma 1 know that τ optD ∈ (cid:16) − (cid:0) N − N − D (cid:1) D , (cid:17) . Define a fixed-point iteration: x n +1 = x n · H ( x n ) + 1 H ( x n ) + 1 for n = 0 , , , . . . . With the following theorem we guaranteethat τ optD for M > can be obtained by this fixed-pointiteration. Theorem 5.
For any initial guess x ∈ (1 − ( N − N − D ) D , ,the sequence x , x , x , . . . converges to the fixed point τ optD for < M < N .Proof. Define g ( x ) := x ( H ( x ) + 1) H ( x ) + 1 on the domain (0 , . By (7) and Theorem 4, the equation g ( x ) = x has a unique solution at x = τ optD in (0 , .Therefore, the case that x = τ optD obtains the fixed point τ optD since g ( τ optD ) = τ optD . So we consider x = τ optD in whatfollows.Since g ( x ) = x has a unique solution at τ optD ∈ (0 , , toprove the sequence { x n } ∞ converges to τ optD , it suffices toshow that ( x < g ( x ) < τ optD , for x ∈ (1 − ( N − N − D ) D , τ optD ); τ optD < g ( x ) < x, for x ∈ ( τ optD , . (15)As proved in Appendix thatdd x g ( x ) > for x ∈ (0 , , (16) g ( x ) is increasing on (0 , . This implies that ( g ( x ) < τ optD , for x ∈ (1 − ( N − N − D ) D , τ optD ); τ optD < g ( x ) , for x ∈ ( τ optD , . (17)Since, in the interval (0 , , H ( x ) = H ( x ) only when x = τ optD and dd x H ( x ) < dd x H ( x ) by Lemma 2 and Lemma 3,we have < H ( x ) < H ( x ) for x ∈ ( τ optD , . Then, g ( x ) = x H ( x ) + 1 H ( x ) + 1 < x, for x ∈ ( τ optD , . (18)Following the same argument, we have H ( x ) > H ( x ) for x ∈ (1 − ( N − N − D ) D , τ optD ) . By the result in Lemma 3 that dd x H ( x ) > and the L’Hospital’s Rule that lim x → + H ( x ) = − , we further have H ( x ) > H ( x ) > − for x ∈ (1 − ( N − N − D ) D , τ optD ) . Then, g ( x ) = x H ( x ) + 1 H ( x ) + 1 > x, for x ∈ (1 − ( N − N − D ) D , τ optD ) . (19) Therefore, (15) can be derived combining (17)–(19), and thusthe result follows.
UNTIME O PTIMIZATION
In the previous section, the optimal transmission probabilityhas been derived, however, the analysis requires knowing inadvance the number of users, N , which may be unavailablein some practical scenarios of distributed networks. To copewith this restriction, we in this section develop a distributedalgorithm to estimate N for M > , by which each user cantune the transmission probability to obtain an SDP close tothe theoretical limit.Consider a tagged user. Recall the variable Y that de-notes the number of packets transmitted by the other N − users in a time slot. By the distribution of Y in (1), for ≤ i < i ≤ M , we obtain the following equation: P ( Y = i ) P ( Y = i − P ( Y = i ) P ( Y = i − (cid:0) N − i (cid:1)(cid:0) N − i − (cid:1) τ i + i − (1 − τ ) N − − i − i (cid:0) N − i (cid:1)(cid:0) N − i − (cid:1) τ i + i − (1 − τ ) N − − i − i = (cid:0) N − i (cid:1)(cid:0) N − i − (cid:1)(cid:0) N − i (cid:1)(cid:0) N − i − (cid:1) = i ( N − i ) i ( N − i ) . (20)It then directly follows that: N = i ( i − i ) i P ( Y = i ) P ( Y = i − P ( Y = i ) P ( Y = i − − i + i . (21)As P ( Y = i ) P ( Y = i − P ( Y = i ) P ( Y = i − is locally measurable if a user isequipped with an array with at least i antennas [27], [28],we find that (21) provides a linear function for the taggeduser to estimate N without priori knowledge of other accessparameters.We assume that the tagged user knows there are at most N max users in the network, but the actual number of users ischanging and unknown. An update interval is defined as ablock of consecutive of L time slots. We require the taggeduser to update the transmission probability at the beginningof the n + 1 th update interval, according to the followingtwo necessary estimates of the network status.(i) µ n : the estimated value of P ( Y = i ) P ( Y = i − P ( Y = i ) P ( Y = i − at the endof the n th update interval;(ii) N n : the estimated value of N at the end of the n thupdate interval.The tagged user initially guesses that there are N = N max users, and sets µ = i ( N − i ) i ( N − i ) . The transmissionprobability for the first update interval is obtained withgiven N , by Theorem 4 and a fixed-point iteration. Duringthe n th interval for n = 1 , , . . . , we describe the proposedalgorithm as follows:(i) Update µ n at the end of the n th update interval : To evaluate P ( Y = i ) P ( Y = i − P ( Y = i ) P ( Y = i − at runtime, the tagged user is requiredto record A i,n , the number of the slots during the n thupdate interval in which i users are simultaneouslytransmitting and the tagged user is not transmitting,for i = i − , i , i − and i . Let µ n be the measure of P ( Y = i ) P ( Y = i − P ( Y = i ) P ( Y = i − during the n th update interval, and its value is calculated by: µ n = A i ,n · A i − ,n A i ,n · A i − ,n To avoid sharp changes in the estimated value, thetagged user further applies an Exponential MovingAverage filter as follows: µ n = δ · µ n − + (1 − δ ) · µ n where δ ∈ [0 , is a memory factor.(ii) Update N n at the end of the n th update interval : By (21),the tagged user obtains N n = D i ( i − i ) i µ n − i + i E where h x i is the integer closest to x .(iii) Update the transmission probability at the beginning ofthe n + 1 th update interval : Update the transmissionprobability with given N n , by Theorem 4 and a fixed-point iteration.To avoid harmful measure of µ n due to occasional fluctua-tions of the network status, (i) if µ n is found to be smallerthan µ , a contradiction to the fact that (20) must be equalto or larger than µ as M < N ≤ N , the tagged usersets µ n = µ ; (ii) if µ n is found larger than i ( M +1 − i ) i ( M +1 − i ) ,a contradiction to (20) as N > M , the tagged user sets µ n = i ( M +1 − i ) i ( M +1 − i ) ; and (iii) if µ n has an invalid value as A i ,n · A i − ,n = 0 , the tagged user sets µ n = µ n − . Remark 4:
It should be noted that, although the pro-posed algorithm for estimating N is devised for the slot-ted ALOHA in saturation assumption, it can apply alsoto unsaturated traffic, provided that the estimation targetbecomes the average number of competing users rather thanthe total number of users; and can also apply to τ -persistentCSMA, provided that the length of a slot may vary due todifferent channel status: idle, collision or success. IMULATION R ESULTS
In this section, to demonstrate the accuracy of the derivedanalytical results and the effectiveness of our proposedtuning algorithm, we present simulation results with respectto different network parameters obtained from Monte Carlosimulation developed in Matlab. In the following, we willanalyze the performance under stationary scenarios when N is known, and then under dynamic scenarios when N isunknown. Each simulation point in stationary conditionsrepresents the average value over 10 independent runs,each of which consists of slots. The results in dynamicconditions are from a single representative simulation runfor 500 updated intervals. N is known Fig. 2 and Fig. 3 show the optimal transmission probabilityand the maximum SDP as a function of N for differentvalues of M and D , respectively. We see a good agree-ment between analytical and simulation results in all thescenarios. Notice that the results for D = 1 can be seen asthe throughput maximization issue investigated in [10]. In
10 15 20 25 30 35 400.020.040.060.080.10.120.140.160.18 The number of users, N O p t i m a l t r an s m i ss i on p r obab ili t y D=1D=5D=10D=20D=50 M =2, analysis M =2, simulation (a) M = 2
10 15 20 25 30 35 400.050.10.150.20.250.30.350.4 The number of users, N O p t i m a l t r an s m i ss i on p r obab ili t y D=1D=5D=10D=20D=50 M =5, analysis M =5, simulation (b) M = 5
10 15 20 25 30 35 4000.10.20.30.40.50.60.7 The number of users, N O p t i m a l t r an s m i ss i on p r obab ili t y D=1D=5D=10D=20D=50 M =8, analysis M =8, simulation (c) M = 8 Fig. 2. The optimal transmission probability as a function of N for different values of M and D .
10 15 20 25 30 35 4000.10.20.30.40.50.60.70.80.9 The number of users, N M a x i m u m s u cc e ss f u l de li v e r y p r obab ili t y D=1D=5D=10D=20D=50 M =2, analysis M =2, simulation (a) M = 2
10 15 20 25 30 35 4000.10.20.30.40.50.60.70.80.91 The number of users, N M a x i m u m s u cc e ss f u l de li v e r y p r obab ili t y D=1D=5D=10D=20D=50 M =5, analysis M =5, simulation (b) M = 5
10 15 20 25 30 35 400.10.20.30.40.50.60.70.80.91 The number of users, N M a x i m u m s u cc e ss f u l de li v e r y p r obab ili t y D=1D=5D=10D=20D=50 M =8, analysis M =8, simulation (c) M = 8 Fig. 3. The maximum SDP as a function of N for different values of M and D . Fig. 2, as expected, we observe that the optimal transmis-sion probability becomes smaller when N increases. Thisis because that more users attempt to access the channeland the contention level becomes severer. We also see theoptimal transmission probability becomes larger when M increases or D decreases. The reason is that a user needsto be more aggressive in accessing the channel if moreconcurrent packets can be successfully received or the userwants to successfully send out a packet within a shorterdelivery deadline. In Fig. 3, as expected, the curves showthat a smaller N , a larger M or a larger D leads to a largervalue of the maximum SDP. In particular, we note that theoptimal transmission probability for D > is smaller thanthat for D = 1 for given N and M . This phenomenonindicates that the throughput maximization degrades theSDP for D > , and in turn the maximization of SDP for D > degrades the throughput performance. N is unknown By setting i = M and i = 2 in (21), we analyze theperformance of the proposed tuning algorithm when thenumber of users sharply changes. In detail, 20 users arealways active from 1st to 500th updated interval, and 20new users are active only from the 101st updated intervalto the 400th updated interval. Each user knows there areat most N max = 100 users, and initially guesses there are N = 100 users. D = 1 We start by discussing the performance for D = 1 . Fig. 4and Fig. 5 show the estimated number of users and SDP for M = 5 with different L and δ , respectively. To characterizeindividual behavior and improve the readability, we onlypresent the real trajectories of two representative users inthe same run.We observe from Fig. 4 that, in all the cases, when theactual number of users is changed to 20, the estimate rapidlyadapts to changes and keeps a high level of accuracy; whenthe actual number of users is changed to 40, the estimate isstill able to capture changes within a few update intervals,but the slop increases, i.e, the accuracy of the estimationdegrades. This phenomenon is because that as the actualnumber of users increases, the fluctuation in measuring µ n is amplified in estimation by using (21).As expected, we observe from Fig. 5 that the usersachieve SDPs close to the theoretical limit at runtime by tun-ing the transmission probability according to the estimatednumber of users as shown in Fig. 4. Moreover, even whenthe oscillation of estimation is high, we find that the SDP stillkeeps relatively stable. This phenomenon is due to the factthat the SDP is less sensitive to the change in transmissionprobability as the number of users increases.To evaluate the fairness of the proposed algorithm, wefurther study mean and variance of SDP among all activeusers at different stages. As shown in Table 1, we find T he e s t i m a t ed nu m be r o f u s e r s One of initially active usersOne of new usersOne of initially active users [2]One of new users [2]Actual value (a) L = 50000 , δ = 0 . T he e s t i m a t ed nu m be r o f u s e r s One of initially active usersOne of new usersOne of initially active users [2]One of new users [2]Actual value (b) L = 20000 , δ = 0 . Fig. 4. The estimated number of users for M = 5 and D = 1 when the number of users sharply changes. S u cc e ss f u l de li v e r y p r obab ili t y One of initially active usersOne of new usersOne of initially active users [2]One of new users [2]Theoretical maximum value (a) L = 50000 , δ = 0 . S u cc e ss f u l de li v e r y p r obab ili t y One of initially active usersOne of new usersOne of initially active users [2]One of new users [2]Theoretical maximum value (b) L = 20000 , δ = 0 . Fig. 5. Individual SDP for M = 5 and D = 1 when the number of users sharply changes. the mean value is very close to the theoretical maximumvalue for each stage with a tolerance of at most 5%, andthe variance value is also very small. This result indicatesthat the proposed algorithm can enable every active userto achieve near-optimal performance in dynamic scenarios.Notice that the transient period is a dominant factor todegrade the performance.As the case with D = 1 can be viewed as runtimeoptimization of throughput, to show the proposed algo-rithm is more effective in a variety of dynamic scenarios,we consider the approach in [10] for comparison purposes.The method therein recursively updates the transmissionprobability with an estimated N , and then uses the measureof E [ X ] E [ X ] ( X is the number of users involved in each success-ful transmission slot) to estimate new N necessarily relyingon the adopted transmission probability. As shown in Fig. 4and Fig. 5, one sees their estimate keeps a very high level ofaccuracy when 20 new users are inactive, but is not able tokeep track of the change well when 20 new users are active.This phenomenon is because when two groups of users become active at different time instants, they may adoptdifferent transmission probabilities, and then have biasedestimate of N . Whereas, our algorithm does not require anyaccess parameter input, and hence would not lead to biasedestimate in such a scenario. D > Fig. 6 shows the real trajectories of the estimated number ofusers of two representative users for M = 5 and D = 20 .Similar to the case of D = 1 as shown in Fig. 4, the estimateis able to capture changes within a few update intervals,but the accuracy degrades when the actual number of usersincreases.Fig. 7 shows the real trajectories of individual SDPs oftwo representative users for M = 5 and D = 20 . We observethat both users are able to adapt to different changes in thenumber of users, and achieve SDPs close to the theoreticallimit. In addition, to evaluate the fairness issue, we in Table 2compare the mean and variance of individual SDP amongall active users against the theoretical maximum SDP for Tuning parameters Interval index Theoretical maximum value Mean in our simulation Variance in our simulation L = 50000 , δ = 0 . −
100 0 . . . −
400 0 . . . · − −
500 0 . . . · − L = 20000 , δ = 0 . −
100 0 . . . −
400 0 . . . −
500 0 . . . TABLE 1Mean and variance of individual SDP among all active users at different stages for M = 5 and D = 1 . D = 20 . We confirm that the average performance of eachuser is near-optimal at every stage. ONCLUSION
In this paper, we investigated the impact of MPR capability M and delivery deadline D on the optimal transmissionprobability that maximizes the SDP of the slotted ALOHAprotocol in a communication channel shared by N userswith saturated traffic. We have obtained the optimal trans-mission probability for any ≤ M < N and any D ≥ , andshown it can be computed by a fixed-point iteration. Then,we developed an adaptive tuning algorithm for maximizingthe SDP when N is unknown and time-varying. Simulationresults show the proposed algorithm is effective in a varietyof dynamic scenarios.As a special case of our work, the maximization of theSDP when D = 1 can be viewed as the maximization ofindividual throughput. It should be pointed out that [10] isa first attempt to deal with this issue, however, the optimaltransmission probability therein was obtained by necessar-ily assuming dd τ T ( τ ) < N − for any ≤ M < N in theinterval < τ < . Therefore, the throughput maximizationfor an M -user MPR channel is first completely addressed inthis paper. A PPENDIX
Proof of (16) . We first havedd x g ( x ) = x d H ( x ) d x + H ( x ) + 1( H ( x ) + 1) · (cid:16) H ( x ) + x d H ( x ) d x (cid:17) . Since, for x ∈ (0 , , dd x H ( x ) > by Lemma 2 and H ( x ) > by definition, to prove dd x g ( x ) > it suffices to show that H ( x ) + x d H ( x ) d x =1 − D x (1 − x ) D − (1 − (1 − x ) D ) > , (22)for x ∈ (0 , .Let W ( x ) := D x (1 − x ) D − (1 − (1 − x ) D ) . Observe thatdd x W ( x ) = 2 D x (1 − x ) D − (1 − (1 − x ) D ) (cid:16) (1 − Dx ) (cid:0) − (1 − x ) D (cid:1) − x (1 − x ) D (cid:17) . Since − Dx − (1 − Dx + x )(1 − x ) D = 1 − Dx − (1 − Dx + x )(1 − x ) D < − Dx − (1 − Dx + x ) (23) = − x < , where (23) is due to < (1 − x ) D < , we have dd x W ( x ) < for x ∈ (0 , , i.e., W ( x ) is decreasing. Moreover, by theL’Hospital’s Rule we have lim x → + W ( x ) = 1 . This concludes that W ( x ) < for x ∈ (0 , , which implies(22). Hence we complete the proof. A CKNOWLEDGEMENTS
This work was partially supported by the National Natu-ral Science Foundation of China (No. 11601454, 61472190,61501238), the Open Research Fund of National MobileCommunications Research Laboratory, Southeast University(No. 2017D09, 2017D04), and the Natural Science Founda-tion of Fujian Province of China (No. 2016J05021). R EFERENCES [1] N. Abramson, “The ALOHA system: Another alternative for com-puter communications,” in
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