Optimal Slotted ALOHA under Delivery Deadline Constraint for MultiplePacket Reception
Abstract
This paper considers the slotted ALOHA protocol in a communication channel shared by users. It is assumed that the channel has the multiplepacket reception (MPR) capability that allows the correct reception of up to () timeoverlapping packets. To evaluate the reliability in the scenario that a packet needs to be transmitted within a strict delivery deadline () (in unit of slot) since its arrival at the head of queue, we consider the successful delivery probability (SDP) of a packet as performance metric of interest. We derive the optimal transmission probability that maximizes the SDP for any and any , and show it can be computed by a fixedpoint iteration. In particular, the case for (i.e., throughput maximization) is first completely addressed in this paper. Based on these theoretical results, for reallife scenarios where may be unknown and changing, we develop a distributed algorithm that enables each user to tune its transmission probability at runtime according to the estimate of . Simulation results show that the proposed algorithm is effective in dynamic scenarios, with nearoptimal performance.
1 Introduction
1.1 Motivation
Since Abramson’s seminal work [1] in 1970, ALOHAtype protocols have been widely used for initial terminal access or short packet transmissions in a variety of distributed wireless communication systems due to their simplicity. There were extensive studies on the slotted ALOHA under the traditional model of a singlepacket reception (SPR) channel: a packet can be correctly received if there is no other packet transmission during its transmission. However, the SPR has become restrictive due to the advent of multiplepacket reception (MPR) techniques that allow the correct reception of timeoverlapping packets. Hence, there is a natural interest in gaining a clear insight into the fundamental impact of MPR on the behavior of the slotted ALOHA protocol.
Differently from previous studies that dealt with the stability, throughput or delay issue of slotted ALOHA under MPR [2, 3, 4, 5, 6, 7, 8, 9, 10], we in this paper concentrate on achieving maximum reliability in the scenario that a packet needs to be transmitted within a strict delivery deadline (in unit of slot) since its arrival at the head of queue. Such a scenario can be safety message dissemination in vehicular networks [11] or machine tomachine communications in Internet of Things [12]. Some recent work on a similar issue can be found in [13, 14, 15, 16, 17] for an SPR channel, and [20, 18, 19] for a multichannel system.
1.2 Contribution
Along the lines of [6, 7, 8, 9, 10], this paper considers a specific MPR channel, namely the user MPR channel, in which up to timeoverlapping packets can be received correctly. Our key contributions are summarized as follows:

We derive the optimal transmission probability for the reliability maximization when users contend for the channel access. Our work can be seen as a generalization of the work in [13] that only focused on the SPR channel. Moreover, we show that the optimal transmission probability can be computed by a fixedpoint iteration.

As explained in Section 3, the saturation throughput maximization of finiteuser slotted ALOHA [8, 10] can be studied as a special case of reliability maximization that we investigate here. It should be pointed out that Bae et al. [10] obtained the optimal transmission probability for saturation throughput maximization necessarily relying on an unproved technical condition ( is the number of users involved in each successful transmission and is the transmission probability). In this paper, we present analysis to prove that this technical condition always holds for an arbitrary . Hence, the issue of saturation throughput maximization under an user MPR channel is first completely addressed in this paper.

Clearly, deriving the optimal transmission probability requires priori knowledge of . To achieve maximum reliability in reallife scenarios where is unknown and changing over time, built on the theoretical results derived above, we propose a tuning algorithm that allows each user to estimate without requiring any access parameter input, and adjust its transmission probability accordingly at runtime. Through an extensive performance study we show that the tuning algorithm is effective in a variety of dynamic network configurations considered in the paper, with nearoptimal performance.
1.3 Related work
The first attempt to study slotted ALOHA under MPR was made by Ghez et al. [2, 3], in which they proposed the symmetric MPR channel model and analyzed stability properties under an infiniteuser assumption. Naware et al. [4] extended the stability study to finiteuser systems without posing any limitation on the MPR model, and in addition investigated the average delay in capture channels. Luo et al. [5] further established the throughput and stability regions for finite population over a standard MPR channel in which simultaneous packet transmissions are not helpful for the reception of any particular group of packets.
After the aforementioned studies for various generalized MPR channels, the throughput performance of slotted ALOHA over an user MPR channel has received much attention recently. Gau [6, 7] derived the saturation and nonsaturation throughput for finiteuser cases. To demonstrate the capacityenhancement, Zhang et al. in [8] proved that the maximum achievable throughput increases superlinearly with for both finiteuser case with saturation traffic and infiniteuser case with random traffic, and in [9] further showed that superlinear scaling also holds under bounded delaymoment requirements. Following [8], to fully utilize the user MPR channel, Bae et al. [10] derived the optimal transmission probability that maximizes the saturation throughput in the finiteuser case under some unproved technical conditions. To the best of our knowledge, little work has been done to investigate the reliability issue over an user MPR channel. Our paper here is an attempt along this direction.
Under unknown and timevarying operating conditions, developing an estimation algorithm to acquire knowledge of the number of users is of significant importance. Many approaches have been proposed for the SPR channel. The method in [21] estimates based on the channel state in the previous slot, and the schemes in [22, 23] estimate with statistics of consecutive idle and collision slots. However, all of them require that follows a Poisson distribution. To remove the assumption on the distribution of , the algorithm in [24] applies an ARMA filter to estimate relying on the measured collision probability, the algorithm in [25] estimates by the number of idle slots between two successful transmissions, and the algorithm in [26] estimates from the knowledge of number of consecutive idle slots. The extension of the estimation algorithm to the MPR case is rarely reported. We are aware of only one previously proposed algorithm in [10] that estimates according to the collected information on the number of users involved in each successful transmission. However, we find it is ineffective in some dynamic scenarios, which will be shown in Section 5.
The remainder of this paper is organized as follows. In Section 2, we describe the considered system model. In Section 3, we derive the optimal transmission probability that maximizes the reliability for any and any , and show it can be computed by a fixedpoint iteration. In section 4, we propose a tuning algorithm by which each user can achieve a reliability level close to the theoretical limit at runtime. In Section 5, simulation results verify the accuracy of our analytical results and the effectiveness of the proposed tuning algorithm. Finally, Section 6 concludes this paper.
2 System model
As adopted in [13, 8, 10], we develop our analytical model based on the following assumptions:

There are () users with saturated traffic in the network, and all of them are within the transmission range of each other.

The system is limited by user interference and the channel has an MPR capability, which means a packet can be correctly decoded by the receiver if at most other packet transmissions overlap with it at any time, and is unrecoverable otherwise. To avoid some trivial cases, we assume . Specially, corresponds to the SPR channel.

The channel time is divided into time slots of an equal length, and every packet exactly occupies the duration of one time slot.

Each user knows the slot boundaries, and attempts to transmit a packet with a common transmission probability at the beginning of a time slot, .

Every packet is neither acknowledged nor retransmitted, since that an acknowledgement mechanism would incur extra overhead, waiting time and energy cost for short packets, and meanwhile, for some periodic traffic, the content can simply be replaced, and hence there is no need to retransmit an outdated packet.

Every packet should be delivered within a strict delivery deadline () (in unit of slot), which is defined as the duration from the moment of its arrival at the head of the queue to the completion of its transmission.
3 Optimal Transmission Probability
Given any real number and integer , let , called successful delivery probability (SDP), be the probability that a packet will be successfully received within the delivery deadline under the common transmission probability . Consider a tagged user. Let denote the number of packets transmitted by the other users in a time slot. It is easy to see follows a binomial distribution with parameters and , and then for , we have
(1) 
Furthermore, the value can be obtained as:
(2) 
In this section, we aim to obtain the optimal transmission probability for maximizing .
For a given integer , let denote the maximum SDP going through all possible , that is,
Then, define the optimal transmission probability, denoted by , to be the transmission probability such that the SDP achieves , i.e.,
Note that may not be unique by definition.
Remark 1: As refers to the individual saturation throughput defined as the time average of the number of packets successfully transmitted by a user provided that all users have saturated traffic, is indeed the optimal transmission probability maximizing the saturation throughput under MPR, which has been investigated in [10].
Remark 2: When , is the optimal transmission probability maximizing the SDP within the delivery deadline under SPR, which has been derived in [13].
It is easy to see from (2) that, when is fixed, is a continuous function of on the closed interval . Hence, by The Extreme Value Theorem, exists. In the remainder of this subsection, we shall show the uniqueness of , and present how to obtain it.
Define the following semi open interval
We first provide some properties of .
Lemma 1.
For any integers and ,

is a solution of , and

must lie in .
Proof.
We prove these two statements by investigating the monotonicity of . Define
(3) 
and
(4) 
Obviously, and for , and .
By adopting the notation and , the derivative of with respect to can be written as
(5)  
(6) 
It is easy to see that is continuous on the interval .
Since that if , we know the continuous function has a local maximum at , which lies in . As always exists on the interval , by the Fermat’s Theorem, is a solution of .
Let
and
Following the proof of Lemma 1, in (6), is a solution of equation if and only if it is a solution of the following equation:
(7) 
In what follows, we will show that the equation (7) has a unique solution in the interval by investigating the monotonicity of and separately.
Lemma 2.
For , we have if , and otherwise
Proof.
The case for obviously holds as . In the following, we only consider .
We first show that by a known result in [10]. Let
(8) 
By letting , after some algebraic manipulations, we have
(9) 
Since it has been proven in [10] that for and , we have by (9).
Now, we will show that . By the binomial theorem, can be rewritten as
(10) 
where
The proof of is as follows.
For , we have
Then by recursively using the above equation, we have
To prove , by (10), it suffices to show that . Let
By plugging , we have
(11) 
Note that as increases from to , increases from 0 to , and hence . By taking the derivative of with respect to , we have, for ,
(12) 
Then,
(13) 
Hence the result follows. ∎
Lemma 3.
For , we have
Proof.
Taking the derivative of with respect to derives that
So we have
(14) 
Let . We have
which is larger than for . Therefore, is strictly increasing for , which implies that
Hence we complete the proof by (14). ∎
To illustrate that on the interval for any and obtained by Lemma 1 and Lemma 2, a numerical example is presented in Fig. 1, which plots and for the varying , and when .
Now, we are ready to derive the uniqueness of .
Theorem 4.
For any integers and , the equation (7) has a unique solution on in the interval , denoted by , and .
Proof.
Suppose there are two distinct solutions to the equation (7) in . By The Mean Value Theorem, there exists a solution of . However, by Lemma 2 and Lemma 3, we have
This implies a contradiction to . Hence we conclude that the equation (7) has a unique solution on in the interval , which by Lemma 1 promises the uniqueness of , and yields . ∎
Remark 3: In the context of saturation throughput maximization, Bae et al. in [10] derived the under the assumption in the interval . They claimed that they proved for , but could not prove it for any arbitrary due to extremely complex algebraic manipulations. Here, the proof in Lemma 2 has addressed this unsolved question, as . In other words, the issue of saturation throughput maximization under an user MPR channel is first completely addressed in this paper. Moreover, we in Theorem 4 obtained the existence and uniqueness of for any and any without any assumption.
For the case , which implies , it is easy to see from (7) that
For the case , as when , we by Lemma 1 know that
Define a fixedpoint iteration:
for . With the following theorem we guarantee that for can be obtained by this fixedpoint iteration.
Theorem 5.
For any initial guess , the sequence converges to the fixed point for .
Proof.
Define
on the domain . By (7) and Theorem 4, the equation has a unique solution at in . Therefore, the case that obtains the fixed point since . So we consider in what follows.
Since has a unique solution at , to prove the sequence converges to , it suffices to show that
(15) 
As proved in Appendix that
(16) 
is increasing on . This implies that
(17) 
Since, in the interval , only when and by Lemma 2 and Lemma 3, we have for . Then,
(18) 
Following the same argument, we have for . By the result in Lemma 3 that and the L’Hospital’s Rule that
we further have for . Then,
(19) 
Therefore, (15) can be derived combining (17)–(19), and thus the result follows. ∎
4 Runtime Optimization
In the previous section, the optimal transmission probability has been derived, however, the analysis requires knowing in advance the number of users, , which may be unavailable in some practical scenarios of distributed networks. To cope with this restriction, we in this section develop a distributed algorithm to estimate for , by which each user can tune the transmission probability to obtain an SDP close to the theoretical limit.
Consider a tagged user. Recall the variable that denotes the number of packets transmitted by the other users in a time slot. By the distribution of in (1), for , we obtain the following equation:
(20) 
It then directly follows that:
(21) 
As is locally measurable if a user is equipped with an array with at least antennas [27, 28], we find that (21) provides a linear function for the tagged user to estimate without priori knowledge of other access parameters.
We assume that the tagged user knows there are at most users in the network, but the actual number of users is changing and unknown. An update interval is defined as a block of consecutive of time slots. We require the tagged user to update the transmission probability at the beginning of the th update interval, according to the following two necessary estimates of the network status.

: the estimated value of at the end of the th update interval;

: the estimated value of at the end of the th update interval.
The tagged user initially guesses that there are users, and sets . The transmission probability for the first update interval is obtained with given , by Theorem 4 and a fixedpoint iteration. During the th interval for , we describe the proposed algorithm as follows:

Update at the end of the th update interval: To evaluate at runtime, the tagged user is required to record , the number of the slots during the th update interval in which users are simultaneously transmitting and the tagged user is not transmitting, for and .
Let be the measure of during the th update interval, and its value is calculated by:
To avoid sharp changes in the estimated value, the tagged user further applies an Exponential Moving Average filter as follows:
where is a memory factor.

Update at the end of the th update interval: By (21), the tagged user obtains
where is the integer closest to .

Update the transmission probability at the beginning of the th update interval: Update the transmission probability with given , by Theorem 4 and a fixedpoint iteration.
To avoid harmful measure of due to occasional fluctuations of the network status, (i) if is found to be smaller than , a contradiction to the fact that (20) must be equal to or larger than as , the tagged user sets ; (ii) if is found larger than , a contradiction to (20) as , the tagged user sets ; and (iii) if has an invalid value as , the tagged user sets .
Remark 4: It should be noted that, although the proposed algorithm for estimating is devised for the slotted ALOHA in saturation assumption, it can apply also to unsaturated traffic, provided that the estimation target becomes the average number of competing users rather than the total number of users; and can also apply to persistent CSMA, provided that the length of a slot may vary due to different channel status: idle, collision or success.
5 Simulation Results
In this section, to demonstrate the accuracy of the derived analytical results and the effectiveness of our proposed tuning algorithm, we present simulation results with respect to different network parameters obtained from Monte Carlo simulation developed in Matlab. In the following, we will analyze the performance under stationary scenarios when is known, and then under dynamic scenarios when is unknown. Each simulation point in stationary conditions represents the average value over 10 independent runs, each of which consists of slots. The results in dynamic conditions are from a single representative simulation run for 500 updated intervals.
5.1 Stationary Scenarios when is known
Fig. 2 and Fig. 3 show the optimal transmission probability and the maximum SDP as a function of for different values of and , respectively. We see a good agreement between analytical and simulation results in all the scenarios. Notice that the results for can be seen as the throughput maximization issue investigated in [10]. In Fig. 2, as expected, we observe that the optimal transmission probability becomes smaller when increases. This is because that more users attempt to access the channel and the contention level becomes severer. We also see the optimal transmission probability becomes larger when increases or decreases. The reason is that a user needs to be more aggressive in accessing the channel if more concurrent packets can be successfully received or the user wants to successfully send out a packet within a shorter delivery deadline. In Fig. 3, as expected, the curves show that a smaller , a larger or a larger leads to a larger value of the maximum SDP. In particular, we note that the optimal transmission probability for is smaller than that for for given and . This phenomenon indicates that the throughput maximization degrades the SDP for , and in turn the maximization of SDP for degrades the throughput performance.
5.2 Dynamic Scenarios when is unknown
By setting and in (21), we analyze the performance of the proposed tuning algorithm when the number of users sharply changes. In detail, 20 users are always active from 1st to 500th updated interval, and 20 new users are active only from the 101st updated interval to the 400th updated interval. Each user knows there are at most users, and initially guesses there are users.
5.2.1 The cases with
We start by discussing the performance for . Fig. 4 and Fig. 5 show the estimated number of users and SDP for with different and , respectively. To characterize individual behavior and improve the readability, we only present the real trajectories of two representative users in the same run.
We observe from Fig. 4 that, in all the cases, when the actual number of users is changed to 20, the estimate rapidly adapts to changes and keeps a high level of accuracy; when the actual number of users is changed to 40, the estimate is still able to capture changes within a few update intervals, but the slop increases, i.e, the accuracy of the estimation degrades. This phenomenon is because that as the actual number of users increases, the fluctuation in measuring is amplified in estimation by using (21).
As expected, we observe from Fig. 5 that the users achieve SDPs close to the theoretical limit at runtime by tuning the transmission probability according to the estimated number of users as shown in Fig. 4. Moreover, even when the oscillation of estimation is high, we find that the SDP still keeps relatively stable. This phenomenon is due to the fact that the SDP is less sensitive to the change in transmission probability as the number of users increases.
To evaluate the fairness of the proposed algorithm, we further study mean and variance of SDP among all active users at different stages. As shown in Table I, we find the mean value is very close to the theoretical maximum value for each stage with a tolerance of at most 5%, and the variance value is also very small. This result indicates that the proposed algorithm can enable every active user to achieve nearoptimal performance in dynamic scenarios. Notice that the transient period is a dominant factor to degrade the performance.
As the case with can be viewed as runtime optimization of throughput, to show the proposed algorithm is more effective in a variety of dynamic scenarios, we consider the approach in [10] for comparison purposes. The method therein recursively updates the transmission probability with an estimated , and then uses the measure of