Joint Optimization of Preamble Selection and Access Barring for Random Access in MTC with General Device Activities

The emerging Internet-of-Things (IoT), whose key idea is to connect everything and everyone by the Internet, has received increasing attention in recent years. Machine-type communications (MTC) are expected to support IoT services and applications, such as home automation, smart grids, smart healthcare, and environment monitoring [2, 3]. Long Term Evolution (LTE) cellular networks offer the most natural and appealing solution for MTC due to ubiquitous coverage. Specifically, the Third-Generation Partnership Project (3GPP) has developed radio technologies, such as LTE-M [4] and Narrowband IoT (NB-IoT) [5], to enhance existing LTE networks and to provide new solutions inherited from LTE, respectively, for better serving IoT use cases. In addition, 3GPP proposes to the International Telecommunications Union (ITU) that LTE-M and NB-IoT should be integrated as part of the fifth generation (5G) specifications [6].

In LTE-M and NB-IoT, devices compete in a random access channel (RACH) to access the base station (BS) through the random access procedure [7]. Specifically, each active device randomly selects a preamble from a pool of available preambles according to a preamble selection distribution, and transmits it during the RACH. The BS acknowledges the successful reception of a preamble if such preamble is transmitted by only one device. In[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], the authors consider the random access procedure and study the effect of preamble selection under certain assumptions on the knowledge of device activities. Specifically,[8, 9, 10, 13, 16, 17, 18] assume that the number of active devices is known; [15, 14] assume that the distribution of the number of active devices is known; [11, 12] assume that the statistics of the data queue of each device are known. In[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], preambles are selected according to a uniform distribution, the average throughput [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], average access efficiency [16] and resource consumption [10] are analyzed and the number of allocated preambles is optimized to maximize the average throughput [10, 17] or access efficiency [16]. Notice that optimal solutions are obtained in [10, 16, 17]. In [19, 20, 21], the active devices are assumed to be known. Under this assumption, each active device is allocated a preamble if the number of preambles is greater than or equal to the number of active devices, and the preambles are allocated to a subset of active devices otherwise.

When many active devices attempt to access a BS simultaneously, a preamble is very likely to be selected by more than one device, leading to a significant decrease in the probability of access success. In this scenario, access control is necessary. One widely used access control method is the access barring scheme [22], which has been included in the LTE specification in [7]. In[8, 9, 10, 11, 12, 13, 14, 15, 16], the authors consider access barring, besides random access procedure with preambles selected according to a uniform distribution. Specifically, the access barring factor is optimized to maximize the average throughput [8, 9, 10, 11, 12, 13, 14] or access efficiency [16], or to minimize the number of backlogged devices [15]. Note that optimal solutions are obtained in [15, 8, 13, 10, 14, 16], an approximate solution is obtained in [9], and asymptotically optimal solutions for a large number of devices are obtained in [11, 12].

There exist two main types of MTC traffic, i.e., event-driven (or event-triggered) traffic and periodic traffic [23, 24]. In event-driven MTC (e.g., smart health care and environment sensing such as rockfall detection and fire alarm), one device being triggered may increase the probability that other devices in the vicinity trigger in quick succession, or many devices are simultaneously triggered in response to a particular IoT event. Hence, device activities can be dependent. Besides, in periodic MTC, devices have their own periods, and hence device activities can be considered independent for tractability (if the periodic behaviors of devices are not tracked). Intuitively, effective preamble selection and access barring should adapt to the statistical behaviors of device activities. Specifically, if devices are more likely to activate simultaneously, letting them always select different preambles can avoid more collisions; if devices are less likely to activate simultaneously, letting them always select the same preamble can avoid more collisions. However, most existing random access schemes simply adopt a uniform preamble selection distribution, irrespective of the underlying device activity distributions [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. They hence may yield far from optimal access efficiency. This motivates us to model device activities as multiple Bernoulli random variables following an arbitrary multivariate Bernoulli distribution, which can reflect both dependent and independent device activities [25, 26, 27, 28], and optimize preamble selection and access barring according to the multivariate Bernoulli distribution [27]. To our knowledge, [27] is the first work that considers a general joint device activity distribution and attempts to optimize the preamble selection distributions and access barring factors of all devices. More specifically, in [27], the authors assume that a perfect joint device activity distribution is known, maximize an approximation of the average throughput, which captures the active probabilities of every single device and every two devices, and develop a heuristic algorithm to tackle the challenging nonconvex problem. The approximation error and the heuristic algorithm may yield a nonnegligible loss in access efficiency. Therefore, it is critical to explore more effective algorithms adaptive to statistics of device activities. Furthermore, notice that most existing works assume that the number of active devices [8, 9, 10, 13, 16, 17], the distribution of the number of active devices [14, 15], the statistics of the data queue of each device [11, 12], the active devices [19, 20, 21], or the device activity distribution [27] is perfectly known. However, in practice, such assumptions are hardly satisfied. Hence, it is critical to consider the optimization of preamble selection and access barring under imperfect information of device activities or even historical samples of device activities.

This paper investigates the joint optimization of preamble selection and access barring for IoT devices in MTC, whose activities are modeled by an arbitrary multivariate Bernoulli distribution, which can reflect both dependent and independent device activities. Specifically, we consider three cases of the joint device activity distribution, i.e., the case of perfect general joint device activity distribution (where the joint device activity distribution has been perfectly estimated), the case of imperfect joint device activity distribution (where the estimation error of the joint device activity distribution lies within a known deterministic bound), and the case of unknown joint device activity distribution (where activity samples generated according to the joint device activity distribution are available). The optimization-based solutions are generally applicable for any dependent and independent device activities. Our main contributions are summarized below.

• •

  In the case of perfect joint device activity distribution, we formulate the average throughput maximization problem, which is nonconvex with a complicated objective function. Based on the block coordinate descend (BCD) method [29], we develop an iterative algorithm, where most block coordinate optimization problems are solved analytically, and a low-complexity iterative algorithm, where all block coordinate optimization problems have closed-form optimal points, to obtain stationary points of the original problem and an approximate problem, respectively. Furthermore, we characterize an optimality property of a globally optimal point of the original problem.

• •

  In the case of imperfect joint device activity distribution, we formulate the worst-case average throughput maximization problem as a robust optimization problem, which is a max-min problem. By duality theory and successive convex approximation (SCA) [30], we develop an iterative algorithm to obtain a Karush-Kuhn-Tucker (KKT) point of an equivalent problem of the max-min problem. Based on the BCD method, we also develop a low-complexity iterative algorithm, where all block coordinate optimization problems are solved analytically, to obtain a stationary point of an approximate problem of the max-min problem.

• •

  In the case of unknown joint device activity distribution, we formulate the sample average throughput maximization problem, which can be viewed as a nonconvex stochastic optimization problem. Based on mini-batch stochastic parallel SCA [31], we develop an efficient parallel iterative algorithm where the approximate convex problems in each iteration are solved analytically, to obtain a stationary point.

• •

  Finally, we show that for dependent and independent device activities, the proposed solutions achieve significant gains over existing schemes in all three cases by numerical results. Besides, the performance of each low-complexity algorithm is close to that of its counterpart designed for solving the original problem.

The key notation used in this paper is listed in Table I.

We consider the uplink of a single-cell wireless network consisting of one BS and $K$ devices. Let $\mathcal{K}\triangleq\{1,2,...,K\}$ denote the set of $K$ devices. We consider a discrete-time system with time being slotted and assume that devices activate independently and identically over slots within a certain period. In each slot, a device can be either active or inactive. Thus, the activities of the $K$ devices in a slot can be modeled as $K$ Bernoulli random variables, following an arbitrary multivariate Bernoulli distribution [28, 25, 26]. Generally, the $K$ Bernoulli random variables (representing the activities of the $K$ devices) can be dependent or independent. Let $x_{k}\in\{0,1\}$ denote the activity state of device $k$, where $x_{k}=1$ if device $k$ is active and $x_{k}=0$ otherwise. Let binary vector ${\bf x}\triangleq(x_{k})_{k\in\mathcal{K}}\in\mathcal{X}$ denote the activity states of the $K$ devices, where $\mathcal{X}\triangleq\{0,1\}^{K}$. The joint device activity distribution (assumed invariant over the considered period) is denoted by ${\bf p}\triangleq(p_{\bf x})_{{\bf x}\in\mathcal{X}}$, where $p_{\bf x}$ represents the probability of the activity states of the $K$ devices being $\bf x$. Note that whether the device activities are dependent or independent depends on the values or forms of $\mathbf{p}$’s components. According to the nonnegativity and normalization axioms, we have

In this paper, we consider $p_{\mathbf{0}}<1$.¹¹1$p_{\mathbf{0}}=1$ is not an interesting or practical scenario. We do not pose any additional assumption on $\mathbf{p}$. Thus, all analytical and optimization results, relying on $\mathbf{p}$ or its related quantities, are general.

In a slot, each active device tries to access the BS. Congestion may occur when many active devices require to access the BS at the same time. We adopt an access barring scheme for access control [22]. In particular, at the beginning of each slot, all active devices independently attempt to access the BS with probability $\epsilon$, where

Here, $\epsilon$ is referred to as the access barring factor for all devices and will be optimized later.²²2The average throughput can be improved when devices have different access barring factors. To avoid resource starvation for each device and maintain fairness, we consider identical access barring factors for all devices in the optimizations for simplicity. The proposed solution framework can be extended to handle different access barring factors. That is, the access barring scheme is parameterized by the access barring factor $\epsilon$.

We adopt the random access procedure, which consists of four stages, i.e., preamble transmission, random access response, scheduled transmission, and contention resolution [3]. We focus only on the first stage, which mainly determines the success of access [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 27]. Consider $N$ orthogonal preambles, the set of which is denoted by $\mathcal{N}\triangleq\{1,2,...,N\}$. Specifically, at the first stage, each device that attempts to access the BS independently selects a preamble out of the $N$ preambles to transmit. The probability that device $k$ selects preamble $n$ is denoted by $a_{k,n}$, which satisfies

Let ${\bf a}_{k}\triangleq(a_{k,n})_{n\in\mathcal{N}}$ denote the preamble selection distribution of device $k$. Let ${\bf A}\triangleq({a}_{k,n})_{k\in\mathcal{K},n\in\mathcal{N}}$ denote the distributions of the $K$ devices. The $k$-th column of $\bf A$ is ${\bf a}_{k}$. Note that for all $k\in\mathcal{K}$, the random preamble transmission parameterized by ${\bf a}_{k}$ reduces to the preamble transmission in the standard random access procedure [7] when $a_{k,n}=\frac{1}{N}$, $n\in\mathcal{N}$. Furthermore, note that for all $k\in\mathcal{K}$, the considered preamble selection is generally random and becomes deterministic when $a_{k,n}\in\{0,1\},n\in\mathcal{N}$. We allow ${\bf a}_{k}$, $k\in\mathcal{K}$ to be arbitrary distributions to maximally avoid the collision.

If a preamble is selected by a single device, this device successfully accesses the BS [11]. Then, the average number of devices that successfully access the BS at activity states $\bf x$ in a slot is given by [27]

In this paper, we consider the following three cases of joint device activity distribution and introduce the respective performance metrics.
Perfect joint device activity distribution: In this case, we assume that the joint device activity distribution has been estimated by some learning methods, and the estimation error is negligible. That is, the exact value of $\bf p$ is known. We adopt the average throughput [27]

as the performance metric, where $T({\bf A},{\epsilon},{\bf x})$ is given by (4). The proof for (6) can be found in Appendix A. Note that for all $\mathcal{K}^{\prime}\subseteq\mathcal{K}$, $\sum\nolimits_{{\bf x\in\mathcal{X}}}$ $p_{\bf x}\prod\nolimits_{k\in\mathcal{K}^{\prime}}{x}_{k}$ represents the probability that all devices in $\mathcal{K}^{\prime}$ are active and can be computed in advance.
Imperfect joint device activity distribution: In this case, we assume that the joint device activity distribution has been estimated by some learning methods with certain estimation errors. For all ${\bf x}\in\mathcal{X}$, let $\hat{p}_{\bf x}$ denote the estimated probability of the device activity states being $\bf x$, and let $\Delta_{{\bf x}}\triangleq p_{\bf x}-\hat{p}_{\bf x}$ denote the corresponding estimation error. Note that $\Delta_{{\bf x}},{\bf x}\in\mathcal{X}$ satisfy $\sum\nolimits_{{\bf x}\in\mathcal{X}}\Delta_{{\bf x}}=0$, and $|\Delta_{{\bf x}}|\leq\delta_{\bf x},{\bf x}\in\mathcal{X}$ for some known estimation error bounds $\delta_{{\bf x}}\in[0,1)$, ${\bf x}\in\mathcal{X}$. Assume that $\hat{p}_{\bf x}$, $\delta_{{\bf x}},{\bf x}\in\mathcal{X}$ are known to the BS, but neither $\bf p$ nor $\Delta_{\bf x},{\bf x}\in\mathcal{X}$ is known to the BS. Thus, the BS knows that the exact joint activity distribution satisfies $\bf p\in\mathcal{P}$, where

with $\underline{p}_{\bf x}\triangleq\max\{\hat{p}_{\bf x}-\delta_{\bf x},0\}$ and $\overline{p}_{\bf x}\triangleq\min\{\hat{p}_{\bf x}+\delta_{\bf x},\ 1\}$, for all ${\bf x}\in\mathcal{X}$. Note that $\mathcal{P}$ reflects $\mathbf{p}$ to a certain extent. It can be easily verified that $\mathcal{P}$ shrinks to $\{\mathbf{p}\}$ as the estimation error bounds $\delta_{\mathbf{x}}$, $\mathbf{x}\in\mathcal{X}$ decrease to $0$. We adopt the worst-case average throughput

as the performance metric, where $\bar{T}({\bf A},{\epsilon},{\bf p})$ is given by (6). Note that $\bar{T}_{\text{wt}}({\bf A},{\epsilon},\mathcal{P})\leq\bar{T}({\bf A},{\epsilon},{\bf p})$ with the equality holds if and only if $\delta_{\mathbf{x}}=0$, $\mathbf{x}\in\mathcal{X}$.
Unknown joint device activity distribution: In this case, we assume no direct information on the joint device activity distribution, but $I$ samples of the device activity states, generated according to $\mathbf{p}$, are available. The $I$ samples reflect $\mathbf{p}$ to a certain extent via the empirical joint probability distribution, an unbiased estimator of $\mathbf{p}$. It can be easily verified that the mean squared error decreases with $I$. Denote $\mathcal{I}\triangleq\{1,2,...,I\}$. For all $i\in\mathcal{I}$, the $i$-th sample is denoted by ${\bf x}_{i}$. We adopt the sample average throughput

as the performance metric, where $T({\bf A},{\epsilon},{\bf x})$ is given by (4). Note that $\bar{T}_{\text{st}}({\bf A},{\epsilon})$ is a random variable with randomness induced by the $I$ samples and $\mathbb{E}[\bar{T}_{\text{st}}({\bf A},{\epsilon})]=\bar{T}({\bf A},{\epsilon},\mathbf{p})$.

As illustrated in the introduction, overlooking the joint device activity distribution $\mathbf{p}$ (or related quantities, such as $\mathcal{P}$ and $\mathbf{x}_{i}$, $i\in\mathcal{I}$) may yield ineffective preamble selection and access barring designs. This can be further seen from the following example.

Motivated by Example 1, in Section III, Section IV, and Section V, we shall optimize the preamble selection distributions $\bf A$ and access barring factor $\epsilon$ to maximize the average, worst-case average, and sample average throughputs for given $\mathbf{p}$, $\mathcal{P}$, and $\mathbf{x}_{i}$, $i\in\mathcal{I}$ in the cases of perfect, imperfect, and unknown joint device activity distributions, respectively, as shown in Fig. 2. In what follows, we assume that the optimization problems are solved at the BS given knowledge of $\mathbf{p}$, $\mathcal{P}$, and $\mathbf{x}_{i}$, $i\in\mathcal{I}$, and the solutions are then sent to all devices for implementation.⁵⁵5In practice, $\mathbf{p}$ may change slowly over time. For practical implementation, we can re-optimize the preamble selection and access barring using the proposed methods when the change of $\mathbf{p}$ is sufficiently large, and the obtained stationary point for an outdated $\mathbf{p}$ generally serves as a good initial point for the latest $\mathbf{p}$.

In this section, we consider the average throughput maximization in the case of perfect joint device activity distribution. First, we formulate the average throughput maximization problem, which is a challenging nonconvex problem. Then, we develop an iterative algorithm to obtain a stationary point of the original problem. Finally, we develop a low-complexity iterative algorithm to obtain a stationary point of an approximate problem.

In this section, we consider the worst-case average throughput maximization in the case of imperfect joint device activity distribution. First, we formulate the worst-case average throughput maximization problem, which is a challenging max-min problem. Then, we develop an iterative algorithm to obtain a KKT point of an equivalent problem. Finally, we develop a low-complexity iterative algorithm to obtain a stationary point of an approximate problem.