Dynamic Transmission Rate Control for Multi-Interface IoT Devices: A Stochastic Optimization Framework

Recent advances in the Internet of Things (IoT) technologies have enabled ubiquitous smart devices to sense and process various kinds of data. However, these innovations also raise the concern of efficient data transmission. Tackling the above issue is nontrivial since the resource constraints and environmental randomness in IoT require a lightweight transmission scheme while guaranteeing system stability. In this paper, we formulate the transmission scheduling problem of multi-interface IoT devices as a concave optimization, aimed at accommodating the randomness of the IoT environment within the network capacity. By applying the Lyapunov optimization technique, we divide the stochastic problem into a series of low-complex subproblems, which can be individually solved per time slot, and develop a dynamical control algorithm that does not require a priori knowledge such as link states. Theoretical analysis shows that our algorithms nicely bound the average queue length and are asymptotically optimal. Finally, extensive simulation results verify the theoretical conclusions and validate the effectiveness of the proposed algorithm.


Introduction
With the ubiquitously deployed smart devices, the Internet of Things (IoT) technology has been facilitating the intelligence of residential daily activities by providing advanced services for transportation, agriculture, industrial manufacturing, etc. [1][2][3]. However, the continuous growth of IoT applications with the proliferation of various devices has resulted in an unprecedented explosion of network data traffic [4]. According to related report [5], the global IoT cellular traffic is expected to grow to 1.7 exabytes per month by 2022, a twofold increase over 2020. This huge amount of data traffic poses a critical challenge to the current networks [6,7], making it impractical to provide transmission guarantees.
Recent innovations of scalable communication systems can empower the IoT devices to connect to heterogeneous networks concurrently, enabling IoT devices to transmit data through different networks in parallel [8]. In particular, an IoT device equipped with multiple interfaces can use multi-ple channels simultaneously in the physical link layer [9] and deliver packets through Multipath TCP (MPTCP) in the network layer [10]. As a result, the IoT devices with multiple data flows can be treated as a transmission scheduler with a many-to-many traffic pattern. Namely, an IoT device can adopt different kinds of data and category them into different types of flows, depending on priorities or requirements. Then, the transmission scheduler can choose to deliver these data flows through either one link or multiple links. In this regard, the IoT device performs like an inputqueued switch for data transmission. Moreover, this design can also bring other advantages for IoT applications, such as performance improvement and scalability support [11].
The novel transmission paradigm introduced by [8] also raises the concerns of transmission scheduling for achieving the desired transmission rate. For example, how many packets of different flows should be transmitted through which link, and how to adjust transmission rates for each flow with system stability guaranteed. These concerns are further complicated with IoT devices' mobility, as it brings dynamical communication environments and stochastic link conditions. Few efforts like [12] consider the stochastic multipath IoT scenarios, as it is not easy to determine the optimal scheduling schemes. The essential reason lies in two aspects. On the one hand, foresightedly optimize the scheduling problems requires the full knowledge of environmental information. However, because of the randomness of packet arrival rates and link states, it is almost impractical to apply this kind of solution to IoT scenarios. Although predicting the network state may be an alternative approach, it is inefficient in facing emergencies, i.e., traffic bursts, which would result in system instability. On the other hand, most IoT devices are constrained with limited computation and energy resources, incapable of operating complex algorithms. Besides, it is also unnecessary to spare a large number of resources for determining data transmission decisions. Thus, efficient online scheduling solutions with low computation complexity are more suitable for IoT devices.
This paper introduces a novel stochastic optimization framework for multi-interface IoT devices to simultaneously achieve optimal transmission scheduling. We first formulate the transmission scheduling problem as a stochastic optimization with the objective of maximizing the long-term transmission utility within the network capacity. Then, we leverage the Lyapunov optimization technique to develop a low-complexity scheduling algorithm. Our main contributions are summarized as follows.
(1) We characterize the multi-interface IoT devices with multiple traffics as a multiqueue model and decouple the scheduling process into two subproblems: admission control and output rate control. Then, we formulate the transmission scheduling problem as a stochastic concave optimization, which includes the randomness of network states and system stability constraint (2) By leveraging the drift-plus-penalty framework, we divide the proposed stochastic problem into three deterministic subproblems, which can be further separated based on their linearly coupling characteristics. Then, we propose a low-complex transmission scheduling algorithm and prove that it can provide an upper bound of the queue length and achieve a ½ OðVÞ, Oð1/VÞ trade-off between the queue length and the transmission utility (3) Extensive simulation results demonstrate the efficiency of our algorithms, which explicitly outperform the benchmarks in terms of system stability, average delay, and network utility The rest of this paper is organized as follows. Section 2 reviews the related works. Section 3 describes the considered model and formulates the scheduling problem. In Section 4, we develop the scheduling algorithm and present the theoretical analysis. Experimental results are shown in Section 5, and the conclusion is given at last in Section 6.

Related Works
As transmission rate control plays a crucial issue in the research literature of communication, there have been extensive efforts in developing customized schemes for diverse network paradigms in different aspects. For example, to deal with the variation traffic requirements, the authors of [13] consider the situation that the resources can be dynamically shifted between cells and develop a dynamic resource allocation protocol. For device to device communication systems, the authors of [14] try to maximize the weighted sum transmission rate and use a two-step approach to solve the nonconvex mixed-integer problem of resource allocation and subchannel assignment. In [15], the authors focus on the information-centric network and present a multipath-aware ICN rate-based congestion control algorithm to calculate per-link rates for the multipath scenario. The authors of [16] address the bandwidth sharing issues in a softwaredefined network and design a distributed resource allocation algorithm that can provide a trade-off between fairness and cost. For data center networks, the authors of [17] formulate the multiple rate control issue as a convex optimization problem and use their proposed transmission protocol to achieve efficient bandwidth allocation.
In addition, there are also many research works jointly considering the assignment of transmission and other kinds of resources. In [18], the authors study the trade-off between data rate performance and energy consumption in heterogeneous networks and introduce an energy-efficient scheduling scheme to improve system performance. The authors of [19] propose a low-complexity algorithm by difference-of-convex programming to simultaneously optimize service level selection and transmission resource allocation in mobile edge computing systems. By jointly considering task assignment, transmission, and computing resources allocation, the authors of [20] propose a multilayer data flow process system that can provide low latency services for real-time applications. The authors of [21] consider a three-node relay system and provide analytical solutions to the proposed optimization problem for power assignment and relay location. The authors of [22] formulate a delay-sensitive data offloading algorithm to optimize the computing and communication resources to minimize the execution delay and transmission delay concurrently for fog networks.

System Model and Problem Formulation
3.1. Multi-Interface System Model. In this paper, we consider the transmission scheduling problem in a typical multiinterface IoT scenario. Each IoT device is equipped with multiple antennas and can deliver data through different links concurrently. Their collected data are categorized into different types, forming transmission flows, respectively. The transmission scheduler operates at the IoT device and makes transmitting decisions according to current network condition. Hence, the multi-interface IoT system can be viewed as a model of a single node with multiple uplink channels. Table 1 summarizes the notations used in this paper.

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To facilitate the analysis of the above model and deal with the time-varying link states, we assume that the system operates in discrete time with unit time slots t ∈ f0, 1, 2 ⋯ g. At every time slot, packets randomly enter the transmission scheduler. We define α m ðtÞ as the amount of packets of flow m (in the unit of packet number) that arrive at time t. For link n, we use μ n ðtÞ to denote its maximum allowable rate, the number of packets that can be transmitted at time t. To make our model practical, we only assume α m ðtÞ and μ n ðtÞ are independent and identically distributed (i.i.d) over time and rate-convergent, which means that equations (1) and (2) hold with probability 1. Note that, in our work, the scheduler does not need to know the average arrival rate α m and transmission rate μ n previously.
In addition, we assume a maximum arrival rate α m,max and a maximum transmission rate μ n,max , regardless of the time and the channel state, so that Our goal in this work is to design a dynamic, optimal algorithm for the transmission scheduler to make the following decisions strategically: (1) scheduling decision: how many packets of flow m should be transmitted through link n at each time slot? (2) Rate control: how does the device allocate the transmission rate while ensuring system stability?
We next propose a multiqueue model to characterize the scheduling problem and then develop an optimization framework to solve the first problem. After that, we introduce a virtual queue to cope with the second issue.

Queue Model and Optimization Objective
3.2.1. Queue Model. According to the above system, we assume that the transmission scheduler holds multiple queues for each transmitting link and flow, respectively, as shown in Figure To ensure system stability, we use the definition of queue stability in [23], which is given as follows.
By definition, the queue length can be bounded by a positive constant, implying that the average queuing latency is also constrained.
Additionally, the admission decision a m ðtÞ is made subject to the constraint, α m ðtÞ = ∑ n a m n ðtÞ, implying that all packets should be admitted. Similarly, u n ðtÞ must satisfy 0 ≤ ∑ m u m n ðtÞ ≤ μ n ðtÞ, which means that the delivered packet rate could not exceed the link capacity at any time.

Optimization Objective.
To introduce the optimization objective, we define the time-average admitted packets on link n, a n . a n = lim After that, we introduce a continuous, concave, and nondecreasing utility function, Φ n ð a n Þ, to represent our optimization target. According to [23], this kind of utility function can be used to measure network fairness. Intuitively, this utility can be the profit gained by transmitting packets through link n or the reciprocal of transmission cost of link n. An example of Φ n ð·Þ can be given by Φ n ðxÞ = log ð1 + xÞ, which is also used in our simulation experiments. Hence, to achieve long-time utility maximization, we choose to maximize the weighted sum of all transmission utilities. Then, the formulated scheduling problem can be as follows.
max 〠 n ω n Φ n a n ð Þ, ð7aÞ where ω n is the weight factor which can be used to dominate the transmission fairness. Additionally, we assume that Φ n ð·Þ satisfies the Lipschiz condition, where D is a constant. Intuitively, the above optimization problem is an integer programming problem. It is hard to derive the optimal solution directly. Moreover, it requires the full knowledge of state information of all time, which is almost impossible in the realistic environment.
In the next section, we will leverage the Lyapunov optimization technique to decompose the scheduling problem and provide an online algorithm to approximate the optimal solution with system performance guaranteed.
All queues Q m n t ð Þ are stable, ð8eÞ where Ω denotes the state space given by (7a) and (7b) that arbitrary stationary algorithms can achieve. According to [24], the auxiliary variables, γðtÞ, are introduced to decouple the variables from the optimization objective. Besides, in this paper, the auxiliary variables can also simplify the optimization problem by replacing multiple variables with a single variable.
To satisfy (8c), we introduce a virtual queue Y n ðtÞ for each link n, with an arrival rate of γ n ðtÞ and a departure rate of ∑ m a m n ðtÞ, which evolves as According to [23], if this virtual queue is stable, the timeaverage value of ∑ n a m n ðtÞ is greater than or equal to the timeaverage value of γ n ðtÞ, which ensures (8c), and so the optimization objective will also be large.

Problem Decomposition via Drift-plus-Penalty
Minimization. Next, we solve the problem ((8a), (8b), (8c), (8d), and (8e)) via the drift-plus-penalty framework, aimed at developing a dynamic scheduling algorithm that can preserve the queue stability and solution optimality. Let ΘðtÞ ≜ ½QðtÞ, YðtÞ be the collective vector of all Q m n ð tÞ and Y n ðtÞ queues. Then, we define the following quadratic Lyapunov function.
Denote ΔðΘðtÞÞ = LðΘðt + 1ÞÞ − LðΘðtÞÞ as the one-step conditional Lyapunov drift. According to [23], we can determine our scheduling policy by minimizing the following drift-plus-penalty expression at each time slot.
where V is a nonnegative penalty parameter that will affect the utility delay trade-off. Instead of minimizing equation (11), our algorithm minimizes its upper bound, which has the following expression.
where B is a positive constant satisfying As a m n ðtÞ, u m n ðtÞ, and γ n ðtÞ are all bounded, it is easy to prove that such a constant b always exists for all t. For example, an upper bound of B can be defined as 1/2ð1 + jMjÞðjN jjMjα 2 max + jNjμ 2 max Þ), where α max and μ max denote the maximum value of the arrival and transmission rate, respectively.
Our dynamic algorithm is given to make scheduling decisions by minimizing the right hand of equation (12) in each time slot t, based on the observed queue states, Q m n ðtÞ and Y n ðtÞ. Furthermore, by observing the form of equation (12), the scheduling decisions for a m n ðtÞ, u m n ðtÞ, and γ n ðtÞ are linearly coupled. Thus, we can decompose the scheduling problems into the following three subproblems auxiliary variable selection, admission control, and transmission rate allocation.
(1) Auxiliary Variable Selection. The auxiliary variables can be chosen by minimizing the following expression: As γðtÞ is linearly coupled, γ n ðtÞ can be separable with each other. Besides, given that Φ n ðγ n ðtÞÞ is continuous and concave, we can provide a closed-form solution of (14), (2) Admission Control. By omitting the terms containing a m n ðtÞ, the admission control can be determined by solving the following problem: In the case of the separable objective, problem (15) can be divided into m independent subproblems, each for one flow. The optimal solution can be as follows.
Intuitively, the transmission rate allocation algorithm always trends to serve the longest queue.
The scheduling problem can be described by Algorithm 1. At every time slot t, the scheduler receives the packets, a m ðtÞ, and observes the current states of all queues, Q m n ðtÞ and Y n ðtÞ . Then, it makes scheduling decisions by solving (14), (15), and (17). After that, it updates the virtual queues Y n ðtÞ according to equation (9) and the actual queues Q m n ðtÞ according to equation (4), with the derived γ n ðtÞ, a m n ðtÞ, and u m n ðtÞ.

Performance Analysis.
Here, we provide the performance of our dynamic algorithm, with respect to the queue length and utility. Theorem 1. Suppose the problem ((8a), (8b), (8c), (8d), and (8e)) is feasible, all queues are initially set to be zero, and the above algorithm is used in each time slot t with a fixed V; then: (a) All queues Q m n ðtÞ are stable for all t. Specifically, the following equation holds: where D is the Lipschiz constant, R represents the region of γ, ε is a positive constant, and jWj denotes the 1-norm of the weight factor.
Proof. According to Theorem 4.5 in [23], the feasibility of the problem ((7a), (7b), (7c), and (7d)) implies that for any δ, δ > 0, there exists a scheduling policy c γ n ðtÞ, c a m n ðtÞ, and c u m n ðt Þ, which yields Using δ = 0 and plugging the above into equation (12) yield Summing up the above equation of all t, t ≤ T, and dividing both side by T, we can have

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As Φ n ð·Þ is concave and nondecreasing, based on Jensen's inequality, the following equation can be derived.
Plugging the above equation into equation (22) and taking liminf T⟶∞ of both sides prove part (b).
To prove part (a), we can use c γ n ðtÞ = Ef∑ m c a m n ðtÞg. Then, according to the queue stability constraint, we have the following equations.
Plugging these two equations in equation (12), we can have Rearranging the terms of the above equation and using telescoping yield Consider that Φ n ð·Þ satisfies the Lipschitz condition. Then, the following equation can be derived.
Using this in equation (26) and dividing both sides by εT, we can derive for all t > 0 Finally, taking limsup T⟶∞ of both sides proves part (a). Theorem 1 demonstrates that the control parameter V dominates the performance of our algorithm. In particular, the upper bound of the queue length grows linearly with V. This situation implies we can choose a small V to achieve a short queue length, which corresponds to a short queuing delay according to Little's theory. Besides, it also confirms that the gap between the achieved utility and the optimal one is bounded by Oð1/VÞ. The gap can be made arbitrarily small by increasing V, which declares that increasing V can approximate the optimal solution of (8a), (8b), (8c), (8d), and (8e). Overall, we can conclude that our algorithm achieves the trade-off between the queuing delay and the transmission utility with ½OðVÞ, Oð1/VÞ. This feature shows that improving the transmission utility is at the expense of increasing the queuing delay.
Additionally, consider that (14), (15), and (17) are independent. All these subproblems can be solved in parallel. This situation indicates that the overall complexity of our algorithm depends on that of each subproblem. Furthermore, by observing their forms, the complexity of solving (14),   (17) is related to the number of data flows and links, M and N. In particular, the computation complexity is bounded by OðMNÞ. Normally, N is constant in practical scenarios, and M varies with the number of data types, which is relatively small. Thus, the complexity of our algorithm is acceptable.

Simulation Results
This section presents the simulation results of the proposed algorithm, named myopic algorithm. For comparison, we also simulate two other heuristic algorithms as benchmarks: (1) weighted round robin: the scheduler admits and sends packets of different flows in turn and the time slice of each flow is determined based on its average arrival and transmission rates, and (2) weighted random: the scheduler admits and transmits packets of different flows according to a given probability distribution, proportional to the average arrival and transmission rates.

Experimental Setting.
In this work, we consider a 5 × 3 wireless transmission scheduler with five input ports, each admitting one certain flow, and three output ports, each delivering packets through one wireless link [8]. Hence, there   Wireless Communications and Mobile Computing are fifteen queues Q m n ðtÞ, representing that packets arrived from input port m must be delivered to output port n, for m ∈ f1, 2, 3, 4, 5g and n ∈ f1, 2, 3g. The arrival processes follow uniform distributions, i.i.d. over time slots, varying from 60 to 240. Considering that link states are time-varying, transmission rates of all links are arbitrarily distributed, i.i.d. over time slots with average rates f30,90,150g. We use log ð1 + xÞ as the utility function for all links and, respectively, set the maximum arrival rate and transmission rate to twice the average rates of the two. We implement a software scheduler with Python and deploy all algorithms in it. To better evaluate the proposed algorithms, we run these three algorithms simultaneously with the same instantaneous arrival and transmission rates. Note that all simulations are over 5000 time slots, and each data point in the figures is averaged based on 10 times independent runs.

System
Performance. In this section, we verify the system performance with respect to queue length, average delay, and achieved utility, with V = 100. Figure 2 plots the queue length of the three algorithms over time slots. The results confirm that our algorithm outperform the other two algorithms. In particular, the weighted random algorithm varies dramatically over time, while the other algorithms are relatively stable. This is because only the weighted random algorithm schedules the packet randomly, without considering the environment state. Definitely, it can only guarantee that the queue length cannot grow indefinitely. Figure 3 compares the average delay between the three algorithms. We can see that the myopic algorithm achieves the lowest average delay, followed by myopic weighted round robin, and the weighted random algorithm shows the worst performance. This situation is consistent with the queue length. Little's theory, N = λT, can provide a reasonable explanation that the average delay grows proportionally with the arrival rate.
In our simulation, we also measure the achieved utility, shown in Figure 4, calculated by the number of packets. Obviously, all these three algorithms converge to a stable value, and our proposed algorithms show a better performance than the benchmarks. The former situation demonstrates the system's stability. The latter is because our proposed algorithms leverage the utility function as targets while the benchmarks can only be designed to ensure system stability. Moreover, the weighted random algorithm converges the slowest. Similar to the queue length, this phenomenon is also induced by the weighted random algorithm's randomicity.

Impact of Algorithm Parameters.
In this section, we investigate the impact of the arrival rate and V on system performance. Figure 5 reveals how the queue length, average delay, and achieved utility vary with the arrival rate. Figure 5(a) shows that the queue length increases with the increase in λ. In particular, the weighted random algorithm grows the fastest, followed by the weighted round robin algorithm, and our proposed algorithm has the shortest queue lengths. Figure 5(b) shows the same trend as Figure 5(a). These two subfigures demonstrate that our proposed algorithm can always provide queuing delay guarantees. As for Figure 5(c), it reveals that the achieved utility also grows as λ rises. However, the increasing rates of the proposed algorithm decrease with λ. This trend can be explained as follows.
When λ is small, the primary objective is to maximize the utility. However, when λ is large, system stability becomes essential. Otherwise, the queue length will grow infinitely. As a comparison, the weighted round robin and weighted random algorithms show an increasing trend almost linearly. Figure 6 shows the average queue length under different values of V. Obviously, for all cases, the queue length converges to be stable. In particular, we can see that, when V     Wireless Communications and Mobile Computing grows, the stable queue length also increases. Besides, the increasing rates of the stable queue length and V have a linear approximation relationship. Figure 7 depicts the average utility under different values of V. Similar to the queue length, all average utilities finally remain stable and increase with the growth of V. However, the growth rate of the average utility is inversely proportional to that of V. Hence, these two figures together confirm the correctness of the aforementioned theorems.
Another observation from these two figures is that a larger V results in a higher convergence rate. As shown in Figure 6, the convergence time of V = 300 is more than 1500 time slots, which is almost three times that of V = 50. Moreover, in Figure 7, when V = 300, the average utility remains stable after the 3000th time slots, and when V = 50, the algorithm reaches its stable value around the 1000th time slots. Therefore, it is necessary to choose V carefully when deploying our algorithms in practical systems.

Conclusion
In this paper, we investigate the transmission scheduling in IoT. We propose a generic optimization problem and solve it via the Lyapunov optimization technique. Both theoretical proofs and simulation results confirm that our approach can achieve the optimal utility while guaranteeing system stability and constraining average delay. In the future, we will continue our work in two aspects. (1) Extend our algorithms into more scenarios, i.e., delay-sensitive scenarios, where the delay requirement of a specific flow must be satisfied for all times. (2) Improve the performance of our proposed algorithm, i.e., investigating how to choose parameters dynamically to provide real-time transmission resource allocation.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.