Context-aware Container Orchestration in Serverless Edge Computing Peiyuan Guan Dept. of Informatics University of Oslo Oslo, Norway peiyuang@ifi.uio.no Chen Chen Dept. of Computer Science and Technology University of Cambridge Cambridge, UK cc2181@cam.ac.uk Ziru Chen Dept. of Electrical and Computer Engineering Illinois Institute of Technology Chicago, USA zchen71@hawk.iit.edu Lin X. Cai Dept. of Electrical and Computer Engineering Illinois Institute of Technology Chicago, USA lincai@iit.edu Xing Hao School of Information Science and Technology Northwest Univesity Xi’an, China xhao@nwu.edu.cn Amir Taherkordi Dept. of Informatics University of Oslo Oslo, Norway Amirhost@ifi.uio.no Abstract—Adopting serverless computing to edge networks benefits end-users from the pay-as-you-use billing model and flexible scaling of applications. This paradigm extends the boundaries of edge computing and remarkably improves the quality of services. How- ever, due to the heterogeneous nature of computing and bandwidth resources in edge networks, it is challeng- ing to dynamically allocate different resources while adapting to the high burstiness and concurrency in serverless workloads. This article focuses on serverless function provisioning in edge networks with the aim of optimizing end-to-end latency, where the challenge lies in jointly allocating wireless bandwidth and computing resources among heterogeneous computing nodes. We devise a context-aware learning framework that adap- tively orchestrates a wide spectrum of resources and jointly considers them to avoid resource fragmentation. Extensive simulation results justify that the proposed algorithm saves over 95% of converge time, and the total delay is almost the same level as the state of the art. Index Terms—Serverless Computing, Edge Comput- ing, Resource Management I. Introduction The progressive development of Internet-of-Things (IoTs) and mobile devices has produced massive Internet traffic to cloud data centers, dramatically increasing the network pressure in backbone networks. To cope with this issue, many IoT applications can be offloaded from remote clouds to edge servers, reducing the traffic to remote clouds and improving the end-to-end latency. Such a near-data paradigm largely mitigates the transmission problem that inhibits the deployment of delay-sensitive applications, e.g., online gaming, health care and autonomous vehicles [1]–[3]. Recently, serverless computing [4], also known as Function-as-a-Service (Faas), endows edge computing with new inspirations. Serverless applications are encapsulated Fig. 1. Example of serverless edge computing in functions that are triggered by user events. It hands over the server management to service providers and enables developers to focus only on application development [5]. Developers upload these functions to the serverless plat- form and use the API server to execute their computation. Given the promising prospect of serverless computing, many research works believe that bringing serverless com- puting to the network edge will realize new flexibility, efficacy and scalability. A number of research works have been proposed to integrate the serverless paradigm into edge computing, e.g., [6], [7]. However, existing solutions are not directly applicable because several inherent features of serverless edge computing have been overlooked. First, edge servers are usually equipped with different amounts of commu- nication and computation resources. Such heterogeneity requires dynamic and adaptive resource allocation because the service latency can be bounded by a certain type of resource, e.g., communication bandwidth and processing capacity. Second, the size and arrival pattern of serverless workloads can vary significantly due to the burstiness and concurrency over time, the resources must adapt to these changes in serverless workloads. Since more and more base stations are increasingly equipped with computing capacity, as depicted in Figure 1, service requests can be assigned to base station 2 if base station 1 is overloaded, aiming to achieve faster data transmission and processing. In other words, how to jointly select the base stations and allocate bandwidth resources for low-transmission latency is non-trivial but complex. Besides request assignment, extra efforts are still needed to allocate processing capacity for serverless functions on each edge server. Since both computing resources are highly constrained at the network edge [8], an efficient orchestration policy is vitally important for low-latency applications. In this paper, we propose a serverless container place- ment and resource allocation framework for latency- sensitive applications, addressing the challenges mentioned above in serverless edge computing. First, we formulate the problem as an Integer Programming problem with several constraints. After that, we propose a context-aware neural network (CANN) based on the results of MIDACO [9], which is a solver for numerical optimization problems, and compare the total delay and converge time with MIDACO and genetic algorithm. Extensive numerical tests justified that MIDACO can obtain the global optimal fast and ro- bustly on hundreds of benchmarks [10]. Hence, we compare our algorithm with the results of MIDACO, justifying that our algorithm can achieve comparable performance. Our main contributions are listed as follows. 1) We formulate a request distribution problem as a Mixed Integer Linear Programming problem that jointly considers transmission latency and processing latency with constraints of bandwidth and comput- ing resources. We prove the NP-hardness of the problem and propose an online competitive algo- rithm for request distribution to solve the problem in polynomial time. 2) We propose an NN, trained by the results of MI- DACO, that jointly considers the bandwidth and computing resources, dynamically adapting to the heterogeneity, burstiness, and high concurrency of serverless computing. 3) We conduct extensive experiments based on real- world datasets. To evaluate the performance, we compare the proposed algorithms with the best solu- tion obtained by MIDACO, justifying that the pro- posed algorithm achieves a comparable performance for the total delay and the times of winning while the converge time decreases by about 95%. The remainder of the paper is organized as follows. Section II gives an overview of the related work before presenting the system model and the problem in section III and IV, respectively. After that, we present the proposed algorithm, the experiments and the conclusion. II. Related Work A number of works investigate the container placement problem, optimizing various objectives such as latency, system cost and etc. Shang et al. [6] formulate a container placement and flow routing problem by considering the heterogeneity be- tween edge nodes and the overhead of adopting serverless platform. Extensive simulation results justify the efficacy of the proposed online algorithm by jointly considering delay, operating cost and data availability. Hu et al. [11] investigate the request scheduling problem in the context of Vehicle-Infrastructure Collaboration. By using layer sharing and container sharing, this paper reduces the long-term system cost consisting of transmission cost, preparation cost, retention cost, computation cost and transfer cost. Xiao et al. [8] study the cold-start problem in serverless edge computing, aiming to minimize the system cost incurred by cold-start, transmission and container caching. Sahraei et al. [12] propose an approach to improve the CPU utilization in Meta hyperscale private cloud. The proposed approach defers the invocations of delay- tolerant functions to off-peak hours and adopt a TCP- like congestion control policy to regulate the function execution. Wang et al. [13] adopt Markov decision process to model the service deployment problem in 6G networks, aiming to optimize the overall latency at a lower cost. The paper uses a greedy algorithm to find service deployment in a multi-layer edge network where services are assigned to the nearest ancestor devices in the routing tree. There are some other works investigating container placement in serverless computing. ServerMore [14] pro- poses colocation of serverless applications with serverful VMs. By this means, ServerMore improves the resource utilization by up to 245% with a minimal degradation of latency. Pan et al. [15] resolve the container placement and retention problem by mapping it to the classic ski-rental problem. The proposed online algorithm opportunistically distributes request by jointly considering resource capacity and network latency. However, none of the aforementioned work has consid- ered the inherent nature of serverless workloads such as the various size of jobs and the resource contention in wireless transmission. By jointly considering those dynamics with the topology in edge computing, we set our work apart from existing approaches. III. System Model We use v ∈ V to represent the set of edge nodes which is equipped with a processing capacity Cv. Each edge node is a basestation that can process serverless requests. Also, each edge node is equipped with a certain amount of wireless bandwidth which is denoted by Bv. We use k ∈ K to denote a service request that requires a type k container. For each service request k, we use Lk to denote the size of the job. Also, we use a binary variable xv k,t to represent the request k is assigned to edge node v in time slot t when xv k,t = 0 and vice versa. Let ck,t denote the required amount of hardware resources for request k in time slot t. Also, we use Cv to represent the total amount of hardware resources in edge node v. Moreover, we use bk,t and Bv to denote the deployment cost of running service request k and the total budget allocated to edge node v, respectively. In modeling the wireless communication scenario, the path-loss function as detailed in Equation 1 and sourced from Giordani et al. [16] is utilized to assess the impact of distance on each service request k. PLi = 38.77 + 16.7 ∗ log10 dv k + 18.2 ∗ log10 fk (1) In Equation 1, dv k is the distance (the units are meters) between service request k and the edge node v, whereas f is the frequency (we set this parameter as 5.9 GHz in this work) of the transmission signal. The units of path loss computed by Equation 1 are decibels (db). Path loss measurements obtained from Equation 1 are expressed in decibels (dB). The coefficients 38.77, 16.7, and 18.2 are empirical values referenced from Giordani et al. [16]. Equation 2 determines the accepted signal power, mea- sured in decibel-milliwatts (dBm), using Pk to represent the signal power, which is 21 dBm according to the wireless standard detailed in [16]. P ′ k = Pk − PLk. (2) It is important to recognize that both dB and dBm represent logarithmic scales; thus, P ′ k is measured in dBm. To convert P ′ k into milliwatts (mW), one should use Equa- tion 3. P ′′ k = 10 P ′ k 10 (3) allows the computation of the signal-to-noise ratio (SNR) for each service request using Equation 4 SNRv k = P ′′ k N0 ∗ bv k (4) where N0 is the power of environment noise, and it equals 10−11.4 mw [17]. The Shannon equation (see Equation 5) calculates the transmission speed TSv k for service request k to edge node v, using the Signal-Noise Ratio (SNRv k) and Bandwidth (bv k, specified in MHz). TSv k = bv k ∗ log2 (1 + SNRv k) (5) To obtain the transmission time for request k, divide the data size Lk for each service by the transmission speed TSv k . TT v k = Lk TSv k . (6) It is worth noting that the bandwidth allocation of service requests is highly relevant to the distances be- tween the service requests and edge nodes. For short distances represented by a small dv k, a large bandwidth bv k is generally not required. However, increasing bv k becomes important to decrease the maximum transmission time when dv k is significant. TABLE I Symbols and Variables Symbols Description G = (V, E) Physical network graph V Set of edge nodes E Set of links K Set of container types T Set of time intervals ck The required hardware resource of type k container Cv The hardware capacity of node v Bv The total bandwidth for edge node v Lk The size of job for request k dv k The distance between request k and node v Variables xv k,t Binary variable whether request k is assigned to node v in time slot t bv k The bandwidth allocated to request k at node v IV. Problem Formulation In this section, we formulate the request scheduling problem as an Integer Linear Programming problem and prove its NP-hardness. All symbols and variables are listed in Table I. We consider two categories of latency that are important to the system performance, e.g., transmission latency and processing latency. According to [18], [19], We formulate the average trans- mission latency as follows. Dtran = Lk TSk v xv k,t (7) where Lk denotes the size of a job k. We use TSk v to represent the transmission rate. When a container processes a user request, a consider- able amount of time is needed which is proportional to the size of the job. The processing time of a user request in time slot t is given by: Dproc = Lk pk,t xv k,t (8) where pk,t denotes the allocated processing capacity. Let xv k,t represent whether the request k is distributed to base station v. The overall latency of a user request can be formulated as: Dtotal = Dtran + Dproc (9) Problem We provide the mathematical model of the container placement and allocation problem with band- width and processing capacity constraints, aiming to op- timize the total latency of requests. min max( ∑ k∈K ∑ v∈V ∑ t∈T Dtotal) (10) s.t. ∑ k∈K bv kxv k,t ≤ Bv, ∀v ∈ V, ∀t ∈ T (11)∑ k∈K ckxv k,t ≤ Cv, ∀v ∈ V, ∀t ∈ T (12)∑ k∈K ∑ v∈V xv k,t = 1, ∀t ∈ T (13) where bk,t denotes the allocated bandwidth of request k. Bv represents the total amount of bandwidth of base station v. Also, we use ck,t to represent the amount of allocated processing capacity to run a container for request n. Let Cv denote the total amount of processing capacity at base station v. Constraint 11 guarantees that the allocated bandwidth of these containers must not exceed the total bandwidth capacity. Constraint 12 guarantees that the allocated pro- cessing capacity must not exceed the total processing capacity. Constraint 13 ensures that one request is only allocated once. A. Proof of NP-Hardness We show that the Generalized Assignment Problem (GAP), which is known to be NP-hard, can be reduced to the proposed problem. The GAP problem is distributing a number of K jobs to a set of J agents with minimized cost. For our problem, let cn be the size of the task. Let each edge node v represent an agent j equipped with a resource capacity of Cv. Then, assigning a job to an agent becomes assign a serverless request to an edge node, aiming to minimize the end-to-end latency which can be mapped to the cost of GAP problem. Thus, the GAP problem is a special case of our problem and hence our problem is NP- hard. V. Request Scheduling A. MIDACO MIDACO, a solver utilizing the Ant Colony algorithm, is suitable for commercial use in various numerical op- timizations. It effectively handles continuous non-linear (NLP), discrete/integer (IP), and mixed integer (MINLP) optimization challenges [20]. To solve the proposed problem, we formulate a set of internal constraints that any feasible solution must also obey. These internal constraints are shown in Equation 14. G(0) = sum(bv 0) − Bv G(1) = sum(bv 1) − Bv ... G(i − 1) = sum(bv i−1) − Bv G(i) = Cv − sum(cv i ) G(i + 1) = Cv − sum(cv i+1) ... G(2i − 1) = Cv − sum(cv 2i−1), (14) where G(0) to G(i-1) equals 0, and other G()≥0. B. Genetic Algorithm For this research, a custom Genetic Algorithm (GA) was designed to benchmark against other heuristic op- timization methods. This GA combines bandwidth {bv k} with binary variables {xk, tv} to construct chromosomes. Initially, the GA generates a set of random parent chro- mosomes and pairs each to mate—randomly choosing a ’father’ and ’mother’ for crossover. The crossover occurs at a randomly chosen splice point, where father’s genes up to the splice point are combined with the mother’s genes from that point onward. For instance, a progeny chromosome at splice point s inherits the father’s genes bv 1 through bv s and the mother’s genes bv s+1 through bv n, where n is the chromosome’s length. Post-crossover, the collective bandwidth may deviate from the limit. This is rectified by adjusting the genes so that the child’s total bandwidth aligns with the preset constraint. Specifically, for the discrepancy ∆ = sum{child bi}−total bandwidth, we alter a randomly chosen gene in the child’s chromosome by incrementing or decrementing 1 depending on whether ∆ is negative or positive, respectively. This adjustment is repeated for each unit of discrepancy, minus one. After crossing, we perform mutation to introduce chro- mosome diversity. We choose a random number µ ∈ (0, 1) and use it to mutate the child items. For bandwidth, if 0.3 < µ < 0.75, we do not mutate. If µ ≥ 0.75, we left- rotate the items one position. If µ ≤ 0.5, subtract 1 from the maximum item and add 1 to the minimum item in child bi. C. CANN Both MIDACO and Genetic are heuristic algorithms, so a common issue for them is the converge time, i.e., how long do they cost to produce a stable solution. Nor- mally, this process is counted by seconds, even minutes. Obviously, it is not practical to deploy such heuristic algorithms for the real system due to the fast dynamics. To solve this issue, we introduce a two-layer LSTM model to accelerate the process, which is trained by the results of MIDACO. With adequate training, the CANN is able to orchestrate with satisfied total delay and fast response time. To achieve the best performance, we always use the best result of MIDACO as training data, i.e., MIDACO 50000 in the experiment section. 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 To tal De lay (s) G e n e t i c , 5 0 0 0 r o u n d s G e n e t i c , 5 0 0 0 0 r o u n d s M I D A C O , 5 0 0 0 r o u n d s M I D A C O , 5 0 0 0 0 r o u n d s C A N N Fig. 2. Total delay of various settings VI. Experimental Evaluation A. Experiment Setup The experiments were conducted on a system equipped with an AMD Ryzen 7 5800H processor, clocked at 3.20 GHz, and featuring 16GB RAM and 4 cores. We utilized MIDACO Version 6.0, licensed commercially, for mixed- integer programming and developed a custom genetic algorithm tailored for the research presented in this paper. To assess the solvers’ effectiveness, 100 scenarios with randomized service request locations were generated. In each scenario, there are 20 requests. Each request’s hard- ware capacity needs are randomly assigned from a uniform range of 50 MHz to 150 MHz, and their proximity to the edge node follows a uniform distribution from 30 m to 200 m. There are 2 edge nodes, and each of them has 100 MHz bandwidth for data transmission and 1.5 GHz hardware capacity for data processing. We evaluate 5 settings in the experiments. Genetic 5000 and 50000 means there are 5000 and 50000 generations in the Genetic algorithm respectively. MIDACO 5000 and 50000 means there are 5000 and 50000 times of evaluations in the algorithms respectively. For CANN, we use the output of MIDACO 50000 as the training lables. B. Simulation results MIDACO [9] is a solver for numerical optimization problems by using evolutionary hybrid algorithms. Ex- tensive numerical tests proved that MIDACO can achieve the global optimal solution on the majority of problems robustly [10]. Hence, we adopt MIDACO to approach the near-optimal solutions of the proposed problem. In Fig. 2, the Genetic algorithm demonstrates a pro- nounced variability in delay outcomes contingent on the number of rounds. When operating with 5000 rounds, the Genetic algorithm incurs a substantial delay of approxi- mately 1.5 seconds. However, when the number of rounds 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 Re sp on se Ti me (s) G e n e t i c , 5 0 0 0 r o u n d s G e n e t i c , 5 0 0 0 0 r o u n d s M I D A C O , 5 0 0 0 r o u n d s M I D A C O , 5 0 0 0 0 r o u n d s C A N N Fig. 3. Response time of various settings is increased to 50000, there is a noteworthy decrease in to- tal delay, reducing to about 0.42 seconds. This reduction of 1.08 second signifies an enhanced performance attributed to a more exhaustive search capability, allowing the algo- rithm to explore and refine solutions more effectively over a larger number of generations. Conversely, the MIDACO solver exhibits a lesser degree of improvement in delay reduction when comparing its two scales of operation. At 5000 rounds, the MIDACO solver records a delay of around 1.17 second, which marginally decreases to just below 0.5 second at 50000 rounds. This minimal decrease suggests that while MIDACO benefits from increased rounds, the scale of improvement is sig- nificantly tapered, indicating a potential saturation in efficiency gains beyond a certain number of rounds. Since both MIDACO and genetic are heuristic, it is essential to compare their response time to CANN. Fig. 3 provides a detailed comparative analysis of convergence times associated with the Genetic algorithm, MIDACO solver, and CANN across different scales of computational effort. Analyzing the data, the Genetic algorithm at 5000 gen- erations manifests a convergence time of approximately 2.0 seconds, indicating moderate efficiency under constrained iterations. However, when the generations are increased to 50000, the convergence time reduces significantly to about 0.5 seconds, a fourfold decrease. This substantial improvement highlights the Genetic algorithm’s capability to optimize solutions more effectively with greater com- putational leeway, suggesting its suitability for complex problems where more extensive solution exploration is feasible. In contrast, the MIDACO solver displays a convergence time of around 1.5 seconds at 5000 evaluations. When the evaluations are escalated to 50000, the time slightly decreases to 1.0 seconds. The CANN model shows a 0 1 0 0 2 0 0 3 0 0 4 0 0 Wi n T im s f or To tal De lay G e n e t i c , 5 0 0 0 0 r o u n d s M I D A C O , 5 0 0 0 r o u n d s M I D A C O , 5 0 0 0 0 r o u n d s C A N N Fig. 4. How many times the various settings win for total delay convergence time slightly below 1.0 second, outperforming the other algorithms at their lower operational settings and closely matching the MIDACO at its higher setting. Fig. 4 indicates the details about the championship of various settings for the total delay based on 1000 samples. It is notable that Genetic 5000 never gets a championship. For the MIDACO, it is possible to be the winner in some circumstances. The number of championships for MIDACO 50000 and CANN is almost the same, around 420. VII. Conclusion In this paper, we have investigated the container place- ment problem in serverless edge computing. By jointly considering wireless bandwidth and node capacity alloca- tion, we have proposed a context-aware NN to reduce the end-to-end latency. Experimental results have shown that the CANN achieves comparable performance on the total delay and times of winning, whereas the converge speed is much better than MIDACO and Genetic, especially when the iteration of heuristics increases dramatically. Acknowledgment This work was supported by the Norwegian Research Council under Grant 262854/F20 (DILUTE project). References [1] Dewant Katare, Diego Perino, Jari Nurmi, Martijn Warnier, Marijn Janssen, and Aaron Yi Ding. A survey on approximate edge ai for energy efficient autonomous driving services. IEEE Communications Surveys & Tutorials, 25(4):2714–2754, 2023. [2] Chen Chen, Lars Nagel, Lin Cui, and Fung Po Tso. B-scale: Bottleneck-aware vnf scaling and flow routing in edge clouds. In 2022 IEEE Symposium on Computers and Communications (ISCC), pages 1–6, 2022. [3] Chen Chen, Lars Nagel, Lin Cui, and Fung Po Tso. Distributed federated service chaining for heterogeneous network environ- ments. In Proceedings of the 14th IEEE/ACM International Conference on Utility and Cloud Computing, UCC ’21, New York, NY, USA, 2021. Association for Computing Machinery. [4] Yongkang Li, Yanying Lin, Yang Wang, Kejiang Ye, and Chengzhong Xu. Serverless computing: State-of-the-art, chal- lenges and opportunities. IEEE Transactions on Services Com- puting, 16(2):1522–1539, 2023. [5] Chen Chen, Manuel Herrera, Ge Zheng, Liqiao Xia, Zhengyang Ling, and Jiangtao Wang. Cross-edge orchestration of serverless functions with probabilistic caching. https://doi.org/10.48550/ arXiv.2310.04185, 2023. [6] Xiaojun Shang, Yingling Mao, Yu Liu, Yaodong Huang, Zhen- hua Liu, and Yuanyuan Yang. Online container scheduling for data-intensive applications in serverless edge computing. In IEEE INFOCOM 2023 - IEEE Conference on Computer Communications, pages 1–10, 2023. [7] Chen Chen, Lars Nagel, Lin Cui, and Fung Po Tso. S-cache: Function caching for serverless edge computing. In Proceedings of the 6th International Workshop on Edge Systems, Analytics and Networking, EdgeSys ’23, page 1–6, New York, NY, USA, 2023. Association for Computing Machinery. [8] Ke Xiao, Song Yang, Fan Li, Liehuang Zhu, Xu Chen, and Xiaoming Fu. Making serverless not so cold in edge clouds: A cost-effective online approach. IEEE Transactions on Mobile Computing, pages 1–14, 2024. [9] Midaco-solver. http://www.midaco-solver.com/. [10] Matthias Gerdts Martin Schlüter and Jan-J. Rückmann. A numerical study of midaco on 100 minlp benchmarks. Opti- mization, 61(7):873–900, 2012. [11] Shihong Hu, Zhihao Qu, Bin Tang, Baoliu Ye, Guanghui Li, and Weisong Shi. Joint service request scheduling and container retention in serverless edge computing for vehicle-infrastructure collaboration. IEEE Transactions on Mobile Computing, pages 1–14, 2023. [12] Alireza Sahraei, Soteris Demetriou, Amirali Sobhgol, Haoran Zhang, Abhigna Nagaraja, Neeraj Pathak, Girish Joshi, Carla Souza, Bo Huang, Wyatt Cook, Andrii Golovei, Pradeep Venkat, Andrew Mcfague, Dimitrios Skarlatos, Vipul Patel, Ravinder Thind, Ernesto Gonzalez, Yun Jin, and Chunqiang Tang. Xfaas: Hyperscale and low cost serverless functions at meta. In Proceed- ings of the 29th Symposium on Operating Systems Principles, SOSP ’23, page 231–246, New York, NY, USA, 2023. Associa- tion for Computing Machinery. [13] Luying Wang, Anfeng Liu, Neal N. Xiong, Shaobo Zhang, Tian Wang, and Mianxiong Dong. Sd-srf: An intelligent service de- ployment scheme for serverless-operated cloud-edge computing in 6g networks. Future Generation Computer Systems, 151:242– 259, 2024. [14] Amoghavarsha Suresh and Anshul Gandhi. Servermore: Op- portunistic execution of serverless functions in the cloud. In Proceedings of the ACM Symposium on Cloud Computing, SoCC ’21, page 570–584, New York, NY, USA, 2021. Association for Computing Machinery. [15] Li Pan, Lin Wang, Shutong Chen, and Fangming Liu. Retention- aware container caching for serverless edge computing. In IEEE INFOCOM 2022 - IEEE Conference on Computer Communi- cations, pages 1069–1078, 2022. [16] Marco Giordani, Takayuki Shimizu, Andrea Zanella, Takamasa Higuchi, Onur Altintas, and Michele Zorzi. Path loss models for v2v mmwave communication: Performance evaluation and open challenges. In 2019 IEEE 2nd Connected and Automated Vehicles Symposium (CAVS), pages 1–5. IEEE, 2019. [17] Yue Li, Mohammad Ghasemiahmadi, and Lin Cai. Uplink co- operative transmission for machine-type communication traffic in cellular system. In 2016 IEEE 84th Vehicular Technology Conference (VTC-Fall), pages 1–5. IEEE, 2016. [18] Jinke Ren, Guanding Yu, Yunlong Cai, and Yinghui He. Latency optimization for resource allocation in mobile-edge computation offloading. IEEE Transactions on Wireless Communications, 17(8):5506–5519, 2018. [19] David Williams. Probability with martingales. Cambridge Univ. Press, 1991. [20] Mixed integer distributed ant colony optimization, 2023. http: //www.midaco-solver.com/ Accessed Sep 2023.