A Digital Twin (DT) may protect information that is considered private to its associated physical system. For a mobile device, this may include its mobility profile, recent location(s), and experienced channel conditions. Online schedulers, however, typically use this type of information to perform tasks such as shared bandwidth and channel time slot assignments. In this paper, we consider three transmission scheduling problems with energy constraints, where such information is needed, and yet must remain private: minimizing total transmission time when (i) fixed-power or (ii) fixed-rate time slotting with power control is used, and (iii) maximizing the amount of data uploaded in a fixed time period. Using a real-time federated optimization framework, we show how the scheduler can iteratively interact only with the DTs to produce global fractional solutions to these problems, without the latter revealing their private information. Then dependent rounding is used to round the fractional solution into a channel transmission schedule for the physical systems. Experiments show consistent makespan reductions with near-zero bandwidth/energy violations and millisecond-order end-to-end runtime for typical edge server hardware. To the best of our knowledge, this is the first framework that enables channel sharing across DTs using operations that do not expose private data.

A digital twin (DT) may protect information that is considered private to its associated physical system. For a mobile device, this may include its mobility profile, recent location(s), and experienced channel conditions. Online schedulers, however, typically use this type of information to perform tasks such as shared bandwidth and channel time slot assignments. In this article, we consider three transmission scheduling problems with energy constraints, where such information is needed, and yet must remain private: minimizing total transmission time when: 1) fixed-power or fixed-rate time slotting with power control is used and 2) maximizing the amount of data uploaded in a fixed time period. Using a real-time federated optimization framework, we show how the scheduler can iteratively interact only with the DTs to produce global fractional solutions to these problems, without the latter revealing their private information. Then, dependent rounding is used to round the fractional solution into a channel transmission schedule for the physical systems (PSs). Experiments show consistent makespan reductions with near-zero bandwidth/energy violations and millisecond-order end-to-end runtime for typical edge server hardware. To the best of our knowledge, this is the first framework that enables channel sharing across DTs using operations that do not expose private data.

A digital twin (DT) is a software version of a physical system (PS) that interacts with other objects on its behalf. In order to do so, changes in the PS must be communicated to the DT in a timely fashion, and this updating is referred to as DT synchronization. This paper addresses the Minimum Synchronization Period (MSP) problem in vehicular networks, which seeks to place DTs on execution servers (ESs) so as to minimize the maximum synchronization period for all physical systems and their DTs (PS-DT pairs), while satisfying communication and computation requirements. A novel solution is proposed by modelling the MSP problem as a multi-commodity quickest flow evacuation problem, which treats the synchronization data and processing as flow network inputs to be evacuated in the shortest possible time. Transmission and computation components are represented as network flows with linear edge delays, which enables the use of well-known techniques to find the quickest flow solution. To ensure that each DT is placed at a single execution server, an unsplittable flow rounding procedure is used that assigns DTs to servers without significantly increasing the synchronization objective. Simulation results demonstrate the quality of the MSP solutions produced by our algorithm using the optimal fractional solution as a lower bound for the optimal integral solution.

A Digital Twin (DT) is a virtual representation of a real physical system (PS) that interacts with other objects on its behalf. In these interactions, the Age of Information (AoI) is a key performance measure that is dependent on the DT's current network server placement. To maintain acceptable AoI performance as the system evolves, the DT location may have to be moved, which is referred to as Digital Twin migration. In this paper we consider the problem of DT migration in a vehicular system, focusing on minimizing the time-averaged AoI. In this type of system, it is difficult to maintain acceptable AoI performance due to the speed of the vehicles, which can result in frequent abrupt handoffs between different cellular domains. This makes the question of when to initiate DT migration an important one. The problem is formulated as a Markovian stopping problem and an optimal online algorithm is proposed using dynamic programming and the statistics of vehicular motion. A more computationally intensive adaptive version of this algorithm is also proposed where the dynamic programming tables are recomputed at each time step. A best-in-expectation algorithm is introduced that gives sub-optimal AoI performance but is more computationally efficient than in the optimal version. These algorithms are also compared to heuristics that do immediate migration and migration at handoff. An offline algorithm is formulated that provides a lower bound on the average AoI that is attainable. Performance results show that the proposed algorithm can significantly improve the efficiency of Digital Twin migrations compared to the other approaches while guaranteeing the minimized time-averaged AoI.

In federated learning (FL), clients update their local machine learning models using private data that is not to be shared with others. In each update period, the local models are then shared with a central server that maintains a global model that is used by all the clients. In this paper we consider the problem of scheduling and bandwidth assignment for vehicles that share a wireless communication channel during the FL. The objective is to minimize the update period duration so that global model updates can occur as quickly as possible. This is done by creating a transmission schedule and a fractional bandwidth assignment for each FL update period. The problem is modeled as a mixed-integer nonlinear program (MINLP) and since the problem is NP-complete, approximation algorithms are introduced that yield near-optimal solutions. This is done by doing a binary search on the update duration using a fractional relaxation and then by applying different dependent rounding procedures to obtain valid solutions. A variety of simulation results are presented that demonstrate the excellent performance of the proposed solutions when compared to the results obtained by an optimum direct solver on the same inputs.

This article introduces algorithms for static task class partitioning in mobile computation offloading (MCO). The objective is to partition a given set of task classes into two sets that are either executed locally by the mobile device (MD) or those classes that are permitted to contend for remote edge server (ES) execution. The goal is to find the task class partition that gives the minimum mean MD power consumption subject to task completion deadlines. This article generates these partitions for both soft and hard task completion deadlines. Two variations of the problem are considered. The first assumes that the wireless and computational capacities are given and the second generates both capacity assignments subject to an additional resource cost budget constraint. The proposed partitioning algorithms are based on heuristic class ordering methods. This article introduces two class ordering methods, a simpler one based on a task latency criterion, and an hierarchical version that first sorts and groups classes based on a mean power consumption criterion and then orders the task classes within each group based on a task completion time criterion. A variety of simulation results are presented that demonstrate the excellent performance of the proposed solutions for both given and optimized network resource assignments.

Digital twins (DTs) are virtual implementations of real physical systems (PSs) that interact with other objects on their behalf. Each PS periodically communicates with its digital twin so that the state of the DT is always sufficiently current. Using these updates, a DT can provide features that represent the real behavior of its PS using models that yield differing levels of system accuracy. In this paper, we study the DT model selection problem in wireless networks where the DTs of multiple PSs are hosted at an edge server (ES). The accuracy obtained from a given model is a function of its required amount of PS input data, the updating frequency, and the amount of computational capacity needed at the ES. The objective is to maximize the minimum achieved accuracy among the requested features by making appropriate model selections subject to wireless channel and ES resource availability. The problem is first formulated as an NP-complete integer program. The paper then uses relaxation and dependent rounding, and introduces a polynomial time approximation algorithm to obtain good solutions. A variety of simulation results are presented that demonstrate the excellent performance of the proposed solution.

In mobile computation offloading (MCO), mobile devices (MDs) can choose to either execute tasks locally or to have them executed on a remote edge server (ES). This paper addresses the problem of assigning both the wireless communication bandwidth needed, along with the ES capacity that is used for the task execution, so that task completion time constraints are satisfied. The objective is to obtain these allocations so that the average power consumption of the mobile devices is minimized, subject to a cost budget constraint. The paper includes contributions for both soft and hard task completion deadline constraints. The problems are first formulated as mixed integer nonlinear programs (MINLPs). Approximate solutions are then obtained by decomposing the problems into a collection of convex subproblems that can be efficiently solved. Results are presented that demonstrate the quality of the proposed solutions, which can achieve near optimum performance over a wide range of system parameters.

Digital twins (DTs) are virtual implementations of physical systems (PSs) and can represent the states of the PSs in realtime. In order to update the DTs with changes in their corresponding PSs, the PSs should regularly send their state information data to the DTs. Each DT must be assigned to an execution server (ES) that processes the forwarded data from its corresponding PS. The output is then made available to applications that are operating at an Internet cloud server. In this article, we consider the problem of DT placement such that the maximum data request–response delay experienced by the application over all PSs is minimized, subject to maximum data age target constraints at the DTs and the application server. The problem is first formulated as an integer quadratic program (IQP) and then transformed into a semidefinite program (SDP). The problem is NP-complete. Since exact polynomial solutions are unavailable, several practical polynomial-time approximation algorithms are introduced. The algorithms are designed to give solutions with different tradeoffs between the accommodation of the application input timing latency and the achievement of data age targets.

This paper proposes a method of joint wireless network and job service allocation for use with mobile computation offloading where task completion times have deadline constraints. In this design, mobile devices (MDs) may execute a computational task locally or offload the task through a wireless network for execution on an edge server (ES). The network owner offers to lease wireless communication channels at a given set of base stations along with edge server capacity that is used for job execution. The objective is to obtain a wireless and service capacity allocation that minimizes the total energy consumption of the mobile devices, subject to a cost budget constraint and constraints on the delay incurred by offloaded task execution. The design is first formulated as a mixed integer nonlinear programming problem. An approximate solution is then obtained by decomposing it into a collection of convex subproblems that can be efficiently solved. Results are presented that demonstrate that the proposed solution achieves near optimum performance over a wide range of system parameters.

Digital twin (DT) has attracted a lot of attention from both industry and academia since it was proposed over a decade ago. A DT can be viewed as a virtual implementation of a real physical system (PS) and used as a representation of the PS for various applications. Despite the great potential of DTs in various fields, implementing DTs to obtain the desired functionality is not always straightforward. Specifically, accurate real-time synchronization between the features at a PS and its DT is essential for the DT to represent the PS. In this case, appropriate networking support is a key component to enable future DT development and applications. Currently, the research on DTs from a networking standpoint is still at an early stage, and only limited work has been done on DT implementation in practical systems. To fill this gap, this article investigates networking-related issues for DTs. Based on the existing literature, a feature-based method is provided for describing the desired properties and quality of DTs from the networking perspective. A stage-based implementation framework is presented for creating large-scale DTs for complex PSs by considering various networking constraints. Networking-related challenging issues and open research topics are discussed at the end.

This paper considers mobile computation offloading when concurrent local execution (CLE) is used to enforce task execution time constraints. This mechanism can ensure that hard task deadlines are satisfied regardless of any randomness induced by the wireless channel, network or cloud servers. In this type of system, mobile device energy may be reduced by segmenting a task upload into multiple parts, rather than doing a conventional contiguous task upload. The paper uses this mechanism to adapt to changes in channel conditions during the offload. Unlike the contiguous task offload case however, the upload initiation times of each part must be determined dynamically. This is done while ensuring that hard task deadlines are always satisfied. In this paper, the multi-part computation offloading case is considered. In multi-part offloading, the task to be offloaded is partitioned into K upload parts before any offload initiation decisions are made. In this case, current channel state information is incorporated into the offload decisions, and the system must always satisfy a hard task execution time constraint using concurrent local execution. The paper considers the case for Markovian wireless channels. A provably energy-optimal online computation offloading algorithm (MuliOpt) is introduced for multi-part offloading. MultiOpt is shown to be optimal using Markovian decision process stopping theory. Since the computational complexity of MultiOpt can be significant, simpler and more computationally efficient heuristics, which also respect the hard task execution deadline, may be used. The paper introduces two such heuristics, the Immediate Offloading, and Multi Threshold algorithms. The mobile energy use of MultiOpt is compared to these heuristics, as well as to local execution without offloading and an offline energy bound. Simulation results show that MultiOpt performs significantly better when compared to the proposed heuristics, as well as when K increases.

This paper considers mobile computation offloading where task completion times are subject to hard deadline constraints. Hard deadlines are difficult to meet in conventional computation offloading due to the stochastic nature of the wireless channels involved. Rather than using binary offload decisions, we permit concurrent remote and local job execution when it is needed to ensure task completion deadlines. The paper addresses this problem for homogeneous Markovian wireless channel models. An online energy-optimal computation offloading algorithm, OnOpt, is proposed. Its energy optimality is shown by constructing a time-dilated absorbing Markov process and applying dynamic programming. Closed form results are derived for general Markovian processes, and the Gilbert-Elliott channel model is used to show how the particular structure of the Markov chain can be exploited in computing optimal offload initiation times more efficiently. It is shown that job completion time probabilities can be computed recursively, which leads to a significant reduction in the computational complexity of OnOpt. The performance of the proposed algorithm is compared to three others, namely, Immediate Offloading, Channel Threshold, and Local Execution. Performance results show that the proposed algorithm can significantly improve mobile device energy consumption compared to the other approaches while guaranteeing hard task execution deadlines.

This paper considers the problem of algorithmic efficiency in mobile computation offloading with Concurrent Local Execution (CLE). Online energy optimal algorithms can be developed when CLE is used to guarantee hard task deadlines while offloading over Markovian wireless channels. Unfortunately, these algorithms often have a high computational complexity, which prohibits their use in online mobile implementations. Three algorithms are introduced to reduce this complexity: Markovian Compression (MC), Time Compression (TC) and Preemption Using Continuous Offloading (Preemption-CO). MC and TC reduce the state space of the offloading Markovian process, by using a novel notion of geometric similarity, or by running an optimal online offloading algorithm in periodic time steps. In Preemption-CO, while a task is offloaded preemptively, the offloading decision at every time-slot is based on non-preemptive calculations. Our simulations show that, by applying these methods, the running times of the algorithms can be significantly reduced without suffering unreasonable performance degradation compared with the optimal energy performance.

Multi-decision mobile computation offloading occurs when a task to be remotely executed is uploaded in separate parts. Since the upload is partitioned, separate decisions are needed to determine the best time to initiate each upload. The multi-decision problem is considered for the case where execution completion times are subject to hard deadline constraints and where task offloads occur over a Markovian wireless channel. An online energy-optimal computation offloading algorithm, Multiopt (Multi-decision online Optimum), is introduced, whose optimality is proven using Markovian stopping theory. The paper presents results using the Gilbert-Elliott channel model, where task completion time probabilities can be efficiently computed using Dynamic Programming. Although the proposed algorithm is proven to be energy optimal, its performance is also compared to four others, namely, Immediate Offloading, Channel Threshold, Local Execution, as well as optimal single-part offloading. Results show that the proposed algorithm can significantly improve mobile device energy consumption compared to the other approaches while guaranteeing hard task execution deadlines.

This paper considers the problem of vehicular roadside unit (RSU) placement so that the sum of capital expenditure (CAPEX) and operating expenditure (OPEX) costs is minimized. The minimum cost solution considers these two cost contributions jointly when making RSU placement decisions. The input to the placement process consists of historical vehicular traffic traces and a set of candidate site locations from which RSU placements are chosen. An integer linear program (ILP) formulation is first given that computes the minimum cost placement based on the input traffic traces and candidate locations. A practical algorithm is then introduced that solves a relaxed version of the ILP and uses a novel rounding procedure to obtain real RSU placements. The algorithm takes into account the energy costs incurred by vehicular requests when the latter are scheduled using a minimum energy online scheduler. Performance results are presented that show that the proposed algorithm performs well compared to the case where placements are done without jointly considering both CAPEX and OPEX cost components.

This paper considers the problem of capacity augmentation in energy efficient road-side unit (RSU) deployments. RSU placements for a road network, and a set of vehicular traffic flow design traces are used as inputs. The objective is to find an RSU radio capacity assignment that minimizes the long-term operating expenditure costs subject to meeting packet deadline constraints with a given packet loss rate target. A procedure, referred to as the capacity augmentation (CA) algorithm, is proposed that iterates over the RSUs, selecting candidates for capacity augmentation based on their packet loss rate sensitivities. A variety of results are presented that characterize and compare the performance of the CA Algorithm using a greedy online packet scheduler. In particular, we show how to counterbalance the lack of causality in designing the RSU network when it is used for the online scheduling of incoming transmission requests. The comparisons include those using energy-optimal offline scheduling obtained by solving an integer linear program (ILP) formulation. It is shown that the CA Algorithm is an efficient way to assign RSU radio capacity that achieves the desired performance objectives.

This paper considers the problem of vehicular roadside unit (RSU) placement so that the sum of capital expenditure (CAPEX) and operating expenditure (OPEX) costs is minimized. The minimum cost solution considers these two cost contributions jointly when making RSU placement decisions. The input to the placement process consists of historical vehicular traffic traces and a set of candidate site locations from which RSU placements are chosen. An integer linear program (ILP) formulation is first given that computes the minimum cost placement based on the input traffic traces and candidate locations. A practical algorithm is then introduced that solves a relaxed version of the ILP and uses a novel rounding procedure to obtain real RSU placements. The algorithm takes into account the energy costs incurred by vehicular requests when the latter are scheduled using a minimum energy online scheduler. Performance results are presented that show that the proposed algorithm performs well compared to the case where placements are done without jointly considering both CAPEX and OPEX cost components.

Smart traffic scheduling can be used to reduce downlink roadside unit (RSU) energy use in green vehicular roadside infrastructure. In this paper, we consider the problem of downlink schedule generation when the RSU-to-vehicle radios use a variable bit rate (VBR) air interface. We first present offline scheduling formulations that provide lower bounds on the energy required to fulfill vehicle requests. An integer linear program is introduced that can be solved to find optimal offline VBR time slot schedules. We then prove that this problem is NP-complete by a reduction from the well-known Santa Claus problem. Two flow-graph-based models are then used to solve the minimum energy VBR scheduling problem. The first uses generalized flow (GF) graphs that represent time slots as individual graph nodes. The second uses time-expanded graphs (TEGs) that model the temporal evolution of the system. Both of these models can be used to compute lower bounds on energy performance and provide the basis for energy-efficient online schedulers. The first scheduler introduced, i.e., First-Come First-Serve (FCFS), is very simple and treats all vehicles equally and in the order of arrival. Since the time spent in energy-favorable locations decreases with higher vehicle speed, the second scheduler, i.e., Fastest First (FF), gives priority to faster moving vehicles. The greedy GF (G-GF) and greedy TEG (G-TEG) schedulers are then introduced, which are motivated by the two flow-graph-based models. The proposed algorithm performance is examined under different traffic scenarios, and they are found to perform well compared with the lower bound. Our results show that the less computationally intensive algorithms, i.e., FCFS and FF, can perform well under light load, but G-GF and G-TEG, while more complex, can provide near-optimal energy consumption and with reasonable demand dropping rates. The results also show that the G-GF and G-TEG algorithms are much more fair than the simpler algorithms in heavy-load situations.