Joint Resource Scheduling for AMR Navigation Over Wireless Edge Networks

The future of autonomous systems will rely on the usage of wireless time-sensitive networks to connect mobile cyberphysical systems, such as Autonomous Mobile Robots (AMRs), to Edge compute platforms to offload computationally intensive workloads necessary to complete tasks. In the case of AMRs, due to their mobility, the offloading of expensive processes such as localization and tracking methods to the Edge computing infrastructure must also be done over dynamic wireless networks. In larger scale systems, the network and compute resource requirements can quickly become prohibitively large due to network traffic and heavy workloads and tight deadline requirements for proper execution of time-critical tasks. In this paper, we formulate the problem of jointly allocating network and compute resources for time sensitive systems as the state of the wireless channel changes over time. By characterizing a compute model for AMR workloads, we further demonstrate how the network and compute scheduling decisions can be serialized, thus making the optimal scheduling problem significantly more tractable, via the incorporation of a compute-utility aware network cost function. Simulation results of AMR systems in a Wi-Fi network demonstrate substantial gains over baseline scheduling methods in total resource efficiency.


I. INTRODUCTION
Future cities, factories and enterprises are envisioned to feature large fleets of autonomous robotic systems performing various time-critical tasks [1]. Key to this vision is the autonomous mobile robot (AMR), which is capable of navigation around complex spaces using on-board sensing modalities [2]. Perception functions and path planning algorithms necessary for AMR control are computationally heavy tasks that can be more efficiently completed by leveraging a more powerful Edge compute infrastructure for real-time processing [3], [4], [5], [6]. Due to the mobility of AMRs, however, it is necessary to use wireless networks to communicate state and control information between the sensors, Edge compute, and the AMR controller. Doing so further complexity to the problem, as wireless communications can add significant delay to the data flow when insufficient network resources are utilized.
Managing the delay is critical in the operation of cyberphysical systems like AMRs because the control duty cycle operates on a fixed period. For wireless Edge systems, this can include both managing delay in the wireless network as well as managing delay in the Edge computing system through proper scheduling and resource management. Timesensitive networks have recently become an active area of research [7], [8] and standards [9], [10]. A wide variety of scheduling algorithms have been developed with the aim of guaranteeing desired latency performance over wireless networks such as 5G [11] and Wi-Fi [12], [13], [14]. Likewise, Edge compute resource management methods have also been developed to meet targeted latency requirements [15], [16], [17], [18] by optimizing compute resource parameters such as power, cores, and bandwidth. However, such approaches tend to rely on general compute resource-delay characterizations and not incorporate workload-specific models for relevant functions used in, e.g., AMR navigation tasks [19], [20], [21], [22].
While both the network and compute layers can be optimized for time sensitive performance independently to meet a total end to end flow deadline, the time varying nature of wireless channels resulting from fast fading phenomena makes this highly resource inefficient. This is because during periods in which the wireless channel provides poor signal quality, more wireless resources are needed to meet a delay requirement that could otherwise be compensated for by making the computation at the Edge faster. Such is the motivation behind joint resource scheduling between the wireless network and Edge compute systems [23], [24], [25], [26], [27]. Joint resource scheduling of network and compute seeks to optimize a joint utility of the network and computational resources. In the case of time sensitive systems, the optimization problem also can include a utility or constraints on the end-to-end, or roundtrip, latency incurred by the data flow as it traverses the network and computational nodes [24]. These problems can often be challenging to solve due to complexities of the problem structure, requiring heuristic approximate solutions, or the practical limitations of joint decision-making across the network and Edge orchestration resource allocation engines, which requires colocation and fused intelligence between the two [28], [29].
In this paper we address the problem of joint scheduling of network and compute resources over the wireless Edge system for a series of AMR navigation tasks. The average speed of an AMR requires certain perception frame rates, which in turn sets a fixed duty cycle of the AMR navigation control loop. This duty cycle in turn enforces a tight end-to-end latency constraint on the data flow, which most critically features a heavy network traffic load of camera data in the uplink and a heavy computational task of AMR localization at the Edge. While wireless communication models are sufficiently generic in representing the resource-delay tradeoff for any arbitrary traffic profile, it is known that different computational workloads can have drastically different responses to changes in various compute resource parameters. As such, properly addressing the joint scheduling problem for AMR systems necessitates an empirical modelling of the resource-delay tradeoff of a standard SLAM method, which we perform via the high-level CPU bandwidth control [30], [31]. The compute delay model we uncover in our analysis proves critical in establishing key technical simplification for the joint scheduling optimization. Namely, we establish the optimality of a serialized scheduling architecture, in which for each scheduling instance, network resources can be allocated without any regard to compute resource allocation through the adoption of a so-called "compute-utility aware" network utility function. Following this, the compute resource allocation can be optimally solved by incorporating the output of the network resource allocation problem. Such an architecture greatly simplifies the joint optimization by decomposing into two simpler problems, while further simplifying the decision-making architecture in a practical deployment by decoupling the network scheduling and compute orchestration decisions.
The rest of the paper is organized as follows. We begin in Section II by formulating the time-sensitive joint scheduling problem for generic wireless network and compute systems. In this problem formulation, we seek to minimize the aggregate cost of a network and compute scheduling decision while meeting unique end-to-end delay constraints for each system/data flow. The network delay is heavily dependent upon time-varying wireless channel conditions. We briefly detail in Section II-A how the time sensitive networking (TSN) standards can be applied to wireless networks such as Wi-Fi to ensure time-bounded data delivery in the wireless network, a core component of our solution. In Section III we proceed to outline the function and data flow of the implementing an AMR navigation task over the wireless Edge. The network traffic profile is dominated by RGB and depth images sent from on-robot cameras to the Edge (Section III-A), while the computation profile is dominated by the SLAM workload running at the Edge. The latter is analyzed via a compute bandwidth-delay profile over a variety of Edge platforms to establish an approximately inverse affine model for the delay experienced by running the SLAM workload with varying compute bandwidths (Section III-B).
In Section IV, we incorporate the empirical compute delay model for SLAM workloads into the joint scheduling problem. Most critically, the inverse affine model allows us to establish a key result regarding the optimality of a serialized scheduling algorithm (Proposition 1). Such a result allows us to formulate a new optimal network scheduling problem that can be solved prior to considering any compute resource allocation, which can be subsequently determined after network scheduling has been completed. In Section IV-A we detail how the compute-aware network scheduling problem is solved for the Wi-Fi time-sensitive network, in scenarios both with and without additional background traffic. Lastly, in Section V we present a wide array of numerical simulations that demonstrate the improved resource efficiency of the proposed solution for AMR navigation problems.

II. JOINT SCHEDULING FOR TIME SENSITIVE SYSTEMS
We consider a cyberphysical system of m AMRs sharing a wireless network and an Edge computing platform. Each AMR is assumed to have limited computation capacity relative to its application workload requirements, and thus must offload its primary workload to be executed on the remote Edge platform. To do so, application state from each system information must be sent over the wireless network via uplink to an access point (AP) and forwarded to the Edge compute infrastructure, where the application state is then used to perform the relevant workload. The output of this workload, e.g. state or control information, is communicated back to the system in the downlink. The end to end data flow for each system constitutes a closed loop control system, which generally operates with a fixed duty cycle, and is illustrated in  Due to the sharing of both communication and compute resources across flows, the goal is to allocate network radio resources and compute resources via scheduling policies to meet latency requirements for each flow. Denote by N ∈ N and C ∈ C the network and Edge compute scheduling decisions, respectively. Time-varying conditions of the wireless medium caused by fading will, moreover, impact the latency of a particular network scheduling decision. We therefore denote by h ∈ H m a set of wireless channel fading states, with element h i ∈ H denoting the channel state for the device associated to AMR i. This network state is observable via the use of pilot signals and is thus considered in performance modelling and resource allocation decisions. From here, we define the function τ n i : N × H → R + that, for a given network scheduling action N and under channel conditions h returns the total communication latency of an application frame in flow i. We assume this latency includes both the latency in the uplink and downlink of all network packets associated with a single application-level frame.
While not subject to physical effects such as fading, the Edge compute system performance also in practice may be subject to random variations. In particular, we define an Edge compute state denoted by θ ∈ that reflects a dynamic state, e.g. CPU utilization availability, of the Edge system. Like the network fading state, we consider this to be an observable state of the Edge compute platform via system queries. We accordingly define the function τ c i : C × → R + that returns the latency in processing the application frame at the Edge under a compute scheduling C under state θ .
We proceed to define the optimal joint scheduling problem under a set of current channel conditions h and flow latency requirements T 1 , . . . , T m . To do so, we must consider the cost of a given scheduling decision. We define then the cost functions f : N → R and g : C → R to reflect the cost of the network and compute scheduling actions, respectively. We assume that network and compute resource costs are independent of one another, and therefore the total cost can be represented as a simple summation of these two individual costs. The optimal joint scheduling problem for time sensitive systems can be then expressed as where N * (h, θ ) and C * (h, θ ) are the optimal network and compute scheduling actions under a set of wireless channel states h and Edge platform state θ . Observe that, while network and compute performance are only directly impacted by their own states h and θ , respectively, the optimal scheduling decisions at each layer are ultimately functions of both states due to their coupling via joint design in (1). The deadline constrained joint resource scheduling problem in (1) covers a wide range of wireless Edge network systems, which can be specified further via the definition of the network scheduling decision set N and compute scheduling decision C. As an example, consider that concurrent wireless network transmissions are divided across n orthogonal frequency bands. Then, the associated resource set would include a set of binary matrices N ⊂ {0, 1} m×n that indicate which flows are scheduled in which frequency band. Likewise, an example compute resource allocation framework may include assigning an integer number of CPU cores and a percentage of total cache across workloads, which can be represented by the set C ⊂ (Z m + , P m ), where Z m + and P m represent the set of m-dimensional nonnegative integer and probability vectors, respectively.
The complexity involved in solving (1) is heavily dependent both on the resource scheduling sets defined by N and C as well as the cost and latency functions. In generic cases, undesirable properties such as non-convexity may make finding optimal solutions intractable, with heuristic approaches instead used to find approximate solutions. In this paper, we demonstrate how the joint scheduling problem can be addressed in a typical wireless Edge network and compute system and under the requirements and models present in key industrial use case of the controlling a fleet of AMRs via remote SLAM algorithms at the wireless Edge. In this case, our empirical modelling provides a way in which we can solve (1) with low complexity. Before detailing the modelling and traffic profiles of the AMR use-case, we proceed to describe the network features necessary to perform time sensitive scheduling in practical Wi-Fi networks.

A. TSN & WI-FI SCHEDULING
Time-Sensitive Networking (TSN) refers to a family of standards defined by the IEEE 802.1 TSN task group that aims to enable deterministic networking based on standard connectivity technologies, such as Ethernet and Wi-Fi, for time-sensitive and best-effort applications in the same network [8], [9]. TSN capabilities include time synchronization protocol (e.g. 802.1AS), traffic scheduling (802.1Qbv) and other network management tools. Recent advances in Wi-Fi and 5 G has enabled the extension of TSN features over wireless links, as demonstrated in [13], [32]. In particular, the latest Wi-Fi 6/6E generation include new scheduling capabilities as defined by the 802.11ax specification that enables the Access Point to trigger multiple uplink transmissions in OFDMA (or multi-user) mode.
Without lack of generality, we assume a Wi-Fi based wireless Edge network enhanced with TSN capabilities to support the AMR scenario described previously. Wireless devices (AMRs and APs) are time synchronized using 802.1AS time synchronization over 802.11 Fine Timing Measurement (FTM) protocol, which enables applications to synchronize their data generation (e.g. synchronous control loop) and provide a deterministic traffic profile with high precision. The AP is responsible for scheduling resources using the 802.11ax trigger-based OFDMA mode in alignment with the periodicity and latency deadlines for each traffic flow. For uplink flows, the AP sends a Trigger frame allocating resources units (RUs) to clients within a multi-user Physical layer Protocol Data Unit (PPDU), which is followed by a multi-user "acknowledgement" in the downlink direction. For downlink flows, the AP includes RU allocation and other PHY configuration parameters in the control header of the downlink multi-user PPDU which carries data for multiple clients within corresponding RUs.

III. USE-CASE REQUIREMENTS FOR REMOTE SLAM/NAVIGATION
In this section, we detail the network and computational requirements needed to control and perform navigation tasks for an AMR over the wireless Edge. The basic navigation task can be represented as a closed-loop control system, with a series of task-specific functions for sensing, localization, and control [2]. To operate the AMR over the wireless Edge, such functions can be distributed across the agent and Edge computing platforms, which are detailed in the flow chart presented in Fig. 2. First, note in the left hand side of the image that functions such as obstacle avoidance can be done locally onboard the robot [33]. On the other hand, note in the right hand side of the image that, for AMR navigation and control, the Edge provides two primary functions: (i) 3D environment mapping and path planning and (ii) simultaneous localization and mapping (SLAM), or tracking. Function (i), i.e. mapping and path planning, sensor information from multiple AMRs to construct a 3D map of the environment, and subsequently, a so-called occupancy grid, to dictate where the AMRs can safely move around the environment. This occupancy grid information is used in conjunction with the goal locations to outline the proper paths to be taken by each AMR, which is represented as a series of waypoints that trace the desired path. Mapping and path planning is performed at the Edge due to both its high computational complexity and its possible need FIGURE 2. AMR navigation functional pipeline. The solid line indicates the primary control flow, with compressed camera images sent from the AMR to the Edge for SLAM processing (ORB-SLAM2). The AMR position is processed through the path planner to send relative position and waypoint information to the on-board AMR controller to compute linear and angular velocity commands. An additional control flow, denoted by the dashed line, concerns the generation of mapping and path planning. This is only performed at the beginning of task operation and is not of concern for dynamic resource allocation.
to aggregate sensor data and goal position information across multiple AMRs to avoid collisions. We stress that, while this is a computationally intensive procedure, it only needs to be done once at the beginning of the task or ocassionally when AMRs request a new path plan. Indeed, once waypoints are generated, they can be communicated back to each AMR and stored locally. As such, function (i) does not generally require dynamic resource allocation and is therefore not of focus in this work (this is denoted by the dashed line in Fig. 2).
Function (ii), on the other hand, concerns the dynamic updates and local tracking of AMR positions within the map in what are called SLAM methods, highlighted in blue in Fig. 2. SLAM methods are necessary to extract AMR position from RGB and depth camera data to feed back to the AMR [22], [34]. As illustrated on the bottom portion of Fig. 2, the relative AMR position is used by the controller to determine linear and angular velocity commands for the AMR. Thus, the sending of camera state information from the agent to the Edge, the processing of the SLAM workload to obtain position information, and the sending of position information to the onboard controller of the AMR, constitute the primary closed loop control system considered in this paper (this is denoted by the solid line). The blue shaded region between the local agent and Edge processing blocks denote the wireless medium over which state information is exchanged. As will be detailed in the following subsection, the uplink of the camera data to the Edge provides the bulk of the wireless network traffic. The joint resource allocation between network and compute resources for the AMR navigation task, therefore, is concerned primarily with the management of such resources needed to send uplink camera data and process position information via the SLAM workload, respectively.

A. NETWORK TRAFFIC PROFILE
In this subsection, we detail the network traffic profile utilized to perform remote SLAM functions at the wireless Edge. As previously discussed, the primary traffic of interest for real-time operation of an AMR at the wireless Edge is used to run SLAM methods, which is summarized in Table 1. We assume in our setting that the cameras capture images at 25 frames per second (fps), implying a navigation control duty cycle and latency requirement of T i = 40 ms for each AMR. The uplink traffic is composed of the image data coming from the RGB-D camera, including both RGB and depth images. The raw image data is immense in size, so it is assumed that any practical implementation of offloading image data will incorporate compression techniques to reduce the data size. While a variety of image compression techniques exist for RGB-D, for simplicity we incorporate JPEG compression schemes for RGB data and the lossless PNG compression for depth data [6], [35], [36]. To determine the compression rates typical for such data, we leverage the MuJoCo environment [37] to construct representative RGB-D images and empirically evaluate compression rates. We thus assume JPEG and PNG can compress RGB and depth images at a rate of 1:30 and 1:20, respectively.
The downlink traffic, on the other hand, consists solely of AMR position and waypoint information at 40 bytes, each. Compared to the heavy volume of data in the uplink, observe that the downlink data are short packets. It is therefore reasonable to focus our attention on uplink traffic when planning wireless network resource allocation decisions, as downlink packets can be easily sent for all AMRs with low latency and minimal resource utilization. In particular, small downlink latency impacts the round trip communication latency model for AMR i by decomposing the full network latency as τ c with τ c,d being some small, fixed, duration for sending downlink packets. Note that, for notational simplicity and without loss of generality, for the remainder of the paper we remove reference to τ c,d in our scheduling approach and consider τ c i (N) as a function of uplink network schedule N.

B. COMPUTE MODELLING OF ORB-SLAM2
As previously described, the primary compute workload of an AMR navigation task to be performed by the Edge is the SLAM localization and tracking function, which uses camera images to localize the position of the AMR in the environment to be used for computing steering and acceleration commands.
The so-called ORB-SLAM2 tracking algorithm [34] is a state of the art, but computationally intense, method for performing the localization and tracking function. We have performed empirical modelling of compute performance of ORB-SLAM2 method as it relates to the total compute latency for a number of different compute platforms. The compute resource we dynamically optimize is given the CPU Quota percentage, otherwise referred to as compute bandwidth. While physical limitations of a CPU, such as number of cores, technically limit the degree to which compute resources can be simultaneously shared across multiple workloads, known methods such as the Linux-based CFS scheduler [30], [31], perform "virtual" multi-tasking through careful time-domain multiplexing across multiple processes to achieve an effective controlled bandwidth allocation. We represent the CPU bandwidth allocation as a scalar α i ∈ (α min , ∞) for each workload i, with α min ≥ 0 as a minimum bandwidth allocation and the full CPU scheduling decision given by full vector C = α := [α 1 , . . . , α m ]. Observe that while quota percentage of bandwidth is in fact limited to the range (α min , 1], without loss of generality we allow α to values greater than 1, which here indicates the use of additional CPUs.
In Fig. 3 we demonstrate the empirical characterization on three compute platforms. Note that, for protecting raw measurement data, we scale the y-axis by a constant value. Observing the relationship between CPU bandwidth and latency, we utilize an approximate inverse affine model as where r > 0 is a platform-specific hyperparameter that characterizes the relative performance of a particular Edge platform. In Fig. 4, we demonstrate the fit of the proposed model in (2) with respect to the empirical modelling of the Xeon platform for r = 0.008 and α min = 0.05. We stress that the empirical characterization in (2) and illustrated in Figs. 3 and 4 represents an instantaneous characterization of the compute bandwidth-delay performance. However, as mentioned in the formulation of (1), this performance is not deterministic and in practice is subject to random fluctuations in CPU resource availability due to background processes. We thus utilize the random compute state  θ ∈ (0, 1] to represent the current CPU resource availability across the entire Edge system which, according to [31], scales the effective compute bandwidth allocated to workload i by a multiplicative factor. This provides us our complete statedependent compute model for the ORB-SLAM workload as a function of α i and θ as

IV. OPTIMAL SCHEDULING FOR REMOTE SLAM
We proceed in this section to utilize the empirical compute model for the AMR navigation task to derive the optimal joint scheduling policy under the strict deadline requirements necessary for AMR operation. Recall for (3) that the compute latency for the SLAM workload is approximately inversely linear with respect to the CPU bandwidth allocated to the workload. Supposing we then restrict our compute resource allocation model to CPU bandwidth allocation, we obtain the following latency constraint for each flow i, The flow-level latency constraint in (4) for the SLAM compute model in fact provides a desirable form of the joint scheduling problem-namely, one in which we can perform the optimization of the network and compute schedules in a serialized manner. This is possible due to the fact that, under certain conditions, we can precisely express the optimal compute bandwidth allocation α * (h, θ ) as a function of a given network allocation N and associated state conditions h and θ . We derive the form of the optimal resource scheduling policy under the stated assumptions in the following theorem. Proposition 1: Consider the joint resource scheduling problem for time-sensitive systems presented in (1) in which allocated compute resources are given by the CPU bandwidth, i.e. C = α and C = R m ++ , in conjunction with the workload latency model for the AMR-SLAM algorithm in (3). Under the assumption that the compute cost function g(α) is monotonically increasing in α, then the optimal CPU bandwidth α * i (h, θ ) for workload i under channel conditions h can be expressed as where the optimal network scheduling policy N * (h, θ ) is given by the solution to the following program, Proof: To begin, consider the AMR latency constraint in (4). Given that the compute latency is represented as an inverse affine function of α i for each flow i, we can rearrange the terms in each constraint as From the fact that g(α) is monotonically increasing function of CPU bandwidth allocation α and, further, that we define α i may take any positive value in R ++ , it follows that, for any network allocation N, the optimal α * i should satisfy the constraint (7) with equality such to minimize the resource cost g(α).
Given that the latency constraint in (1) can be satisfied with equality, we can replace the constraint by writing α as a function of N. Substituting the left hand side of (7) into the cost function in (1), we obtain a description of the optimal network allocation as given in (6). The optimal compute allocation is then recovered by substituting the solution of (6) into the left hand side of (7) and replacing the inequality with equality, giving us the result in (5).
The result in Proposition 1 is critical in that it allows us to serialize the allocation decisions across the wireless network and compute levels, rather than perform such optimizations simultaneously, which is often a complex or intractable task. In particular, given that the optimal bandwidth allocation can be determined directly from the optimal network schedule in (5), we can directly optimize the network schedule in (6) independently of any compute allocation decision. This is highly beneficial both from the view of complexity-the optimal solution can be obtained rather tractably-and from the view of practical implementation-the network AP can operate independently of compute orchestrator with limited signaling requirements (see Fig. 5).
To understand the expressions in Proposition 1, observe in (5) that the optimal compute allocation is determined to exactly meet the end to end latency deadline given the network latency resulting from the network scheduling policy. Meanwhile, observe in (6) that the network schedule is optimized over a modified cost function, which includes the indirect cost of the associated compute allocation. We refer to this as the compute-utility-aware network cost function At each scheduling instance t = 0, 1, . . ., the serialized allocation process is illustrated in Fig. 5 and outlined in Algorithm 1 given current network fading state h t and compute utilization state θ t . First, the wireless scheduling agent (i.e. the AP) measures the current channel conditions h t while the Edge compute system queries its own compute utilization state θ t . The latter of these is signaled back to the AP, or network scheduler, which minimizes the compute-utility-aware network scheduling problem in (6) to derive the network scheduling decision. Second, the associated network delays τ n i (N) are signaled back to the Edge compute scheduling agent (i.e. Orchestrator) to be used in compute allocation. The optimal compute bandwidth allocation is then obtained via  (6): 4: Report network latency d i,t = τ n i (N) ∀i to Edge orchestrator. 5: for i = 0, . . . , m do 6: Determine compute bandwidth allocation via (5): 7: end for 8: end for 9: Return: Optimal allocation decisions (5). Observe in Fig. 5 that, while state information is signaled between the network and compute schedulers, they otherwise operate in an independent manner without any direct joint decision making. The joint scheduling process described in Algorithm 1 is generic with respect to the wireless network and its mode of resource scheduling. Observe in (6) that we define the optimal network scheduling problem over both a generic loss function f (·) and a generic wireless network resource N ∈ N . The exact form of the optimal network schedule, as well as the complexity in obtaining it, will depend on the specifics of how the wireless network and resource scheduling system is defined. We proceed in the following subsection to demonstrate the optimal algorithm over a Wi-Fi network designed for industrial use-cases using the capabilities of wireless time sensitive networking as described in Section II-A.

A. OPTIMAL SCHEDULING OVER WI-FI TSN
As previously described, new features in next generation Wi-Fi 802.11 standards and compatible enhancements provided by TSN 802.1 standards allows for fully scheduled operation by the AP. In particular, in the 802.11ax standard, a triggerbased OFDMA architecture is used to schedule packets for multiple users across frequency and time to meet desired performance goals. In this section, we consider the AMR/SLAM network traffic profile into a Wi-Fi scheduling architecture, and derive the resulting scheduling algorithm to minimize the compute-utility-aware network cost in (8).
To begin, we consider the problem of single user network scheduling with fixed network bandwidth W (for Wi-Fi, typically 20, 40, or 80 MHz). In the single user case, the AP schedules uplink packet transmissions across time-slotted PPDUs, with each PPDU allocating the full bandwidth to a single user packet. Recall that the SLAM application generates compressed camera images at a fixed frame rate, with each frame arriving periodically at the network queue at each discrete time index t. The application level frames are large in size so are segmented into P Wi-Fi packets for each user i = 1, . . . , m. However, because application level processing in ORB-SLAM is limited by the time to receive the final packet [34], standard "fairness"-based packet scheduling methods, such as round-robin or proportional fair scheduling, will lead to high network latency τ n i (N, h t ) for all flows and are generally ill-advised in this context.
It therefore suffices to consider scheduling decisions at each time t an ordered sequence of scheduled users N = (ω 1 , . . . , ω m ), with ω s ∈ {1, . . . , m} denoting the user scheduled in slot s. In each slot s, we assume that all P packets from user ω s are transmitted in direct sequence. The time taken for a packet transmission for user i determined by the MCS, or data rate, achieved under channel condition h i t and is denoted bŷ τ (h i t ). 1 Denoting a sequence of scheduled users up to time s as ω −s = (ω 1 , . . . , ω s−1 ), the time to complete the final packet transmission in slot s from the beginning of the sequence is then given by the aggregated transmission durations, i.e.
Observe in (9) that the time for each user transmission is given by the time to transmit P user-packets. The total network latency τ n i (N, h t ) for a user i is determined by the sequence of scheduled users as From (9) and (10) we can derive the optimal network scheduling decision as defined in (6). In this paper we will consider two scenarios: (i) the entire network bandwidth is dedicated for AMR traffic and there is therefore no "resource cost," and (ii) the network bandwidth is shared with additional best-effort devices, thus implying a cost of utilizing a larger share of network bandwidth for AMR traffic. For both cases, we consider the compute resource cost to be simply the aggregate CPU bandwidth g(α) = m i=1 α i . In scenario (i), because the full bandwidth is available and there is no inherent resource cost for one user scheduling sequence over another, we may assume f (ω 1 , . . . , ω m ) ≡ 0. The optimal packet scheduling sequence is then given by the solution to the following minimization problem, 1 Note that, for notational simplicity, we assume all segmented packets are of equal size and thus transmission duration is determined solely by data rate. Observe in (11) that, due to the absence of network resource cost, we can in fact ignore the impact of the compute utilization state θ t , further simplifying the decision making architecture from Fig. 5. The optimization of (11) over sequences of scheduled users is, in any case, a non-linear assignment problem, and finding the exact solution is intractable. Given the reduced dimensionality of the problem, we can however use greedy assignment methods to find approximate solutions. In greedy methods, we schedule users one at a time, where in slot s the next user ω s is selected to have its packets scheduled such to minimize the incremental cost as determined by its channel state h ω s t , latency deadline T i , and relative workload complexity r. The greedy scheduling algorithm for the dedicated bandwidth scenario (i) is fully outlined in Algorithm 2. Note in Step 4 that we may consider the incremental cost solely in terms of the next scheduling sequence element ω s , as the prior sequence elements (ω 1 , . . . , ω s−1 ) and their resulting transmission durations have already been determined by the greedy algorithm. Thus at each slot s, we select the user that adds the smallest additional cost given by the expression in Step 4.
In the second considered scenario (ii), we assume that the wireless network is shared with so-called "best-effort" traffic, which has no priority and only scheduled when resource is available. Here, we consider dynamic network bandwidth allocation for each user rather than dedicate the entire band to the AMR traffic. In this way, unused bandwidth is available to be utilized by background, best-effort traffic. Compared to the scenario previously considered and detailed in Algorithm 2, dynamic bandwidth allocation expands the network resource scheduling decision N to include network bandwidth utilization factors β = [β 1 , . . . , β m ], where β i ∈ (0, 1] denotes the percentage of network bandwidth utilized to transmit the frame for user i. Because larger network bandwidth utilization implies decreased performance of best-effort traffic, we further associate a total aggregate network resource cost f (ω 1 , . . . , ω m , β) = ρ m i=1 β i , where ρ ∈ R + is a nonnegative weighting parameter that balances compute and network resource costs.
Given the additional network resource cost associated with wireless bandwidth utilization, the goal in this scenario is to select both the sequence of users to schedule, and the network bandwidth allocated to each user transmission to minimize the combined network and compute resource costs f (N) and g(C). To formulate this expanded version of the network scheduling problem, observe that the fraction of network bandwidth β i utilized by user i scales the transmission duration asτ Accordingly, the total network latency τ n i (N, h t ) for a user i follows from (10) and (12) in the case in which network bandwidth is dynamically allocated so that it may be shared with background network traffic. The resulting optimal computeutility-aware network scheduling allocation decision is then given by As in the prior scenario (i), solving (13) exactly is intractable and we resort to a greedy algorithm to schedule users and allocate bandwidth. The resulting greedy algorithm for the shared network scenario is outlined in Algorithm 3. Observe in Steps 4-5 that, here we evaluate the incremental cost for each considered user ω as the minimum cost over potential bandwidth allocations β ∈ (0, 1] considering the additional resource cost factor ρ. In Step 7, we select the next scheduled user ω s in the sequence as that which minimizes the optimal incremental cost and allocate the associated optimal bandwidth β ω s . Relative to the fixed bandwidth scenario detailed in Algorithm 2, the scheduling complexity necessarily increases as it becomes necessary to solve per-user optimization problems in Step 4 to properly schedule users and allocate optimal network bandwidth. However, we point out that each sub-optimization problem is over a scalar variable β and can generally be solved in a closed-form expression. With Algorithm 2 and 3 we have defined methods to determine optimal compute-utility-aware user scheduling and network bandwidth allocation strategies in Wi-Fi networks for a fixed bandwidth and dynamic bandwidth scenario, respectively. Either algorithm may be used to solve Step 4 in Algorithm 1 to solve the network scheduling problem at the network intelligence layer. As detailed in Fig. 5, the outputs of these methods are used to determine network latencies (Algorithm 1, Step 5), which are used to determine the compute bandwidth allocation decision (Algorithm 1, Step 6-8). We proceed to simulate the performance of the proposed serialized joint network and compute scheduling method for AMR navigation systems in the following section. Before Calculate incremental cost and associated bandwidth allocation .
8: end for 9: Return: Network allocation decisions proceeding, we conclude with a brief remark regarding bandwidth allocation in modern Wi-Fi networks. Remark 1: Note in (13) we assume that network bandwidth can be allocated as any fractional unit β i ∈ (0, 1] of the total bandwidth. As mentioned in Section II-A, the usage of OFDMA in modern Wi-Fi networks is in fact limited to so-called resource units (RUs) of fixed size, e.g. 2,4,8,20 MHz. For a network with large bandwidth, e.g. W = 80 MHz, we can assume that a unit of 2 MHz provides sufficient granularity to approximately allocate per-user bandwidth as a continuous variable. For smaller bandwidths, e.g. W = 20 MHz, we can replace in Algorithm 3 Step 4, the optimization over β ∈ (0, 1] with a search over a discrete set of permissible fractional units, e.g. β ∈ {0.1, 0.2, . . . , 1}.

V. SIMULATION ANALYSIS
In this section, we simulate the performance of the joint scheduling algorithm for managing the Edge network and compute resources for a series of AMR navigation tasks. For each setting, we compare the performance against a number of baseline methods in terms of resource efficiency. That is, we determine the amount of total compute or network resources needed to meet the flow-level latency requirements for all AMRs in the system. The AMRs move at an average speed of 1 m/s with a camera frame rate of 25 fps. The AMR control duty cycle, and thus the end to end latency requirement, is therefore given T i = 0.04 seconds for all i = 1, . . . , m. The network is modeled as a Wi-Fi 6 network with necessary TSN capabilities, namely fully scheduled triggered uplink and downlink operation and proper time synchronization, and a channel bandwidth of W = 80 MHz. The channel states h t are determined using IEEE Channel Model E for indoor environments. Path loss is determined by the location of AMRs relative to the access point with additional fast fading components drawn i.i.d. from the standard Rayleigh distribution at each scheduling instance. The traffic profile for the uplink camera traffic is given by the Table 1.
The Edge compute system, on the other hand, is modelled as a series of shared Xeon platforms, as shown previously in Fig. 4. To incorporate random compute utilization states θ t , we use the logit-normal distribution to model these proportional values in (0,1]. In particular, for these simulations, we assume θ t is drawn from a logit-normal distribution with parameters location μ = 1 and scale σ = 1.78. Such a density function places the mode at θ = 1, with the density gradually decreasing with θ . This implies that there is often full compute utilization available (i.e. θ = 1), and with high utilization availability more likely than low utilization availability.
To demonstrate the improved performance of the proposed joint scheduling method, in each case we also simulate the performance of a number of baselines that either optimize the resources of the network, compute, or neither of the two. In particular, the compared scheduling methods are given as follows: r Joint optimization: The proposed scheduling algorithm in Algorithm 1, with network scheduling determined by Algorithm 2 or 3 and compute scheduling via 5.
r Min latency network + Static compute: Network scheduling is optimized in a greedy fashion to minimize the total network latency, with compute resources being fixed as the minimum needed to meet all latency deadlines over the entire period of operation.
r RR + Dynamic compute: Network scheduling is given by round-robin (RR) scheduling, with compute resources dynamically optimized via (5).
r RR + Static compute: Network scheduling is given by round-robin (RR) scheduling, with compute resources being fixed as the minimum needed to meet all latency deadlines over the entire period of operation. We first simulate a setting with m = 3 AMRs sharing a Wi-Fi network with no consideration for background traffic. That is, we utilize Algorithm 2 for compute-utility-aware network scheduling with no penalty for utilizing all network bandwidth for AMR traffic. In Fig. 6, we plot the empirical CDF of the total CPU resource cost to achieve the latency requirement for all AMR systems. We further document in Table 2 the CPU resource requirement to meet the latency requirement 99% of the time. As can be seen in both figures, the joint optimization algorithm proposed in this work substantially increases the resource efficiency with respect to baseline methods that dynamically optimize only CPU or network scheduling, with overall a 13% improvement in compute resource requirements over the static configuration. While dynamic compute allocation, represented in the yellow line in Fig. 6, can provide   some efficiency gains relative to a static compute resource configuration, these gains are ultimately hindered by the inefficiencies in the wireless network scheduling. Indeed, channel inefficient methods such as RR scheduling provides small room for improvement by dynamic compute scheduling. It is thus highly beneficial for the wireless network scheduling to be "compute-utility aware," as done in the proposed methodology.
In the next setting, we consider a larger scale scenario with m = 6 AMR agents. The empirical CDF of the compute resource requirements to meet latency requirements is presented in Fig. 7, with the 99% percentile resource requirement displayed in Table 3. In the larger scale setting, we indeed see a similar trend as in the smaller setting, with joint optimization of network and compute resources improving the compute resource efficiency relative to the baselines with an overall increase of 36% over the static configuration. This demonstrates that with larger systems, there is even more opportunity to optimize efficiency via joint optimization of network and compute resources. In the final simulation, we simulate the performance in the network setting in which background traffic is present in the network, and thus network bandwidth utilization is penalized; recall (13) and the associated Algorithm 3 for the network scheduling method in this scenario. In Table 4 we show the 95th percentile of CPU resource needed to meet latency deadlines and the average network bandwidth utilized across all uplink transmissions for the scenario with m = 6 AMRs. We display the results for various values of the network resource cost parameter ρ, with increasing values placing a higher cost on network bandwidth usage. As can be seen in the table, by increasing ρ, we utilize less total network bandwidth but at a cost of greater CPU resource usage. This result is expected but gives insight into how compute and network resource utilization can be traded off with one another through careful tuning of this cost parameter.

VI. CONCLUSION
In this paper, we considered the problem of controlling a series of AMRs over the wireless Edge in a factory environment. Most critical to the operation of AMRs at the wireless Edge is the transmission of high volume camera data over wireless links and the processing of SLAM functions at the Edge. To optimize the total resource efficiency across the wireless network and Edge platform, we formulate a joint resource scheduling problem with time-sensitivity constraints necessary for AMR operation. The compute bandwidth-delay tradeoff of SLAM functions is empirically modelled to derive an approximate analytic representation of compute processing delays. Incorporating this compute model into the timesensitive joint scheduling problem, we formally establish the optimality of a serialized scheduling architecture, in which network schedules can be determined prior to compute resource allocation using a compute-utility aware scheduling utility. This architecture allows for more simplistic algorithms to find joint scheduling solutions and a more natural intelligence architecture between the network and compute scheduling units. We further derive greedy algorithms for compute-utility aware wireless scheduling in Wi-Fi networks with time sensitive networking (TSN) capabilities. Lastly, we perform a series of numerical simulations that demonstrate the improved resource efficiency performance of the proposed scheduling method against baselines.
Future research directions on this topic should address considering a more generic compute research allocation model beyond the CPU bandwidth modelled considered in this paper. Moreover, compute-utiltiy-aware network scheduling methods may be examined beyond the greedy approach considered in this work to consider lower complexity or better performing algorithms.