# Scheduling of Industrial Control Traffic for Dynamic RAN Slicing with Distributed Massive MIMO +
## 4 System Model  
The notation we use is summarized in Table 2. We have a set \(\mathcal{K}=\{1,2,\ldots,K\}\) of nodes (end devices) needing to communicate with the serving CSPs. Communication takes place in frames, where the same pattern of time slots (the frame) is repeated, forming a periodic schedule (see Figure 3). This schedule must satisfy the traffic requirements of all of the end devices. The frame length \(T\), that is, the number of slots in the frame, is a decision variable able to be chosen so as to satisfy the traffic requirements. In the problem formulations, we will assume that the maximal frame length is equal to a given parameter \(S\) that is large enough to accommodate a frame that satisfies all nodes' traffic demands. Thus, we assume that the maximal frame is composed of the slots \(i\in\mathcal{S}\), where \(\mathcal{S}=\{1,2,\ldots,S\}\), and all other (feasible) frames are composed of slots \(\{1,2\ldots,T\}\), where \(T\leq S\). Determining a suitable value for \(\mathcal{S}\) is not entirely straightforward and impacts the performance of the optimization solver. This is a question we will discuss further in Section 7. Each node \(k\in\mathcal{K}\) has three traffic requirements: an uplink demand rate \(\hat{h}(k)\), a downlink demand rate \(\hat{h}(k)\), and a maximum number of slots \(d(k)\) between two consecutive slots in which a pilot is allocated to \(k\). The uplink and downlink demands are traffic rates expressed in slots per frame slot, that is, a number of slots in which the device must transmit (uplink) or receive (downlink) per slot in the frame in order to satisfy the demand. For example, if a node \(k\) has an uplink demand rate \(\hat{h}(k)\) of 0.2, then for every five slots  
Figure 3: One time slot lasts for the coherence time of the wireless channel, and contains a number of resources over frequency and time. For instance, if orthogonal frequency division multiplexing (OFDM) is used, each resource is one symbol long in time and one subcarrier wide in frequency. Some of the resources in a slot are used for pilot signals, each of which is assigned to one user (colored resources). The rest of the slot is used for user data, and during this time, all users transmit or receive simultaneously and are separated by spatial multiplexing. A series of consecutive slots, each with end devices assigned to pilots, forms a frame, and the same frame is repeated during the scheduling window. The scheduler calculates a new schedule (frame structure) as the end of the scheduling window approaches, and this new schedule is then activated when the new scheduling window begins. in the frame, \(k\) must transmit in one slot. If the frame lengths were determined to be, say, \(20\), then \(k\) must transmit in at least \(4\) slots during the frame in order to satisfy its uplink demand. The downlink demand rate works similarly. Here, we do not consider different modulation and coding schemes yielding different data rates, but rather consider that \(\hat{h}(k)\) and \(\hat{h}(k)\) can be satisfied with a fixed number of slots for a given node \(k\) and frame length. Finally, \(d(k)\) gives the maximum number of slots that can elapse between two consecutive transmissions by the device. For example, if \(d(k)=10\), then \(k\) must transmit (or receive) at least once every \(10\) slots. This ensures a given frequency of sending and receiving control updates. Depending on the type of controller and the type of actuator, different devices will have different allowable periods for their control. In the following section, we provide our optimization formulations based on the above system model. We will first consider the case where the frame length is fixed. This leads to a simpler problem formulation, and is applicable in cases where external constraints enforce a specific frame length, for example, if a frame should be equal to the length of the network's transmission time interval (TTI).