Mixed-Criticality Scheduling with Multiple Radio Interfaces

Abstract

In this chapter, we introduce the nodes with multiple radio interfaces (MRI) into mixed-criticality industrial wireless networks. When an error occurs or transmission demand changes, the MRI nodes can switch their transmission mode, changing to a high-criticality configuration to meet the system’s new demand. Hence, we first propose a heterogeneous MRI system model. Based on this model, we propose a Slot Analyzing Algorithm (SAA) that guarantees system schedulability by reallocating slots for each node after replacing conflict nodes with MRI nodes. By considering both system schedulability and cost, SAA also reduces the number of MRI nodes. Then, we propose a Priority Inversion Algorithm (PIA) that improves the schedulability by adjusting slot allocations before replacing conflict nodes with MRI nodes. By reducing the use of MRI nodes, PIA achieves better performance than SAA when the system is in the high-criticality mode.

5.1 Background

Due to the high real-time and reliability requirements of industrial systems, traditional cyber-physical systems cannot be applied directly [1,2,3,4]. To guarantee the requirements of ICPSs, we introduce multiple radio interface (MRI) nodes into ICPSs. A traditional network node is usually equipped with only one antenna, while an MRI node with two antennas, can both receive and send packets simultaneously [5,6,7]. By replacing conflict nodes with MRI nodes, multiple flows can transmit without delays. If we were to replace all the conflict nodes with MRI nodes, that would eliminate transmission conflicts in the system; consequently, when the number of MRI nodes is sufficient, we can guarantee system schedulability. However, considering the energy consumption and cost of MRI nodes, it is advantageous to be able to guarantee the network schedulability using fewer MRI nodes [8,9,10,11,12]. Hence, there are two main problems in mixed-criticality industrial wireless networks, (1) reducing the number of MRI nodes required to guarantee the network schedulability and (2) analyzing the network schedulability under different criticality modes. To address these problems, we propose two algorithms, a Slot Analyzing Algorithm (SAA) and a Priority Inversion Algorithm (PIA), to improve the network schedulability. SAA first allocates slots without considering transmission conflicts and replaces conflict nodes with MRI nodes to guarantee future schedulability in low-criticality mode. In contrast, PIA first optimizes slot allocation to reduce the number of transmission conflicts; then, we replace conflict nodes with MRI nodes only when the system cannot be scheduled. The contents of this chapter are as follows:

• 1. We first propose an algorithm to obtain the candidate node set (CNSA), which indicates the key nodes that affect system performance, and then propose SAA to improve system schedulability. SAA can guarantee that a system can be scheduled when it satisfies Theorem 5.1. Furthermore, SAA also reduces the number of MRI nodes through slot analysis.

• 2. We propose PIA to reduce the number of transmission conflicts before replacing the overlap nodes with MRI nodes. When a flow cannot be scheduled, priority inversion occurs when it satisfies Theorem 5.3.

• 3. We analyze system schedulability in high-criticality mode and prove that PIA achieves better schedulability than SAA in Theorem 5.4.

• 4. Simulation results show that our algorithms can improve the schedulability of industrial wireless networks with MRI nodes and that PIA achieves a better performance than SAA in the high-criticality mode.

5.2 System Model

We consider a heterogeneous network consisting of field devices (both MRI nodes and traditional sensor nodes), one centralized controller and one gateway. In this section, firstly, we propose a network model, and then we introduce MRI nodes into mixed-criticality networks. Finally, we apply a fixed priority (FP) scheduling scheme in industrial wireless networks.

5.2.1 Network Model

The system model is based on WIA-PA [13]. There are several features as follows: (1) Time Division Multiple Access (TDMA); (2) Route and Spectrum Diversity and (3) Handling Internal Interference. The number of channels is denoted as m, and there are N fixed nodes in our system.

Industrial systems have higher requirements of real-time performance and reliability. Here, we introduce MRI nodes into industrial systems. MRI nodes are promising for wireless transmissions because they can improve the average user spectral efficiency [14]. By mounting multiple antennas on a single sensor node, the node can receive and transmit simultaneously (or improve the packet acceptance ratio when all the antennas work on the same channel).

Hence, our model includes two types of sensor nodes: traditional nodes and MRI nodes. Each traditional node in our system is equipped with a half-duplex omni-directional radio transceiver whose status can alternate between transmitting and receiving. In contrast, an MRI node has two working modes: (1) when the antennas work on different channels, the node can both receive and transmit packets from several paths (the transmission capacity of an MRI node depends on the number of antennas it has); and (2) when the antennas work on the same channel, the node can reduce signal interference and improve the acceptance ratio because several antennas receive a packet from the same flow. Each packet in our system is transmitted through the network under source routing.

Figure 5.1 shows an example of an MRI node in the first mode (where antennas work on different channels) to address a transmission conflict in which F ₁ and F ₂ send packets to the same destination simultaneously. We introduce MRI to solve this issue. The flows can be scheduled by replacing V 3 with an MRI node. MRI nodes can also be used as a method to increase the packet acceptance ratio when two or several antennas receive packets on the same channel. For the second mode of an MRI node (where the antennas work on the same channel), when the links around V 3 have poor signal quality (such as from co-channel interference, intermodulation interference, spurious emissions or adjacent-channel interference), node V 3 can switch its antenna receiving frequencies to the same channel to improve the acceptance ratio. However, considering that the power consumption and cost of MRI nodes is considerably higher than the cost of a normal node, we cannot deploy MRI nodes throughout the entire system.

Fig. 5.1

An example

5.2.2 Mixed-Criticality System

Due to the introduction of MRI nodes, our mixed-criticality system model differs from other models [15,16,17,18,19]. The flow is a periodic end-to-end communication between a source and its destination. There are n flows in our system, denoted by F = {F ₁, F ₂, …, F _n}. F _i is characterized by < t _i, d _i, ξ, c _i, p _i > , a number between 1 ≤ i ≤ n, where the period is t _i, the deadline is d _i, the criticality level is ξ (we focus on a dual-criticality system {LO, HI}) in which ξ = 2, meaning the system has two critical levels. The superframe as the lowest common multiple of the flow periods is denoted as T. Initially, the system works at a low critical level, and the MRI nodes receive packets on different channels. The system switches to the high critical level to enhance system reliability, and the MRI nodes reconfigure their antennas to receive high critical flow packets on the same channels when the accident occurs. Here, c _i is the number of hops from a source to a destination, and the routing path is p _i. When the system critical level switches from low to high, the MRI node’s work pattern changes from receiving on different channels to receiving on the same channels to improve the schedulability of high critical flows. In addition, the characteristics of high-criticality flows switch to high-criticality mode. In this chapter, the system reduces the sampling period of a high-criticality flow to ℵ _i t _i to improve the sampling rate of high critical flow, where ℵ _i is a value that satisfies $\frac {c_{i}}{t_{i}} \leq \aleph _{i}<1$. To simplify the calculation we assume that d _i = t _i. Hence, ℵ _i satisfies $\frac {c_{i}}{d_{i}} \leq \aleph _{i}<1$. We model the duration of the mode switch as γ, which is used to calculate the delay of packets delivered during the mode change.

In the beginning, packets are transmitted in low critical mode. To provide more resources, each MRI node’s antennas work on different channels. The system switches when the accident occurs. To enhance system reliability in high-criticality mode, each MRI node’s antennas are reconfigured to work on the same channel.

5.2.3 Fixed Priority Scheduling

We provide a summary of fixed priority scheduling to analyze the schedulability of systems in this subsection. FP scheduling is a commonly adopted scheme in practice for cyber-physical systems and real-time CPU scheduling [20]. Each job priority is pre-allocated by the network controller, and transmissions are scheduled based on this priority.

We assume that the priority of each flow is the same as its number. That is, F ₁ has the highest priority, and F _n has the lowest priority in the system. There are two types of delays in industrial wireless sensor systems: (1) Channel contention and (2) transmission conflicts, the definitions can be obtained in [21].

We define a system as schedulable when all the flows in a system can be scheduled (reach the destination before its deadline). Then the definition of network schedulability is whether or not all flows in a network are schedulable. It is worth noting that when the system is in high-criticality mode, we no longer focus on the schedulability of low-criticality flows. When we repeat Z experiments, and only z experiments the emergency flow can be scheduled, then, the schedulability ratio is $\frac {z}{Z}$.

5.3 Problem Statement

Given the flow set F, a wireless network, and the FP scheduling policy, our objective is to use MRI nodes to improve the schedulability of mixed-criticality industrial networks. We first analyze the schedulability of mixed-criticality industrial wireless networks. Initially, the system is working in low-criticality mode. To improve system schedulability, we introduce MRI technology to provide more resources (the resources such as slots and channels are increased since MRI nodes can receive and send packets simultaneously). Hence, we replace nodes at several intersections with MRI nodes and propose a Priority Inversion Algorithm to guarantee that the system can be scheduled. However, when the accident occurs, we must guarantee high-criticality flows schedulability; consequently, the system switches to high-criticality mode, and the transmission mode of MRI nodes changes. We then analyze the schedulability of our method to evaluate the quality of PIA. The challenges in this situation are listed below.

• 1. When the system is deployed, we can easily determine flows that miss their deadlines. However, each flow’s schedulability is interrelated with others. Thus, how should we decide which nodes should be replaced with MRI nodes?

• 2. As described in the previous section, the power consumption and cost of an MRI node are much higher than those of a normal node. Therefore, it is unreasonable and not cost-effective to deploy many MRI nodes when the system can be scheduled. Hence, the problem of determining the smallest number of MRI nodes that can meet the system requirements in low-criticality mode is another challenge.

• 3. The MRI nodes’ transmission mode changes when the system switches to high-criticality mode. Therefore, the scheduling algorithm also needs to consider schedulability in high-criticality mode.

• 4. We need to analyze the system’s schedulability under our proposed method.

5.4 Scheduling Algorithms

In this section, we first study the issue of how to improve system schedulability with a small number of MRI nodes. First, we identify the nodes that may be replaced with MRI nodes to define candidate nodes as follows:

Definition 5.1 (Candidate Node)

We define a candidate node as a node at which transmission conflicts (intersection or overlap nodes) can occur. As Fig. 5.1 shows, the paths of F ₁ and F ₂ intersect at V 3; consequently, a transmission conflict may occur at this node. Thus, V 3 is a candidate node.

Flows may conflict when they have path overlaps. To facilitate candidate node identification, we assume that, at most, one part of a path overlaps between two flows. As Fig. 5.2 shows, two periodic flows transmit in a network that conflict at the second and third slots in the first period. There are two methods for addressing transmission conflicts. The first method is to reallocate the slots for each node after replacing the conflict nodes with MRI nodes (as Fig. 5.2 show, the system can be scheduled when we replace node V 5 with an MRI node, where CH is the abbreviation of Channel); the other method is to adjust slot allocation, as shown in Fig. 5.3. By rationally allocating each flow’s transmission slots, we can reduce the number of MRI nodes and the signal interference caused by the coexistence of multiple channels, which can also improve the schedulability when the wireless network is in high-criticality mode. However, this is unsuitable for a system that cannot be scheduled (in this example, the system cannot be scheduled if the deadline of F _i is 4).

Fig. 5.2

The analysis of conflicts. (a) Routing. (b) Superframe. (c) Scheduling

Fig. 5.3

An example for un-schedulable conflict

Therefore, for a system in which each link’s transmission slot has been allocated, we propose the slot analyzing algorithm to improve system schedulability under a traditional FP policy. When the system cannot be scheduled by MRI nodes, SAA re-allocates system resources to guarantee system schedulability (increase the transmission speed of unscheduled data flows). By considering the characteristics of mixed-criticality systems and MRI nodes, we then propose the PIA algorithm, which adjusts slot allocations before replacing intersection nodes with MRI nodes (the second method to address transmission conflict). By optimizing slot allocation, PIA can guarantee system schedulability with fewer MRI nodes. To guarantee the schedulability of the network, PIA will also re-allocate slots when the system cannot be scheduled. It is important to note that initially both the SAA and PIA algorithms work in low-criticality mode. We will analyze the schedulability of these two algorithms later.

5.4.1 Finding Candidate Nodes

A network consists of numerous nodes. In this subsection, we study how to select candidate nodes to guarantee the system can be scheduled. Transmission conflicts occur at the path overlaps of flows. As Fig. 5.4 shows, there are two types of overlaps (without considering the flow’s direction).

Fig. 5.4

Transmission conflict

Lemma 5.1

When flow paths have an overlapping region, the overlap nodes are candidates for MRI nodes. We denote the candidate node set as Λ, and the node set on each flow’s path can be denoted as λ _i.

Proof

Transmission conflict can obviously only occur at an overlapping region. We can solve this issue by MRI nodes. We can account for the conflicts caused by the first type as shown in Fig. 5.1. For the transmission conflict caused by the second type, as Fig. 5.4 shown, there are two nodes (V 1 and V 2) on both F ₁ and F ₂ involved in the second type of overlap. When this type of transmission conflict occurs, we can improve the schedulability by replacing both V 1 and V 2 with MRI nodes. Hence, when flow paths have an overlapping region, the overlap nodes are candidates for MRI nodes. □

Hence, we propose a Candidate Node Searching Algorithm (CNSA) to search for the set of candidate nodes as follows:

Algorithm 5.1 Candidate node searching algorithm

We search the candidate nodes by traversing the flow paths of the entire system. Nodes on more than one flow path are added to the candidate node set Λ. The time complexity of CNSA is O(F ²).

5.4.2 Slot Analyzing Algorithm

After obtaining the candidate node set, we reduce the number of nodes in this set to reduce the system’s cost. Because not all the candidate nodes can experience transmission conflicts, we propose the slot analyzing algorithm to analyze the schedulability of the network under a traditional FP policy in one superframe (a superframe is the lowest common multiple of t _i, i ∈ F). SAA improves system schedulability by replacing some of the nodes in Λ with MRI nodes.

Since SAA is used after the CNSA, we do not consider the slot allocations for each flow; we just replace conflict nodes with MRI nodes when a transmission conflict occurs.

When the system cannot be scheduled, we need to improve the schedulability using MRI nodes. Because MRI nodes can both receive and send packets in a single time slot, we first allocate slots for each node without considering transmission conflicts (the network controller allocates the channel for each transmission using the FP scheduling policy). Then, we replace conflict nodes with MRI nodes to guarantee system schedulability. Obviously, it is both unnecessary and not cost-effective to replace all the nodes in the candidate node set with MRI nodes. Therefore, SAA reduces the number of MRI nodes in Λ through slot analysis. The pseudo code of the SAA algorithm is as follows:

Algorithm 5.2 Slot analyzing algorithm

We reallocate the transmission time slots for each node by the schedulability of each flow (lines 1–8). If the flow cannot be scheduled, we find the intersection nodes and reallocate the slots for this flow without considering transmission conflicts. Otherwise, we retain the original allocation. Then, we analyze the transmission slot for each node in Λ (lines 9–23). When a node in Λ has more than one transmission at the same time slot, that means a transmission conflict occurs at this node, and we need to replace the node with an MRI node (lines 9–13). When the system also cannot be scheduled, SAA then re-allocates slots to accelerate the transmission speed of unscheduled flows until the flows can be scheduled (lines 14–21). Finally, SAA updates λ _i and returns Λ—the set of nodes that need to be replaced (lines 22–23). The theorem is as follows:

Theorem 5.1

A network can be scheduled with SAA when the number of channels is no less than the number of flows (m ≥ n).

Proof

When k flows conflict at node A, we denote the flow with the highest priority as F ₁, and the flow with the lowest priority as F _k. F ₁ transmits first and cannot be delayed at node A. The other flows must wait on F ₁,consequently, this generates delay. When the number of channels is no less than the number of flows, no delay is caused by transmission conflicts. Hence, flow F _i, i ∈ k will miss its deadline and cannot be scheduled when c _i + del _i > d _i, where del is the delay slots. Using SAA, we can eliminate the delays caused by transmission conflicts. k flows can transmit simultaneously when m > n because the number of hops in each flow’s transmission c is no larger than its deadline d. Thus, we can guarantee all the flows can be scheduled and, therefore, the system can be scheduled using SAA when the number of channels is no less than the number of flows. □

5.4.3 Priority Inversion Algorithm

In this subsection, we study how to reduce the number of MRI nodes by the second method, the PIA scheduling algorithm, which performs optimal allocation of resources before the system runs. Through optimal allocation, PIA can decrease the number of transmission conflicts at intersection nodes and further reduce the number of required MRI nodes. As Fig. 5.5 shows, V 5 does not need to be replaced by an MRI node when d _i = 6 in Fig. 5.2a.

Fig. 5.5

Transmission conflict

Initially, PIA allocates slots based on the traditional FP policy. If the system cannot be scheduled, the network controller obtains each flow’s arrival time at its destination, which can be denoted as r _i. Then, we can obtain the following theorem,

Theorem 5.2

When F _i and F _j conflict at node set V = V ₁, V ₂…, V _h (i < j), we denote the length of the overlap as Len(ij), and its conflict delay is D(ij). The schedulability of F _i is unaffected by the priority inversion of F _i and F _j when it satisfies:(1) r _i + D(ij) ≤ d _i ;(2) V  is the last part of the overlapping nodes of F _i.

Proof

The priority of F _i is higher than that of F _j based on the assumption that i < j. When transmission conflict occurs at node set V = V ₁, V ₂…, V _h, F _j must be delayed by F _i. We can accelerate the transmission speed by transmitting F _j first when the schedulability of F _i does not be changed. We introduce the result proposed by Saifullah in [22], in which the upper bound Δ(i, j) of the delay that F _j can experience from an instance of F _i is

$$\displaystyle \begin{aligned} {} \Delta(ij)=\sum_{k=1}^{\delta(ij)}Len_{k}(ij)-\sum_{k'=1}^{\delta'(ij)}(Len_{k'}(ij)-3), \end{aligned} $$

(5.1)

where δ(ij) is the number of path overlaps, Len _k(ij) is length of the kth overlap on F _i and F _j, δ′(ij) is the number of path overlaps larger than 3. Hence, the problem is transformed into one of calculating the delay F _i must bear while waiting for F _j to transmit. By simplifying Eq. (5.2), we can obtain the delay in V  as follows:

$$\displaystyle \begin{aligned} {} D(ij)=\begin{cases} Len(ij), & Len(ij)<4, \\ Len(ij)-3, & Len(ij) \geq 4. \end{cases} \end{aligned} $$

(5.2)

When V  is the last part of the overlap nodes of F _i, no other factors can cause additional delay to F _i. Hence, the schedulability of F _i is unaffected by the priority inversion of F _i and F _j when it satisfies:(1) r _i + D(ij) ≤ d _i; and (2) V  is the last part of the overlap nodes of F _i. □

Based on Theorem 5.2, PIA accelerates the unscheduled flow until it can be scheduled or no longer influences the other flow’s schedulability. However, Theorem 5.2 is not suitable for all overlaps. When F _i involves several parts of overlaps, we can only guarantee the acceleration of the low priority flow on the last overlap. For the flow’s intersections with F _i in the other parts of the overlaps, we define an Effective Overlap Region(EOR) as follows:

Definition 5.2 (Effective Overlap Region)

When F _i and F _j conflict in node set V = V ₁, V ₂…, V _h (i < j), the overall node set V  is the effective overlap region. If F _i and F _j have an overlap but do not conflict, then V  is not an effective overlap region.

Hence, Theorem 5.2 can be extended as follows:

Theorem 5.3

When F _i and F _j conflict at node set V = V ₁, V ₂…, V _h (i < j), the schedulability of F _i is unaffected by the priority inversion of F _i and F _j when it satisfies:(1) r _i + D(ij) ≤ d _i ; (2) it does not change the character of non-EORs after V ; and (3) V  is the last EOR of F _i.

Proof

Based on Theorem 5.2 we extend priority inversion to the last part of the EOR of F _i. When the operation satisfies (1) r _i + D(ij) ≤ d _i; (2) it does not change the character of non-EORs after V ; and (3) V  is the last part of the EOR of F _i. Then, the schedulability of F _i and its intersection flows are unaffected by the wait for F _j to transmit. Hence, when F _i and F _j conflict at node set V = V ₁, V ₂…, V _h (i < j), the schedulability of F _i is unaffected by the priority inversion of F _i and F _j when it satisfies:(1) r _i + D(ij) ≤ d _i; (2) it does not change the character of non-EORs after V ; and (3) V  is the last part of the EOR of F _i. □

To guarantee the second constraint condition in Theorem 5.3, we denote the number of overlaps after the last EOR in F _i as o _i. For any flow k, when F _k and F _i have a path overlap after the last EOR in V , we denote this overlap path as o _ik. Hence, the fact that the second condition in Theorem 5.3 can be translated into the priority inversion operation does not affect the schedulability of F _i and F _k. Obviously, F _k and F _i cannot conflict when the packets generated by the two flows arrive such that o _ik satisfies

$$\displaystyle \begin{aligned} \|r_{i}^{o_{ik}}-r_{k}^{o_{ik}}\|>D{ij}, \end{aligned} $$

(5.3)

where $r_{i}^{o_{ik}}$ is the time at which the packet generated by F _i reaches o _ik (it is easy to obtain o _ik from the network controller). When the arrival time difference of F _i and F _k is larger than Dij, o _ik is not an EOR. By updating $r_{i}^{o_{ik}}, i\in F$, we can determine whether to invert the priority of F _i and F _j.

Based on the above discussion, PIA can reduce the number of transmission conflicts before adding MRI nodes to the system. Furthermore, PIA can also improve the schedulability of networks under a high-criticality mode. When the system cannot be scheduled under PIA, the nodes at flow intersections will be replaced by MRI nodes to guarantee the schedulability of the network. The pseudo code of PIA is as follows,

Algorithm 5.3 Priority inversion algorithm

PIA first allocates slots under the FP policy (line 1) and initializes the number of MRI nodes (line 2). Then, it judges whether the system can be scheduled or not. If the system can be scheduled, it returns τ, otherwise, PIA accelerates the transmission speed of unscheduled flows by Theorem 5.3. When the flows do not satisfy the conditions, PIA replaces the overlap nodes with MRI nodes, similar to SAA, and increments τ by one (lines 3–16).

When the system performs the priority inversion operation at the last EOR, the conflict is removed. Hence, that part of the overlap is no longer an EOR. The system repeats this process until either the system can be scheduled or no flow satisfies Theorem 5.3.

5.4.4 Algorithm Analysis in High-Criticality Mode

In this subsection, we analyze the schedulability of SAA and PIA. Since both SAA and PIA can be scheduled by adding MRI nodes in low-criticality mode, we analyze only the schedulability in high-criticality mode.

Theorem 5.4

PIA has a higher schedulability than SAA when the system is in high-criticality mode.

Proof

When the system switches to high-criticality mode, low-criticality flows are abandoned and the high criticality flow period changes to $\aleph _{i} t_{i}, \frac {c_{i}}{t_{i}} \leq \aleph _{i}<1$. Because t _i = d _i, the flow must arrive at its destination before ℵ _i d _i. We analyze the schedulability by a discussion of classification. The flows are classified into two categories: (1) those in which the flow does not overlap other high-criticality flows and (2) those in which the flow overlaps other high-criticality flows. For F _i in the first category, the schedulability of both SAA and PIA are identical. When a flow is not blocked by other flows it can be scheduled when

$$\displaystyle \begin{aligned} r_{i}\leq \aleph_{i} d_{i}. \end{aligned} $$

(5.4)

For F _j in the second category, F _j is delayed by other flows in D(j) slots. Then, F _j can be scheduled when

$$\displaystyle \begin{aligned} r_{j}+D(j)\leq \aleph_{j} d_{j}. \end{aligned} $$

(5.5)

We denote the number of MRI nodes in SAA and PIA as $\tau _{j}^{SAA}$ and $\tau _{j}^{PIA}$, respectively. Because PIA reduces the number of transmission conflicts before replacing intersection nodes with MRI nodes, $\tau _{j}^{SAA} \geq \tau _{j}^{PIA}$. By Theorem 5.2, we can obtain

$$\displaystyle \begin{aligned} D(j)=\sum_{k=1}^{\tau_{j}}D(ij), \end{aligned} $$

(5.6)

when F _i and F _j conflict, $D(ij)=\begin {cases} Len(ij), & Len(ij)<4 \\ (Len(ij)-3), & Len(ij) \geq 4 \end {cases}$, which is determined only by the length of the overlap. Then, we can obtain the relationship between SAA and PIA as follows:

$$\displaystyle \begin{aligned} r_{j}^{PIA}+D(j)^{PIA}\leq r_{j}^{SAA}+D(j)^{SAA}. \end{aligned} $$

(5.7)

Hence, PIA has a higher schedulability than SAA when the system is in high-criticality mode. □

5.5 Performance Evaluations

We evaluate the performance of our proposed methods by experiments. We compare our approaches with the traditional FP algorithm without MRI nodes. We compare both the acceptance rate and the number of MRI nodes for each criticality mode. We use the acceptance rate to represent the schedulability of a network. When all flows can be scheduled, the acceptance rate is 1; otherwise, it is 0. To control the workload of the entire system, the simulations use the UUniFast algorithm, which can make the flows neither pessimistic nor optimistic for the analysis [23]. All algorithms are implemented in C language. These programs run on a Windows machine with 3.2 GHz CPU and 8 GB memory. Some simulation parameters are summarized in Table 5.1.

Table 5.1 Simulation parameters

5.5.1 Low-Criticality Mode

We first compare the performances of the algorithms in low-critical mode. As Fig. 5.6(a) shown, the system acceptance rate is decreased by the FP scheduling policy (n=15, N=50, m=16), which occurs because the idle resources decrease as the system utilization increases. The latency tolerance of a packet is reduced along with the idle resources. The network can be scheduled under both SAA and PIA in any situation by increasing the number of MRI nodes when m ≥ n. Figure 5.6b shows the relationship between the number of MRI nodes and utilization. Obviously, both SAA and PIA can reduce the number of MRI nodes under the premise that the system is schedulable. Because PIA optimizes slot allocation before the node replacement operation, PIA uses fewer MRI nodes than SAA. However, none of the curves have an obvious tendency because system utilization involves not only the flow period (t) but also the number of transmission hops (c). We need to regenerate the transmission path for each flow to satisfy the system utilization requirements. Hence, the number of candidate nodes goes up and down. Because an MRI node is selected from the candidate node set, the number of MRI nodes is always less than the number of candidate nodes.

Fig. 5.6

Relationship between the acceptance rate/the number of MRI nodes and system utilization. (a) Acceptance rate. (b) The number of nodes

We repeat this simulation for the situation in which all flows can be scheduled as shown in Fig. 5.7. The number of candidate nodes is fixed when we increase system utilization by only adjusting the flow period (because there is only one test in this simulation, and the flow transmission paths do not vary). Initially, no MRI nodes exist in the system because they can be scheduled without any MRI nodes. However, when we increase the system utilization, transmission conflict occurs. To guarantee the schedulability of the system, we need to add MRI nodes to the system. In addition, the number of MRI nodes required by PIA is always less than the number required by SAA. This occurs because by optimizing slot allocation, PIA can reduce the number of transmission conflicts. Hence, the number of MRI nodes required by PIA is always less than the number required by SAA.

Fig. 5.7

Relationship between the number of MRI nodes and system utilization

The relationship between the acceptance rate/number of MRI nodes and the number of flows is shown in Fig. 5.8 (U=0.3, N=50, m=16). The acceptance rate is reduced along with the number of flows under the FP scheduling policy. The acceptance rates of SAA and PIA are reduced when the number of flows becomes larger than the number of channels. This result occurs because when m < n, delays are caused by channel contention. PIA achieves a better performance than SAA when n > 16 because it uses fewer MRI nodes. In addition, both the number of candidate nodes and the number of MRI nodes increase as the number of flows increases. This result occurs because the number of intersections increases when the number of flows increases, causing more transmission conflicts in the system. Thus, the system requires more MRI nodes to guarantee its schedulability, and the number of MRI nodes always satisfies τ ^FP > τ ^SAA > τ ^PIA.

Fig. 5.8

Relationship between the acceptance rate/number of MRI nodes and the number of flows. (a) Acceptance rate. (b) The number of nodes

Figure 5.9 shows the relationship between the acceptance rate/number of MRI nodes and the number of nodes (U=0.3, n=15, m=16). All these results indicate that SAA and PIA can guarantee the schedulability of the system. The number of MRI nodes in both SAA and PIA is no larger than the number of candidate nodes regardless of the conditions. In addition, when the number of nodes increases to 90, the number of MRI nodes in PIA and SAA is identical because PIA transmutes into SAA when it cannot resolve the conflict.

Fig. 5.9

Relationship between the acceptance rate/number of MRI nodes and the number of flows. (a) Acceptance rate. (b) The number of nodes

5.5.2 High-Criticality Mode

When the system switches to high-critical mode, the number of MRI nodes is no longer our main concern (the number of MRI nodes is the same as in low-critical mode). Instead, we are more concerned with the performances of SAA and PIA in high-critical mode.

Figure 5.10 shows the relationship between the system acceptance rate and system parameters such as the number of nodes and the number of flows. In these conditions, we set $\aleph =0.9>\max \{\frac {c_{i}}{t_{i}},i \in F\}$. If $\max \{\frac {c_{i}}{t_{i}}, i\in F\} \geq 0.9$, we will regeneration the system. Figure 5.10 illustrates (as Theorem 5.4 proves) that the acceptance rate in PIA is always better than in SAA.

Fig. 5.10

Relationship between the acceptance rate and system parameters in high-critical mode. (a) Acceptance rate. (b) The number of nodes

Figure 5.11 shows the relationship between the system acceptance rate and system utilization. To study the relationship between these two elements, we increase system utilization by adjusting only the flow period. Subsequently, we obtain the same result as Fig. 5.10, which also verifies the correctness of Theorem 5.4.

Fig. 5.11

Relationship between the number of MRI nodes and system utilization in high-critical mode

5.6 Summary

In this chapter, we first introduce MRI nodes into mixed-criticality networks. Then, we analyze the transmission paths and obtain the candidate node set. Next, based on the characteristics of MRI nodes, we propose SAA and PIA to guarantee the network schedulability in low-criticality mode. By considering system cost, these two algorithms help to reduce the number of MRI nodes used. Finally, we analyze the schedulability of these two algorithms when the system switches to high-criticality mode. The simulation results show that our scheduling algorithms perform better than existing scheduling policies.