Network Delay

An additional important characteristic of routing-scheduling policies is their resulting delay in the packet delivery. While the average delay is proportional to the average number of queued packets in the network, we also study this measure explicitly. In order to do this, and under the assumption of first-in first-out queues, we compute the number of time slots it takes for a packet to be delivered to a sink node. We plot in Figure 6.10 the resulting histogram. In average, the number of time slots it takes to deliver a packet to a sink node is 4.04 for the SSBP-EH policy, while it is 5.36 for the SBP-EH policy.

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gy

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SBP-EH

SSBP-EH

Figure 6.9: Average energy balance in the network, given by the expression 1_tPt_l=0 P i∈N P k∈K  ei[l]−Pj∈Nir k ij[l]  .

This is about a 1 time slot of difference between the policies. Taking a more detailed look at the histogram, we can see that the distribution for the SSBP-EH is very similar to the one of the SBP-EH, but with a 1 time slot shift to the left. As already seen in Fig. 6.5, the more aggressive behavior of the SSBP-EH policy leads to an overall reduction in the network queues. These smaller queues result in a reduction of the waiting time of packets at each hop, which results in a smaller delivery delay.

6.6

Conclusions

In this chapter, we have generalized the stochastic family of backpressure policies to energy harvesting networks. Different from other works, which are based on Lyapunov drift notions, we have resorted to duality theory. This has allowed us to study the problem under a framework based on the correspondence between queues and Lagrange multipliers. Under this framework, we have proposed two policies, (i) SBP-EH, an easy to implement policy where nodes track the difference between their queue multipliers and the ones of their neighbors. The pressure is further reduced by the battery multipliers as the stored energy decreases. Then, the transmit decision is to transmit the flow with the highest pressure. And (ii) SSBP-EH, a probabilistic policy with improved performance and convergence guarantees, where nodes track the pressure in the same way as SBP- EH but perform an equalization in the form of an inverse waterfilling. This results in a probability mass function for the routing-scheduling decision, where a sample of this

6.6 Conclusions 103

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Figure 6.10: Histogram of packet elapsed time before reaching a sink node.

distribution is then taken to decide the transmission. For both policies, we have studied the conditions under which energy causality and queue stability are guaranteed, which we have also verified by means of simulations. The numerical results show that given feasible data and energy arrivals, both policies are capable of stabilizing the network. Overall, the SSBP-EH policy shows improvements in queued packets, stabilization speed and delay with respect to the SBP-EH policy. Furthermore, when compared to non-EH policies, the SSBP-EH policy shows to have an asymptotically vanishing gap.

Chapter

7

Random Access Policies for

Wireless Networked Control

Systems

In this chapter, we study wireless networked control systems with energy harvesting sen- sors. Multiple sensors share a wireless medium over which they transmit measurements to their respective controllers, and due to the shared medium, packet collisions occur if sensors transmit simultaneously. To alleviate this problem, we propose random access policies that satisfy a required control performance on each control loop, while also sat- isfying the energy constraints imposed by the energy harvesting process. The optimal scheduling policy is shown to follow a simple thresholding operation. Moreover, we pro- vide a stochastic dual method for their computation, which is shown to be decoupled across sensors. Finally, we verify numerically the properties of the proposed policy.

7.1

Introduction

Wireless networked control systems are rapidly becoming prevalent in the modern world. They are present in smart homes, robotic automation, smart transportation, industrial plants and more. A critical component of these wireless control systems are the sensing devices. These sensor nodes measure the state of the system and transmit their obser- vations over a wireless channel. However, due to the uncertain nature of the wireless channel, the choice of communication policy critically affects the closed loop performance of the control system. The sensors share the wireless communication medium and there- fore one should aim for an efficient use of this resource in a way that meets the control performance requirements.

When dealing with classically powered sensors, the scheduling problem in wireless networked control systems has been previously studied in the literature. The most com-

mon approach to this problem is the design of centralized scheduling policies. In this setup, there exists an overseeing entity specifying which sensor is allowed to transmit at a given time slot, in order to avoid collisions between packets. These type of policies might be static [118,119] or of a more dynamic nature, where centralized decisions can be taken based on plant state information [120] or wireless channel conditions [121], among others.

7.1.1 Contribution

In this work, we study the design of random access policies for sensor nodes powered by energy harvesting. Different from previous works [119–121], we design decentralized scheduling policies. This is more in line with the random access policy presented in [122]. However, these policies are designed for traditionally powered systems and are not necessarily stable when the sensor nodes are powered by energy harvesting. In contrast, our goal is to design channel access policies such that all control loops satisfy their control performance requirements and the power consumption satisfies the energy causality constraints imposed by the energy harvesting process. To this end, we use a control performance abstraction which allows us to translate the control performance requirements to successful transmission probabilities of the random access scheme. Under ergodic assumptions on the channel states and energy harvesting process, we propose a simple dynamic threshold scheduling policy which accounts for the channel as well as the battery state of the sensor. Furthermore, the optimal scheduling policies are computed by means of a stochastic dual method. Finally, we numerically verify the behavior of the proposed policies.