As introduced above, this chapter considers a centralized scheduling problem for a wireless network with nodes equipped with HESSs, as illustrated in Figure 5.1. To simplify the discussion, it is assumed that the batteries are non-rechargeable and that they do not suffer from energy leakage in the time scale of interest. Instead, the capacitors are rechargeable and can potentially leak energy. This a reasonable approximation of practical scenarios, since non-rechargeable batteries typically suffer from self-discharge over a time-scale much larger than that of capacitors (see, e.g., [63]). It is also assumed that the EH and leakage processes are independent and identically distributed (i.i.d.) across nodes and time-slots, and that they are modeled as binary random processes. That is, in each slot a node either harvests a unit of energy (to be defined below) or not, and similarly for the leakage process.
Main contributions of this chapter. For the full state information scenario, considered in Section 5.3, the scheduling problem is formulated as the maximization of the network lifetime, and it is shown to reduce to a stochastic shortest path (SSP) problem [64], which is a special instance of a Markov decision process (MDP). Under the assumption that the system is symmetric, so that the statistics of the EH and leakage processes at the nodes are equivalent, optimal scheduling policies are obtained in two limiting cases: a) harvesting-only model, in which the amount of energy leaked is negligible; and b) leakage-only model, in which the amount of energy harvested is negligible. As it will be discussed, these two limiting scenarios are useful approximations of situations in which the capacitors tend to be close to full or close to empty most of the time, respectively.
An optimal policy in the harvesting-only scenario is shown to select in each slot the K nodes with the largest energy stored in their capacitors (when available). Instead, for the leakage-only model, the optimal policy selects in each slot the K nodes with the smallest non-zero energy in their capacitors. An easily computable
performance upper bound on the network lifetime is also proposed for the general scenario, which can be used as a performance reference when the size of the network makes the numerical computation of the optimal policy intractable. It is then shown that, when the FC schedules only one node in each slot (i.e., K = 1), the computation of the network lifetime can be decomposed into separate contributions due to batteries and due to capacitors, and, based on this result, an algorithm that enables the computation of the network lifetime with reduced complexity is proposed.
In the partial state information case, considered in Section 5.4, finding the optimal scheduling policies explicitly is more challenging than in the full state information case, and the numerical computation of optimal policies is generally intractable. Therefore, based on the insights obtained from the analysis of the full state information scenario, two heuristic policies that can be easily implemented in practical systems are proposed. Moreover, to improve on these policies, opportunistic feedback schemes are considered, in which each node with a sufficiently large energy in its capacitor opportunistically provides additional information to the FC over a dedicated transmission resource. It is then shown in the numerical results in Section 5.5 that this limited-feedback approach has the potential to greatly improve the lifetime performance.
Related work: The lifetime of battery-powered wireless networks was studied in [65], where the problem of scheduling a subset of battery-powered nodes in a wireless network, subject to fading channels, was tackled by resorting to a SSP formulation. In [65] the nodes are equipped with non-rechargeable batteries (i.e., they have no EH capabilities), while the transmission power of each node is adapted to the channel quality in each slot. A similar system setting is considered in [66], where the emphasis is instead on the development of distributed access protocols based on the channel state information and the residual energy information at each node. The work in this chapter differs from [65, 66] in that the energy availability at
the nodes keeps changing even when nodes are not scheduled due to EH and energy leakage. However, the impact of fading is not considered here. A relevant reference for HESS systems is [62] (see also references therein), where the problem of routing in wireless networks operated by nodes equipped by a HESS is considered. Reference [62] also provides a review of the properties of batteries and capacitors and their trade-offs.
5.2 System Model
This chapter considers a wireless network in which a FC is tasked with collecting data packets from a set of M nodes, labeled as U1, U2, ..., UM, under the constraint that K packets must be collected in each time-slot (see Figure 5.1 and Figure 5.2). To this end, in each slot t, of duration T , a subset U(t) ⊆ {U1, ..., UM} of |U(t)| = K nodes is selected for transmission. Each node has a new packet to transmit at each slot. It is assumed that nodes’ transmissions take place over orthogonal communication resources (e.g., frequencies) so that they do not interfere with each other. It is also assumed that the total duration of the communication between the nodes and the FC in each slot is fixed and equal to Tc (see Figure 5.2), where Tc is generally assumed to be much smaller than the slot duration T , i.e., Tc ≪ T for reasons that will be clarified below (see Figure 5.2). Moreover, the FC’s scheduling commands and the nodes’ packets are considered to be received without error by the intended destinations.