Traducciones de Fray Antonio Figueras, O P.

Aware Scheduling

We propose a novel distributed energy-aware scheduling algorithm guaranteeing ultra reliability based on matching theory. Matching theory is a powerful mathe- matical tool that is decentralized [74]. Here, matching theory is used to formally model the tunable drift-plus-penalty optimisation problem in (6.24), as a one-to- many matching game using costs. Thus, the one-to-many matching game models the interactions between a number of CHs desiring to be served by one UAV. Hence, the game considers two disjoint sets of players, i.e., a CH set A and a UAV set

K, where players from each set have ranked preferences over players in the other set, and vice versa. In the context of the studied network, each UAV k ∈ K will be matched to a subset of CHs Skh ⊆ Akh, Akh ⊆ A, while each CH g ∈ Akh will

be matched to a single UAV k ∈ K. The proposed one-to-many matching game is defined as follows:

6.3 One-to-Many Matching for Ultra-Reliable Energy-Aware Scheduling 149

Definition 6.1 Given two disjoint finite sets of UAVs K and aggregators A, the

ultra reliability and energy-aware scheduling algorithm for UAVs in a clustered IoT networks, can be defined as a one-to-many matching function Φ :K ∪ A → K ∪ A

such that for all k ∈ K and g ∈ A: 1. Φ(k) = Skh ⊆ A and |Φ(k)| ≤µk,

2. Φ(g)∈ Kand |Φ(g)| ∈ {0,1}, 3. Φ(g) =k, if and only if g ∈Φ(k).

From Definition 6.1, the first condition states that each UAVk can be matched to a subset of CHsSkh, withµk being the quota of UAV k, which is the maximum

number of CHs that UAV k can serve, and thus, |Φ(·)| denotes the set cardinality. Depending on the CH subset that UAV k will serve, the quota will not always be the same, and thus, is not fixed across all UAVs and decision epochs. The second condition states that each CHg ∈ Akh can be matched to at most one UAV

k ∈ K, i.e., the quota for each CH is set to 1. The fourth condition states that if UAV k is matched to CH g, then CH g is also matched with UAV k. In the case that Φ(g) = g, then CHg is matched to itself, which means that CH g is not assigned/matched to any UAV.

6.3.1

Utility Functions and Preference Relations of UAVs

and CHs

The one-to-many matching game can be defined by G _{= {K,A,≻k,≻g}, where}

≻k and ≻g refer to the preference relations for UAVs and CHs, respectively. A

preference relation a defined as a complete, reflexive and transitive binary relation between the players in the opposite set. Thus, the preference relation for a UAV

≻k depends on the subset of CHs it wishes to serve, and the queue length of each

CH to be served. UAVk ∈ KservingS is preferred to servingS′_{, which is denoted}

as S ≻k S′. On the other hand, a preference relation for a CH ≻g depends on the

amount of transmission power required to send the data to a UAV, as well as the duration of dwelling time the UAV will collect the data. CH g prefers to be served by UAV k over UAV k′_{, which is defined as k ≻}

g k′. To derive the preference

relations for the two disjoint sets of players, we propose individual utility functions for each set based on the optimisation problem in (6.24).

150 Utilizing UAVs in Mission Critical M2M Communications

To enable ultra reliable communications in an IoT network, each UAVk aims to select a subset of CHsSkh ⊆ Akh that will maximise the number of packets sent

from each CHg ∈ Skh to UAVk, while also guaranteeing that the maximum queue

length of all CHs are finitely bounded. The proposed utility function for UAVs is a linear weighted function, where the weights are the queue lengths of the served CHs, and the virtual queue lengths. Given matching Φ and the transmit power vector ph of all CHs, we define the utility function of UAV k serving a subset of

CHs Skh at decision epoch h, as follows:

Uk(Skh,Φ) =

X

g∈Skh

qghdgk,h−z1hQh−z2hQ2h, (6.33)

where z1h and z2h are given by (6.20) and (6.21) respectively; and dgk,h is given

by (6.2) which is a function of the dwelling time Tgk,h and the CH transmit power

pgh. In (6.33), as the queue length of CHg increases, the UAV serving this CH will

allocate more dwelling time to this CH, in order to reduce its queue length and guarantee rate stability and ultra reliability.

Meanwhile, each CH g aims to maximise its transmission packet rate to UAV

k, while minimising its total energy consumption. Given matching Φ we define the utility function of CH g ∈ Sk,h being served by UAV k at decision epoch h, as

follows: Ug(k,Φ) = X t∈Tgk,h  log2 1 + pghβξgk,t Bσ2 Dg −vpgh  τ. (6.34)

Based on the utility function in (6.34), each CH g aims to minimise its trans- mission power based on the following optimisation problem:

max pgh X t∈Tgk,h  log2 1 + pghβξgk,t Bσ2 Dg −vpgh  τ, (6.35) Subjected to: Rth ≤log2 1 + pghβξgk,t Bσ2 ! , t∈ Tgk,h, (6.36) 0≤pgh ≤Pmax. (6.37)

6.3 One-to-Many Matching for Ultra-Reliable Energy-Aware Scheduling 151

and CHs at decision epoch h is given by:

Skh ≻kSkh′ ⇔ Uk(Skh,Φ) > Uk(Skh′ ,Φ), (6.38)

k≻g k′ ⇔ Ug(k,Φ) > Ug(k′,Φ), (6.39)

for all Skh,Skh′ ⊆ A and k, k′ ∈ K. The proposed matching game considered strict

preference relations for each UAV and CH. Thus, from (6.38) each UAV ranks a subset of CHs by giving a preference to the set of CHs that provides the highest utility. Moreover, each CH can use (6.39) to rank the UAVs by giving a preference to a UAV that provides the highest utility.

6.3.2

Stable Matching Analysis of Distributed Ultra Reli-

able and Energy-Aware Scheduling Algorithm

To solve the proposed problem in (6.24) as a one-to-many matching game, we use the concept of two-sided stable matching to find a suitable matching between UAVs and CHs [203]. A two-sided stable matching for the proposed game is defined as [203]:

Definition 6.2 A UAV and CH pair (k, g)6∈Φ is said to be a blocking pair of the

matching Φ, if and only if k ≻g Φ(g) and g ≻k Φ(k). A matching Φ is stable, if

there is no blocking pair.

Hence, under a stable matching Φ, the set of UAVs can guarantee that the CH queue lengths will be bounded by a finite value, that is, satisfying the ultra-reliable and low-latency constraints for mission critical IoT communications. In order to solve a one-to-many matching problem, most of the prior works [75,204] have relied on the so-called deferred acceptance (DA) algorithm [203, 205]. However, in our proposed one-to-many matching game, we cannot directly apply the DA algorithm. This is due to the UAV utility being a function of the maximum queue length across all CHs in the network Qh, i.e., Qh is a function of all CH queues, and thus, is

a dependant variable. Hence, each UAV not only considers which set of CHs to match with, but also that the set of CHs satisfy their ultra reliability constraint. In addition, the quota of each UAV changes in each decision epoch, where the DA algorithm assumes a fixed quota all the time. Therefore, our proposed one- to-many matching game can be defined as a one-to-many matching game with

152 Utilizing UAVs in Mission Critical M2M Communications

externalities [153, 206–210]. Due to the externalities in the proposed matching game, the DA algorithm cannot be used to converge to a two-sided stable matching. Hence, we propose a new matching algorithm to solve the one-to-many matching game with externalities, and use the concept oftwo-sided-stable matching as defined in Definition 6.2. At the stable convergence of the one-to-many matching game with externalities, no UAV or CH will have incentive to change from their current matching, without causing a decrease in their utility function, respectively, i.e., a network-wide stability.