Time and Energy Constrained Large-Scale IoT Networks

Time and Energy Constrained Large- Scale IoT Networks: The Feedback Dilemma

Item Type Preprint

Authors Emara, Mostafa Lotfy;Kouzayha, Nour Hicham;ElSawy, Hesham;Al-Naffouri, Tareq Y.

Eprint version Pre-print

Publisher arXiv

Rights This is a preprint version of a paper and has not been peer reviewed. Archived with thanks to arXiv.

Download date 2024-01-22 19:43:29

Link to Item http://hdl.handle.net/10754/691090

arXiv:2304.04232v1 [cs.IT] 9 Apr 2023

Time and Energy Constrained Large-Scale IoT Networks: The Feedback Dilemma

Mostafa Emara,Member, IEEE,Nour Kouzayha, Member, IEEE,Hesham ElSawy,Senior Member, IEEE, and Tareq Y. Al-Naffouri,Senior Member, IEEE

Abstract—Closed-loop rate adaptation and error-control de- pends on the availability of feedback, which is necessary to maintain efficient and reliable wireless links. In the 6G era, many Internet of Things (IoT) devices may not be able to support feedback transmissions due to stringent energy constraints. This calls for new transmission techniques and design paradigms to maintain reliability in feedback-free IoT networks. In this context, this paper proposes a novel open-loop rate adapta- tion (OLRA) scheme for reliable feedback-free IoT networks.

In particular, large packets are fragmented to operate at a reliable transmission rate. Furthermore, transmission of each fragment is repeated several times to improve the probability of successful delivery. Using tools from stochastic geometry and queueing theory, we develop a novel spatiotemporal framework to determine the number of fragments and repetitions needed to optimize the network performance in terms of transmission reliability and latency. To this end, the proposed OLRA is bench- marked against conventional closed-loop rate adaptation (CLRA) to highlight the impact of feedback in large-scale IoT networks.

The obtained results concretely quantify the energy saving of the proposed feedback-free OLRA scheme at the cost of transmission reliability reduction and latency increment.

Keywords—Rate adaptation, Stochastic geometry, Spatio- temporal analysis, Markov chains, Open-loop and closed-loop control.

I. INTRODUCTION

The fifth generation (5G) and beyond wireless systems are foreseen to support massive Internet of Things (IoT) deploy- ments [1], [2]. There is currently unprecedented proliferation of IoT devices, which are expected to exceed 5 billions by 2025 to enable ubiquitous monitoring and smart automation of industrial systems, smart grids, precision agriculture, intelli- gent transportation, remote healthcare, and public safety verti- cals [3]. The heterogeneity of the IoT use-cases along with the massive numbers of IoT devices call for novel transmissions techniques to accommodate such unrelenting traffic demand while satisfying the diverse quality of service (e.g., reliability, latency, energy consumption, etc.) constraints [4], [5].

Energy conservation is among the fundamental challenges for the operation of IoT networks. Due to the massive numbers of IoT devices, it is overwhelming to monitor, recharge, and/or replace their batteries [6], [7]. Being a major source of energy consumption, wireless transmissions should be minimized

M. Emara, N. Kouzayha, and T. Y. Al-Naffouri are with King Abdullah University of Science and Technology, Thuwal, Saudi Arabia (e-mail: mostafa.emara@kaust.edu.sa; nour.kouzayha@kaust.edu.sa;

tareq.alnaffouri@kaust.edu.sa).

H. ElSawy is with the School of Computing, Queen’s University, Ontario, Canada (e-mail: hesham.elsawy@queensu.ca).

where applicable to conserve the scarce energy resource. In this context, eliminating the feedback channel is a sought solution to conserve the energy of IoT receivers. Transmission acknowledgments are examples of wireless feedback overhead sent by receivers to notify transmitters about the status of packet reception. The acknowledgments are not only utilized to re-transmit failed packets but also to create a closed-loop control scheme to adapt transmission rate and power [8]. The fundamental role of feedback in wireless systems calls for innovative solutions to counter the impact of its absence, which is important to balance the trade-off between energy conserva- tion and IoT network performance in terms of reliability and latency.

Packet fragmentation and packet repetition are two viable solutions that can be utilized to maintain reliability in the absence of feedback channel. In a time slotted systems, large packets should be fragmented to operate at a reliable rate, where one fragment is transmitted in each time slot. The work in [9], [10] shows that the fragment size should be care- fully adjusted to balance the trade-off between transmission reliability and latency. On the other hand, packet repetition capitalizes on the temporal diversity to improve reliability, where the transmitter sends multiple copies of the same packet over different time slots to increase the chances of successful delivery [11], [12]. The number of repetitions is also a critical design parameter to balance the trade-off between transmission reliability and latency.

Utilizing both packet fragmentation and repetition, this paper proposes a novel open-loop rate adaptation (OLRA) scheme for reliable transmissions in feedback-free large-scale IoT networks. To design and assess the performance of the proposed OLRA, we develop a novel spatiotemporal mathe- matical framework using tools from stochastic geometry and queueing theory. To this end, two variations of the OLRA are proposed and compared to the conventional closed-loop rate adaptation (CLRA) benchmark. Hence, the impact of feedback presence/absence is investigated and the trade-off between transmission reliability, latency, and energy consumption is characterized.

A. Related Work

Developing mathematical frameworks to analyze the perfor- mance of large-scale time-slotted IoT networks is an attractive research topic that has been widely studied in recent years [7], [13]–[17]. Spatiotemporal models are considered to jointly account for the massive spatial existence of IoT devices and

the temporal traffic flow per device [14], [18]–[20]. From the temporal perspective, the queuing theory is utilized to conduct the microscopic analysis that accounts for the packets departure/arrival and the devices activities. From the spatial perspective, the macroscopic analysis relies on stochastic geometry that models the aggregate interference among active devices operating on the same channel due to the shared nature of the wireless channel [13].

The recent developed spatio-temporal models have gained popularity in characterizing the transmission reliability in terms of packet successful delivery probability, latency, scal- ability, and stability of large-scale IoT networks [19], [21]–

[27]. For instance, the stability and the scalability of the random access in IoT networks are characterized in [19], [22]–

[25]. The transmission latency of a downlink IoT network is evaluated in [26], [27]. A spatiotemporal analysis of an uplink network is conducted in [21]. One of the main shortages of the spatiotemporal models developed in [14], [19], [21]–[27] is the fact that, these models ignore the packet deadline which is a critical parameter for time-constrained IoT networks. On the other side, the spatiotemporal models in [28]–[30] characterize the IoT network with hard-packet deadlines while considering asynchronous periodic traffic in [28], multi-cast traffic in [29]

and multi-stream traffic in [30], respectively. However, non of the spatiotemporal models in [14], [19], [21]–[28], [28]–[30]

applies packet fragmentation nor time diversity (i.e., packet repetition) techniques to improve the transmission reliability of the IoT network.

Among the developed spatiotemporal models that charac- terize the IoT network, specific interest is devoted to highlight the gains and trade-offs of the rate adaptation and repetition techniques in terms of extended coverage and increased la- tency. For instance, the spatio-temporal model proposed in [9]

highlights the effect of static and dynamic rate adaptation in IoT networks. The works in [31], [32] characterize the delay and reliability in ultra-reliable and low-latency (URLLC) IoT networks with time diversity. However, the packet deadline constraint is ignored in [9], [31], [32]. Furthermore, the authors apply the frame repetition technique in [33]–[35] to improve the IoT coverage.

Apart from temporal analysis, a spatial framework con- sidering hard-packet deadlines, is developed to evaluate the average performance of frame repetition techniques in ter- restrial IoT-cellular networks [33], [34] and IoT-over-satellite networks [35]. To sum up, none of the available studies in the literature have developed a spatiotemporal analytical framework for large-scale IoT networks with rate adaptation and repetition taking into account time and energy constraints which is the main focus of this paper.

B. Contributions and Organization

This article provides a spatiotemporal framework that uti- lizes stochastic geometry and queuing theory to characterize the transmission reliability and latency trade-off of the OLRA and CLRA schemes in time and energy-constrained large-scale IoT networks. Moreover, the impacts of closed-loop feedback and open-loop feedback-free are also captured. An absorb- ing Markov chain (MC) is utilized to capture the temporal

dimension and model the traffic flow of the IoT network with packet deadline constraint. To the best of the authors’

knowledge, this is the first work that provides a spatiotemporal mathematical analysis of large-scale IoT networks with rate adaptation, repetition, and feedback. The main contributions of this paper are summarized as follows.

• We develop a novel spatiotemporal model to characterize the OLRA scheme in which the packet is divided into equal-size fragments. Due to the absence of feedback, packet fragments are sent multiple times irrespective of the decoding process status at the test receiver. Two different schemes, namely, OLRA with fixed rate (OLRA- FR) and OLRA with variable rate (OLRA-VR) are con- sidered.

• A spatiotemporal model is developed for the benchmark CLRA scheme in which feedback messages are sent from the receiver to its paired transmitter to control its data queue, which is important to quantify the impact of feedback.

• Using the matrix analytical method (MAM) [36], the transmission reliability defined by the packet success delivery probability, the transmission latency, and the energy consumed by the receiver in packet decoding are mathematically expressed.

• The trade-off between reliability, latency, and energy consumption is addressed. Moreover, the effect of feed- back presence/absence on the system performance is investigated.

The rest of this paper is organized as follows. Section II introduces the system model. The CLRA, OLRA-FR, and OLRA-VR transmission schemes are defined in Section III.

Section IV presents the temporal, spatial, and performance metrics analysis for the considered transmission schemes.

Section V explains the numerical results. Finally, the paper is concluded in Section VI.

II. SYSTEMMODEL

A. Spatial Parameters

We focus on a pair of IoT transmitter/receiver (Tx/Rx) devices separated by a distance Ro. The impact of other coexisting IoT devices is captured via a heterogeneous Poisson field (HPF) of interferers. An arbitrary realization of the HPF can be modeled by a marked Poisson process (Ψ,V). The locations of the interfering IoT devices are abstracted by a Poisson point process (PPP)Ψof intensityλ. A set of marks V = {1,2,· · ·, V} of an arbitrary density function fv(v), independent of the devices locations, are used to reflect the different types of the coexisting IoT devices. Each device of markv∈Vhas a transmission powerpvand an activity factor αv. Hence, the IoT device is active and can interfere with the intended transmission with probabilityαv and is idle with probability1−αv. The locations and types of the interfering devices are assumed static once realized due to the short time slot assumption that prevents tangible changes in the locations or types of devices.

Without loss of generality, a test receiver located at the origin is considered to analyze the network performance. The

(a) Network snapshot

Tx1 Rx1

h

Data

S Data

dn=P{SIR> βn}

(b) Individual Tx/Rx activity

Tx Rx

c ba ✄❈

d¯n

t= 1

Tx Rx

c ba dn

t= 2

Tx Rx

cb dn

t= 3

Tx Rx

c dn

t= 4

NACK

ACK

ACK

X

(c) CLRA transmission scheme

Tx Rx

a1 a2 a3 b1 b2 b3

... ✄❈

d¯n t= 1

Tx Rx

a2 a3 b1 b2 b3

... dn

t= 2

Tx Rx

a3 b1 b2 b3 t= 3 ...

Sleep-mode

Tx Rx

On-mode b1

b2 b3 c1

... dn

t= 4

Tx Rx

b2 b3 c1 c2 t= 5 ...

Sleep-mode

Tx Rx

b3 c1 c2 c3 t= 6

Sleep-mode

Tx Rx

c1 c2

c3 ✄❈

d¯n t= 7

On-mode

Tx Rx

c2

c3 ✄❈

d¯n t= 8

Tx Rx

c3

dn

t= 9 X

(d) OLRA transmission scheme

Fig. 1: (a) Snapshot of the network. Nodes represent transmitters, squares denote receivers, and dashed lines represent Tx/Rx links. The test Tx/Rx is depicted in red color while the IoT interferers are in blue color. (b) Tx/Rx activity. Transmitters have buffers for the packet fragments. The fragment is successfully delivered with probabilitydn. (c) CLRA transmission scheme in the case of packet successful delivery. (d) OLRA transmission scheme in the case of packet successful delivery (packet of n= 3fragments, indexed by{a,b,c}and fragment repetition of 3times, indexed by {x1, x2, x3} forx∈ {a,b,c}).

power-law path loss model is assumed in which the power of transmitted signals decays with the distancerat the rater^−η, where η > 2 is the path-loss exponent. Moreover, Rayleigh fading channels of unit mean power fading is considered, which is independent of different locations and types of IoT devices.

B. Temporal Parameters and Main Performance Metrics A time-slotted system is considered with deterministic traf- fic arrival, where a packet is generated every T time slots.

Transmitters with non-empty buffers are required to send packets of length L bits with transmission rate Rn. Hence packets are divided into n ≤ T equal fragments of length

⌈_nT^L

s⌉, where the transmission and decoding of a fragment are occurred within a single time slot of duration Ts. The transmission rate Rn can be expressed as

Rn= L

nTs =Wlog₂(1 +βn), (1) where W is the frequency bandwidth and βn = 2^RnW −1 = 2^{nW TsL} −1 is the signal to interference ratio (SIR) detection

threshold required to correctly decode the fragment at the in- tended receiver. A packet due-time ofT time slots is assumed to represent hard transmission deadlines. Such assumption is convenient for time-constrained IoT applications that require fresh updates or measurements. The packet success delivery (PSD) is achieved by correctly decoding all fragments within the packet deadlineT. A fragment is successfully delivered by the test receiver if the received SIR is larger than the detection threshold βn. Thus, the fragment success delivery (FSD) probability, denoted bydn, is defined asdn =P{SIR≥βn}.

According to (1), dividing the packet into more fragments enhances the transmission reliability as it leads to a lower detection threshold βn that is more likely to be satisfied.

This, however, expands the packet transmission over multiple time slots and may increase the overall decoding latency.

In this work, we adopt the PSD probability and the PSD mean latencyas the main performance metrics to assess and compare the different proposed schemes. Moreover, theenergy consumptionof the IoT receiver during the decoding process is also considered to highlight the impact of feedback and fragment repetition on energy-constrained IoT networks.

III. TRANSMISSIONSCHEMES

To enhance the PSD probability of the IoT network, we consider a fragmentation and repetition mechanism, according to which each fragment is transmitted several times. The repe- tition mechanism is defined based on the adopted transmission policy and whether the feedback exists or not. In this paper, we consider two different schemes, namely, closed-loop rate adaptation (CLRA) andopen-loop rate adaptation (OLRA).

In the CLRA scheme, a feedback signal is transmitted from the receiver to control fragments retransmissions. In the OLRA scheme, the feedback is absent, and hence, the transmitter de- cides on the number of repetitions. The transmission/decoding procedure for each scheme is explained in the sequel.

1) Closed Loop Rate Adaptation (CLRA) Scheme: In this scheme, the intended receiver attempts to decode the first transmitted fragment. In the case of successful decoding, the receiver sends an acknowledgment (ACK) message via an error-free feedback channel to the transmitter to drop this fragment from its queue and send the subsequent fragment in the next time slot. Otherwise, the receiver sends a negative ACK (NACK) asking for the retransmission of the fragment.

The transmitter in the NACK case keeps the same fragment at the head of its queue and persists in sending it until receiving the ACK from the receiver or reaching the maximum number allowed for retransmissions. The PSD and failure events of the CLRA scheme are defined as follows.

Definition 1 (CLRA PSD). A packet sent at rate Rn is successfully delivered if the receiver correctly decoded all the n fragments within theT time-slots period.

Definition 2 (CLRA Packet Delivery Failure). At any instant t, if the remaining time slots are insufficient to complete the transmission of the pending fragments related to the same packet, a CLRA failure event occurs. To save energy, a sleeping trigger signal is sent by the receiver to the transmitter to drop the packet and both of them switch to sleep mode until the next packet generation.

2) Open Loop Rate Adaptation (OLRA) Scheme: In this scheme, the transmitter sends several copies of the same packet fragments regardless of the decoding status at the paired receiver. For the sake of maximizing the PSD, we assume that the transmitter exploits all the available T time slots for fragments repetition¹. Specifically, each fragment is sent κ=⌊T /n⌋times, where nis the total number of fragments.

Since κ has to be an integer, we differentiate between two OLRA schemes, namely, OLRA with fixed-rate (OLRA-FR) andOLRA with variable-rate (OLRA-VR), based on how the remaining τ = mod (T, n) time slots are exploited. In the OLRA-FR scheme, the transmitter randomly selects τ < n fragments to be sent one more time each. Therefore, each of the selected fragments is sentκ+ 1times while the others are repeatedκtimes. As all the fragments are sent with the same transmission rateRn, the OLRA-FR scheme is described as a fixed rate OLRA scheme. On the other hand, the OLRA-VR scheme retransmits the packet with a new transmission rate

1This will impose more energy burden on the transmitter, but relieve the receiver from feedback.

Rτ > Rn in the remaining time slots τ. Thus, the packet is divided into τ fragments and retransmitted again. The PSD and failure events of the OLRA schemes can be defined as follows.

Definition 3 (OLRA PSD). A packet sent at a transmission rateRn is successfully delivered if at least one copy of each fragment is correctly decoded within the packet deadlineT. Definition 4 (OLRA Packet Delivery Failure). In the case of decoding failure for all copies of any fragment, an OLRA failure event occurs. To save energy, the receiver stops decod- ing the subsequent fragments and switches to sleep mode to conserve energy. It worth mentioning that the transmitter does not switch to sleep mode since it is unaware of the receiver status.

Fig. 1 depicts a network snapshot in which nodes represent the IoT transmitters, squares denote the receivers, and dashed lines are for IoT Tx/Rx links. Transmitters are equipped with buffers to store packets fragments. A fragment is successfully decoded with probabilitydn that depends on the interference experienced at the test receiver. The first input first out (FIFO) discipline is assumed for packet service. Fig. 1 also offers a pictorial illustration of the transmission policies for CLRA and OLRA schemes in the case of packet successful delivery. The figure shows a packet ofn= 3fragments, indexed by{a,b,c}

and fragment repetition of 3 times, indexed by {x1, x2, x3} forx∈ {a,b,c}.

The CLRA and OLRA schemes can be modeled using discrete-time absorbing Markov chains (MC) with two absorb- ing states, namely success and timeout (failure) states. The absorbing MC is fully characterized by the transition matrix P that tracks the decoding attempts of the fragments of a packet until it is eventually absorbed to either the success or failure states. The transition matrix P depends on the FSD probabilitydn. The FSD of packet fragments varies according to the realization of the HPF of interfering devices, which is assumed fixed once realized. To account for the different IoT network realizations, we consider the meta distribution of the FSD probability. We then group the realizations that would lead to an FSD probability of a range ±_2M¹ into the same class, denoted hereafter as FSD class, whereM is determined based on the needed accuracy. Thus, dn,m denotes the FSD probability of a fragment sent from a device that belongs to the mth FSD class with a transmission rate Rn. Given dn,m∀m, the transition matrix P^(m)for each FSD class can be formulated for the CLRA and OLRA schemes.

IV. ANALYSIS

This section develops the mathematical frameworks to eval- uate the performance of the CLRA, OLRA-FR, and OLRA- VR schemes. We start with the temporal analysis of the absorbing MC that characterizes each scheme. Hence, we construct the transition matrix P^(m) for a device belonging to the mth FSD class in terms of the successful decoding probability dn,m for the different considered transmission schemes. Using the matrix analytical method (MAM) [36], the PSD probability, average decoding latency, and the receiver

energy consumptionare mathematically obtained. Finally, the SIR meta distribution of the FSD is obtained for the different M classes to characterize the macroscopic network spatial analysis and provide an expression for the FSD probability dn,m,m∈ {1,2, ...M}.

A. Temporal Analysis

As previously mentioned, the considered schemes can be modeled by a discrete-time absorbing MC to track the frag- ments decoding attempts until reaching the final state where the packet is either successfully delivered or discarded due to elapsed deadline. Consider a device belonging to themth FSD class that transmits a packet with rate Rn, its corresponding MC can be mathematically represented with the transition matrix P^(m) formulated as [36, Section 3.6]

P^(m)=

Q^(m) H^(m)

0 I

=























Q^(m)₁ 0 0 . . . 0 H^(m)₁ 0 Q^(m)₂ 0 . . . 0 H^(m)₂

... . .. ... ... ... ... 0 0 0 . . . Q^(m)_T−2 H^(m)_T−2 0 0 0 . . . 0 H^(m)_T−1

0 0 0 . . . 0 I





















 , (2)

where Q^(m) denotes the transient matrix describing the at- tempts handled by the test receiver to decode the transmitted fragments before absorption andH^(m)denotes the absorbing matrix and captures the probability that the packet is either successfully delivered or discarded due to elapsed deadline.I is an identity matrix of size2×2representing the success and failure absorbing states. The rows in (2) depict the progressing time evolution of the packet until absorption. The matrixQ^(m)_t represents the transition between time-slotstandt+ 1.H^(m)_t captures the absorption probability at time slot t. Next, we focus on formulating the matrix P˜^(m) presented in (2) and consisting of onlyQ^(m) andH^(m)for the CLRA and OLRA schemes. Without notation abuse, we drop the subscripts in dn,m andd¯n,m= 1−dn,m in the hereafter matrices.

1) Temporal Analysis of the CLRA Scheme: To facilitate the exposition of the transition matrix P˜^(m)_CLRA of the CLRA scheme, we first present an illustrative example for the trans- mission of a packet with n = 3 fragments and due-time T = 8 time slots. We then generalize P˜^(m)_CLRA for different transmission rates and due-times. Fig. 2 depicts the absorbing MC of the considered illustrative example. The three fragments of the packet are denoted as {a, b, c} and are retransmitted several times according to the feedback of the receiver. Fig. 2 starts with an idle state to represent the empty buffer at the test transmitter at instant t= 0. Att= 1, fragmenta is sent and its first decoding attempt is handled at the test receiver. This is represented by a transition with probability1 between time slots t= 0 andt= 1. The decoding attempt either succeeds with probability dm,n or fails with probabilityd¯m,n. If frag- ment ais successfully decoded, the transmitter drops it from

the queue and switches to transmit fragmentbin the next time slot. Otherwise, the transmitter persists in sending fragmenta during the subsequent time slots. The retransmission/decoding trials of fragmenta proceed until it is successfully delivered or the maximum number of retransmissions is reached. If the later event happens, the packet is discarded. Note that a packet is discarded at instant t if the remaining (T −t) time slots are insufficient to transmit all the pending fragments. When this happens, both transmitter and receiver switch to sleeping mode until the next packet generation. The same strategy is followed for the other fragments until the CLRA PSD event, defined in Definition 1, occurs. According to this policy, the instantt= 2carries either the2^nd decoding trial for fragment aafter a single failure or the1^stattempt to decode fragmentb after the successful delivery of fragmenta. Then,t= 3places one of the following4 possibilities:

• The3rd trial to decode fragmentaafter two consecutive failures.

• The 1st trial to decode fragment b after one failure followed by a single success in decoding fragmenta.

• The 2nd decoding attempt of fragment b after the suc- cessful delivery of fragmenta followed by a failure of decoding fragmentb.

• The1st trial to decode fragmentcafter two consecutive successful decoding attempts for fragmentsaandb.

It is worth noting that at least n time slots are required for the CLRA packet delivery event to succeed. This is visualized in Fig. 2 by following the diagonal transitions from a to b thenc, where the success event occurs, att= 3. On the other side, the packet discarding event is due to successive failures in decoding the transmitted fragments. This is also manifested in Fig. 2 by the absorption into the timeout state, for example, at t = 6 after 6 successive decoding failures of fragment a.

The detailed structure of the transition matrixP˜^(m)_CLRA for the absorbing MC in Fig. 2 is given by (3).

P˜^(m)_CLRA=h

Q^(m)_CLRA H^(m)_CLRA i

=

a b a b c a b c a b c a b c b c c S t-out



































































































































 t= 1 a d d¯

a d d¯ 0 t= 2 b 0 ¯d d

a d d¯ 0 0 0

t= 3 b 0 ¯d d 0 0

c 0 0 ¯d d 0

a d d¯ 0 0 0

t= 4 b 0 ¯d d 0 0

c 0 0 ¯d d 0

a d d¯ 0 0 0

t= 5 b 0 ¯d d 0 0

c 0 0 ¯d d 0

a d0 0 d¯

t= 6 b d d¯ 0 0

c 0 ¯d d 0

t= 7 b d 0 d¯

c d¯ d 0

t= 8 c 0 d d¯

(3)

The structure of P˜^(m)_CLRA in (3) follows the general transition matrix structure in (2). The transient matrixQ^(m)_CLRAconsists of

0 a a a a a a

b b b b b b

c c c c c c S

t-out

t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 Absorbing

states Idle

state

1 dn,m

d¯n,m

Fig. 2: The absorbing MC of the CLRA scheme. The packet consists ofn= 3fragments, denoted by{a,b,c}, and its due-time T = 8.

the non-zero diagonal submatrices Q^(m)_t , t={1,2,· · ·, T − 2} and zero submatrices of appropriate sizes, which are left blanked. For 1 ≤ t < n, it can be noticed that Q^(m)_t is a fat matrix of dimension t×(t+ 1), thus its size gradually grows with time evolution. Forn≤t < T−n+ 1,Q^(m)_t is a square matrix with unchanged dimension ofn×n. Finally, for T−n+1≤t≤T−2,Q^(m)_t is a thin matrix of dimension(T− t+1)×(T−t), hence it gradually declines with time progress.

We denote byQG^(m)_t ,QU^(m)_t andQD^(m)_t the fat, square and thin Q^(m)_t matrices in the three mentioned time intervals. On the other hand, the absorbing matrix H^(m)_CLRA consists of two- columns submatricesH^(m)_t , t={1,2,· · ·, T −1} to capture the packet absorption into the success and failure absorbing states. It can be shown that the first possibility for a packet to be absorbed into the success state occurs att=nand into the timeout state at t=T−n+ 1, which complies with Fig. 2.

The transition matrix P˜^(m)_CLRA in (3) describing the absorbing MC of the CLRA scheme for n = 3 fragments and packet due-time T = 8. Lemma 1 provides the general form of the transition matrix of the CLRA schemeP˜^(m)_CLRAfor an arbitrary tuple (n, T).

Lemma 1(CLRA Transition Matrix). The transition matrix P˜^(m)_CLRAdescribing the absorbing MC for a generic packet sent from an m-FSD class IoT device consists of the submatrices Q^(m)_t,CLRA andH^(m)_t,CLRA for an arbitrary tuple(n, T), which are given by

Q^(m)_t,CLRA=

















 QG^(m)_t

t×(t+1) 1≤t < n, QU^(m)_t

n×n

n≤t < T −n+ 1, QD^(m)_t

(T−t+1)×(T−t) T−n+ 1≤t≤T−2, (4)

H^(m)_t,CLRA=

























 0

t×2 1≤t < n,

HU^(m)_t

n×2 n≤t < T −n+ 1, HD^(m)_t

(T−t+1)×2

T−n+ 1≤t≤T−2, h

dn,m d¯n,m

i

t=T−1,

(5)

whereX

i×j denotes a matrixX of size(i×j)and the nota-

tions{G, U, D}in{QG^(m)_t ,QU^(m)_t ,QD^(m)_t }and{HU^(m)_t , HD^(m)_t } refer to{growing, unchanged, and declining}-size matrices, respectively. The elements of QG^(m)_t ,QU^(m)_t and QD^(m)_t are given by:

QG^(m)_t (i, j) =QU_t^(m)(i, j) =







d¯n,m, forj=i, dn,m, forj=i+ 1, 0, otherwise,

(6)

QD_t^(m)(i, j) =







dn,m, fori=j, d¯n,m, fori=j+ 1, 0, otherwise.

(7)

Similarly, the elements of HU^(m)_t andHD^(m)_t are given by

HU_t^(m)(i, j) =

(dn,m, fori=n, j= 1

0, otherwise, (8)

HD^(m)_t (i, j) =







dn,m, fori= 1, j= 2

d¯n,m, fori=T−t+ 1, j= 1 0, otherwise.

(9)

2) Temporal Analysis of the OLRA-FR Scheme: The OLRA-FR scheme implements packet repetition with rate adaptation to improve the transmission reliability of feedback- free IoT networks. In this scheme, the packet is divided into n fragments and transmitted with the rate Rn several times.

In particular, given thatκ=⌊^T_n⌋andτ= mod (T, n), each fragment is sentκtimes while the remainingτ time slots are exploited to transmit each of the randomly selected τ < n fragments one more time. In other words, each fragment is sent nrep = κ+¹ǫ times, where ¹ǫ is an indicator function which is equal to 1 if the fragment is sent (κ+ 1) times of probability ^τ_n and 0 if the fragment is sent κ times of probability (1−_n^τ). Consequently, the test receiver has nrep

possible decoding chances for each fragment.

Fig. 3 shows the absorbing MC of OLRA-FR scheme for a given tuple (n= 3,T = 11), thereforeτ = 2 andκ= 3.

For illustration, the figure assumes that fragments{a, c} are selected to be sent κ+ 1 = 4 times while fragment b is sent κ = 3 times. The subscript i in ai denotes the ith transmission/decoding attempt of fragment a. As shown in Fig. 3, the PSD is conditioned by the successful decoding of

0 a₁ a₂ a₃ a₄ b₁ b₂ b₃ c₁ c₂ c₃ c₄

S

t-out

LS LS LS LS LS

t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10 t=11 Absorbing states

Idle state

Fragmentais sentκ+ 1times Fragmentbis sentκtimes Fragmentcis sentκ+ 1times 1

dn,m

d¯n,m

Fig. 3: The absorbing MC of the OLRA-FR scheme. The packet consists of n = 3fragments, denoted as {a, b, c} and the packet deadline T = 11. So,κ=⌊T /n⌋= 3, andτ = mod (T, n) = 2. Hence, each fragment is sentκtimes whileτ = 2 fragments are randomly chosen to be sent one more time. We assume that {a, c} are the selected fragments. The subscript in xi, x∈ {a, b, c} denotes theith decoding attempt.

all thenfragments. For any fragment, if all thenrepdecoding attempts failed, the packet is discarded and the receiver goes into sleeping mode until the reception of the next packet. In contrast, if any decoding attempt of a fragment, except the last fragment, succeeded, the receiver has to wait until receiving the subsequent fragment and follow the same procedure. This waiting is represented by a transition into a success logic state, denoted by LS, in Fig. 3. While for the last fragment, the successful decoding of any attempt directly leads to packet absorption into the success state. Fig. 3 reveals that, in the OLRA-FR scheme, a similar pattern exists for all fragments, except for the last fragment, which should be reflected in the structure of the transition matrix P^(m)_OLRA-FR that characterizes the absorbing MC. Thus, the transition matrix P^(m)_OLRA-FR of the OLRA-FR scheme in Fig. 3 is given by

P˜^(m)_OLRA-FR=h

Q^(m)_OLRA-FR H^(m)_OLRA-FR i

=

a2LS a3LS a4LS b1b2LS b3LS c1c2c3c4 S t-out



























































































































































 t= 1 a₁ d d¯

a2 d d¯ t= 2 LS 0 1

a3 d d¯

t= 3 LS 0 1

a₄ d 0 d¯

t= 4 LS 1 0 0

t= 5 b1 d d¯

b2 d d¯

t= 6 LS 0 1

b3 d 0 d¯

t= 7 LS 1 0 0

t= 8 c₁ d¯ d 0

t= 9 c2 d¯ d 0

t= 10 c2 d¯ d 0

t= 11 c2 d¯d d¯

(10)

where LS in the columns and rows labels denotes the success logic state shown in Fig. 3. The transition matrix P˜^(m)_OLRA-FR in (10) shows similar transient and absorbing matrices for fragments a and b that differ from those of fragment c. The

dashed horizontal lines separate thenrepsubmatrices that track the decoding trials of each fragment. The following lemma provides the general form of the transition matrix of the OLRA-FR schemeP˜^(m)_OLRA-FR for arbitrary tuple (n, T).

Lemma 2 (OLRA-FR Transition Matrix). The transition matrix P˜^(m)_OLRA-FR describing the absorbing MC for a generic packet sent from an m-FSD class IoT device consists of the submatricesQ^(m)_t,OLR-FR and H^(m)_t,OLRA-FR for an arbitrary tuple (n, T), which are given by

Q^(m)_t,OLR-FR=



































hd¯n,m dn,m

i

, t=i nrep+ 1,

"

d¯n,m dn,m

0 1

#

, i nrep+ 2≤t≤(i+ 1)nrep−1,

"

dn,m

1

#

, t= (i+ 1)nrep,

d¯n,m, (n−1)nrep+ 1≤t≤T−2, h

dn,m d¯n,m

i

, t=T−1,

(11)

H^(m)_t,OLR-FR=

























 0

2×2, i nrep+ 1≤t≤(i+ 1)nrep−1,

"

0 d¯n,m

0 0

#

, t= (i+ 1)nrep, h

dn,m 0i

, (n−1)nrep< t≤T−1, h

dn,m d¯n,m

i

, t=T−1,

(12) wherei∈[0,1,· · ·, n−2].

3) Temporal Analysis for the OLRA-VR Scheme: The OLRA-VR scheme implements fragment repetition and vari- able rate adaptation. In this scheme, the IoT transmitter sends the packet κ= ⌊^T_n⌋ times with a transmission rate Rn and uses the remainingτ = mod (T, n) time slots to transmit the packet with an incremented rate Rτ > Rn. Therefore, each fragment is sent κ times and the last τ time slots are exploited to re-transmit the packet after dividing it intoτ < n fragments.

Fig. 4 shows the absorbing MC of the OLRA-VR scheme for a given tuple (n = 3 , T = 11) where κ= 3 and τ =

0 a1 a2 a3 b1 b2 b3 c1 c2 c3 f1 f2

S

t-out

LS LS LS LS

LF LF LF LF LF LF

t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10 t=11 Absorbing

states Idle

state

Fragmentais sentκtimes Fragmentbis sentκtimes Fragmentcis sent κtimes

Packet is divided to τ < nfragments 1

d_n,m d¯_n,m dτ,m

d¯_τ,m

Fig. 4: The absorbing MC for OLRA-VR scheme. The packet consists of n= 3 fragments, denoted as{a, b, c}, and packet deadlineT = 11. Thus,κ=⌊T /n⌋= 3andτ= mod (T, n) = 2. Hence, each fragment is sentκtimes with rateRn =R3

while the packet is resent one-more time within the τ residue time slots with rateRτ =R2 (i.e., packet is segmented into 2 fragments, denoted by {f1, f2}).

2. Hence, the test transmitter divides the packet into n = 3 equal-size fragments, denoted by{a, b, c}and each fragment is transmittedκ= 3times with rateR3. To exploit the remaining slots, the packet is divided into τ = 2 fragments, denoted by {f1, f2}, and re-transmitted with a transmission rateR2> R3

during the last τ = 2time slots. For any fragment except the last one, the receiver tries to decode it from its received κ copies. If a fragment is successfully decoded, the receiver has to wait until the reception of all the subsequent fragments. The waiting event is captured by a success logic state, denoted by LS. The successful decoding of any copies of the last fragment directly leads to the packet successful delivery. When all the attempts of a fragment fail, the receiver waits until the packet is re-transmitted in the lastτtime slots with the increased rate Rτ> Rn. The waiting for retransmission is represented by a transition into a failure logic state, denoted by LF. Within the lastτtime slots, the failure in decoding any of theτfragments leads to packet delivery failure, and the PSD is conditioned on the successful decoding of all theτ fragments.

The transition matrix that characterizes the absorbing MC of the OLRA-VR scheme given in Fig. 4 for a device that belongs to the mth FSD class, is given in (13) at the top of the next page. Note that dn anddτ in (13) represent the FSD probabilities dn,m and dτ,m of a device that belongs to the m-th class receiving a packet with rates Rn andRτ, respectively. The transition matrix in (13) shows a similar decoding pattern for fragments j ∈ {2,3,· · · , n−1}, which is different from the patterns of the first fragment, the last fragment before re-transmission, and the fragments transmitted in the last τ time slots. The dashed horizontal lines in (13) separate the submatrices that track the decoding attempts of each fragment. Consequently, the general form of the transition matrix P˜^(m)_OLRA-VR characterizing the absorbing MC of the OLRA-VR scheme at an arbitrary tuple(n, T)can be provided in the following Lemma.

Lemma 3 (OLRA-VR Transition Matrix.). The transition matrixP˜^(m)_OLRA-VRdescribing the absorbing MC for an arbitrary tuple(n, T)is defined by the transient matrixQ^m_t,OLRA-VR, and the absorbing matrixH^(m)_t,OLRA-VRgiven in (14) and (15) at the bottom of next page.

B. Performance Metric Analysis

In this section, the main performance metrics of the CLRA, OLRA-FR, and OLRA-VR schemes are formulated given the transition matrices provided in Lemmas 1, 2, and 3. By referring to the MAM, PSD probability and mean latency for a device in themth FSD class are expressed in the following theorem.

Theorem 1. Let the vector A^(m)∈ {A^(m)s , A^(m)_f }define the probability that a generic packet of a device that belongs to the mth FSD class is eventually absorbed into the success or timeout (failure) states, respectively, whereA^(m)_f = 1−A^(m)s . In addition, letD^(m)∈ {Ds^(m), D_f^(m)} denote a scaled mean latency (delay) to absorption for a device in the mth FSD class, where ^D

(m) s

A^(m)_s and ^D

(m) f

A^(m)_f are, respectively, the packet mean latency, in time slots, for absorption into success and timeout states. Thus,A^(m) andD^(m)can be defined as [36]

A^(m)=H^(m)₁ +

T−2

X

i=2 i−1

Y

t=1

Q^(m)_t

!

×H_i^(m), (16)

D^(m)=H^(m)₁ +

T−2

X

i=2 i−1

Y

t=1

iQ^(m)_t

!

×H_i^(m). (17)

whereH_i^(m)andQ^(m)_t are the

Proof:According to (2), the transition probability within the decoding attempts before the absorption into a final state is captured by the transient matrix Q^(m) and the probability of absorption is captured by H^(m). Hence, the probability that the receiver handles t consecutive decoding attempts is given byQt

i=1Q^(m)_i . Similarly, the probability that a generic packet is absorbed after exactly t time slots is given by Qt−1

i=1Q^(m)_i H^(m)_t . Therefore, by applying the law of total probability and accounting for theT- time slot packet deadline, Theorem 1 is proved.

The overall energy consumption by a generic test receiver that belongs to an m-FSD class in packet delivery depends on the adopted transmission policy whether OLRA or CLRA schemes. In the OLRA scheme, the test receiver consumes energy in packet fragments reception and decoding. While for CLRA, the receiver consumes extra energy in feedback

P˜^(m)_OLRA-VR= h

Q^(m)_OLRA-VR H^(m)_OLRA-VR i

=

a₂ LS a₃ LS LF b₁ LF b₂ LS LF b₃ LS LF c₁ LF c₂ LF c₃ f₁ f₂ S t-out







































































































































































 t= 1 a1 d¯n dn

a2 d¯ndn

t= 2 LS 0 1

a₃ d¯n dn

t= 3 LS 0 1

LF 1 0 0

t= 4 b1 0 d¯ndn

LF 1 0 0

t= 5 b2 0 d¯ndn

LS 0 0 1

LF 1 0

t= 6 b3 d¯n dn

LS 0 1

LF 1 0 0 0

t= 7 c1 0 d¯n dn 0

LF 1 0 0 0

t= 8 c2 0 d¯n dn 0

LF 1 0 0

t= 9 c3 d¯n dn 0

t= 10 f1 dτ 0 d¯τ

t= 11 f2 0 dτ d¯τ

(13)

ACK/NACK signaling transmission. Let Er andEackdenote, respectively, the energy consumption by the test receiver in a fragment reception and decoding, and feedback acknowledg- ment, which are given by [37], [38]

Er=pcrTs,

Eack= (ηpt+Pct)Tack, (18) where pcr is the RF circuit power consumption at the re- ceiver, consisting of the mixer, frequency synthesizer, low noise amplifier (LNA), active filter, intermediate frequency amplifier (IFA), and the analogue to digital converter (ADC) andPct denotes the RF circuit power consumption of the test receiver’s transceiver in transmitting the ACK/NACK feedback messages.Pctincludes the power consumption of the digital to analogue converter (DAC), active filter, frequency synthesizer, mixer, and IFA. Finally, η is the power amplifier conversion factor of value at least 1. Tack is the the feedback message duration, which is significantly shorter than the fragment durationTsowing to the short ACK/NACK messages. Having the energy consumption in a fragment reception/decoding and acknowledgement, the total energy consumption in packet

delivery depends on the mean latency for packet absorption, whether to success or time-out states. The total energy con- sumption by a generic receiver that belongs to anm-SDF class is given by the following Corollary.

Corollary 1. The overall energy consumption by anm-FSD class device for the CLRA and OLRA transmission schemes is given by

E^(m)_OLRA=Er

D_s,OLRA^(m) +D_f,OLRA^(m)

, (19)

E_CLRA^(m) = (Er+Eack)

D_s,CLRA^(m) +D^(m)_f,CLRA

. (20) Proof:According to the OLRA transmission scheme, the test receiver consumes energy in packet fragments reception and decoding. While for the CLRA scheme, the feedback ACK/NACK signaling consumes extra energy. Therefore, the energy consumption of the OLRA scheme in a fragment delivery isEr while for CLRA is(Er+Eack). Consequently, the overall energy consumption is obtained by multiplying the energy consumption in a single attempt by the weighted number of time slots the receiver spends in packet delivery

Q^(m)₁ =d¯n,m dn,m

, Q^(m)_2≤t≤κ=

d¯n,m dn,m

0 1

, Q^(m)_t=iκ+1=

1 0 0 0 dn,m d¯n,m

, Q^(m)iκ+2≤t<(i+1)κ=





1 0 0

0 1 0

0 dn,m d¯n,m





Q^(m)_t=(i+1)κ=





1 0

0 1

d¯n,m dn,m



, Q^(m)(i+1)κ+1≤t<nκ= 1 0

0 d¯n,m

, Q^(m)_t=nκ= 1

d¯n,m

, Q^(m)nκ+1≤t<T−2 =dτ,m (14)

H^(m)₁ =0

1×2, H^(m)_{2≤t≤(n−1)κ}=0

2×2, H^(m)(n−1)κ+1≤t≤nκ=

0 0 dn,m 0

, H^(m)_{nκ+1≤t≤T−1}= 0 dτ,m

H^(m)_T =

dτ,m d¯τ,m

(15)