key: cord-0442072-nm5e5fv3
authors: Abd-Elmagid, Mohamed A.; Dhillon, Harpreet S.
title: Age of Information in Multi-source Updating Systems Powered by Energy Harvesting
date: 2021-09-15
journal: nan
DOI: nan
sha: ef9e7e3e31fce0e35aaeb2978136496dddb1f4a3
doc_id: 442072
cord_uid: nm5e5fv3

This paper considers a multi-source real-time updating system in which an energy harvesting (EH)-powered transmitter node has multiple sources generating status updates about several physical processes. The status updates are then sent to a destination node where the freshness of each status update is measured in terms of Age of Information (AoI). The status updates of each source and harvested energy packets are assumed to arrive at the transmitter according to independent Poisson processes, and the service time of each status update is assumed to be exponentially distributed. Unlike most of the existing queueing-theoretic analyses of AoI that focus on characterizing its average when the transmitter has a reliable energy source and is hence not powered by EH (referred henceforth as a non-EH transmitter), our analysis is focused on understanding the distributional properties of AoI in multi-source systems through the characterization of its moment generating function (MGF). In particular, we use the stochastic hybrid systems (SHS) framework to derive closed-form expressions of the average/MGF of AoI under several queueing disciplines at the transmitter, including non-preemptive and source-agnostic/source-aware preemptive in service strategies. The generality of our results is demonstrated by recovering several existing results as special cases.

including healthcare, factory automation, autonomous vehicles, and smart homes, to name a few [1] . As a concrete example of healthcare applications, large-scale IoT deployments could be useful in containing pandemics through efficient monitoring and contact/infection tracing [2] . The performance of these applications highly depends upon the freshness of the information status at the destination node about its monitored physical process(es). Because of that, the main design objective of such real-time status update systems is to ensure timely delivery of status updates from the transmitter node to the destination node. To measure the freshness of information at the destination node, the authors of [3] introduced the concept of AoI which accounts for the generation time of each status update (which was ignored by conventional performance metrics, specifically throughput and delay). In particular, for a queuing-theoretic model in which status updates are generated at the transmitter node according to a Poisson process, AoI was defined in [3] as the time elapsed since the latest successfully received status update at the destination node was generated at the transmitter node.

As will be discussed next in detail, the queueing-theoretic analyses of AoI have mostly been focused on the characterization of its average in the case of having a non-EH transmitter.

However, it is infeasible to ensure the availability of a reliable energy source at the transmitter node in many practical IoT scenarios. For instance, a transmitter node could represent an aggregator deployed at a hard-to-reach place in a large-scale IoT network, where it is impractical to replace or recharge the energy battery at the aggregator [4] . To enable a sustainable operation of real-time status update systems in such scenarios, EH has been considered as a promising solution for powering the transmitter nodes (majority of which are low-power nodes). While there are a handful of prior works analyzing AoI for the system in which a transmitter node is powered by EH, their analyses have been limited to the evaluation of its average and that too to the special case where the transmitter has a single source that generates status updates about a single physical process. Motivated by this, we provide the first queueing-theoretic analysis of the distributional properties of AoI for a generic setup in which an EH-powered transmitter has multiple sources which generate status updates about multiple physical processes.

For systems in which a non-EH transmitter has a single source that generates status updates about some physical process, referred to as single-source systems, the authors of [3] first derived a closed-form expression of the average AoI under first-come-first-served (FCFS) queueing discipline. The average of AoI or peak AoI (an AoI-related metric introduced in [5] to capture the peak values of AoI over time) is then characterized under several queueing disciplines in a series of subsequent prior works [5] - [11] . Further, a handful of recent works aimed to characterize the distribution (or some distributional properties) of AoI/peak AoI [12] - [17] . While AoI has been extensively analyzed in single-source systems, the prior work on the analysis of AoI in multi-source systems has been fairly limited [18] - [27] . Note that a multi-source system refers to the setup where a non-EH transmitter has multiple sources generating status updates about multiple physical processes. The average AoI was characterized for the M/M/1 FCFS queueing model in [18] , the M/G/1 FCFS queueing model in [19] , and the M/M/1 FCFS with preemption in waiting queueing model (where the transmitter has a buffer that only keeps the latest generated status update from each source) in [20] . The authors of [21] and [22] analyzed the average AoI under scheduled and random multiaccess strategies for delivering the status updates generated from different sources at the transmitter. The average peak AoI was derived for the M/G/1 lastcome-first-served (LCFS) queueing model with (without) preemption in service in [23] (in [24] ), and for the priority FCFS and LCFS queueing models (where the sources of information are prioritized at the transmitter) in [25] . Further, the distributions of AoI and PAoI were numerically characterized for various discrete time queues in [26] , and for a probabilistically preemptive queueing model in [27] where a new arriving status update preempts the one in service with some probability. Different from [5] - [27] , our focus in this paper is on the analytical characterization of distributional properties of AoI in the case where the transmitter has multiple sources of information and is powered by EH.

The analyses of the above works were mainly based on identifying the properties of the AoI sample functions and applying geometric arguments, which often involve convoluted calculations of joint moments. This has motivated the authors of [28] and [29] to build on the SHS framework 1 in [30] , and derive promising results allowing the use of the SHS approach for the queueingtheoretic analyses of AoI. Following [28] , [29] , the SHS approach was then used to evaluate the average AoI for a variety of queueing disciplines in [31] - [33] , and the MGF of AoI for a two-source system with status update management in [34] . Compared to the analyses of [31] - [34] that considered a non-EH transmitter, the analysis of AoI using the SHS approach becomes much more challenging when we consider an EH-powered transmitter. This is due to the fact that the joint evolution of the battery state at the transmitter and the system occupancy with respect to the status updates has to be incorporated in the process of decision-making (i.e., the 1 A detailed description of the SHS will be provided in Section III. I   A SUMMARY OF THE QUEUEING THEORY-BASED ANALYSES OF AOI IN THE EXISTING LITERATURE.   A non-EH transmitter  EH-powered transmitter   Single-source  Multi-source  Single-source  Multi-source Average of AoI/peak AoI [3] , [5] - [11] [18]- [25] , [28] , [31] - [33] [35]- [38] This paper Distribution/distributional properties of AoI/peak AoI [12] - [17] [26], [27] , [29] , [ For the case where the transmitter is powered by EH, there are a handful of prior works [35] - [39] analyzing AoI by applying geometric arguments [35] , [36] , and by using the SHS approach and with [source-agnostic (LCFS-PS)/source-aware (LCFS-SA)] preemption in service queueing

disciplines. An arriving status update at the transmitter preempts the one being served (regardless of its generating source index) under the LCFS-PS queueing discipline, whereas the preemption in service under the LCFS-SA queueing discipline only occurs when the two status updates (the arriving one and the one being served) are generated from the same source. In our analysis, the harvested energy packets/status updates generated from each source are assumed to arrive at the transmitter according to a Poisson process, and the service time of each status update is assumed to be exponentially distributed. For this setup, our main contributions are listed next.

A novel analysis for deriving the average/MGF of AoI associated with each source at the destination. We use the SHS framework to first derive closed-form expressions of the average AoI of each source for each of the considered queueing disciplines. We then extend our analysis to understand the distributional properties of AoI through the characterization of its MGF under each queuing discipline. These results allow us to gain useful insights about the achievable AoI performance by each of the considered queueing disciplines. For instance, we analytically characterize the gaps between the achievable average AoI performances by the considered queueing disciplines as functions of the system parameters. Further, using the MGF of AoI expressions, we also characterize the relationship between the achievable second moments of AoI by the considered queueing disciplines.

Asymptotic results demonstrating the generality of the derived expressions. We demonstrate that as the aggregate generating rate of status updates from all the sources other than the source of interest approaches zero, the average AoI expressions derived in this paper reduce to their counterparts in [37] and [39] for single-source systems with an EH-powered transmitter, and the derived MGF of AoI expressions reduce to their counterparts in [39] . We further demonstrate that as the arrival rate of harvested energy packets at the transmitter node becomes large, the derived AoI results converge to their counterparts in [5] and [28] for single-source and multi-source systems with a non-EH transmitter, respectively.

System design insights. Our numerical results provide several useful system design insights.

For instance, they show that the achievable AoI performance by each queueing discipline improves with the increase in either the battery capacity or the arrival rate of harvested energy packets at the transmitter node. They also show that the superiority of the LCFS-PS queueing discipline over the LCFS-WP and LCFS-SA queueing disciplines in terms of the achievable AoI performance comes at the expense of having unfair achievable average AoI values among different sources. Further, they reveal that as the number of sources increases, the LCFS-SA queueing discipline becomes more effective (compared to the LCFS-PS) in achieving fairness between the achievable AoI performances by different sources. Finally, the results demonstrate the importance of incorporating the higher moments of AoI in the implementation/optimization of multi-source real-time status updates systems rather than just relying on its average.

We consider a real-time status update system in which an EH-powered transmitter node observes N physical processes, and sends its measurements to a destination node in the form of status update packets. As shown in Fig. 1 , the transmitter node contains N sources and a single server; each source generates status updates about one physical process, and the server delivers the status updates generated from all the sources to the destination. In particular, each status update packet generated by source i carries some information about the value of the i-th physical process and a time stamp indicating the time at which that information was measured. This system setup can be mapped to many scenarios of practical interest, such as an IoT network in which an aggregator (represents the transmitter in our model) delivers measurements sensed/generated by the N IoT devices (represent the sources) in its vicinity to a destination node.

Source i is assumed to generate status update packets at the transmitter node as a rate λ i Poisson process. Further, the transmitter harvests energy in the form of energy packets such that each energy packet contains the energy required for sending one status update packet from any of the sources to the destination node [35] - [38] . In particular, the harvested energy packets are assumed to arrive at the transmitter according to a Poisson process with rate η, and are stored in a battery queue of length B packets at the server (for serving the update packets generated by the different sources). Given that the transmitter node has at least one energy packet in its battery queue, the time needed by its server to send a status update packet is assumed to be a rate µ exponential random variable [3] , [5] , [6] . Let ρ = λ µ and β = η µ respectively denote the server utilization and energy utilization factors, where λ = N i=1 λ i . Further, we have ρ i = λ i µ , λ −i = N j=1, j =i λ j , and ρ −i = λ −i µ . We quantify the freshness of information status about each physical process at the destination node (as a consequence of receiving status update packets from the transmitter node) using the concept of AoI. Formally, AoI is defined as follows [3] . Definition 1. Let t i,k and t i,k denote the arrival and reception time instants of the k-th update packet of source i at the transmitter and destination, respectively. Further, define L i (t) to be the index of the source i's latest update packet received at the destination by time t, i.e., L i (t) = max{k|t i,k ≤ t, ∀k}. Then, the AoI associated with the physical process observed by source i at the destination node (referred henceforth as the AoI of source i) is defined as the following random process

(1)

For the above system setup, we analyze the AoI performance at the destination under three different queueing disciplines for managing update packet arrivals at the transmitter node. These queueing disciplines are described next.

• LCFS-WP queueing discipline: Under this queueing discipline, a new arriving update packet at the transmitter (from any of the sources) enters service upon its arrival if the server is idle (i.e., there are no status update packets in the system) and the battery contains at least one energy packet; otherwise, the new arriving update packet is discarded.

• LCFS-PS queueing discipline: When the server is idle, the management of a new arriving update packet under this queueing discipline is similar to the LCFS-WP one. However, when the server is busy, a new arriving update packet replaces the current packet being served and the old packet in service is discarded.

• LCFS-SA queueing discipline: This queueing discipline is similar to the LCFS-PS one with the only difference that a new arriving update packet preempts the packet in service only if the two packets (the new arriving packet and the one in service) are generated from the same source.

Note that according to the LCFS-PS queueing discipline, status updates of a source i with a small λ i are more likely to be preempted in service by status updates of a source j with λ j λ i . Since this issue is resolved under the LCFS-SA queueing discipline by only allowing preemption in service between the status updates generated from the same source, we expect that the LCFS-SA queueing discipline will be more effective (compared to the LCFS-PS) in achieving fairness between the achievable AoI performances by different sources (as will be demonstrated in Section VI). As already conveyed, we consider that an energy packet contains the amount of energy required for sending one status update to the destination. Therefore, we assume that the length of the energy battery queue reduces by one whenever a status update is successfully transmitted to the destination. Further, with regards to the EH process, we consider that the transmitter can harvest energy only if its server is idle 2 . This case corresponds to the scenario where the transmitter is equipped with a single radio frequency (RF) chain and a single antenna, and thus can either transmit a status update or harvest energy at a certain time instant.

Our goal is to analytically characterize the AoI performance of each source at the destination node as a function of: i) the rates of generating status update packets by the N sources {λ i },

ii) the rate of harvesting energy packets η, iii) the rate of serving status update packets µ, and iv) the finite capacity of the energy battery queue B, at the transmitter node. Unlike most of the analyses of AoI in the existing literature which were focused on deriving its average, our analysis is focused on deriving distributional properties of AoI through the characterization of its MGF. To derive the MGF of AoI for the considered queueing disciplines at the transmitter node (presented in Subsection II-B), we resort to the SHS framework in [30] , which was first tailored for the analysis of AoI by [28] and [29] . In the following, we provide a very brief 3 introduction of the SHS framework, which will be useful in understanding our AoI MGF analysis in the subsequent sections. The SHS technique is used to analyze hybrid queueing systems that can be modeled by a combination of discrete and continuous state parameters. In particular, the SHS technique models the discrete state of the system q(t) ∈ Q = {1, · · · , m} by a continuous-time finite-state Markov chain, where Q is the discrete state space. This continuous-time Markov chain governs the dynamics of the system discrete state that usually describes the occupancy of the system, e.g., q(t) represents the numbers of status update and energy packets in the system for our problem. On the other hand, the evolution of the continuous state of the system is described by a continuous process x(t) = [x 0 (t), · · · , x n (t)] ∈ R 1×(n+1) , e.g., x(t) models the evolution of the age-related processes in our system setting.

A transition l ∈ L from state q l to state q l (in the Markov chain modeling q(t)) occurs due to the arrival of a status update/energy packet or the delivery of a status update to the destination (i.e., the departure of a status update from the system), where L denotes the set of all transitions. Since the time elapsed between arrivals/departures is exponentially distributed, a transition l takes place with a rate λ (l) δ q l ,q(t) , where the Kronecker delta function δ q l ,q(t) ensures that l occurs only when the discrete state q(t) is equal to q l . As a consequence of the occurrence of transition l, the discrete state of the system moves from state q l to state q l , and the continuous state x is reset to x according to a binary reset map matrix

. Different from ordinary continuous-time Markov chains, an inherent feature of SHSs is the possibility of having self-transitions in the Markov chain modeling the system discrete state. In particular, although a self-transition keeps q(t) unchanged, it causes a change in the continuous process x(t).

Now, we define some useful quantities for the characterization of the MGF of AoI at the destination node using the SHS technique. Denote by π q (t) the probability of being in state q of the continuous-time Markov chain at time t.

denote the correlation vector between q(t) and x(t), and v s q (t) = [v s q0 (t), · · · , v s qn (t)] ∈ R 1×(n+1) denote the correlation vector between q(t) and the exponential function e sx(t) , where s ∈ R.

Thus, we can respectively express π q (t), v q (t) and v s q (t) as

According to the ergodicity assumption of the continuous-time Markov chain modeling q(t)

in the AoI analysis [28] , [29] , the state probability vector π(t) = [π 0 (t), · · · , π m (t)] converges uniquely to the stationary vectorπ = [π 0 , · · · ,π m ] satisfyinḡ

where L q = {l ∈ L : q l = q} and L q = {l ∈ L : q l = q} denote the sets of incoming and outgoing transitions for state q, ∀q ∈ Q.

Using the above notations, it has been shown in [29, Theorem 1] that under the ergodicity assumption of the Markov chain modeling q(t), if we can find a non-negative limitv q =

then:

, converges to the following stationary vector:

• There exists s 0 > 0 such that for all s < s 0 , v s q (t) converges tov s q that satisfies v s q l∈Lq

is a binary matrix whose elements are constructed aŝ

Further, the MGF of the state x(t), which can be obtained as E[e sx(t) ], converges to the following stationary vector:

From (7) and (10), when the first element of the continuous state x(t) represents the AoI at the destination node, the expectation and the MGF of AoI at the destination node respectively converge to:

It is clear from [29, Theorem 1] (stated in Section III) that in order to use (8) to derive the MGF of AoI at the destination, one needs to find a non-negative limitv q (∀q ∈ Q) satisfying (6) , which directly characterizes the average AoI as observed from (7). Thus, we first show in this section that this condition holds for the three queueing disciplines considered in this paper, which will immediately lead to the average AoI characterization for each queueing discipline.

Afterwards, we extend our analysis in the next section to derive the MGF of AoI.

Without loss of generality, we consider that source 1 is the source of interest in the AoI analysis in the sequel. The AoI performance of the other sources can then be obtained using the same expressions derived for source 1, as will be clear shortly. While analyzing the AoI of source 1, the status update packets associated with the other sources are generated according to a Poisson process with rate λ −1 = N j=2 λ j . Using the notations of the SHS approach (presented in Section III), the continuous process x(t) in each queueing discipline is given by

where x 0 (t) represents the value of the source 1's AoI at the destination node at time instant t (i.e., ∆ 1 (t)), and x 1 (t) indicates the value that the source 1's AoI at the destination will become if the existing update packet in the system completes its service at time instant t (i.e., the packet is delivered to the destination at t). Recall from Section III that as long as there is no change (due to the arrival/departure of an update/energy packet) in the discrete state q(t), we have

i.e., the elements of the age vector x(t) increase linearly with time.

The continuous-time Markov chain modeling the discrete state of the system q(t) ∈ Q under the LCFS-WP queueing discipline is depicted in Fig. 2 . Each state in Q represents a potential combination of the number of update packets in the system and the number of energy packets in the battery queue at the server. For instance, a state q = (e q , u q ) indicates that the system has u q status update packets and the energy battery queue at the server contains e q energy packets.

Note that since the system can have at most one status update packet at any time instant in the LCFS-WP queueing discipline and there is no need to track the source index from which the update packet in service was generated, we have u q ∈ {0, 1}. In particular, u q = 0 indicates that the system is empty and hence the server is idle, and u q = 1 indicates that the server is serving the existing update packet in the system. Since the battery queue at the server has a capacity of B packets, we have e q ∈ {0, 1, · · · , B}. We denote the set of states in the i−th row of the Markov chain by r i . Further, Table II Further, to ensure that the value of ∆ 1 (t) does not change when this new arriving update packet is received by the destination, we set the second component of xA 4k−1 to x 0 , i.e., the value of the source 1's AoI at the arrival instant of this new update packet.

• l = 4k: This subset of transitions occurs when the update packet in service is delivered to the destination. When the update packet received at the destination belongs to source 1, the AoI of source 1 is reset to its age; otherwise, the AoI of source 1 does not change.

Note that the latter case is achieved by setting the second component of xA 4k−1 to x 0 . In addition, since the system becomes empty after the occurrence of this transition, the second component of the age vector x(t) becomes irrelevant, and thus its corresponding value in the updated age vector xA 4k is 0. Now, in order to obtainv q satisfying (6), the steady state probabilities {π q } and the vector v q l A l (associated with each transition l in L) need to be computed. The calculations ofv q l A l , l ∈ L, are listed in Table II , and {π q } are given by the following proposition. 

Proposition 1. The steady state probabilities {π q } can be expressed as

where 1 ≤ k ≤ B andπ 1 is given bȳ

, otherwise.

Proof: The expressions in (13)-(15) follow from solving the set of equations in (5) . A detailed proof can be found in Appendix A of [39] .

Having the steady state probabilities {π q } in Proposition 1 and the set of transitions L in Table II , we are now ready to derivev q satisfying (6) as well as to characterize the average value of ∆ 1 (t) in the following theorem.

Theorem 1. Under the LCFS-WP queueing discipline, there exists a non-negative limitv q , ∀q ∈ Q, satisfying (6) and the average AoI of source 1 is given by

where the set {c 0 , c 2 , · · · , c 2B } is defined as

Proof: See Appendix A.

Note that the average AoI performance for source i ∈ {2, 3, · · · , N } can be obtained directly using (16) by replacing λ 1 with λ i (which results in replacing

as well). This argument applies to all the results derived in this paper for source 1.

For the single source case where ρ −1 = 0 and ρ = ρ 1 ,

Note that the expression of WP ∆ 1,1 in (18) 

Note that the expression in (19) . FIG. 3 Table III refers to the event of having a new arriving update packet at the transmitter node while its server is serving another update packet. According to the mechanism of the LCFS-PS queueing discipline, the status update that is currently being served will be discarded, and the new arrival will enter service upon its arrival. From (5), we note that the self-transitions do not impact the values of the steady state probabilities {π q }, and hence {π q } in this case can be obtained using Proposition 1. That said, the average value of ∆ 1 (t) is provided in the next theorem.

Theorem 2. Under the LCFS-PS queueing discipline, there exists a non-negative limitv q , ∀q ∈ Q, satisfying (6) and the average AoI of source 1 is given by

where the set {c 0 , c 2 , · · · , c 2B } is defined as in (17) .

Proof: See Appendix B. 

Note that the expression of PS ∆ 1,1 in (21) 

Note that the expression in (22) 

Since the set {c 0 , c 2 , · · · , c 2B } contains positive real numbers, we observe from (23) Under the LCFS-SA queueing discipline, the continuous-time Markov chain modeling the discrete state of the system q(t) ∈ Q is depicted in Fig. 4 . Recall that according to the mechanism of the LCFS-SA queueing discipline, a new arriving update packet preempts the packet in service only if the two packets are generated from the same source. Thus, the discrete state of the system needs to not only account for the number of update packets in the system (as it was the case for the LCFS-WP and the LCFS-PS queueing disciplines) but also track the index of the source which generated the current packet in service. Because of that, we observe from Fig. 4 that for a state q = (e q , u q ), we have u q ∈ {0, 1, · · · , N }. In particular, u q = 0 indicates that the system is empty and hence the server is idle, and u q = i indicates that there is an update packet in service and the index of its generating source is i. Further, due to the finite capacity of the battery queue at the server, we have e q ∈ {0, 1, · · · , B}. Table IV presents impact on the values of both q(t) and x(t). While the description of most transitions in Table   IV is similar to the description of their counterparts in Tables II and III, there are some key differences. The first difference is that the second component of the updated age vector xA l in Table IV is set to 0 (rather than x 0 as in Tables II and III) when a new update packet is generated from source i ∈ {2, 3, · · · , N } at the time when the system is empty (i.e., transition l = (3N + 1)k − 3N + i) or another packet generated from source i is being served (i.e., transition l = (3N + 1)k − 2N + i). This is because under the LCFS-SA queueing discipline, we know the index of the source that generated the packet in service, and hence we can safely set the irrelevant second component of xA l to 0 when such transitions l occur. The second difference, which has a similar interpretation to the first one, is that the first component of the updated age vector xA l in Table IV is set to x 0 (rather than x 1 as in Table II ) when an update packet generated from source i ∈ {2, 3, · · · , N } is delivered to the destination (i.e., transition l = (3N + 1)k − N + i). We start the analysis by characterizing the steady state probabilities {π q } in the following proposition.

Proposition 2. The steady state probabilities {π q } can be expressed as

where 1 ≤ k ≤ B, 1 ≤ i ≤ N , andπ 1 is given by (15) .

Proof: The expressions in (24) and (25) follow directly from the solution of (5). Now, using Table IV and Proposition 2, the average AoI is obtained in the following theorem. FIG. 4 (2 

Theorem 3. Under the LCFS-SA queueing discipline, there exists a non-negative limitv q , ∀q ∈ Q, satisfying (6) and the average AoI of source 1 is given by

wherev 10 is given bȳ

where the set {c −1 ,c 0 , · · · ,c B−1 } is defined as

Proof: See Appendix C. 

which indicates that lim

Remark 2. Note that from Theorems 1, 2 and 3, we have

Since the set {c −1 ,c 0 , · · · ,c B−1 } contains positive real numbers, we observe from (30) 

This section is dedicated to the analysis of the MGF of AoI under each of the queueing disciplines considered in this paper.

According to Theorem 1, there exists a non-negative limitv q , ∀q ∈ Q, satisfying (6), under the LCFS-WP queueing discipline. Thus, the MGF of AoI can be evaluated using (8) as in the following theorem, where the calculations required to solve the set of equations (i.e.,v s q l A l and π q l 1Â l , l ∈ L) in (8) are listed in Table II .

Theorem 4. The MGF of AoI of source 1 for the LCFS-WP queueing discipline is given by

wheres = s µ andv s 10 is given bȳ v s 10 =

where the set {c s 0 , c s 2 , · · · , c s 2B } is defined as

Proof: See Appendix D. 

where θ can be expressed as

, otherwise. queueing discipline can be respectively expressed as

, otherwise.

(37)

ζ 0 = ρ 3 8ρ 3 + 36ρ 2 + 28ρ + 15 , ψ 7 = ρ 2 1 ρ + ρ 1 ρ 2 − 1 + (1 + ρ) 3 , ψ 6 = 4ρ 2 1 ρ 2 + 4ρ 1 ρ ρ 2 − 1

Proof: The expressions in (37) and (38) 

where d k ds k denotes the k-th derivative with respect tos.

Based on Theorem 2, the MGF of AoI under the LCFS-PS queueing discipline is derived in the following theorem by solving the set of equations in (8) using the calculations in Tables II   and III .

Theorem 5. The MGF of AoI of source 1 for the LCFS-PS queueing discipline is given by

wherev s 10 is given byv

where the set {c s 0 , c s 2 , · · · , c s 2B } is defined as in (34) .

Proof: See Appendix E. 

where θ is given by (36) .

Corollary 9. When B = 2, the first and second moments of AoI of source 1 under the LCFS-PS queueing discipline can be respectively expressed as

otherwise.

(44)

Based on Theorem 3, the MGF of AoI under the LCFS-SA queueing discipline is derived in the following theorem by solving the set of equations in (8) using the calculations in Table IV .

Theorem 6. The MGF of AoI of source 1 for the LCFS-SA queueing discipline is given by

wherev s 10 is given bȳ v s 10 =

where the set {c s −1 ,c s 0 , · · · ,c s B−1 } is defined as

Proof: See Appendix F. Corollary 11. When B = 2, the first and second moments of AoI of source 1 under the LCFS-SA queueing discipline can be respectively expressed as

, otherwise.

(49) ζ 9 = 12ρ 2 ,ζ 8 = ρ 2 (43ρ + 22) ,ζ 7 = ρ 2 (18ρ + 4) ,ζ 6 = ρ 2 24ρ 2 + 74ρ + 8 ,ζ 5 = 2,ζ 4 = 6ρ 2 + 5ρ + 4,ζ 3 = 2 (4ρ + 1) ,ζ 2 = ρ 8ρ 5 + 20ρ 4 + 3 ,ζ 1 = ρ 3 16ρ 3 + 62ρ 2 + 61ρ + 6 ,ζ 0 = ρ 3 8ρ 3 + 36ρ 2 + 54ρ + 15 ,ψ 7 = −ρ 3 1 (3 + 2ρ) + ρ 2 1 ρ 3 + 4ρ 2 + 2ρ − 2 + ρ 1 (1 + ρ) 2ρ 2 + 5ρ + 1 + (1 + ρ) 3 ,ψ 6 = −4ρ 3 1 ρ (3 + 2ρ) + 4ρ 2 1 ρ ρ 3 + 4ρ 2 + 2ρ − 2 + 4ρ 1 ρ (1 + ρ) 2ρ 2 + 5ρ + 1 + 4ρ

11ρ + 13 ,ψ 2 = 2ρ 4 1 ρ 3 − ρ 3 1 ρ 3 2ρ 3 + 9ρ 2 + 4ρ − 4 + ρ 2 1 ρ 3 3ρ 4 + 10ρ 3 − 3ρ 2 − 8ρ + 2 + 2ρ 1 ρ 4 3ρ 3 + 12ρ 2 + 7ρ − 2 + 3ρ 5 (1 + ρ) (3 + ρ) ,ψ 1 = −2ρ 3 1 ρ 6 + ρ 2 1 ρ 6 (1 + 3ρ) + ρ 1 ρ 6 (7 + 6ρ) + 

In this section, we study the impact of the system design parameters on the achievable AoI performance under each of the three queueing disciplines considered in this paper. We use µ = 1 in all the figures. In Fig 5, capacity B on the achievable pairs of average AoI (∆ 1,1 , ∆ 2,1 ) when N = 2 and ρ is fixed.

We observe from Fig. 5 that the AoI performance of each queueing discipline improves with increasing B or β until it converges to its counterpart with a non-EH transmitter (as stated in Corollaries 2, 4 and 5) . This happens since increasing B or β decreases the likelihood that the battery queue is empty upon the arrival of a new status update at the transmitter when the server is idle, and hence increases the likelihood of delivering new arriving updates to the destination.

In Fig. 6 , we compare the three queueing disciplines studied in this paper when N = 2 in terms of: i) the achievable average AoI pairs (∆ 1,1 , ∆ 2,1 ) in Figs. 6a and 6d We first show the existence of a non-negative limitv q , ∀q ∈ Q, satisfying (6), and then we obtain the average AoI of source 1 using WP ∆ 1,1 = q∈Qv q0 . The set of equations in (6) can be expressed as

From (53),v 2B,0 can be expressed as

where c 2B = λ. Substituting k = B − 1 in (52),v 2B−2,0 can be expressed as

wherev 2B+1,1 andv 2B,0 were respectively substituted from (54) and (55), and c 2B−2 = η 1 − λ −1 c 2B + λ. Repeated application of (52) gives

After substitutingv 31 from (54) into (50) and then solving (50) and (58),v 10 is given bȳ

where c 0 = η 1 λ −1 − 1 c 2 . Note that the set {c 0 , c 2 , · · · , c 2B } contains positive real numbers, and hence we observe from (59) thatv 10 ≥ 0. Thus, from (57) and (58), we deduce that v q0 ≥ 0, q ∈ r 1 . As a result, we observe from (54) thatv q0 ≥ 0 andv q1 ≥ 0, q ∈ r 2 . Finally, from (50)-(53), one can easily see thatv q1 ≥ 0, q ∈ r 1 , and hence there exists a non-negative limitv q , ∀q ∈ Q, satisfying (50)- (54) . Now, we proceed with evaluating the average AoI of source 1. Summing (50)- (53) gives

Further, by summing the set of equations in (54), we have

Thus, the average AoI of source 1 can be evaluated as

where step (a) follows from substituting q∈ r 2v q0 from (61) into (63), and step (b) follows from substituting q∈ r 1v q0 using (60) and (62) into (63) . The final expression of WP ∆ 1,1 in (16) can directly be obtained by substitutingv 10 from (59) into (63) .

By inspecting Fig. 3 , we observe that the set of equations in (6) corresponding to the states in r 1 are still given by (50)- (53) . Regarding the states in r 2 , we have

where 1 ≤ k ≤ B. Similar to the procedure in (55)- (59) in Appendix A, repeated application of (52) gives

Thus, from (50), (64) and (66),v 10 can be expressed as v 10 =π

Recalling that the set {c 0 , c 2 , · · · , c 2B } contains positive real numbers, we deduce from (50)- (53) and (64)- (67) that there exists a non-negative limitv q , ∀q ∈ Q, satisfying (6). Further, the average AoI of source 1 can be evaluated as follows. We first note that q∈ r 1 /{1}v q0 and q∈ r 2v q0 can be expressed as in (60) and (61), respectively. In addition, summing the set of equations in (64) gives

By solving (60), (61) and (68), we get

(69) From (69), the average AoI of source 1 can be evaluated as

The expression of PS ∆ 1,1 in (20) can be obtained by substitutingv 10 from (67) into (70).

We first note from Table IV that the second component of the vectorv q l A l , ∀l ∈ L q and q ∈ Q, is 0. Thus, we observe from (6) thatv q1 ≥ 0, ∀q ∈ Q. The existence of a non-negative limitv q , ∀q ∈ Q, satisfying (6) is then tied with havingv q0 ≥ 0, ∀q ∈ Q. The set of equations in (6) corresponding to the states in r 1 can be expressed as

Further, the set of equations in (6) corresponding to the states in r i+1 , 1 ≤ i ≤ N , can be expressed as q 2+i+k(N +1) , 0 ≤ k ≤ B − 1 : µv 2+i+k(N +1),0 = λ iv2+k(N +1),0 +π 2+i+k(N +1) .

By noting thatv 3+k(N +1),1 =π

can be rewritten as

where 1 ≤ k ≤ B −2, and N +2+(k+1)(N +1) (73) 

where the set {c h } is defined in (28) . The expression ofv 10 in (27) can be obtained by solving (71) and (78) while noting thatv 31 =π 3 µ+λ 1 and µ N +4 j=4v j0 = λ −1v20 + N +2 j=4π j . Since the set {c −1 ,c 0 , · · · ,c B−1 } contains positive real numbers, we havev 10 ≥ 0. Therefore, from (75), (77) and (78), we observe thatv q0 ≥ 0, ∀q ∈ Q, and hence there exits a non-negative limit v q , ∀q ∈ Q, satisfying (6). In the following, we evaluate the average AoI of source 1. By 

where q∈ r 2v q1 = q∈ r 2π q µ + λ 1 . From (79) and (80), we get

Hence, the average AoI of source 1 can be obtained as 

where step (a) follows from (79) and (81). This completes the proof.

Using Table II , the set of equations in (8) can be expressed as 

Similar to Appendix B, We first note that the set of equations in (8) corresponding to the states in r 1 are given by (83)-(86), and hence q∈ r 1v s q0 can be expressed as in (88). Regarding the states in r 2 , we have q 2k+1 , 1 ≤ k ≤ B : (µ − s)v s 2k+1,0 = λv s 2k,0 , (λ + µ − s)v s 2k+1,1 = λ 2 (v s 2k,0 +v s 2k+1,0 ) + λ 1 (π 2k +π 2k+1 ).

We observe from (92) that q∈ r 2v 

where step (a) follows from substituting (89) and substituting it into (94).

Finally,v s 10 in (41) can be obtained by following similar steps as in (65)-(67).

Using Table IV , the set of equations in (8) corresponding to q ∈ r 1 can be expressed as 

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