Age of information of a server with energy requirements

Motivation

The Age of Information has been introduced recently as a metric to measure the freshness of information in applications in which a monitor at the destination is interested in having timely updates about a process of interest in the source. There is a wide range of applications in which the Age of Information is a useful metric of the performance of the system. For example, in autonomous driving, the location, orientation, and speed of the rest of the vehicles in the route must be as recent as possible for an optimal system behavior. Most of the articles that study the Age of Information model the network where the communication network connecting the system whose status is of interest and the monitor that tracks the status of that system as a queueing network. In this work, we analyze the Age of Information of a queueing system with energy requirements, that is, we consider that the communication system needs some energy to deliver the status updates of the source to the monitor.

In the last decades, there has been a big increase of energy usage that come from renewal sources, such as solar or wind and this growth will continue in the near future. This kind of energy sources are known to be very volatile. For instance, the amount of energy generated by solar or wind generators depend on the weather of the region where the generators are located. Therefore, in a system that run on renewal energy, it might happen the generation of energy when it is not required or the lack of energy when the system needs to work. A solution to this problem that has been presented in the last years consists of considering energy harvesting models. In fact, in these models, it is considered that energy can be stored in a battery or in some other device to be used afterwords if this energy is not required at the moment where it is generated. In this work, we consider that the energy is generated according to a random process (which models its generation by renewal sources) and it can be harvested.

Related Work

The Age of Information metric has been introduced in Kaul et al. (2011) to analyze the freshness of information that a monitor receives from the status updates of a process of interest though a communication channel. More precisely, it is defined as the time elapsed from the generation of the last successfully delivered status update. The authors in Kaul, Yates & Gruteser (2012) consider that the channel by which the status updates are sent to the monitor is a M/M/1 queue, a M/D/1 queue and a D/M/1 queue. The authors in Kam, Kompella & Ephremides (2014) and Kam et al. (2015) consider a channel that consists of a M/M/2 queue. These articles put in evidence that the policies that minimize well-known metrics of queueing theory, such as delay or throughput, are not optimal for the Age of Information. Therefore, since then, there is an active number of researchers that have been interested in analyzing the Age of Information and applying it in a wide range of applications such as content caching Kam et al. (2017a), Yates et al. (2017), Zhong, Yates & Soljanin (2018), ad-hoc networks Kam et al. (2017b), wireless camera networks He, Dan & Fodor (2018, 2019), vehicle -assisted IoT networks Jia et al. (2019), broadcast wireless networks Kadota et al. (2016), Hsu, Modiano & Duan (2017) and Kadota et al. (2018) and the optimal design of freshness-aware IoT Abd-Elmagid, Pappas & Dhillon (2019). The authors in Chaintreau, Le Boudec & Ristanovic (2009) present an age-based model, but in a different context that the Age of Information; in fact, they consider a network with moving nodes of different classes and analyze the effect of gossiping in the occupancy measure of this network

We would like to remark that energy harvesting has been also studied in the context of the Age of Information metric. For instance, the authors in Farazi, Klein & Brown (2018) consider a M/M/1 queue with infinite buffer and last-come-first-served discipline. Another related work is Gindullina, Badia & Gündüz (2020) where the authors study the scheduling that minimizes the average Age of Information in a system with energy harvesting. Our work differs from these ones since we compute an analytical expression of the average Age of Information of a system in a M/M/1/1 queue where the battery is modelled as a M/M/1/1 queue. For a detailed presentation of the recent work about this metric we refer to the following surveys (Sun et al., 2019; Kosta et al., 2017a).

The analytical study of the Age of Information faces a problem to calculate the average Age of Information, one needs to compute the expected value of the product between the random variable of the interarrival times and of the sojourn time, which are dependent random variables. This makes the computation of the average Age of Information a very challenging task for many models. As a result, some authors have been interested in studying other metrics different from the average Age of Information, such that the Peak Age of Information (Costa, Codreanu & Ephremides, 2016), the non-linear age penalty function (Kosta et al., 2017b, 2020) and the age of incorrect information (Maatouk et al., 2020). Recently, the authors in Yates & Kaul (2019) present the Stochastic Hybrid System (SHS) method as a technique to compute the average Age of Information without requiring the computation of the aforementioned difficult expression. We describe this methodology more in detail in “SHS for the Average Age of Information” since it is the technique we use in this article to compute the average Age of Information of our model.

The queueing system we analyze is a particular case of the Energy Packet Network (EPN) model where the queues M/M/1/1 queues. The EPN model was introduced by Gelenbe and his colleagues (Gelenbe, 2011, 2012; Gelenbe & Ceran, 2015) and it is based on G-networks Gelenbe (1991), Gelenbe (1993a, 1993b), Chao, Miyazawa & Pinedo (1999) and Fourneau & Gelenbe (2017) (although not all EPN models are related to G-networks, see for instance, Gelenbe (2014) and Abdelrahman & Gelenbe (2016). Therefore, there is a large number of EPN models where the existence of a product form of the steady-state distribution of jobs in the queues follows from that of G-networks. The importance of the existence of the product form expression is that it allows to optimize the system for some utility function like losses or response time (Gelenbe & Ceran, 2016; Fourneau, Marin & Balsamo, 2016) and the energy distribution (Gelenbe & Zhang, 2019; Zhang, 2019) as well as to design networks (Kadioglu & Gelenbe, 2016; Gelenbe & Abdelrahman, 2018; Doncel & Fourneau, 2019) and data-centers Fourneau (2020) with energy harvesting. We refer to Ray (2019) for a recent survey of the EPN literature. In the EPN model, energy is represented by packets of discrete units called energy packets that model a certain number of Joules, whereas the workload is represented by data packets. Each station of the network consists of a data queue where data packets are stored and an energy queue (or battery) to keep the energy. In such a model, the authors in Gelenbe & Marin (2015) study the influence of the lack of energy in the delay of data packets. As in Kadioglu & Gelenbe (2019), we consider an EPN model where the data packets start that transfer. This means that, when a data packet is served, it is sent to the energy queue and triggers the deletion of an energy packet and the movement of the data packet to the following station, whereas the data packet is lost if it does not find an energy packet. To the best of our knowledge, the average Age of Information of this kind of systems has not been studied so far.

Contributions

In this article, we study the average Age of Information of a queueing system that is formed by two queues: one queue stores the updates of the system of interest and the other stores the energy that is required to deliver the status updates to the monitor. Both queues can handle, at most, one packet. We assume that there are leakages in the energy queue.

The main contributions of this article are summarized as follows:

• We use the SHS technique to study the average Age of Information of a queueing system under consideration. We first observe that the average Age of Information requires to compute some of the variables that are the solution of a system with 8 equations. Then, we prove that, to compute the average Age of Information, it is enough to solve a system with four equations and four variables. Solving the latter system of equation, we provide an explicit expression of the average Age of Information of our model.

• Due to the difficulty of the derived expression, we aim to study approximations of the average Age of Interest of this model. We first study the accuracy of the expression obtained under the assumption that there is always energy available to deliver the updates and we show that this expression is a good approximation when the arrival rate of energy packets is very large and the rate at which energy packets are lost is small. Then, we present another expression where it provides a good approximation of the average Age of Information when the arrival rate of energy packets is large and any value of the rate at which energy packets are lost.

The results of this article provides insides on the impact on the average Age of Information of the presence of energy harvesting devices in which the arrival of the energy follows a random process. This is a particularly interesting problem given that, in the last decades, there has been a huge growth of energy that come from renewable sources. This work also shows the difficulty in obtaining an analytical expression of the average Age of Information for energy harvesting models and illustrate that the study of approximations of such a difficult performance metric is a very promishing research line.

Organization of the paper

The remainder of the article is organized as follows. In “Model Description”, we describe the model we study. In “Exact Computation of the Average Age of Information”, we provide an exact analysis of the average Age of Information of our model, whereas in “Approximation of the Average Age of Information” we present approximations to the derived expression. Finally, we explain the main conclusions of our work in “Conclusions”.

Age of information

We study the transmission of status updates to a monitor. We assume that update i is generated at time s_i and that it is received by the monitor at time $t_i^{{}^{\prime }}$. Let N(t) be the index of the last successfully received update at time t, that is, $$N(t)=max\{i|t_i^{\mathrm{\prime }}\le t\}.$$

From the above definitions, it follows that the generation time of the last received update at time t is given by t_N(t). The Age of Information is defined as $$\mathrm{\Delta }(t)=t-t_{N(t)}.$$

An example of the evolution of Δ(t) is presented in Fig. 1.

We know that the average Δ(t) exists in a stationary ergodic updating system. Under such assumption, it is defined as follows: $$\mathrm{\Delta }=\underset{\tau \to \mathrm{\infty }}{lim}\frac{1}{\tau }{\int }_0^{\tau }\mathrm{\Delta }(t)dt.$$

In this article, we are interested in calculating the average Δ(t) in an energy harvesting model. As we said in the introduction, the analytical study of the above expression in a general setting is known to be an extremely difficult task. In the following section, we describe for completeness the SHS method and how it can be used to calculate the average Age of Information.

SHS for the average age of information

We now present the SHS method and how it is used to compute the average Age of Information for an arbitrary model. For full details about this technique, we refer to Hespanha (2006).

In the SHS, the system is modeled as a hybrid state (q(t), x(t)). Each of the components of this hybrid state is as follows: q(t) belongs to a continuous time Markov Chain and x(t) is a vector evolves according to an ordinary differential equation. The latter contains the current value of the Age of Information as well as the value of the Age of Information if the packets present in the system were delivered to the monitor.

A link l of the Markov Chain represents a transition between two states, which occurs with rate λ^l. In each transition l, the vector x changes to x^′ using a transformation matrix A_l, that is, x′ = x A_l. Therefore, x(t) is a piecewise function.

For each state q, we define b_q as the vector whose elements are zero or one since the time from the generation of the updates that are present in the system increase at unit rate, whereas the updates that are not in the system do not.

We assume the Markov Chain is ergodic and we denote by π_q the stationary distribution of state q. We denote by ${\mathcal{L}}_q$ the set of outgoing links of state q and ${\mathcal{L}}_q^{{}^{\prime }}$ the set of links that get into state q. We now present the following theorem that will be used to characterize the average Age of Information:

Theorem 1 (Yates & Kaul, 2019, Thm 4). Let v_q(i) denote the i-th element of the vector v_q. For each state q, if v_q is a non-negative solution of the following system of equations (1) $${\mathbf{v}}_q\sum _{l\in {\mathcal{L}}_q}{\lambda }^l={\mathbf{b}}_q{\pi }_q+\sum _{l\in {\mathcal{L}}_q^{{}^{\prime }}}{\lambda }^l{\mathbf{v}}_{q_l}{\mathbf{A}}_l$$then the average Age of Information is $\mathrm{\Delta }={\sum }_q{v}_q(0)$.

In “Exact Computation of the Average Age of Information”, we use the SHS technique to characterize the average Age of Information of a model that consists of single server with energy requirements. Prior to that, we describe this energy harvesting model.

Energy harvesting model

We analyze a system formed by one server that stores the updates of the status of a system that is monitored. Updates arrive to the system according to a Poisson process with rate λ. We model the server that handles the status updates as a M/M/1/1 queue, that is, it is a server with exponential service times and without buffer. We allow preemption in updates in service, that is, when an update arrives and there is server is busy, the new packet replaces the update that was getting served.

We consider that the updates require energy to be delivered to the monitor. Energy is represented by packets of discrete units called energy packets that model a certain number of Joules. We consider that energy packets are stored in a battery (or energy queue). We assume that energy packets arrive according to a Poisson process with rate rate α. We consider that the battery can store, at most, one energy packet. We also consider that there are energy leakages, that is, an energy packet in the energy queue can be lost with exponential times. We denote by β the leakage rate of one energy packet in the energy queue.

We now present the dynamics of the system. When a status update is served, it is sent to the energy queue and two things can happen: (i) the energy queue is empty, in which can the status update is lost and (ii) there is a packet in the battery, in which case the update is delivered to the monitor and the energy packet disappears.

The model we study in this article is represented in Fig. 2.

The case α→∞

In this section, we consider that the arrival rate of energy packets tends to infinity and the rest of the parameters are finite. We first note that, when α→∞, π₀₀ and π₀₁ tend to 0, whereas π₁₀ and π₁₁ tends, respectively, to ${\displaystyle \frac{\mu }{\lambda +\mu }}$ and ${\displaystyle \frac{\lambda }{\lambda +\mu }}$. On the other hand, from the first three equations of Eq. (1), it follows that v₀₀(0), v₀₁(0) and v₀₁(1) tend to zero when α→∞. Therefore, to compute the Age of Information in this regime we have to analyze the following system of equations: $$v_{10}(0)(\lambda +\beta )={\pi }_{10}+\alpha v_{00}(0)$$ $$v_{11}(0)(\lambda +\mu +\beta )={\pi }_{11}+\alpha v_{01}(0)+\lambda v_{01}(0)+\lambda v_{11}(0)$$ $$v_{11}(1)(\lambda +\mu +\beta )={\pi }_{11}+\alpha v_{01}(1)$$

We now study α v₀₀(0) when α→∞. Using L’Hopital’s rule, we get that $$\underset{\alpha \to \mathrm{\infty }}{lim}\alpha v_{00}(0)=\mu v_{11}(1)+\beta v_{10}(0)$$

Likewise, we show that α v₀₀(1) tends to β v₀₁(0) and α v₀₁(1) tends to β v₁₁(1) when α→∞. As a result, the above system of equations can be written as follows: (9a) $$v_{10}(0)\lambda ={\pi }_{10}+\mu v_{11}(1)$$ (9b) $$v_{11}(0)(\lambda +\mu )={\pi }_{11}+\lambda v_{01}(0)+\lambda v_{11}(0)$$ (9c) $$v_{11}(1)(\lambda +\mu )={\pi }_{11}.$$

From this expression, we derive the following interesting conclusions. First, we observe that this system of equation does not depend on β. This means that the Age of Information of the EPN model with a single station, when α→∞, does not depend on the leakage rate of energy packets. Another conclusion is related to the comparison of the above expression with the model studied in Section IVB of Yates & Kaul (2019). We observe that, if we consider λ₂ = 0 in their model, that is, that there is a single source, both models coincide. More precisely, if we consider λ₂ = 0 in (48) of Yates & Kaul (2019), we obtain the same system of equation as the one above. Therefore, the average Age of Information of a single station when α→∞ is the same as that of a single server with preemption in service without energy hasvesting and its value can be derived from Theorem 2 of Yates & Kaul (2019).

Proposition 5 The average Age of Information of a single server with energy harvesting when α tends to infinity is $$\mathrm{\Delta }=\frac{1}{\mu }+\frac{1}{\lambda }$$

Proof. See the proof of Theorem 2 of Yates & Kaul (2019).