# G-Networks and the Performance of ICT with Renewable Energy

## Abstract

As a consequence of developing information and communication technology that is playing a significant role in our society and has changed our life dramatically, we witnessed a significant increase in energy consumption in computer systems and networks. Subsequently, energy harvesting technologies with renewable energy are of great interest in the field of computer systems and networks, and thus lead to abundant research which has been carried out to address energy harvesting from different aspects. However, the majority of them focuses on wireless or small-scale networks, which left wired networks with a general structure neglected. We first present a comprehensive systemic review of the trends of overall energy consumption, and energy and quality of service optimization in computer systems and networks. Then, this paper reviews the recent research progress in G-networks and energy packet networks with renewable and intermittent energy from both the system paradigms and the performance optimization and energy reduction algorithms for the wired networks.

## Keywords

ICT Energy harvesting Energy packet networks G-networks QoS optimization Renewable energy## Introduction

The recent growth of information and communication technology (ICT) has led to a rapid increase in energy consumption. The challenge of achieving energy-efficient ICT has, therefore, been extensively explored, including energy harvesting (EH) technologies with renewable sources of energy. The EH technologies are of great interest because they can potentially reduce the dependence on the electricity grid and use “clean” renewable energy in computer systems and networks. However, the intermittent nature of renewable energy makes it challenging to develop and evaluate the energy harvesting model. Thus, abundant research has been carried out to address challenges in energy harvesting from different aspects.

Recently, various communication networks powered by EH have been considered and investigated. Throughput maximization and transmission completion time minimization in single-user channel were studied in [73, 98, 114]. The model of energy cooperation, which can transmit energy between users wirelessly, was introduced in [68]. An optimal policy for multiple access channels with intermittent data and energy arrivals was investigated in [113]. Then, the model was extended to being energy cooperative in [67]. In [4], an online power scheduling policy was characterized to maximize a general utility function. The delay minimization problem was considered in [5]. In [15], a general framework has been developed to maximize the amount of transmitted data with battery capacity constraints and battery imperfections regarding energy leakage. Then, both energy costs in transmission and processing were considered in [96] to maximize throughput. The survey that summarized the recent advances and results of energy harvesting technologies could be found in [66, 88, 107].

Some of the previous work assumed that channel state information and energy harvesting profiles could be perfectly known [73, 98, 114]. Such systems can easily characterize the system state (both the channel state and the energy level), so as to derive the optimal distribution policies. However, the deviation between predicted energy profile and actual output power may enlarge model mismatch in the long term. Hence, recent work is focusing on the stochastic models in which the energy harvesting process is modeled as a random process. On the other hand, most of energy harvesting models consider optimization of quality of service (QoS) and energy efficiency for small-scale networks with one single node or two nodes [68, 73, 98, 114]. Although some work has also considered large-scale networks with energy harvesting, such as ad hoc networks [74] or cellular networks [75], they are mainly wireless networks. More attention should also be given to wired networks with general structures that can consume massive power but have been largely neglected.

The energy packet networks (EPN) [35, 39, 41, 63, 81, 84, 118] is a discrete state-space modeling framework that can analyze the interaction between discrete energy flows and job flows (or packets) in a single system. Energy packets (EPs), jobs and data packets are the customer classes in the EPN. One EP is a fixed amount of energy stored in batteries or energy stores (ESs), each of which is modeled as a “queue of energy”, while jobs or data packets are normal customers (also called positive customers) circulating at workstations (WSs), servers or sensor nodes. Some EPN models also use a diffusion model with continuous flows, e.g., diffusion approximation, instead of discrete EPs or jobs moving at random times [2], whereas the two main approaches to the EPN are both discrete. One main approach to the EPN was first initiated in [38] to consider one single wireless node that can collect energy through energy harvesting and reap data through sensing. Subsequently, an analytical paradigm has been developed to analyze the performance of the energy harvesting wireless sensor node using a Markov chain representation [81, 84]. Additionally, a new product-form solution (PFS) has been found for tandem networks [84]. The work in [38, 81, 84] assumed that transmission is instantaneous, i.e., it takes zero time on the time scales to sense data, and harvest energy. This approach could be of great interest in autonomous digital devices operating with energy harvesting from intermittent sources.

Another approach to the EPN is based on G-networks [23, 31, 47, 70, 71, 94] that is able to consider more general network structures and the service time for both job processing and energy consumption. G-networks with positive and negative customers were first inspired by the research on biophysical neural networks that can communicate through impulse signals with random emitting intervals, known as random neural networks [26, 33]. Impulse signals traveling through the neural networks are analogous to customers traveling through queueing networks. At receiving neurons (servers), impulse signals are recognized as either “excitations” (positive customers) which add one potential of the neuron (the state of the server) or “inhibitions” (negative customers) which cancel one potential of the neuron (the state of the server), or have no effect if potential is zero. A remarkable property of G-networks is the PFS, which is the steady-state joint probability distribution of the number of positive customers at queues. The work of the EPN exploits G-networks to analyze and evaluate a multi-server system’s performance or QoS, for instance, the average response time of jobs [41, 63], energy efficiency and average response time of jobs [118] or energy reserve [41].

The remaining paper is organized as follows. We first summarize the current state of energy usage in computer systems and networks in Sect. 2. Next, Sect. 3 reviews the recent development of energy-efficient ICT with various approaches. Then, we review the theoretical background of G-networks in Sect. 4; the research trends and challenges of the EPN are discussed in Sect. 5. The distinct approaches to the EPN are in Sect. 6. Finally, conclusions are in Sect. 7.

## Energy Consumption in Computer Systems and Networks

The world’s increasing usage of ICT inevitably led to a rise in energy consumption by ICT. Over the last decade, we have seen several reports that discuss how rapidly growing ICT will unsustainably consume a large amount of the overall available energy.

Some of the early estimations of the global carbon dioxide emissions and energy consumption [24, 25, 110] alerted us to the risk that energy consumption related to ICT would grow at a rapid rate. Recently, its carbon footprint has become comparable to that of air transportation, which is at approximately 1.4% of the total emission of carbon dioxide [92]. There are also more rigorous studies [93, 100] which have suggested that ICT consumed approximately 3.9% of global electricity and contributed to approximately 1.3% of global greenhouse gas emissions in 2007. It was also projected that ICT’s electricity consumption would grow to about 3766 TWh by 2020, representing a 156% increase since 2008.

With these early estimations and studies from before 2010, one may easily conclude that ICT would consume a radically increasing amount of the overall world energy in the next decades, which will eventually hinder global energy sustainability. However, [8, 40, 92] evaluated the impact of different sectors of ICT on energy consumption and carbon dioxide emissions and concluded that growth of energy and carbon footprint of ICT has slowed down in the last decade and is now decreasing because of improved energy-efficient material used in devices and “smart system technology”. According to some recent and detailed reports [89, 108] in 2015, ICT electricity consumption in 2012 was 920 TWh or 4.7% of the 19,500 TWh, representing global electricity consumption. In the latest report published in 2018 [92], ICT electricity consumption in 2015 reduced to 805 TWh out of 21,000 TWh global electricity consumption. Comparing 920 TWh with 3766 TWh or 805 TWh with 3766 TWh, we realize there is a significant deviation between the early projected consumption and actual data. In addition, it also suggested that ICT is playing an essential role in increasing the energy efficiency of a large number of activities. We should also consider ICT’s positive impact on the reduction of overall energy consumption, for instance, in the house and transportation, or by smart management of the power grid.

Even though energy consumption of ICT is now decreasing, 805 TWh is a large quantity of electricity, and everything must be done to reduce energy consumption by ICT further.

## Energy and QoS Optimization

Previous work [8, 34, 52] has discussed the development of energy-efficient ICT with various approaches for different system layers. One of the major challenges is to explore the relations among system components and the trade-off that can result in an optimal balance between performance, QoS, and energy consumption. [8].

This challenge has been explored in many aspects. In the rest of this section, we comprehensively review the energy reduction and QoS optimization in ICT from various perspectives: data centres, energy–QoS trade-off in cloud computing, QoS and energy-aware routing, and energy harvesting with intermittent energy sources.

In recent years, massive data centers have become the backbone of the Internet, leading to a huge amount of energy usage. In [86], the author mentioned that data centers used approximately 0.5% of total world electricity consumption in 2005, and that this number increased to 1.1% in 2015 [92]. Although the energy needs of ICT are being fed by the fast-rising demand for data and the increasing amount of installed hardware, data center electricity consumption has been growing moderately since increased Internet traffic, and loads are offset by much increased efficiency including ultra-efficient data centers and the development of “smart system technology”, for instance, cloud computing that shares resources (processor and other hardware) to prevent the servers remaining idle [51, 101, 109].

Regarding ultra-efficient data centers, the well-known Koomey’s law [87] which describes a long-term trend in the development of computing hardware, concludes that the amount of computation per kilowatt-hour has doubled approximately every 1.52 years. This trend was remarkably stable and was faster than Moore’s Law since the 1950s. Re-examined by Koomey in 2011, the doubling period has slowed to every 2.6 years since 2000, from which Moore’s Law has also slowed as the rate of progress would reach saturation. As reported in [102], the energy usage of data centers in the United States can potentially decline by 25% if four-fifths of servers in small data centers are transferred to new hyper-scale ultra-efficient data centers. Presented in [8], ICT energy consumption depends primarily on CPU utilization. However, it is surprisingly hard to achieve high average levels of utilization of typical servers, and the utilization of CPUs is even worse with personal computers and laptops. Moreover, the most common operating mode observed in [7] (10–50% of the maximum utilization levels) unfortunately corresponds to the lowest energy-efficiency region in many current servers. Thus cloud computing, which has attracted much attention, is a promising approach to improve the utilization of data center resources and to save energy. Key ideas include moving services to those servers which operate at the most energy-efficient regime, reducing the number of redundant servers and turning unused servers off, or into “sleeping” mode to achieve energy efficiency. The hotspots generated at data centers should also be measured when a node or server is used excessively, and services or jobs can be transferred to some other nodes with a lower temperature or lower loads to avoid hotspots, and also reduce the overhead expense on cooling.

Because of the development of mobile networks and the IoT, in recent work, energy-aware routing protocols for wireless networks have also drawn much attention. The ad hoc cognitive packet networks in [48] develop an intelligent environment for seeking fast and energy-efficient routes with battery-powered nodes [49]. Topology control was considered in [78], to modify the network graph and optimize QoS. A search algorithm [1] for finding destinations in a random environment with a team of searching packets has explored the interaction of the number of searchers and energy consumption.

However, more attention should also be placed on wired networks that consume massive power and have been largely neglected. A software-based “Energy Management System” for wired packet networks was developed in [59, 60] to reduce energy consumption subject to QoS requirement. In [55], the optimization of a cost function that includes power consumption and QoS metrics was solved using the G-network theory. In other work [56, 57], optimization algorithms were developed to route packets and minimize composite cost functions combining overall network energy consumption and QoS.

Large numbers of heterogeneous digital devices and communication nodes are being incorporated into the Internet [6, 14, 65, 99, 111] to manage cities and various service activities [3, 9, 10, 13, 36, 62, 116], which also leads to more energy consumption by ICT. Such systems must operate autonomously over a long period, and can benefit from collecting energy physically from natural phenomena such as wind, liquid flows, photovoltaic energy, and other renewable energy sources. This has triggered a significant interest in investigations of EH for computer ommunication systems. Much work and research have been conducted to evaluate performance, optimize QoS and energy metrics regarding power consumption and sustainability for such systems [17, 50, 69, 85].

In [90], data transmission with energy harvesting nodes has been characterized by a continuous-time Markov chain, and the optimal online control policy for data was derived by dynamic programming. To reduce the computational complexity of the dynamic programming, in [98], the authors have investigated the simple optimal power transmission for a single channel with a rechargeable battery node. Their work has provided optimal policies to maximize the throughput and minimize the data transmission time. Related work from various perspectives can be found in [15, 73, 96, 114] which assumes, somewhat unrealistically, that channel state information and energy harvesting profiles can be perfectly known, such that statistical knowledge and causal information of energy and channel variation are available to drive the proposed optimal policies.

In other work, a two-user multiple access communication channel [68, 113] was considered, where energy has been discretized in a packet form which can be transferred between users through wireless energy transfer [76], while [67] considered energy harvesting in a two-user cooperative Gaussian multiple access channel. Some of the literature up to 2015 has been reviewed in [107], with emphasis on energy harvesting wireless communication and energy transfer from the perspective of communication and information theory.

Energy harvesting wireless sensor networks are different from general energy harvesting wireless communication systems,because devices need to meet the requirements of data transmission as well as source acquisition, which involves sensing, sampling, and compression, and entails an energy cost compared with that of data transmission. Thus, an optimal energy allocation for the limited energy resource to source acquisition and data transmission is required [64, 72, 79]. The optimal policies that only consider the energy harvesting process and data transmission should be revised to take resource acquisition, for instance, the quality of the measurement taken by the sensor into account. In [11], a single sensor with a signal receiver was considered, where energy harvesting processed in each time slot follows an ergodic stationary process. A case study of multiple sensors is in [106], and sensor reliability was studied in [91]. Moreover, in [97], the optimal resource allocation policy, which is subject to delay constraints, was derived. In [80], it used the energy packet network paradigm (a distinct approach from the EPN paradigms based on G-network theory) to model the wireless sensor networks with energy harvesting, in which a wireless sensor consumes one EP for data sensing and processing and another one EP for data transmission. In [83], The work was extended to the framework that data sensing and processing will consume a variable number of EPs, and data transmission also consumes a variable number of EPs.

The above work focused on point-to-point channels or small-scale systems powered by energy harvesting. Large-scale wireless networks powered by energy harvesting such as mobile ad hoc networks and cellular networks have received some attention recently. The spatial throughput of a mobile ad hoc network with energy harvesting was analyzed in [74]. A queueing model of an energy-efficient wireless base station is presented in [17]. In [75], the authors investigated the coverage of a cellular network, which is affected by the variations in energy sources. Moreover, heterogeneous cellular networks with energy harvesting were modeled in [16].

Recent work has developed the EPN paradigm based on G-network theory [21, 36, 37]. This framework is able to evaluate and analyze the interaction of energy flow and job flow for large-scale wired networks. We note that there is also a distinct approach to the EPN, which does not apply the G-network theory. We review the G-network theory and the EPN approach based on G-networks in the following sections.

## G-Networks

G-networks that are advanced queueing networks with positive and negative customers were first introduced in [27, 29] and were inspired by the study of biophyiscal neural networks [26, 28, 32, 61]. This work started in 1990 and has continued until today. A remarkable and useful property of a G-network is the PFS which is the steady-state joint probability distribution of the number of customers at queues [29, 54]. In addition to ordinary customers or positive customers, G-networks have negative customers which can arrive from outside the networks or can be moved from one queue to another. When a negative customer arrives at a queue, the negative customer can “cancel” a positive customer at this queue. If the queue is empty, the negative customer disappears at which it arrives because negative customers cannot accumulate at queues. A single server system with negative and positive customers was studied in [43]. Stability conditions for G-networks were given in [58]. Multiple class models and relevant work were developed in a series of papers [12, 22, 23, 47, 95]. A special type of customer known as a “trigger” which is able to push an existing customer at a queue to another queue was introduced in [30]. Note that only positive customers can be pushed, because neither negative customers nor triggers can wait at a queue. The G-network model that allows to either trigger a customer from one queue to another or remove a batch of customers at this queue was considered in [31].

There are also other G-network models. For instance, G-networks with “resets” [42, 70], with “adders” [20] and with restarts [22, 23], which have been developed and adopted in a number of applications in computer systems modeling and other fields.

In the rest of this section, we briefly review G-networks with positive and negative customers. Finally, we review G-networks with batch removal which are the fundamental basis of the latest EPN paradigms.

### Random Neural Networks and G-Networks with Positive and Negative Customers

Here we consider an open network with *N* servers with mutually independent service times which are modeled as independent and identically distributed (i.i.d.) random variables that follow an exponential distribution with parameter *r*(*i*) where \(i=1,\dots ,N\). External arrivals at the network can either be positive customers arriving at queue *i* according to a Poisson process with the rate of \(\lambda _i^+\) or negative customers arriving at queue *i* according to a Poisson process with the rate of \(\lambda _i^-\). We assume a customer who has been served leaves queue *i* and moves to queue *j* with probability \(P_{ij}^+\) as a positive customer or with probability \(P_{ij}^-\) as a negative customer. The served customer also can be removed from the network with probability \(d_i\). We note that \(1=d_i + \sum _{j=1}^N P_{ij}^+ +P_{ij}^-\) for all \(i=1,\dots ,N\), because total probability is one. Thus, the transition probability of the Markov chain can be written as \(P_{ij}= P_{ij}^+ + P_{ij}^- \) which represents the customer movements between servers. The queue length at servers is constituted only by positive customers, each of which adds one queue length at which it arrives, while one negative customer that does not require service cancels one positive customer waiting at the queue and reduces queue length by one. The negative customer does not affect the queue if the queue length is zero. Moreover, the queueing discipline is assumed to be the first-come-first-served (FCFS) basis, i.e., customers are served in the order of their arrival.

*i*. Here we recall the PFS of G-networks, which is a steady-state joint probability distribution of the number of positive customers at queues. Let \({\mathbf {K}}(t)=(k_1(t),\dots ,k_N(t))\) be the vector of queue lengths at time

*t*, and \({\mathbf {k}}=(k_1,\dots ,k_N)\) be an arbitrary vector of queue length.

### Theorem 1

*If a unique non-negative solution of the traffic equations given in*(1)

*and*(2)

*exists such that*\(0<q_i<1\)

*for all*\(i=1,\dots ,N\)

*hold, then the following PFS:*

*exists.*

### Proof

Since \(\{ \mathbf{K }(t): t \ge 0 \}\) is a continues-time Markov chain that satisfies the Chapman–Kolmogorov Equation, we can verify that the PFS satisfies the global balance equations. Although it is similar to that in Jackson’s Theorem [77], the proof is more complicated because of the negative customers canceling the customers waiting at queues and the nonlinearity of traffic equations. Since the proof has been given in many G-network papers, we do not provide the proof in this paper. Interested readers can find details about the proof in [29]. \(\square \)

### Remark 1

*i*is

### Remark 2

*i*is the product of the marginal probability distribution, which is

### G-Networks with Batch Removals

- 1.
With probability \(0 \le 1- D_i \le 1\), the arriving signal is “trigger” which instantaneously moves one customer at the head of queue

*i*to queue*j*with probability \(M_{ij}\) where \(1=\sum _{j=1}^N M_{ij}\) for \(i=1,\dots ,N\). - 2.With probability \(D_i\), the arriving signal, which is “negative customer” instantaneously removes a batch of customers with size up to \(X_i\). The size \(X_i\) is a random variable following a given probability distribution \(\pi _{i}(s)\) whereand$$\begin{aligned} \pi _{i}(s)= \Pr [X_i=s],\quad s=1,2,\dots , \end{aligned}$$(7)Thus, the average number of customers removed by one single “negative customer” signal is$$\begin{aligned} \sum _{s=1}^\infty \pi _{i}(s) = 1, \quad \forall i=1,\dots ,N. \end{aligned}$$(8)If the queue length is less than \(X_i\) at time$$\begin{aligned} E[X_i]=\sum _{s=1}^\infty s \pi _{i}(s), \end{aligned}$$(9)
*t*, i.e., \(X_i > k_i(t)\), the queue length becomes zero at the next time step \(t^+\) after service. It can be expressed by$$\begin{aligned} k_i(t^+)= {\left\{ \begin{array}{ll} k_i(t)-X_i, &{}\text { if } k_i(t) \ge X_i,\\ 0, &{}\text { otherwise }. \end{array}\right. } \end{aligned}$$(10)

The external arrival rate of positive customers, represented by the first term of the right side of the equation.

The arrival rate of positive customers from other nodes, some of which are served positive customer moving to node

*i*according to the transition matrix \({\mathbf {P}}^+=[P_{ji}^+]\). Other positive customers moving into node*i*are triggered by the trigger from node*j*according to the transition matrix \({\mathbf {P}}^-=[P_{jm}^-]\). This rate is represented by the second term of the right side of the equations.The arrival rate of positive customers from other nodes at which positive customers are triggered by trigger signals arriving from the external, represented by the third term of the right side of the equation.

The external arrival rate of signals, represented by the first term of the right side of the equation.

The total arrival rate of signals from other nodes where some served positive customers become signals. This rate is represented by the second term of the right side of the equation.

*i*, and also the probability that queue

*i*is not empty. Then we have the function \(f_i(\cdot )\) which is related to the average number of customer that one “negative customer” signal can remove, and it depends on the state of the queue.

The PFS given in Theorem 1 is also valid for G-networks with batch removals. The uniqueness and existence of the PFS is guaranteed if a unique non-negative solution of (13) exists and we have \(0<q_i<1\) for all \(i=1,\dots ,N\). Interested readers can find the proof in [31]

## Energy Packet Networks Derived from G-Networks

The concept of the EPN begins with the integration of adaptive electrical ESs charged by renewable energy sources and distributed consumption systems such as WSs, servers, or sensors. In addition, the EPN paradigms use each EP to represent the fixed amount of energy in Joules, which can also be viewed as a pulse of power that lasts a certain time [36]. The amount of energy in such an EP can be small enough to be close to the smallest energy needs of consumers [36, 39, 41, 84], or large enough to be a significant quantity to power large energy consumers [115]. In the latest research [63, 117, 118], the number of jobs or data packets that can be executed by one single EP is a random variable. Thus if a WS is energy-efficient, it will execute more jobs with a single EP. This relaxed the previous assumption that an EP is the smallest amount of energy needs of consumers, i.e., an EP was used to process only a single job. Thus, the EPN offers a more explicit representation of the interaction between energy flow and job or data flows in computer networks powered by intermittent energy sources. Such EPN paradigms are developed for an environment where renewable sources are common, and energy storage facilities or devices are available [36].

The EPN was first inspired by investigating random arrivals of energy from intermittent energy sources, and random arrivals of jobs or data packets in one single system. Some classes of EPN can be described by G-network theory. Such EPN systems have an important and convenient property which is the existence of a product form of steady-state distribution of jobs in WSs and EPs in the ESs, namely the PFS. With the PFS, it is computationally conventional to evaluate, and so as to improve the QoS and energy efficiency.

It is worthy to notice that there is another approach to the EPN paradigm which is not related to the G-networks. These EPN paradigms, which are not based on G-networks, are associated with various stochastic models. Some EPN use diffusion approximation with continuous flows, instead of discrete EPs or jobs moving at random times [2]. A distinct approach to the discrete EPN in [38, 44, 45, 84] was developed for the devices with “zero service time”. For instance, a device or sensor may process a data packet in nanoseconds or microseconds, and also consume an EP in milliseconds, which are negligibly small.

Note the EPN is not only a theoretical approach. Other independent researches [103, 104, 105] have proposed the “power packet” system, which is a practical hardware-based design for switching power and dispatching data packets simultaneously.

In the following sections, we review the recent development of the EPN paradigms comprehensively. First, we review the initial work of EPN based on G-networks where one EP can be used to process one single job, and its relevant applications and problems. Next, we review the latest model of the EPN, where the number of jobs that can be processed by one single EP is a random variable.

### G-Networks with Negative Customers of the EPN and its Optimizations

The general structure of the EPN^{1} has been considered in [41] which is modeled as a queuing network with a finite set of nodes. The networks has *m* ES nodes denoted as \(E_a\) where \(1 \le a \le m\), and has *n* WS nodes denoted as \(W_i\) where \(1 \le i \le n\). Jobs that must be executed in WSs are ordinary customers in the queueing network. They arrive at one of *n* WS nodes, say \(W_i\) according to the Poisson process at a rate of \(\lambda _i\) jobs/s. WS \(W_i\) may also receive a job from another WS, saying \(W_j\) with probability \(M_{ji}\) after WS \(W_j\) finishing process of that job. The energy from an intermittent external source is discretized to EPs which arrive at one of *m* ES nodes, say \(E_a\) according to Poisson process at a rate of \(\gamma _a\) EPs/s. Similarly, ES \(E_a\) may also transmit one EP to another ES \(E_b\) with probability \(P_{ab}\) to balance energy distribution.

*t*. It is assumed that both WS nodes and ES nodes have a queue with infinity capacity. EPs at ES \(E_a\) are expanded because of energy leakage, or moved to WSs for job processing. When ES \(E_a\) is not empty at time

*t*(i.e., \(L_a(t) > 0\)), it has the following manner:

- 1.
The successive EP leakage times at ES \(E_a\) are modeled as i.i.d. random variables following a common exponential distribution with parameter \(\delta _a\).

- 2.or ES \(E_a\) can forward one EP either to another ES to balance energy distribution, or to one corresponding WS to power job processing. The successive EP forwarding times at ES \(E_a\) are also i.i.d. random variables that follows a common exponential distribution with parameter \(w_a\).
- (a)
With probability \(F_{ai}\), the EP stored in ES \(E_a\) was forwarded to WS \(W_i\) for job processing.

- (b)
As mentioned \(P_{ab}\) is the probability that ES \(E_a\) sends energy to another ES \(E_b\) to balance energy distribution. Thus, we have \(1 = \sum _{i=1}^n F_{ai} + \sum _{b=1}^m P_{ab}\).

- (a)
- 3.
In summary, energy leaks at a rate of \(\delta _a\) EPs/s, or is forwarded to another ES or WSs at a rate of \(w_a\) EPs/s. Thus, the number of EP at \(E_i\) is one less (i.e., \(L_a(t^+) = L_a(t) -1\)) after the average time of \((\delta _a+w_a)^{-1}\) seconds.

*t*(i.e., \(K_i(t) > 0\)), the following happens:

- 1.
With probability \(d_i\), the job that has completed processing is removed from WS \(W_a\) without forwarding to other WSs because the job terminates at \(W_a\).

- 2.
With probability \(M_{ij}\), the job that has completed processing at WS \(W_i\) is forwarded to WS \(W_j\) for further more processing, where \(1=d_i + \sum _{j=1}^n M_{ij}\).

*m*queues are ESs and

*n*queues are WSs). The network has two classes customers (\(C=2\)) where Class 1 refers to the jobs to be processed in the WSs, and Class 2 refers to the EPs to power job processing. The jobs are positive customers at WS queues, and the EPs are positive customers at ES queues. However, one EP becomes a negative customer or a trigger when it arrives at a WS queue. The EP is expended to finish one job processing at one WS, saying \(W_i\). After the job has completed its processing, the job will be forwarded to another WS node, for instance \(W_j\), for more processing with probability \(M_{ij}\) where the EP behaves as a trigger, or the job will be removed from the network with probability \(d_i=1-\sum _{i=j}^n M_{ij}\) where the EP behaves as a negative customer.

- 1.
The \(\varLambda _{1,i}^+\) denotes the total effective arrival rate of jobs at WS \(W_i\). The first term of right-hand side of (15) refers to the job arriving from external. The second term refers to the arrival rate of jobs from other WSs. These jobs have completed processing at other WS, and require more process at WS \(W_i\).

- 2.
The \(\varLambda _{1,i}^-\) and

*r*(1,*i*) denotes the total effective rate of job leaving from WS \(W_i\). The term*r*(1,*i*) refers to the rate of job lost at time out without the need for energy. The term \(\varLambda _{1,i}^-\) refers to the job processing rate, which requires the arrival of EPs from ES nodes. These jobs that have completed processing at WS \(W_i\) are to be forwarded to other WS nodes for more processing, or to be removed without forwarding to other WS nodes. - 3.
The \(\varLambda _{2,a}^+\) denotes the total effective arrival rate of EPs at ES \(E_a\). The first term of right-hand side of (17) refers to the rate of EPs arrives from the external intermittent energy sources. The second term refers to the arrival rate of EPs from other ES nodes for energy balance.

- 4.
The \(\varLambda _{2,a}^-\) and

*r*(2,*i*) denotes the total effective rate of EPs left from ES \(E_a\). The first term of*r*(2,*i*) refers to the rate of EPs that is sent to either other ES nodes or WS nodes. The second term refers to the rate of energy lost because of leakage. Since energy cannot be destroyed, the term \(\varLambda _{2,a}^-\) is null.

### Result 1

*Let*\(k_1,\dots ,k_n\)

*represent the backlogs of jobs to be processed at the WS nodes, and*\(l_1, \dots , l_m\)

*represent the number of EPs stored at the ES nodes. If a unique non-negative solution of the traffic equations in*(15)

*to*(18)

*exists such that*\(q_{1,i},~q_{2,a} \in (0,1)\)

*for all*\(i=1,\dots ,n\)

*and*\(a=1,\dots ,m\)

*hold, then the following PFS*

*exists.*

### Relevant Applications and Optimization Problems

#### Utility Functions and Optimizations

- 1.We may wish to limit the backlog of job at WS queues, and we also want to reserve energy as much as possible in ES queues for unpredictable needs. Thus, a sensible utility function iswhere \(\alpha _i>0\) and \(\beta _a>0\) are weights. This is a sum of the weighted probability that the backlog of jobs exceeds the value \(k_i\) and the weighted probability that the number of EPs is less than \(l_a\). By minimizing this utility function, it is able to reduce the probability of backlog of jobs and increase the probability of more energy reserved at ES nodes. When \(k_i =1\) and \(l_a=1\), the utility function becomes$$\begin{aligned} U_1 = \sum _{i=1}^n \alpha _i q_{1,i}^{k_i} + \sum _{a=1}^m \beta _a \left( 1-q_{2,a}^{l_a}\right) , \end{aligned}$$(22)To minimize this special function, we want to reserve energy as much as possible and reduce the backlog of jobs as many as possible.$$\begin{aligned} U_1^0 = \sum _{i=1}^n \alpha _i q_{1,i} + \sum _{a=1}^m \beta _a (1-q_{2,a}). \end{aligned}$$(23)
- 2.A very similar utility function considers the average response time of jobs waiting at WS nodes rather than the backlog of jobs. It is$$\begin{aligned} U_2 = \sum _{i=1}^n \alpha _i \frac{(\varLambda _{1,i}^-)^{-1}}{1-q_{1,i}} + \sum _{a=1}^m \beta _a \left( 1-q_{2,a}^{l_a}\right) , \end{aligned}$$(24)
- 3.The third utility function considering the throughput of the system and reserved energy needs to be maximized:$$\begin{aligned} U_3 = \sum _{a=1}^m \left( \sum _{i=1}^n \alpha _i q_{2,a} F_{ai} q_{1,i} +\beta _a q_{2,a}^{l_a}\right) , \end{aligned}$$(25)

#### The EPN Used for Mobile Networks with Energy Harvesting

The EPN approach has been used for system analysis of a backhaul multi-hop connection of a wireless mobile network with energy harvesting [39]. The data or other traffic are carried out as data packets (DPs) rather than jobs. These DPs arrive from outside the network in the form of a random process representing the data or other traffic generated by users. DPs stored in data buffers can enter the next node or leave the system. Thus, this EPN has a tandem structure that considers a multi-hop connection traversing the nodes \(N_1, \dots ,N_n\), which can be base stations, routers, or WiFi nodes operating with intermittent harvested energy.

It is assumed that node \(N_1\) receives end-to-end user traffic (voice or SMS) which travels over *n* node to its destination. Thus, this is a mobile connection transmitting DPs along the full path of nodes \(N_1, \dots , N_n\). If a DP cannot be forwarded from node \(N_i\) to node \(N_{i+1}\), \(1\le i \le n-1\) or node \(N_n\) cannot transmit the DP to the external network, the DP is lost and needs to be retransmitted from the first node \(N_1\) with probability *p*. End user generates fresh user traffic at an aggregate rate of \(\lambda \) DPs/s at node \(N_1\) according to a Poisson process. Each node \(N_i\), \(1\le i \le n\) also receives a cross-traffic Poisson flow \(\varLambda _i\) DPs/s according to Poisson process. The external arrival processes are assumed to be independent of each other.

Each node \(N_i\) is powered by an ES charged from intermittent energy sources at rate of \(\gamma _i^H\) EPs/s, or also from the grid at rate \(\gamma _i^G\) EPs/s, so that the total energy supply rate at the nodes is \(\gamma _1, \dots , \gamma _n\), where \(\lambda _i=\lambda _i^H + \lambda _i^G\), \(i=1,\dots ,n\). The energy (EPs) received by a node is first stored in an ES of unlimited capacity. The ES at node \(N_i\) leaks at rate \(\delta _i\), so that the number of EPs stored at the battery is one less after the exponentially distributed time of an average of \(\delta _i^{-1}\). Note that the nodes are efficient that they only use energy when they process and forward a DP, and one EP is the amount of energy consumed by the node to process and forward one DP.

- 1.
The first case considered the type of systems which has been suggested in [112]. The system is very fast so that data buffering is not needed, and the equipment’s power consumption is directly proportional to the load, which goes to “sleep” mode as there is no work to do. Thus, it is assumed that the backhaul network is so fast that no buffering of DPs is needed. The system is also so efficient such that each node only consumes energy when DPs are processing or forwarding, while no energy is consumed when a node is “sleeping”. It assumes one DP is lost when the DP attempts to transit through a node that has run out of energy (i.e., battery is depleted, and its energy source cannot provide power).

- 2.
The second case assumes that nodes store packets until they can be forwarded, and the nodes are kept on even when they have no packets to forward. Thus, only DPs in transit may be lost, while DPs in the memory of a node is preserved.

- 3.
The third case is the “worst case”. It is similar to the second case, while DPs stored in memory will be lost when that node has run out of energy.

### G-Network with Batch Removals of the EPN and Its Optimization Problems

The latest work modified the EPN model by adapting G-networks with batch removals to model a multi-server system consisting of servers of WSs which are powered by an ES that is charged from an intermittent source. It uses energy flow and job flow together to optimize a multi-server system’s performance or QoS, for instance, the average response time of jobs. Throughout the work, it relaxed the assumption of the previous EPN paradigms that one EP can only be used to execute one single job. The number of jobs can be executed by one EP is a random variable of \(X_i\) following general probability distribution.

The EPN consists of *N* WSs and *N* ESs. Jobs that must be executed in the system are modeled as ordinary customers in a queueing network. They arrive at one of the *N* WSs, say WS *i*, at a rate of \(\lambda _i\) jobs/s. Each WS is represented as a queue containing jobs. Jobs first arrive at a given WS *i*. Each WS *i* has an energy storage battery denoted by ES *i*, and there are a total of *N* ESs. EPs arrive from an external intermittent energy source at rate \(\gamma _i\) EPs/s to the ES *i* which can be viewed as a “queue of EPs”.

In the EPN model, the EPs in ES *i* either can be forwarded to the corresponding WS *i* on demand with probability \(d_i\), or moved to another ES *j* with probability \(P_{ij}\) to balance the energy distribution. The jobs in WS *i* can be processed locally with probability \(D_i\) or forwarded to some other WS *j* with probability \(M_{ij}\) for further steps of execution. In this model, \(w_i\) is the rate at which EPs are forwarded from ES *i* to one of the ESs or the corresponding WS. On the other hand, \(\delta _i\) is the loss rate of energy (i.e., leakage) from ES *i*.

It denotes the number of jobs at WS *i* by \(K_i(t)\) and the number of EPs at ES *i* by \(B_i(t)\) at time *t*. It assumes that both the WS queues, and the ES queues (i.e., batteries) are unbounded, i.e., infinite capacity. EPs at ES *i* are expended in the following manner:

*i*is not empty, i.e., \(B_i(t)>0\), ES

*i*will:

- 1.
Either leak energy at some rate \(\delta _i \ge 0\) EPs/s, and after a time of average value \(\delta _i^{-1}\), we will have one less EP at ES

*i*due to energy leakage. The successive EP leakage times for ES*i*are modeled as i.i.d. random variables having a common exponential distribution with parameter \(\delta _i\). - 2.
Or ES

*i*will forward one EP at rate \(w_i\) EPs/s to WS*i*or other ESs. The successive times (that one EP is forwarded) for ES*i*are also modeled as i.i.d. random variables having a common exponential distribution with parameter \(w_i\).

*i*, this EP is used locally by WS

*i*as follows:

- 1.With probability \(0 \le D_i \le 1\), if \(K_i(t)>0\), one EP will be expended to serve a batch of up to \(X_i\) jobs at the WS
*i*in one step. After service, we end up withSince each job may have different energy requirements at WS$$\begin{aligned} K_i(t^+)={\left\{ \begin{array}{ll} K_i(t)-X_i, &{}\text { if } ~K_i(t) \ge X_i,\\ 0, &{}\text { otherwise}. \end{array}\right. } \end{aligned}$$(27)*i*, we assume that the number of jobs that can be processed with a single EP at WS*i*is a random variable. - 2.
Since our purpose is to model different WSs that have different levels of energy efficiency, a single EP is used to process one or more jobs, if there are jobs waiting in the WS queue.

- 3.
With probability \(1-D_i\), if \(K_i(t)>0\), one EP will be used to serve just one job, and then forward that job to another WS

*j*according to the transition probability matrix \({\mathbf {M}}=[M_{ij}]\). As a result, we will have \(K_i(t^+)=K_i(t)-1\) and \(K_j(t^+)=K_j(t)+1\). In the mathematical model, the transition matrix allows the job to return to the same workstation, i.e., the diagonal entries are not null. However, it is not physically meaningful because it wastes one EP to move the job, which is at the head of the WS queue to the tail of the WS queue. Thus, we assume that the diagonal entries of the transition probability matrix are null, such that the jobs cannot return to the same WS. - 4.
If an EP arrives at an empty WS

*i*, i.e., \(K_i(t)=0\), the EP will just be expended to keep the WS in working order (i.e., to keep it on), and no jobs will be processed or moved.

*i*has at least one job (if \(c=1\)) or one EP (if \(c=2\)). For \(i=1,\dots ,N\), we have \(q_{1,i} \ge 0\) and \(q_{2,i}=0\) because EPs cannot wait at WSs; and we have \(q_{2,i+N} \ge 0\) and \(q_{1,i+N}=0\) because jobs cannot wait at ESs.

Because the EPN model we have described is a special case of a G-network with two classes of customers, namely jobs for Class 1 and EPs for Class 2, we can directly apply the PFS given in G-network theory to the EPN model. For this case, i.e., where we model an EPN, each of the queues is either a WS or an ES. WSs only contain Class 1 customers, and ESs only contain Class 2 customers. Therefore, the PFS of the EPN models is

### Result 2

*If the traffic equations given in*(28)

*and*(29)

*have a unique solution such that all*\(q_{1,i}\)

*and*\(q_{2,i+N}\)

*lie between 0 and 1 for*\(i=1,\dots ,N\)

*, the following*PFS

*holds:*

*i*which are, for \(c=1,2\):

#### Optimization Problems

- 1.
Problem 1 investigates how to select the optimal fraction of power that is shared between heterogeneous servers, so as to minimize the average response time of jobs. Problem 1 was solved analytically using the Lagrange multiplier method. In addition, a physically meaningful condition was obtained to guarantee system stability and optimality.

- 2.
Problem 2 investigates how to select the optimal fraction of power that is shared between heterogeneous servers, so as to minimize the average response time of jobs. Problem 2 minimizes the average response time of jobs by dynamically deciding whether to move jobs between servers, so as to balance the workload at each server. Problem 2 is solved numerically through gradient descent.

- 3.
In Problem 3, it considers a cost function that combines both the average response time and the rate of energy loss. It selects the optimal fraction of power shared between heterogeneous servers to minimize the cost function. The optimal solution was obtained by solving a system of equations simultaneously.

- 4.
Problem 4 investigates how to match energy flow into energy buffers and job flow into the corresponding WSs, so as to minimize the average response time of jobs. Through the investigation, the optimal solution must follow a necessary condition that the fraction of energy flow to ES

*i*must align with the fraction of jobs flows WS*i*.

## Distinct Approaches to the Energy Packet Network

### Energy Packet Network with Zero Service Time

A distinct approach to the EPN was first initiated in [38] in which the author considered a single wireless node that can collect energy through energy harvesting and reap data through sensing. In the EPN based on G-networks, we consider “service time” for both energy consumption and job processing. However, in the distinct approach, it considers devices with “zero service time”. For instance, a sensor may process a packet of data in nanoseconds or microseconds, and also consume an EP in milliseconds, which are negligibly small. Such a system is unstable if the data buffer and energy storage have infinite capacities, while an explicit expression for the joint probability distribution of the number of data packets and EPs was derived in [38] when both buffers are finite. Moreover, with respect to such a system, the author provided an analytical analysis for the mathematical model in which data flows and energy flows can be viewed as instantaneously synchronized flows. Developed in [53], the analysis was extended for two-node systems.

Based on the early work of [44, 45], an analytical model was used to analyze the performance of the energy harvesting wireless sensor nodes using a Markov chain representation. A generalized model of a data transmission system with energy harvesting was considered in [82] where a data packet transmission needs more than one EP. It also investigated the probability that a receiver correctly receives a packet in the presence of noise. The work of [80] models the wireless sensor networks that source acquisition (data sensing and processing) consumes one EP, and data transmission also consumes exactly one EP. The work was extended in [83] where a variable number of EPs will be consumed by data sensing and processing, and data transmission, respectively. In [84], the authors derived a new PFS, which is a distinct approach from G-network, for a tandem network of *N* nodes using harvested energy stored in batteries. The effect of battery attacks at the nodes of energy provisioning systems was investigated in [46]. A simple mitigation technique was proposed for a wireless node with a renewable or a replaceable battery to drop a fraction of the traffic to prolong battery lifetime.

### Energy Packet Networks with a Multiple Class Extension and Service Disciplines

Another distinct approach to the EPN considers multiple classes of DPs, where each class of a DP has an independent routing matrix and determines the number of EPs needed to be sent [18, 19]. In this approach, it assumes the DP queue (or WS) is the initiator of the transfer, which means the arrival of a DP at the corresponding ES triggers the movement of EPs (which are consumed). If the energy level at the ES is not enough to power the arrived DP, the DP is lost. Moreover, one of the three following types: the FCFS, the last-come-first-serve with preempt and resume (LCFS-PR) and the processor sharing (PS) is the service discipline for DP queues.

The sufficient conditions for the stability of this EPN approach with an arbitrary topology have been provided. Moreover, such an EPN also possesses the product form of the steady-state distribution of DPs in the queues provided that a stable solution of a fixed point problem exists. Therefore, the distinct approach can be used for a new type of optimization problems for the use of energy based on the fragmentation of data with several routes.

## Conclusion

In the paper, we have briefly surveyed the current state of energy usage in ICT, and also challenges and opportunities corresponding to developing EH technologies on the computer and network systems when EH can be used to replace or complement power supply from the grid, and relevant problems and applications.

Although the growth of energy consumption and the carbon footprint of ICT has decreased, it is still a large quantity of electricity consumption. Thus, a series of researches were carried out to achieve energy-efficient ICT. However, the majority of them focused on wireless or small-scale networks because of the boom of IoT, which resulted in wired networks being largely neglected.

Then, we reviewed a novel EPN-based approach to analyze the interaction of energy flow and job flow in multi-server systems. The approach to the EPN we reviewed in the paper is based on the G-network theory, which is an advanced queueing network with negative customers. We first review the theoretical background of G-network, then we discussed the initial and latest development, challenges and applications of the EPN.

We expect that there will be further work on the development and analysis of energy-efficient ICT, especially the wired networks that operate with renewable and intermittent energy sources. We hope this present paper can provide some insight into the operation and design of such systems.

## Footnotes

- 1.
We slightly abuse the term of “EPN” to refer “EPN based on G-networks” in the following sections.

## Notes

### Compliance with ethical standards

### Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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