This paper introduces a special issue of this journal (Probability in the Engineering and Informational Sciences) that is devoted to G(elenbe)-Networks and their Applications. The special issue is based on revised versions of some of the papers that were presented at a workshop held in early January 2017 at the Séminaire Saint-Paul in Nice (France). It includes contributions in several research directions that followed from the introduction of the G-Network in the late 1980s. The papers present original theoretical developments, as well as applications of G-Networks to Machine Learning, to the performance optimization of energy systems via the novel Energy Packet Networks formalism for systems that operate with renewable and intermittent energy sources, and to packet network routing and Cloud management over the Internet. We introduce these contributions from the perspective of an overview of recent work based on G-Networks.

This paper is devoted to the research of an open Markov queueing network with positive customers and signals, and positive customers batch removal. A way of finding in a non-stationary regime time-dependent state probabilities has been proposed. The Kolmogorov system of difference-differential equations for state probabilities of such network was derived. The technique of its building, based on the use of the modified method of successive approximations combined with a series method, has been proposed. It is proved that the successive approximations converge over time to the stationary state probabilities, and the sequence of approximations converges to the unique solution of the Kolmogorov equations. Any successive approximation can be represented as a convergent power series with infinite radius of convergence, the coefficients of which satisfy the recurrence relations; that is useful for estimations. Model example illustrating the finding of time-dependent state probabilities of the network has been provided.

In this paper, we study a generalization of the classical multi-dimensional Erlang loss model with state-dependent arrival and service rates, in which customers at arrival occupy random quantities of various resources and release them at departure. Total amount of resources allocated to customers cannot exceed predefined maximum levels. There can be two types of customers: positive customers, which occupy positive quantities of resources, and negative customers, which occupy negative quantities of resources. Negative customers increase the amount of resources available to positive customers and therefore decrease blocking of positive customers caused by lack of resources.

We consider open networks of queues with Processor-Sharing discipline and signals. The signals deletes all the customers present in the queues and vanish instantaneously. The customers may be usual customers or inert customers. Inert customers do not receive service but the servers still try to share the service capacity between all the customers (inert or usual). Thus a part of the service capacity is wasted. We prove that such a model has a product-form steady-state distribution when the signal arrival rates are positive.

The random neural network (RNN) is a probabilitsic queueing theory-based model for artificial neural networks, and it requires the use of optimization algorithms for training. Commonly used gradient descent learning algorithms may reside in local minima, evolutionary algorithms can be also used to avoid local minima. Other techniques such as artificial bee colony (ABC), particle swarm optimization (PSO), and differential evolution algorithms also perform well in finding the global minimum but they converge slowly. The sequential quadratic programming (SQP) optimization algorithm can find the optimum neural network weights, but can also get stuck in local minima. We propose to overcome the shortcomings of these various approaches by using hybridized ABC/PSO and SQP. The resulting algorithm is shown to compare favorably with other known techniques for training the RNN. The results show that hybrid ABC learning with SQP outperforms other training algorithms in terms of mean-squared error and normalized root-mean-squared error.

This paper deals with two ideas appeared during the last developing phase in Artificial Intelligence: Reservoir Computing (RC) and Random Neural Networks. Both have been very successful in many applications. We propose a new model belonging to the first class, taking the structure of the second for its dynamics. The new model is called Echo State Queuing Network. The paper positions the model in the global Machine Learning area, and provides examples of its use and performances. We show on largely used benchmarks that it is a very accurate tool, and we illustrate how it compares with standard RC models.

This paper surveys research on mathematical models that predict the performance of digital devices that operate with intermittent energy sources. The approach taken in this work is based on the “Energy Packet Network” paradigm where the arrival of data to be processed or transmitted, and the energy to operate the system, are modeled as discrete random processes. Our assumption is that these devices will capture energy from intermittent ambient sources such as vibrations, heat or light, and capture it onto electrical energy that may be stored in batteries or capacitors. The devices consume this energy intermittently for processing and for wired or wireless transmission. Thus, both the arrival of energy to the device, and the devices workload, are modeled as random processes. Based on these assumptions, we discuss probability models based on Markov chains that can be used to predict the effective rates at which such devices operate. We also survey related work that models networks of such systems.

This paper presents a probability model of an energy-harvesting wireless sensor node, with the objective of linking quality of sensed data to energy consumption and self-sustainability. The model departs from the common energy discretization framework used in the literature, and instead uses a diffusion process modulated by discrete packet arrival and transmission processes for the detailed representation of renewable energy supply, consumption and storage. An analytical–numerical method is developed to compute the average time until the node experiences an outage, due to lack of energy, for a given workload and ambient energy characteristics, battery capacity and initial charge. The results are illustrated with numerical examples.

The concept of Energy Packet Network (EPN) proposed by Gelenbe, is a new framework for modeling power grids that takes distributed energy generation such as renewable energy sources into consideration, and which contributes to modeling the smart grid. Based on G-network theory, this paper presents a simplified model of EPN and formulates energy-distribution as an optimization problem. We analyze it theoretically, and detail its optimal solutions. In addition to using existing optimization algorithms, a heuristic algorithm is proposed to solve for EPN optimization. The optimal solutions and efficacy of the algorithm are illustrated with numerical experiments. Further, we present an EPN with disconnections and a similar optimization problem is investigated. Optimal solutions are presented, and numerical results using the analytic optimal solutions, random solutions, a cooperative particle swarm optimizer and a heuristic algorithm illustrate the power of different approaches for solving energy-distribution problems using the EPN formalism.

G-Network queueing network models, and in particular the random neural network (RNN), are useful tools for decision making in complex systems, due to their ability to learn from measurements in real time, and in turn provide real-time decisions regarding resource and task allocation. In particular, the RNN has led to the design of the cognitive packet network (CPN) decision tool for the routing of packets in the Internet, and for task allocation in the Cloud. Thus in this paper, we present recent research on how to dynamically create the means for quality of service (QoS) to end users of the Internet and in the Cloud. The approach is based on adapting the decisions so as to benefit users as the conditions in the Internet and in Cloud servers vary due to changing traffic and workload. We present an overview of the algorithms that were designed based on the RNN, and also detail the experimental results that were obtained in three areas: (i) traffic routing for real-time applications, which have strict QoS constraints; (ii) routing approaches, which operate at the overlay level without affecting the Internet infrastructure; and (iii) the routing of tasks across servers in the Cloud through the Internet.

The paper provides an analysis of G-network with positive customers and signals when signals arriving to the system move customer to another system or destroy in it a group of customers, reducing their number to a random value that is given by a probability distribution. The signal arriving to the system, in which there are no positive customers, does not exert any influence on the queueing network and immediately disappears from it. Streams of positive customers and signals arriving to each of the network systems are independent. Customer in the transition from one system to another brings the latest some revenue, and the revenue of the first system is reduced by this amount. A method of finding the expected revenues of the systems of such a network has been proposed. The case when the revenues from transitions between network states are deterministic functions depending on its states has been considered. A description of the network is given, all possible transitions between network states, transition probabilities, and revenues from state transitions are indicated. A system of difference-differential equations for the expected revenues of network systems has been obtained. To solve it, we propose a method of successive approximations, combined with the method of series. It is proved that successive approximations converge to the stationary solution of such a system of equations, and the sequence of approximations converges to a unique solution of the system. Each approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which are related by recurrence relations. Therefore, it is convenient to use them for calculations on a PC. The obtained results can be applied in forecasting losses in information and telecommunication systems and networks from the penetration of computer viruses into it and conducting computer attacks.