We introduce the special issue on “Learning, Optimization, and the Theory of G-Networks” of the journal Probability in the Engineering and Informational Sciences that appears in 2019. We first outline some of the applications and developments of G-Networks which motivate the ongoing interest for this area, including some areas which could not be covered in this special issue. We then briefly discuss the contributions presented in the ten papers that are published in this special issue in the context of related work.

The random neural network (RNN) is a mathematical model for an “integrate and fire” spiking network that closely resembles the stochastic behavior of neurons in mammalian brains. Since its proposal in 1989, there have been numerous investigations into the RNN's applications and learning algorithms. Deep learning (DL) has achieved great success in machine learning. Recently, the properties of the RNN for DL have been investigated, in order to combine their power. Recent results demonstrate that the gap between RNNs and DL can be bridged and the DL tools based on the RNN are faster and can potentially be used with less energy expenditure than existing methods.

Falls are a major health concern in older adults. Falls lead to mortality, immobility and high costs to social and health care services. Early detection and classification of falls is imperative for timely and appropriate medical aid response. Traditional machine learning models have been explored for fall classification. While newly developed deep learning techniques have the ability to potentially extract high-level features from raw sensor data providing high accuracy and robustness to variations in sensor position, orientation and diversity of work environments that may skew traditional classification models. However, frequently used deep learning models like Convolutional Neural Networks (CNN) are computationally intensive. To the best of our knowledge, we present the first instance of a Hybrid Multichannel Random Neural Network (HMCRNN) architecture for fall detection and classification. The proposed architecture provides the highest accuracy of 92.23% with dropout regularization, compared to other deep learning implementations. The performance of the proposed technique is approximately comparable to a CNN yet requires only half the computation cost of the CNN-based implementation. Furthermore, the proposed HMCRNN architecture provides 34.12% improvement in accuracy on average than a Multilayer Perceptron.

Artificial Neural Networks (ANNs)-based techniques have dominated state-of-the-art results in most problems related to computer vision, audio recognition, and natural language processing in the past few years, resulting in strong industrial adoption from all leading technology companies worldwide. One of the major obstacles that have historically delayed large-scale adoption of ANNs is the huge computational and power costs associated with training and testing (deploying) them. In the mean-time, Neuromorphic Computing platforms have recently achieved remarkable performance running the bio-realistic Spiking Neural Networks at high throughput and very low power consumption making them a natural alternative to ANNs. Here, we propose using the Random Neural Network, a spiking neural network with both theoretical and practical appealing properties, as a general purpose classifier that can match the classification power of ANNs on a number of tasks while enjoying all the features of being a spiking neural network. This is demonstrated on a number of real-world classification datasets.

Supply chains are fundamental to the economy of the world and many supply chains focus on perishable items, such as food, or even clothing that is subject to a limited shelf life due to fashion and seasonable effects. G-networks have not been previously applied to this important area. Thus in this paper, we apply G-networks to supply chain systems and investigate an optimal order allocation problem for a N-node supply chain with perishable products that share the same order source of fresh products. The objective is to find an optimal order allocation strategy to minimize the purchase price per object from the perspective of the customers. An analytical solution based on G-networks with batch removal, together with optimization methods are shown to produce the desired results. The results are illustrated by a numerical example with realistic parameters.

We use the Energy Packet Network (EPN) to investigate an optimal energy distribution problem for the computer-communication system which is powered by intermittent renewable energy sources. The objective is to find an optimal energy distribution to minimize the proposed cost function which computes penalty costs caused by the overall average response time of jobs and the energy loss. In this EPN system, we consider the energy can be lost through storage leakages, or due to empty workstations which will consume energy even no job needs to be processed. Related numerical examples with different sets of parameter values are presented in the paper to evaluate the system performance and to examine the obtained analytical solution. Then a special case is considered to study the optimal system performance when the total energy harvesting rate is sufficiently large.

We analyze Energy Packet Networks (EPNs) in which the service centers consist of multiclass queues and the Data Packets (DPs) initiate the transfer (i.e., the arrival of a DP at the battery triggers the movement) with multiple energy packet requirements. In other words, a class-k DP in cell i is sent successfully to the next cell if there are $c_i^{(k)}$ energy packets and it is dropped otherwise. Besides, we consider that the queues handling DPs operate under one of the following disciplines: First-Input-First-Output (FIFO), Processor Sharing (PS), or Preemptive Last-Input-First-Output (LIFO-PR). This model is an extension of previously studied EPNs [6,16] where the steady-state distribution of the number of jobs in the queues has a product form. In our model, we show the existence of a product form of the steady-state stationary distribution, where the load of the servers is given by a fixed point expression. We study the existence of a solution to the derived fixed point problem and we provide sufficient conditions for the stability of our model. Finally, we show that, for feed forward EPNs, the load of all the queues can be fully characterized.

The effect of signals on stability, stable throughput region, and delay in a two-user slotted ALOHA-based random-access system with collisions is considered. This work gives rise to the development of random access G-networks, which can model security attacks, expiration of deadlines, or other malfunctions, and introduce load balancing among highly interacting queues. The users are equipped with infinite capacity buffers accepting external bursty arrivals. We consider both negative and triggering signals. Negative signals delete a packet from a user queue, while triggering signals cause the instantaneous transfer of packets among user queues. We obtain the exact stability region, and show that the stable throughput region is a subset of it. Moreover, we perform a compact mathematical analysis to obtain exact expressions for the queueing delay by solving a non-homogeneous Riemann boundary value problem. A computationally efficient way to obtain explicit bounds for the expected number of buffered packets at user queues is also presented. The theoretical findings are numerically evaluated and insights regarding the system performance are derived.

We consider an M/M/1 retrial queue subject to negative customers (called as G-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.

This paper discusses a system of difference-differential equations (DDE) that is satisfied by the time-dependent state probabilities of open Markov queueing networks with various features. The number of network states in this case and the number of equations in this system is infinite. Flows of customers arriving at the network are a simple and independent, the time of customer services is exponentially distributed. The intensities of transitions between the network states are deterministic functions depending on its states.

To solve the system of DDE, we propose a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time to a stationary probability distribution, the form of which is indicated in the paper has been proved. The sequence of approximations converges to a unique solution of the system of equations. Any successive approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for calculations on a computer. Examples of the analysis of Markov G-networks with various features have been presented.

In the paper, we study general Markovian models of loss systems with random resource requirements, in which customers at arrival occupy random quantities of various resources and release them at departure. Customers may request negative quantities of resources, but total amount of resources allocated to customers should be nonnegative and cannot exceed predefined maximum levels. Allocating a negative volume of a resource to a customer leads to a temporary increase in its volume in the system. We derive necessary and sufficient conditions for the product-form of the stationary probability distribution of the Markov jump process describing the system.