In this chapter, we study the economic issues of fresh data trading markets, where the data freshness is captured by Age-of-Information (AoI). In our model, a destination user requests, and pays for, fresh data updates from a source provider. In this work, the destination incurs an age-related cost, modeled as a general increasing function of the AoI. To understand economic viability and profitability of fresh data markets, we consider a pricing mechanism to maximize the source’s profit, while the destination chooses a data update schedule to trade off its payments to the source and its age-related cost. The problem is exacerbated when the source has incomplete information regarding the destination’s age-related cost, which requires one to exploit (economic) mechanism design to induce the truthful information. This chapter attempts to build such a fresh data trading framework that centers around the following two key questions: (a) How should a source choose the pricing scheme to maximize its profit in a fresh data market under complete market information? (b) Under incomplete information, how should a source design an optimal mechanism to maximize its profit while ensuring the destination’s truthful report of its age-related cost information?

Optimization of information freshness in wireless networks has usually been performed based on queueing analysis that captures only the temporal traffic dynamics associated with the transmitters and receivers. However, the effect of interference, which is mainly dominated by the interferers’ geographic locations, is not well understood. This chapter presents a theoretical framework for the analysis of the Age of Information (AoI) from a joint queueing-geometry perspective. We also provide the design of a decentralized scheduling policy that exploits local observation to make transmission decisions that minimize the AoI. To quantify the performance, we derive analytical expressions for the average AoI. Numerical results validate the accuracy of the analyses as well as the efficacy of the proposed scheme in reducing the AoI.

In this chapter, we study the Age of Information (AoI) when the status updates of the underlying process of interest can be sampled at any time by the source node and are transmitted over an error-prone wireless channel. We assume the availability of perfect feedback that informs the transmitter about the success or failure of transmitted status updates and consider various retransmission strategies. More specifically, we study the scheduling of sampling and transmission of status updates in order to minimize the long-term average AoI at the destination under resource constraints. We assume that the underlying statistics of the system are not known, and hence, propose average-cost reinforcement learning algorithms for practical applications. Extensions of the results to a multiuser setting with multiple receivers and to an energy-harvesting source node are also presented, different reinforcement learning methods including deep Q Network (DQN) are exploited and their performances are demonstrated.

In this chapter, we study the value of information, a more comprehensive instrument than the age of information, for shaping the information flow in a networked control system subject to random processing delay. In addition, we establish a connection between these two instruments by presenting a condition under which the value of information is expressible in terms of the age of information. Nonetheless, we show that this condition is not achievable without a degradation in the performance of the system.

Quantifying the multiscale hydraulic heterogeneity in aquifers and their effects on solute transport is the task of this chapter. Using spatial statistics, we explain how to quantify spatial variability of hydraulic properties or parameters in the aquifer using the stochastic or random field concept. In particular, we discuss spatial covariance, variogram, statistical homogeneity, heterogeneity, isotropy, and anisotropy concepts. Field examples complement the discussion. We then present a highly parameterized heterogeneous media (HPHM) approach for simulating flow and solute transport in aquifers with spatially varying hydraulic properties to meet our interest and observation scale. However, our limited ability to collect the needed information for this approach promotes alternatives such as Monte Carlo simulation, zonation, and equivalent homogeneous media (EHM) approaches with macrodispersion approaches. This chapter details the EHM with the macordispersion concept.

This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.

This chapter characterizes the average Age of Information (AoI) for the case of having multiple sources sharing a service facility with a single server. In particular, a simplified explanation of the SHS for AoI approach is provided to calculate the average age of updates of any source at the monitor. This approach is applied to various queueing systems including FCFS, M/M/1*, and M/M/1/2*, and the latter two with and without source priorities.

This chapter serves as an introduction to the concept of conservation and how conservation principles are used in fluid mechanics. The conservation principle is then applied to mass and an equation known as the continuity equation is developed. Various mathematical operations such as the dot product, the divergence, and the divergence theorem are introduced along the way. The continuity equation is discussed and the idea of an incompressible flow is introduced. Some examples using mass conservation are also given.

In this chapter, a concept known as scaling is introduced. Scaling (also known as nondimensionalization) is essentially a form of dimensional analysis. Dimensional analysis is a general term used to describe a means of analyzing a system based off the units of the problem (e.g. kilogram for mass, kelvin for temperature, meter for length, coulomb for electric change, etc.). The concepts of this chapter, while not entirely about the fluid equations per se, is arguably the most useful in understanding the various concepts of fluid mechanics. In addition, the concepts discussed within this chapter can be extended to other areas of physics, particularly areas that are heavily reliant on differential equations (which is most of physics and engineering).

This chapter introduces simple graphical methods to estimate advection velocity and dispersivity of solute migration through soil columns, using one-dimensional ADE presented in previous chapters. Methods of spatial and temporal moments are also introduced for solute concentration breakthroughs in one-dimensional transport and snapshots of the multi-dimensional solute migrations, respectively. Unlike automatic nonlinear regression analysis, these methods use physical insights and analytical solutions to illustrate logical approaches to estimate these parameters. The automatic regression analysis (such as Microsoft Excel introduced in Chapter 1) may find the parameters that fit the solution to the data well. However, the parameter values may not be physically possible if the estimation problem is poorly constrained (see examples in Chapter 11).

This chapter considers an application of age of information called AoCSI in which the channel states in a wireless network represent the information of interest and the goal is to maintain fresh estimates of these channel states at each node in the network. Rather than sampling some underlying time-varying process and propagating updates through a queue or graph, the AoCSI setting obtains direct updates of the channels as a by-product of wireless communication through standard physical layer channel estimation techniques. These CSI estimates are then disseminated through the network to provide global snapshots of the CSI to all of the nodes in the network. What makes the AoCSI setting unique is that disseminating some CSI updates and directly sampling/estimating other CSI occur simultaneously. Moreover, as illustrated in this chapter, there are inherent trade-offs on how much CSI should be disseminated in each transmission to minimize the average or maximum age.

In addition to the continuity equation, there is another very important equation that is often employed alongside the Navier–Stokes equations: the energy equation. The energy equation is required to fully describe compressible flows. This chapter guides the student through the development of the energy equation, which can be an intimidating equation. A discussion on diffusion and its interplay with advection is also included, leading to the idea of a boundary layer. The chapter ends with the addition of the energy equation in shear-driven and pressure-driven flows.