Age of Information in a Cooperative Slotted Aloha Network: Marginal and Joint Distributions

In this paper, we investigate a slotted Aloha cooperative network where a source node and a relay node send status updates of two underlying stochastic processes to a common destination. Additionally, the relay node cooperates with the source by accepting its packets for further retransmissions, where the cooperation policy comprises acceptance and relaying probabilistic policies. Exact marginal steady state distributions of the source and relay Age of Information (AoI) and Peak AoI (PAoI) sequences are obtained using Quasi-Birth-Death (QBD) Markov chain models. Extending this approach, we also obtain the joint distribution of the source and relay AoI sequences out of which one can obtain the steady state distribution of the Squared Difference of the two AoI sequences (SDAoI), which finds applications in network scenarios where not only the timeliness of status updates of each process is desired but also their simultaneity is of crucial importance. In this regard, we numerically obtain the optimal cooperation policy in order to minimize the expected value of SDAoI subject to a constraint on the average PAoI of the relay. Finally, our proposed analytical approach is verified by simulations and the performance of the optimal policy is discussed based on the numerical results.

Age of Information in a Cooperative Slotted Aloha Network: Marginal and Joint Distributions

In this paper, we investigate a slotted Aloha cooperative network where a source node and a relay node send status updates of two underlying stochastic processes to a common destination. Additionally, the relay node cooperates with the source by accepting its packets for further retransmissions, where the cooperation policy comprises acceptance and relaying probabilistic policies. Exact marginal steady state distributions of the source and relay Age of Information (AoI) and Peak AoI (PAoI) sequences are obtained using Quasi-Birth-Death (QBD) Markov chain models. Extending this approach, we also obtain the joint distribution of the source and relay AoI sequences out of which one can obtain the steady state distribution of the Squared Difference of the two AoI sequences (SDAoI), which finds applications in network scenarios where not only the timeliness of status updates of each process is desired but also their simultaneity is of crucial importance. In this regard, we numerically obtain the optimal cooperation policy in order to minimize the expected value of SDAoI subject to a constraint on the average PAoI of the relay. Finally, our proposed analytical approach is verified by simulations and the performance of the optimal policy is discussed based on the numerical results.

Markovian Approach to Stock Price Modelling in the Nigerian Oil and Gas Sector

The study investigates the stock price movement of quoted Nigerian oil and gas firms using the Markovian model. Specifically, the study estimates the change in likelihoods and steady-state distribution of the share prices of the firms to determine the average time spent by the share price to move to another state and the turnover rate of the selected stocks. Markov chain-based stochastic modelling approach was employed by using the daily closing share prices of all the seven oil and gas firms quoted on the Nigerian Stock Exchange from April 2017 to January 2020. The study finds that the transition probabilities and the steady-state distribution of all the firms are stationary at first-order, implying that chain depends on the previous state. The steady-state probabilities of all the firms examined exhibit relatively high price stability in the long run. The study recommends that investors with diverse attitudes to risk-taking can explore the estimated long-run prospect of the investigated stocks in making guided investment decisions.

Communities, Co-ops, and Clubs: Social Capital and Incentives in Large Collective Organizations

We study a continuous-time organization design problem. Each member’s output is an imperfect signal of his underlying effort, and each member’s utility from remaining in the organization is endogenous to other members’ efforts. Monetary transfers are assumed infeasible. Incentives can be provided only through two channels: expulsion following poor performance and respite following good performance. We derive the steady state distribution of members’ continuation utilities for arbitrary values of the initial and maximum continuation utilities and then optimize these values according to organizational objectives. An optimally designed organization can be implemented by associating continuation utilities with a performance-tracking reputation system. (JEL Z13, D23, D86, P13, D82)

The Evolution of Networks and Local Public Good Provision: A Potential Approach

In this paper, we propose a game in which each player decides with whom to establish a costly connection and how much local public good is provided when benefits are shared among neighbors. We show that, when agents are homogeneous, Nash equilibrium networks are nested split graphs. Additionally, we show that the game is a potential game, even when we introduce heterogeneity along several dimensions. Using this result, we introduce stochastic best reply dynamics and show that this admits a unique and stationary steady state distribution expressed in terms of the potential function of the game. Hence, even if the set of Nash equilibria is potentially very large, the long run predictions are sharp.

Infinite Series of Singularities in the Correlated Random Matrices Product

We consider the product of a large number of two 2 × 2 matrices chosen randomly (with some correlation): at any round there are transition probabilities for the matrix type, depending on the choice at previous round. Previously, a functional equation has been derived to calculate such a random product of matrices. Here, we identify the phase structure of the problem with exact expressions for the transition points separating “localized” and “ergodic” regimes. We demonstrate that the latter regime develops through a formation of an infinite series of singularities in the steady-state distribution of vectors that results from the action of the random product of matrices on an initial vector.

Motile cells as probes for characterizing acoustofluidic devices

Dynamically responsive Chlamydomonas reinhardtii algae cells enable real-time assessment of acoustofluidic device performance. The steady-state distribution of these motile cells reflects both the field shape and strength.

The Steady State Distribution for M|M|m|n Model with the Waiting Time Restriction

The queue state in multiprocessor computing systems is an actual problem for the process of optimal scheduling of tasks. In this paper, a system of equations is obtained describing the distribution of the queue for the system in a steady state. The resulting linear system of equations is solved using conventional numerical methods and can be used in schedulers.

Graph of graphs analysis for multiplexed data with application to imaging mass cytometry

Hyper spectral imaging, sensor networks, spatial multiplexed proteomics, and spatial transcriptomics assays is a representative subset of distinct technologies from diverse domains of science and engineering that share common data structures. The data in all these modalities consist of high-dimensional multivariate observations (m-dimensional feature space) collected at different spatial positions and therefore can be analyzed using similar computational methodologies. Furthermore, in many studies practitioners collect datasets consisting of multiple spatial assays of this type, each capturing such data from a single biological sample, patient, or hyper spectral image, etc. Each of these spatial assays could be characterized by several regions of interest (ROIs). The focus of this paper is on a particular application, imaging mass cytometry (IMC), which falls into this problem setup. To extract meaningful information from the multi-dimensional observations recorded at different ROIs across different assays, we propose to analyze such datasets using a two-step graph-based approach. We first construct for each ROI a graph representing the interactions between the m covariates and compute an m dimensional vector characterizing the steady state distribution among features. We then use all these m-dimensional vectors to construct a graph between the ROIs from all assays. This second graph is subjected to a nonlinear dimension reduction analysis, retrieving the intrinsic geometric representation of the ROIs. Such a representation provides the foundation for efficient and accurate organization of the different ROIs that correlates with their phenotypes. Theoretically, we show that when the ROIs have a particular bi-modal distribution, the new representation gives rise to a better distinction between the two modalities compared to the maximum a posteriori (MAP) estimator. We applied our method to predict the sensitivity to PD-1 axis blockers treatment of lung cancer subjects based on IMC data, achieving 92% accuracy. This serves as empirical evidence that the graph of graphs approach enables us to integrate multiple ROIs and the intra-relationships between the features at each ROI, giving rise to an informative representation that is strongly associated with the phenotypic state of the entire image. Importantly, this approach is applicable to other modalities such as spatial transcriptomics.Author summaryWe propose a two-step graph-based analyses for high-dimensional multiplexed datasets characterizing ROIs and their inter-relationships. The first step consists of extracting the steady state distribution of the random walk on the graph, which captures the mutual relations between the covariates of each ROI. The second step employs a nonlinear dimensionality reduction on the steady state distributions to construct a map that unravels the intrinsic geometric structure of the ROIs. We show theoretically that when the ROIs have a two-class structure, our method accentuates the distinction between the classes. Particularly, in a setting with Gaussian distribution it outperforms the MAP estimator, implying that the mutual relations between the covariates and spatial coordinates are well captured by the steady state distributions. We apply our method to imaging mass cytometry (IMC). Our analysis provides a representation that facilitates prediction of the sensitivity to PD-1 axis blockers treatment of lung cancer subjects. Particularly, our approach achieves state of the art results with accuracy of 92%.