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.

Modeling behaviour of wet and dry days in Iran from the perspective of Markov chains

The current study aims to model the behaviour of wet and dry days in Iran using Markov Chain Models. To this end, data related to daily precipitation of 44 synoptic stations for a 25-years interval (1991-2015) was obtained from Iran Meteorological Organization. Then, the Markov features of dry and wet days of Iran including stationary probabilities of dry and wet days occurrence, the expected length of dry periods, the expected length of wet periods, dry-wet spells cycle, return periods for dry or wet episodes and finally, the possibility of occurrence of the continuity of dry days for 5, 10, 15, 20, 25 and 30 days were calculated for all the synoptic stations in a seasonal scale. The results showed that there is the occurrence of dry short continuities (5 and 10 days) in three seasons of autumn, winter and spring with different possibilities all over Iran. However, the possibility of occurrence of long-term dry continuities (more than 20 days) is variable in terms of season and place so that in winter, no possibility of occurrence of this type of continuities is obvious in the northern half of Iran. As in autumn and spring those are the end and beginning of long-term stability conditions of the atmosphere in the upper atmosphere levels of Iran, the possibility of periodical occurrence of 30-days dry days, particularly in the southern half of Iran increases. In addition, the expected return periods for dry days is almost steady for every part of Iran and is in the range between 1 and 2 days. However, the number of return days to a precipitation period does not follow this rule and varies for every part of Iran so that from 2.15 days in autumn to 79 days in spring is variable, pointing to the climate diversity of Iran.

Markov Chains for Computer Music Generation

Random generation of music goes back at least to the 1700s with the introduction of Musical Dice Games. More recently, Markov chain models have been used as a way of extracting information from a piece of music and generating new music. We explain this approach and give Python code for using it to first draw out a model of the music and then create new music with that model.

The Influence of Latent and Chronic Infection on Pathogen Persistence

We extend the classical compartmental frameworks for susceptible-infected-susceptible (SIS) and susceptible-infected-recovered (SIR) systems to include an exposed/latent class or a chronic class of infection. Using a suite of stochastic continuous-time Markov chain models we examine the impact of latent and chronic infection on the mean time to extinction of the infection. Our findings indicate that the mean time to pathogen extinction is increased for infectious diseases which cause exposed/latent infection prior to full infection and that the extinction time is increased further if these exposed individuals are also capable of transmitting the infection. A chronic infection stage can decrease or increase the mean time to pathogen extinction and in particular this depends on whether chronically infected individuals incur disease-induced mortality and whether they are able to transmit the infection. We relate our findings to specific infectious diseases that exhibit latent and chronic infectious stages and argue that infectious diseases with these characteristics may be more difficult to manage and control.