Markov Processes in Continuous Time with Continuous State Space

2017 ◽  
pp. 203-251
1975 ◽  
Vol 7 (01) ◽  
pp. 66-82 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney

We obtain results connecting the distribution of the random variablesYandWin the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1,EYβandEWβconverge or diverge together and regular variation of the tail of one ofY, Wwith non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.


1975 ◽  
Vol 7 (1) ◽  
pp. 66-82 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney

We obtain results connecting the distribution of the random variables Y and W in the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1, EYβ and EWβ converge or diverge together and regular variation of the tail of one of Y, W with non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.


1974 ◽  
Vol 11 (04) ◽  
pp. 669-677 ◽  
Author(s):  
D. R. Grey

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.


1974 ◽  
Vol 11 (4) ◽  
pp. 669-677 ◽  
Author(s):  
D. R. Grey

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.


2021 ◽  
Vol 3 (6) ◽  
Author(s):  
Ogbonnaya Anicho ◽  
Philip B. Charlesworth ◽  
Gurvinder S. Baicher ◽  
Atulya K. Nagar

AbstractThis work analyses the performance of Reinforcement Learning (RL) versus Swarm Intelligence (SI) for coordinating multiple unmanned High Altitude Platform Stations (HAPS) for communications area coverage. It builds upon previous work which looked at various elements of both algorithms. The main aim of this paper is to address the continuous state-space challenge within this work by using partitioning to manage the high dimensionality problem. This enabled comparing the performance of the classical cases of both RL and SI establishing a baseline for future comparisons of improved versions. From previous work, SI was observed to perform better across various key performance indicators. However, after tuning parameters and empirically choosing suitable partitioning ratio for the RL state space, it was observed that the SI algorithm still maintained superior coordination capability by achieving higher mean overall user coverage (about 20% better than the RL algorithm), in addition to faster convergence rates. Though the RL technique showed better average peak user coverage, the unpredictable coverage dip was a key weakness, making SI a more suitable algorithm within the context of this work.


NeuroImage ◽  
2017 ◽  
Vol 162 ◽  
pp. 344-352 ◽  
Author(s):  
Jacob C.W. Billings ◽  
Alessio Medda ◽  
Sadia Shakil ◽  
Xiaohong Shen ◽  
Amrit Kashyap ◽  
...  

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