This paper is concerned with the convergence of a sequence of discrete-time Markov decision processes(DTMDPs)with constraints,state-action dependent discount factors,and possibly unbounded costs.Using the convex analytic approach under mild conditions,we prove that the optimal values and optimal policies of the original DTMDPs converge to those of the"limit"one.Furthermore,we show that any countablestate DTMDP can be approximated by a sequence of finite-state DTMDPs,which are constructed using the truncation technique.Finally,we illustrate the approximation by solving a controlled queueing system numerically,and give the corresponding error bound of the approximation.
Permafrost thickness under identical climates in cold regions can vary significantly because it is severely affected by climate change, topography, soil physical and thermal properties, and geothermal conditions. This study numerically in- vestigates the response of ground thermal regime and talik development processes to permafrost with different thicknesses under a thermokarst lake on the Qinghai-Tibet Plateau. On the basis of observed data and information from a representative monitored lake in the Beiluhe Basin, we used a heat transfer model with phase change under a cylindrical coordinate system to conduct three simulation cases with permafrost thicknesses of 45 m, 60 m, and 75 m, respectively. The simulated results indicate that increases in permafrost thickness not only strongly retarded the open talik formation time, but also delayed the permafrost lateral thaw process after the formation of open talik. Increasing the permafrost thickness by 33.3% and 66.7% led to open talik formation time increases of 83.66% and 207.43%, respectively, and resulted in increases in the lateral thaw duration of permafrost under the modeled thermokarst lake by 28.86% and 46.54%, respectively, after the formation of the open taliks.
