A discrete-time Markov process with a bounded continuous state space is considered. We show that the equilibrium equations on steady-state probability and densities form Fredholm integral equations of the second kind. Then, under a sufficient condition that the transition densities from one state to another state inside the boundaries of the state space can be expressed in the same separate forms, the steady-state probability and density functions can be obtained explicitly. We use it to demonstrate an economic production quantity model with stochastic production time, derive the expressions of the steady-state probabilities and densities, and find the optimal maximum stock level. A sensitivity analysis of the optimal stock level is performed using production time and cost parameters. The optimal stock level decreases with respect to the holding cost and the production cost, whereas it increases with respect to the lost sale cost and the arrival rate.
Keywords: Markov process, continuous state, Fredholm integral equation, steady state probability
@article{RO_2022__56_4_2881_0,
author = {Nakade, Koichi and Karim, Rubayet},
title = {Analysis of a discrete-time {Markov} process with a bounded continuous state space by the {Fredholm} integral equation of the second kind},
journal = {RAIRO. Operations Research},
pages = {2881--2894},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {4},
doi = {10.1051/ro/2022099},
mrnumber = {4471378},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2022099/}
}
TY - JOUR AU - Nakade, Koichi AU - Karim, Rubayet TI - Analysis of a discrete-time Markov process with a bounded continuous state space by the Fredholm integral equation of the second kind JO - RAIRO. Operations Research PY - 2022 SP - 2881 EP - 2894 VL - 56 IS - 4 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2022099/ DO - 10.1051/ro/2022099 LA - en ID - RO_2022__56_4_2881_0 ER -
%0 Journal Article %A Nakade, Koichi %A Karim, Rubayet %T Analysis of a discrete-time Markov process with a bounded continuous state space by the Fredholm integral equation of the second kind %J RAIRO. Operations Research %D 2022 %P 2881-2894 %V 56 %N 4 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2022099/ %R 10.1051/ro/2022099 %G en %F RO_2022__56_4_2881_0
Nakade, Koichi; Karim, Rubayet. Analysis of a discrete-time Markov process with a bounded continuous state space by the Fredholm integral equation of the second kind. RAIRO. Operations Research, Tome 56 (2022) no. 4, pp. 2881-2894. doi: 10.1051/ro/2022099
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