APPLICATION OF RENEWAL REWARD PROCESSES IN HOMOGENEOUS DISCRETE MARKOV CHAIN
DOI:
https://doi.org/10.33003/fjs-2021-0504-785Keywords:
Erlang k distribution, Renewal process, renewal sequence, renewal reward process, homogeneous discrete-time Markov chainAbstract
A renewal process which is a special type of a counting process, which counts the number of events that occur up to (and including) time has been investigated, in order to provide some insight into the performance measures in renewal process and sequence such as, the mean time between successive renewals, ; Laplace-Stiltjes transform (LST) of the mean time, ; the Laplace-Stieltjes transform (LST) of the mean time distribution function, ; Laplace-Stiltjes transform (LST) of fold convolution of distribution function, ; the time at which the renewal occurs, the average number of renewals per unit time over the interval (0, t], and expected reward, . Our quest is to analyse the distribution function of the renewal process and sequence using the concept of discrete time Markov chain to obtain the aforementioned performance measures. Some properties of the Erlang , exponential and geometric distributions are used with the help of some existing laws, theorems and formulas of Markov chain. We concluded our study through illustrative examples that, it is not possible for an infinite number of renewals to occur in a finite period of time; Also, the expected number of renewals increases linearly with time; and from the uniqueness property, we affirmed that, the Poisson process is the only renewal process with a linear mean-value function; and lastly, we obtained the optimal replacement policy for a manufacturing machine which showed that, the exponential property of the lifetime
References
Abdulkareem, J., Pradhan, B., Sulaiman, W., & Jamil, N. (2018). Review of studies on hydrological modelling in Malaysia. Modeling Earth Systems and Environment, 1-29.
Alansi, A., Amin, M., Abdul Halim, G., Shafri, H., & Aimrun, W. (2009). Validation of SWAT model for streamflow simulation and forecasting in Upper Bernam humid tropical river basin, Malaysia. Hydrology and Earth System Sciences Discussions, 6(6), 7581-7609.
Ali, M. H., Bhattacharya, B., & Katimon, A. (2018). Modelling surface runoff in a large-scale paddy field in Malaysia. International Journal of Hydrology Science and Technology, 8(1), 69-90.
Amin, M., Rowshon, M., & Aimrun, W. (2011). Paddy water management for precision farming of rice Current Issues of Water Management: InTech.
Arnell, N., & Reynard, N. (1996). The effects of climate change due to global warming on river flows in Great Britain. Journal of hydrology, 183(3-4), 397-424.
Babel M.S., B. S. P., Walid S.M. . (2014). Climate change and water resources in the Bagmati River, Nepal. RTheor Appl Climatol, 115, 639-654.
Chien, H., Yeh, P. J.-F., & Knouft, J. H. (2013). Modelling the potential impacts of climate change on streamflow in agricultural watersheds of the Midwestern United States. Journal of Hydrology, 491, 73-88.
Chu, X., & Steinman, A. (2009). Event and continuous hydrologic modelling with HEC-HMS. Journal of Irrigation and Drainage Engineering, 135(1), 119-124.
Deni, S. M., Suhaila, J., Zin, W. Z. W., & Jemain, A. A. (2010). Spatial trends of dry spells over Peninsular Malaysia during monsoon seasons. Theoretical and applied climatology, 99(3-4), 357.
Dlamini, N. S., Rowshon, M. K., Amin, M. S. M., Mohd M. S. F., A., F. A., & H., L. S. (2017). “Modeling potential impacts of climate change on streamflow using projections of the 5th assessment report for the bernam river basin, Malaysia. Water (Switzerland), vol. 9, (3), pp. 1–23
Elleuch, M. A., Elleuch, L., & Frikha, A. (2019). A hybrid approach for water resources management in Tunisia. International Journal of Water, 13(1), 80-99.
Goodarzi, M., & Eslamian, S. (2018). Performance evaluation of linear and nonlinear models for the estimation of reference evapotranspiration. International Journal of Hydrology Science and Technology, 8(1), 1-15.
Guo, B., Zhang, J., & Xu, T. (2018). Comparison of two statistical climate downscaling models: a case study in the Beijing region, China. International Journal of Water, 12(1), 22-38.
Huo, A., & Li, H. (2013). Assessment of climate change impact on the stream-flow in a typical debris flow watershed of Jianzhuangcuan catchment in Shaanxi Province, China. Environmental Earth Sciences, 69(6), 1931-1938.
Ismail, H., Kamal, M. R., bin Abdullah, A. F., & bin Mohd, M. S. F. (2020a). Climate-Smart Agro-Hydrological Model for a Large Scale Rice Irrigation Scheme in Malaysia. Applied Sciences, 10(11), 3906.
Ismail, H., Kamal, M. R., Hin, L. S., & Abdullah, A. F. (2020b). Performance of HEC-HMS and ArcSWAT Models for Assessing Climate Change Impacts on Streamflow at Bernam River Basin in Malaysia. Pertanika Journal of Science & Technology, 28(3).
Ismail, H., Kamal, M. R., Jada, D. T., & Sai Hin, L. (2020c). Modelling Future Streamflow for Adaptive Water Allocation under Climate Change for the Tanjung Karang Rice Irrigation Scheme Malaysia. Applied Sciences, 10(14), 4885.
Kabiri, R., Bai, V. R., & Chan, A. (2015). Assessment of hydrologic impacts of climate change on the runoff trend in Klang Watershed, Malaysia. Environmental Earth Sciences, 73(1), 27-37.
Khorram, S., & Vahedi, M. (2018). Evaluating uncertainty in nonlinear hydrological models using the VIC-3D model on the Umeå River basin. International Journal of Water, 12(4), 287-313.
Li, Z., Liu, W.-z., Zhang, X.-c., & Zheng, F.-l. (2009). Impacts of land-use change and climate variability on hydrology in an agricultural catchment on the Loess Plateau of China. Journal of hydrology, 377(1-2), 35-42.
Lian, Y., You, G. J.-Y., Lin, K., Jiang, Z., Zhang, C., & Qin, X. (2015). Characteristics of climate change in southwest China karst region and their potential environmental impacts. Environmental Earth Sciences, 74(2), 937-944.
Ma, Z., Kang, S., Zhang, L., Tong, L., & Su, X. (2008). Analysis of impacts of climate variability and human activity on streamflow for a river basin in the arid region of northwest China. Journal of hydrology, 352(3-4), 239-249.
Mahmood, R., & Jia, S. (2016). Assessment of impacts of climate change on the water resources of the transboundary Jhelum River basin of Pakistan and India. Water, 8(6), 246.
Mahmood, R., Jia, S., & Babel, M. S. (2016). Potential impacts of climate change on water resources in the Kunhar River Basin, Pakistan. Water, 8(1), 23.
Meenu, R., Rehana, S., & Mujumdar, P. (2013). Assessment of hydrologic impacts of climate change in Tunga–Bhadra river basin, India with HECâ€HMS and SDSM. Hydrological Processes, 27(11), 1572-1589.
Moriasi, D. N., Arnold, J. G., Van Liew, M. W., Bingner, R. L., Harmel, R. D., & Veith, T. L. (2007). Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Transactions of the ASABE, 50(3), 885-900.
Paudel, U., Imteaz, M. A., & Matos, C. (2018). Equations for potential water savings through rainwater harvesting for different climatic conditions in Adelaide (Australia). International Journal of Hydrology Science and Technology, 8(1), 91-104.
Purkey, D. R., Huber-Lee, A., Yates, D. N., Hanemann, M., & Herrod-Julius, S. (2007). Integrating a climate change assessment tool into stakeholder-driven water management decision-making processes in California. Water Resources Management,, Vol.21, pp. 315–329 Environment, Vol.113, pp. 356–363
Rost, S., Gerten, D., Bondeau, A., Lucht, W., Rohwer, J., & Schaphoff, S. (2008). Agricultural green and blue water consumption and its influence on the global water system. Water Resources Research, 44(9).
Roy, D., Begam, S., Ghosh, S., & Jana, S. (2013). Calibration and validation of HEC-HMS model for a river basin in Eastern India. ARPN Journal of Engineering and Applied Sciences, 8(1), 33-49.
Sardans, J., & Penuelas, J. (2004). Increasing drought decreases phosphorus availability in an evergreen Mediterranean forest. Plant and Soil, Vol.267,( pp. 367–377).
Schlenker, W., Hanemann, W. M., & Fisher, A. C. (2007). Water availability, degree days, and the potential impact of climate change on irrigated agriculture in California. Climatic Change, 81(1), 19-38.
USACE-HEC. (2016). Hydrology Reference User’s Manual. US Army Corps of Engineers, Davis, CA.
Yan, R., Gao, J., & Li, L. (2016). Streamflow response to future climate and land-use changes in Xinjiang Basin, China. Environmental Earth Sciences, 75(14), 1108.
Yilmaz, A. G., & Shabib, A. G. (2019). Rainfall and air temperature projections for Sharjah City, United Arab Emirates. International Journal of Water, 13(1), 60-79.
Yimer, G., Jonoski, A., & Van Griensven, A. (2009). Hydrological response of a catchment to climate change in the upper Beles river basin, upper blue Nile, Ethiopia. Nile Basin Water Engineering Scientific Magazine, 2, 49-59.
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