A Stochastic Hybrid Systems Framework for Analysis of Markov Reward Models

In this paper, we propose a framework to analyze Markov reward models, which are commonly used in system performability analysis. The framework builds on a set of analytical tools developed for a class of stochastic processes referred to as “Stochastic Hybrid Systems (SHS).” The state space of an SHS is composed of: i) a discrete state that describes the possible configurations/modes that a system can adopt, which includes the nominal (non-faulty) operational mode, but also those operational modes that arise due to component faults, and ii) a continuous state that describes the reward. Discrete state transitions are stochastic, and governed by transition rates that are (in general) a function of time and the value of the continuous state. The evolution of the continuous state is described by a stochastic differential equation, and reward measures are defined as functions of the continuous state. Additionally, each transition is associated with a reset map that defines the mapping between the pre- and post-transition values of the discrete and continuous states; these mappings enable the definition of impulses and losses in the reward. The proposed SHS-based framework unifies the analysis of a variety of previously studied reward models. We illustrate the application of the framework to performability analysis via analytical and numerical examples.National Science Foundation / CMG-0934491Ope

A Stochastic Hybrid Systems Framework for Analysis of Markov Reward Models

In this paper, we propose a framework to analyze Markov reward models, which are commonly used in system performability analysis. The framework builds on a set of analytical tools developed for a class of stochastic processes referred to as “Stochastic Hybrid Systems (SHS).” The state space of an SHS is composed of: i) a discrete state that describes the possible configurations/modes that a system can adopt, which includes the nominal (non-faulty) operational mode, but also those operational modes that arise due to component faults, and ii) a continuous state that describes the reward. Discrete state transitions are stochastic, and governed by transition rates that are (in general) a function of time and the value of the continuous state. The evolution of the continuous state is described by a stochastic differential equation, and reward measures are defined as functions of the continuous state. Additionally, each transition is associated with a reset map that defines the mapping between the pre- and post-transition values of the discrete and continuous states; these mappings enable the definition of impulses and losses in the reward. The proposed SHS-based framework unifies the analysis of a variety of previously studied reward models. We illustrate the application of the framework to performability analysis via analytical and numerical examples.National Science Foundation / CMG-0934491Ope

Penggunaan rantai markov orde dua untuk menganalisis dua Merek penjualan dan persainganair mineral dalam kemasan botol selama pandemic COVID-19 di Kota Medan

Masa pandemi COVID-19 mempengaruhi hasil penjualan air mineral dalam kemasan botol sebab banyak kantor bekerjadari rumah (work from home). Berdasarkan pangsa pasar AQUA menguasai 46,7% pangsa pasar seluruh Indonesia. Sedangkan Le Minerale sebagai salah satu pemain baru, tetap mengempit 3,5 persen pangsa pasar sehingga penelitian ini dilihat pada banyaknya konsumen, peneliti mengambil dua merek air mineral tersebut yaitu AQUA dan Le Minerale. Tujuan penelitian ini adalah untuk mengetahui bagaimana penjualan dan persaingan air mineral dalam kemasan botol selama masa pandemi COVID-19 yang terdapat mengalami penurunan tidak beraktivitas di luar rumah.penelitian ini dilakukan di lima kantor kota Medan yaitu PT Bank Sumut KC Medan Sukaramai, RSUD dr Pirngadi, Bank Mandiri, Kantor Lurah Bantan, PT Kharimantara Indonesia. Dalam penelitian ini menggunakan Metode Rantai Markov Orde Dua. Berdasarkan hasil Penelitian ini Besar peluang peralihan di masa mendatang orde dua yaitu produk AQUA pada September mengalami kenaikan 0,0061 % dan pada bulan berikutnya mengalami penurunan 0,0002 sedangkan produk mineral pada September mengalami penurunan adalah 0,0039 dan pada bulan berikutnya mengalami keadaan tetap. Besar peluang peralihan berdasarkan alasan pelanggan adalah produk AQUA persentasenya lebih besar dari produk Le Minerale yairu 59,8 sedangkan produk Le Minerale persentasenya sebesar 40,4. Pangsa pasar air mineral dalam kemasan botol akan mencapai kondisi stabil pada jangka waktu 3 bulan dimana pangsa pasar pada orde satu AQUA sebesar 63,0 dan Le Minerale sebesar 37,0 sedangkan pangsa pasar air mineral dalam kemasan botol orde dua mencapai stabil pada jangka waktu 3 bulan dimana pangsa pasar orde kedua adalah Produk AQUA adalah 61,0 sedangkan Le Minerale 39,0