M/M/1 Non-preemptive Priority Queuing System with Multiple Vacations and Vacation Interruptions

Non-preemptive priority queue system is a type of priority queue where customers with higher priorities cannot interrupt low priority one while being served. High priority consumers will still be at the head of the queue. This article discusses the non-preemptive priority queue system with multiple working vacations, where the vacation can be interrupted. Customers are classified into two classes, namely class I (non-preemptive priority customers) and class II, with exponentially distributed service rates. Customers will still receive services at a slower rate than during normal busy periods when they enter the system while it is on vacation. Suppose other customers are waiting in the queue after completing service on vacation. In that case, the vacation will be interrupted, and the service rate will switch to the busy period service rate. The model's performance measurements are obtained using the complementary variable method and analyzing the state change equation following the birth and death processes to find probability generating function for both classes. The results of the numerical solution show that the expected value number of customers and waiting time of customers in the queue for both class customers will be reduced when the vacation times rate (θ) and the vacation service rate (μ_0 ) increase

Strong Consistency and Asymptotic Distribution of Estimator for the Intensity Function Having Form of Periodic Function Multiplied by Power Function Trend of a Poisson Process

This manuscript discusses the strong consistency and the asymptotic distribution of an estimator for a periodic component of the intensity function having a form of periodic function multiplied by power function trend of a non-homogeneous Poisson process by using a uniform kernel function. It is assumed that the period of the periodic component of intensity function is known. An estimator for the periodic component using only a single realization of a Poisson process observed at a certain interval has been constructed. This estimator has been proved to be strongly consistent if the length of the observation interval indefinitely expands. Computer simulation also showed the asymptotic normality of this estimato

KAJIAN PENDUGA FUNGSI RAGAM PROSES POISSON PERIODIK MAJEMUK DENGAN TREN FUNGSI PANGKAT

Pada artikel ini dibahas tentang pendugaan fungsi ragam pada proses Poisson periodik majemuk yang mempertimbangkan kehadiran tren fungsi pangkat.  Penulisan artikel ini bertujuan untuk mengonstruksi penduga, memeriksa kekonsistenan penduga, menganalisis bias, ragam dan mean squared error (MSE) asimtotik penduga, serta menentukan ukuran interval pengamatan proses terpendek sehingga nilai dugaan yang diperoleh sudah mendekati parameter yang diduga menggunakan simulasi komputer. Hasil kajian yang telah diperoleh berupa rumusan penduga fungsi ragam, syarat-syarat agar penduga yang dirumuskan kokonsisten, rumusan bias asimtotik, ragam asimtotik dan MSE asimtotik penduga. Berdasarkan hasil simulasi diperoleh bahwa penduga sudah mendekati nilai parameter yang diduga jika panjang interval waktu pengamatan adalah 5500

Confidence Interval for Variance Function of a Compound Periodic Poisson Process with a Power Function Trend

This research is a follow-up research of Utama (2022) on asymptotic distribution of an estimator for variance function of a compound periodic Poisson with the power function trend. The objectives of this research are (i) to formulate a confidence interval for the variance function of a compound periodic Poisson process with a power function trend and (ii) to prove the convergence to 1-α probability of the parameter included in the confidence interval. This research process begins with a review of the existing formulation of the variance function estimator and its asymptotic distribution. Next, the confidence interval for the variance function of the compound periodic Poisson process with a power function trend is formulated and the convergence to 1-α is determined. After obtaining the confidence interval, the research continued by conducting computer simulations to confirmed the results obtained analytically. The results obtained show that the confidence interval for the variance function of a compound periodic Poisson process with a power function trend converges to 1-α both analytically and numerically for different finite time intervals

MODEL STOKASTIK EPIDEMIK SIRS INSIDEN TAK LINEAR DENGAN VAKSINASI

Matematika mempunyai peran penting dalam ilmu kesehatan salah satunya untuk membuat model penyebaran suatu penyakit. Salah satu penyakit yang dapat dibuat modelnya adalah penyakit difteri. Tujuan penelitian ini yakni memodifikasi model matematis difteri yang sudah ada menggunakan model stokastik continuous-time Markov chain (CTMC). Dalam penelitian ini pembahasan difokuskan pada peluang transisi, peluang wabah, dan bilangan reproduksi dasar. Bilangan reproduksi dasar  mewakili jumlah rata-rata individu rentan menjadi terinfeksi karena masuknya satu inividu terinfeksi ke dalam subpopulasi rentan. Jika , maka hasil analisis memperlihatkan bahwa sistem populasi akan mengalami wabah penyakit, sedangkan jika , maka wabah penyakit tidak akan terjadi pada sistem populasi. Pada penelitian ini diperoleh model stokastik penyebaran penyakit difteri dengan dua fungsi yang berbeda yakni fungsi linear  dan fungsi tak linear . Namun, keduanya memberikan hasil yang serupa yakni tidak akan terjadi wabah di dalam sistem ketika . Jika tingkat vaksinasi meningkat, maka bilangan reproduksi dasar menurun. Artinya semakin tinggi tingkat vaksinasi maka penyakit akan hilang di dalam sistem. Fungsi tak linear berpengaruh pada besarnya dan peluang wabah bergantung pada nilai konstanta α yang diberikan. Semakin besar nilai α, maka dan peluang wabah semakin kecil

Interval Kepercayaan Untuk Fungsi Nilai Harapan dan Fungsi Ragam Proses Poisson Periodik Majemuk

Compound cyclic Poisson process have the mean and variance functions. The objective of this paper is to construct confidence intervals for respectively the mean and variance functions of a compound cyclic Poisson process with significance level 0alpha1 and to do a simulation study to observe the probabilities that parameters are contained in the confidence intervals. We do not assume any parametric form for the intensity function except that it is periodic. We consider in the observed there is only one realization of the cyclic Poisson process in a bounded interval. The main results are two theory about confidence intervals for parameters. The simulation shows that the probability values of the observed parameters contained in the confidence intervals are in accordance with the theory

Asymptotic Distribution of an Estimator for Variance Function of a Compound Periodic Poisson Process with Power Function Trend

In this paper, an asymptotic distribution of the estimator for the variance function of a compound periodic Poisson process with power function trend is discussed. The periodic component of this intensity function is not assumed to have a certain parametric form, except it is a periodic function with known period. The slope of power function trend is assumed to be positive, but its value is unknown. The objectives of this research are to modify the existing variance function estimator and to determine its asymptotic distribution. This research begins by modifying the formulation of the variance function estimator. After the variance function is obtained, the research is continued by determining the asymptotic distribution of the variance function estimator of the compound periodic Poisson process with a power function trend. The first result is modification of existing estimator so that its asymptotic distribution can be determined. The main result is asymptotic normality of the estimator of variance function of a compound periodic Poisson process with power function trend

ESTIMATION OF UNEMPLOYMENT RATE USING SMALL AREA ESTIMATION MODEL BASED ON A ROTATING PANEL NATIONAL LABOR FORCE SURVEY

In Indonesia, labor force participation data are collected by Sakernas (National Labor Force Survey). Sakernas is conducted based on a quarterly rotating panel survey. Because of the groups differ according to their time-in-panel and observation strategy, it is possible to the presence of a bias. Besides, there are insufficiency problem of sample size to obtain an adequate precision of direct estimation at the district level. It is necessary to study how to estimate parameter based on a rotating panel survey when sample size is insufficient. Currently, a small area estimation (SAE) model that accomodates the bias component due to the rotation still only assume the effect over time which follows a random walk process, so it is necessary to develop a model that is more general. We propose a SAE model for rotation group level, its combined idea of the time-series multi-level model and the Rao-Yu model. The model will applied to Sakernas data to estimate a quarterly unemployment rate at the district level.Key words : Sakernas, rotating panel survey, time-series multi-level model and Rao-Yu mode

The Application of Modeling Gamma-Pareto Distributed Data Using GLM Gamma in Estimation of Monthly Rainfall with TRMM Data

As a recently developed distribution, the application of Gamma-Pareto is limited to single variable modeling.  A specific transformation of Gamma-Pareto (G-P) yields gamma distribution. Therefore, it is possible to use analysis based on gamma distribution (e.g. GLM) for modeling G-P distributed data.  In this paper we study the application of modeling G-P distributed data using GLM gamma for monthly rainfall which observed in Sukadana Station.  The modeling aims to analyze whether Tropical Rainfall Measuring Mission (TRMM) satellite data is a good estimator for unobserved station’s data.  The transformed of station’s data were considered as response variable in GLM gamma.  The explanatory variable is TRMM data in 9 grids around the station. There are two kinds of modeling i.e. model for whole data and extreme data. The results show that for both data the station’s data are G-P distributed and the transformed data are gamma distributed.  TRMM rainfall data at each grid around the station can be used to estimate the observed data of monthly rainfall. The best model for both data contains dummy variables which correspond to inter quantile data.  The coefficients of dummy variables in the best model may substitute the grouping or the correction in the previous studies