Sains Malaysiana 55(5)(2026): 922-935

http://doi.org/10.17576/jsm-2026-5505-14

 

Finite Mixture of Gamma Interarrival Times and Count Distributions: Computation and Applications

(Campuran Terhingga Lat Ketibaan Gama dan Taburan Bilangan: Pengiraan dan Aplikasi)

 

HARPREET SINGH¹, SENG HUAT ONG^1,2 & CHOUNG MIN NG^1,3,*

 

¹Institute of Mathematical Sciences, Universiti Malaya, 50603 Kuala Lumpur, Malaysia

²Institute of Actuarial Science and Data Analytics, UCSI University, 56000 Kuala Lumpur, Malaysia

³Universiti Malaya Centre for Data Analytics, 50603 Kuala Lumpur, Malaysia

 

Diserahkan: 27 April 2025/Diterima: 4 May 2026

 

Abstract

The Poisson process is the basic renewal process with the exponential distribution as the interarrival time distribution. Many nonexponential interarrival distributions have been investigated by researchers along with the derivation of the corresponding count distributions. In many of these distributions, the computation of the count probabilities may be problematic. This paper proposes a finite mixture of gamma distributions for the interarrival times, with the corresponding count distribution derived for a renewal process, motivated by hospital patient arrivals that include both scheduled and walk-in patients. For the proposed finite gamma mixture model, the inverse Laplace transform technique is advocated to overcome the computational difficulties; this generic method is applicable to nonexponential interarrival time models. Some popular numerical inverse Laplace transforms have been compared. The Den Iseger algorithm is recommended for computing probabilities due to its simplicity, high accuracy, and stability. Analyses of both hospital and nonhospital datasets demonstrate the effectiveness of the proposed finite mixture model in accurately representing count data and providing a better fit compared to other models. This finite gamma mixture model, an extension of hyperexponential mixtures, is useful for resource allocation in healthcare, and other fields like finance and economics.

Keywords: Den Iseger algorithm; finite mixture; inverse Laplace transform; patient arrivals; renewal process

 

Abstrak

Proses Poisson ialah proses pembaharuan asas dengan taburan eksponen sebagai taburan lat ketibaan. Banyak taburan lat ketibaan yang bukan eksponen telah dikaji oleh penyelidik bersama dengan terbitan taburan bilangan yang sepadan. Dalam kebanyakan taburan tersebut, pengiraan kebarangkalian bilangan mungkin menjadi masalah. Makalah ini mencadangkan satu campuran terhingga taburan gama untuk lat ketibaan dengan taburan bilangan sepadan diterbitkan untuk satu proses pembaharuan yang didorong oleh ketibaan pesakit hospital yang merangkumi pesakit berjadual dan pesakit tanpa janji temu. Bagi model campuran gama terhingga yang dicadangkan, teknik jelmaan Laplace songsang disarankan untuk mengatasi kesukaran pengiraan; kaedah generik ini boleh digunakan untuk model lat ketibaan bukan eksponen. Beberapa jelmaan Laplace songsang berangka yang popular telah dibandingkan. Algoritma Den Iseger disyorkan untuk mengira kebarangkalian disebabkan oleh ia mudah, berketepatan tinggi dan stabil. Analisis terhadap set data hospital dan bukan hospital menunjukkan keberkesanan model campuran terhingga yang dicadangkan dalam mewakili data bilangan secara tepat dan memberikan penyuaian yang lebih baik berbanding dengan model yang lain. Model campuran gama terhingga ini yang merupakan perluasan daripada campuran hipereksponen berguna untuk pengagihan sumber dalam bidang penjagaan kesihatan serta bidang lain seperti kewangan dan ekonomi.

Kata kunci: Algoritma Den Iseger; campuran terhingga; jelmaan Laplace songsang; ketibaan pesakit; proses pembaharuan

 

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*Pengarang untuk surat-menyurat; email: ngcm@um.edu.my