Performance Parameters of the FM/FM/1 Queue with a Priori Impatience by the Soft Alpha-Cut Method
DOI:
https://doi.org/10.15408/xm2w8924Keywords:
Fuzzy Markovian queue , A priori impatience, Soft alpha-cut method, Performance indicators , Queueing theoryAbstract
This study investigates the performance parameters of the fuzzy Markovian queueing system FM/FM/1 with a priori impatience using the soft alpha-cut method. The model incorporates the hesitation behavior of customers who decide whether to join a queue based on the expected waiting time. Unlike traditional parametric nonlinear programming approaches that combine multiple fuzzy arithmetics, the proposed method employs a single arithmetic framework based solely on alpha-cuts and interval operations, thereby simplifying and enhancing the computational efficiency. The study develops fuzzy formulations for key performance indicators, including server utilization rate, system throughput, average number of customers in the system and queue, and average waiting and residence times. A numerical application in a banking service context demonstrates the validity and practicality of the approach. The results show that each performance indicator is represented as a fuzzy number characterized by its support, mode, and membership function, allowing greater flexibility in managerial decision-making compared to the classical M/M/1 queueing system. Furthermore, the modal values of the fuzzy indicators coincide with the average values of their classical counterparts, confirming that the classical models are exceptional cases of the fuzzy ones.
Keywords: Fuzzy Markovian queue; A priori impatience, Soft alpha-cut method, Performance indicators, Queueing theory.
Abstrak
Penelitian ini menyelidiki parameter kinerja sistem antrean Markovian fuzzy FM/FM/1 dengan ketidaksabaran apriori menggunakan metode alpha-cut lunak. Model ini menggabungkan perilaku ragu-ragu pelanggan yang memutuskan untuk bergabung dalam antrean berdasarkan perkiraan waktu tunggu. Tidak seperti pendekatan pemrograman nonlinier parametrik tradisional yang menggabungkan beberapa aritmatika fuzzy, metode yang diusulkan menggunakan kerangka kerja aritmatika tunggal yang semata-mata didasarkan pada alpha-cut dan operasi interval, sehingga menyederhanakan dan meningkatkan efisiensi komputasi. Studi ini mengembangkan formulasi fuzzy untuk indikator kinerja utama, termasuk tingkat utilisasi server, throughput sistem, jumlah rata-rata pelanggan dalam sistem dan antrean, serta rata-rata waktu tunggu dan waktu tinggal. Aplikasi numerik dalam konteks layanan perbankan menunjukkan validitas dan kepraktisan pendekatan ini. Hasil penelitian menunjukkan bahwa setiap indikator kinerja direpresentasikan sebagai bilangan fuzzy yang dicirikan oleh fungsi pendukung, modus, dan keanggotaannya, yang memungkinkan fleksibilitas yang lebih besar dalam pengambilan keputusan manajerial dibandingkan dengan sistem antrean M/M/1 klasik. Lebih jauh lagi, nilai modal dari indikator fuzzy bertepatan dengan nilai rata-rata dari padanan klasiknya, yang mengonfirmasi bahwa model klasik adalah kasus luar biasa dari model fuzzy.
Kata Kunci: Antrian Markov fuzzy; a priori impatience; metode soft alpha-cut; indikator kinerja; teori antrian.
2020MSC: 60K25, 03E72.
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