Pamukkale Univ Muh Bilim Derg. 2020; 26(1): 195-202 | DOI: 10.5505/pajes.2019.36974

Analysis of the GI/M/3/K Queueing System by Semi-Markov Process

Hanifi Okan Ýþgüder
Dokuz Eylul University Faculty Of Science, Department Of Statistics, Ýzmir, Turkey

In this study, a queuing system of K-capacity with recurrent entry and three heterogeneous servers has been investigated. In the system discussed, inter-arrival times are independent of one another and have an arbitrary distribution. The service time of each server has an Exponential distribution with parameter μ_k. The customer who enters the system starts to receive service on the server with the lowest index number from the servers that are empty. If all servers are busy on arrival, the incoming customer joins the queue. When the system is at full capacity, the incoming customer leaves the system without receiving any service. The system under consideration was modeled using a semi-Markov process and the embedded Markov chain provided by the semi-Markov process was obtained. Steady-state probabilities and the probability of customer loss were calculated. Additionally, by performing optimization with respect to service discipline and arrival process, the loss probability is minimized. The obtained theoretical results are shown numerically for cases where the inter-arrival times followed Exponential, Erlang, and deterministic distributions.

Keywords: Semi-Markov process, Embedded Markov chain, Loss probability, Heterogeneous servers, Queueing systems

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GI/M/3/K Kuyruk Sisteminin Yarý-Markov Süreciyle Analizi

Hanifi Okan Ýþgüder
Dokuz Eylül Üniversitesi Fen Fakültesi, Ýstatistik Bölümü, Ýzmir, Türkiye

Bu çalýþmada tekrarlý giriþli, K-kapasiteli ve üç heterojen kanallý bir kuyruk sistemi incelenmiþtir. Ele alýnan sistemde geliþlerarasý süreler birbirlerinden baðýmsýz olup rastgele bir daðýlýma sahiptir. Her bir kanalýn hizmet süresi μ_k parametreli üstel daðýlýma sahiptir. Sisteme gelen müþteri boþ olan kanallardan indeks numarasý en düþük olan kanalda hizmet almaya baþlar. Geliþ anýnda bütün kanallar doluysa, gelen müþteri kuyruða katýlýr. Sistem kapasitesi tamamen dolduðu zaman, gelen müþteri hiçbir hizmet almadan sistemden ayrýlýr. Ele alýnan sistem yarý-Markov süreci ile modellenmiþ ve yarý-Markov sürecinin sunulan Markov zinciri elde edilmiþtir. Duraðan durum olasýlýklarý ve müþterinin kaybolma olasýlýðý hesaplanmýþtýr. Ayrýca geliþ sürecine ve hizmet disiplinine göre en iyileme yapýlarak kaybolma olasýlýðý enküçüklenmiþtir. Elde edilen teorik sonuçlar, geliþlerarasý sürelerin daðýlýmý sýrasýyla üstel, Erlang ve deterministik seçilerek sayýsal olarak gösterilmiþtir.

Anahtar Kelimeler: Yarý-Markov süreci, Gömülü Markov zinciri, Kayýp olasýlýðý, Heterojen kanallar, Kuyruk sistemleri

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