GI/M/3/K kuyruk sisteminin Yarı-Markov süreciyle analizi

Bu çalışmada tekrarlı girişli, K-kapasiteli ve üç heterojen kanallı bir kuyruk sistemi incelenmiştir. Ele alınan sistemde gelişlerarası süreler birbirlerinden bağımsız olup rastgele bir dağılıma sahiptir. Her bir kanalın hizmet süresi μ_k parametreli üstel dağılıma sahiptir. Sisteme gelen müşteri boş olan kanallardan indeks numarası en düşük olan kanalda hizmet almaya başlar. Geliş anında bütün kanallar doluysa, gelen müşteri kuyruğa katılır. Sistem kapasitesi tamamen dolduğu zaman, gelen müşteri hiçbir hizmet almadan sistemden ayrılır. Ele alınan sistem yarı-Markov süreci ile modellenmiş ve yarı-Markov sürecinin sunulan Markov zinciri elde edilmiştir. Durağan durum olasılıkları ve müşterinin kaybolma olasılığı hesaplanmıştır. Ayrıca geliş sürecine ve hizmet disiplinine göre en iyileme yapılarak kaybolma olasılığı enküçüklenmiştir. Elde edilen teorik sonuçlar, gelişlerarası sürelerin dağılımı sırasıyla üstel, Erlang ve deterministik seçilerek sayısal olarak gösterilmiştir.

Anahtar Kelimeler:

Yarı-Markov süreci, Gömülü Markov zinciri, Kayıp olasılığı, Heterojen kanallar, Kuyruk sistemleri

Analysis of the GI/M/3/K queueing system by Semi-Markov process

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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