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

Öz

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

Öz

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.

Anahtar Kelimeler

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

Kaynakça

1. Erlang AK. “The theory of probabilities and telephone conversations”. Nyt Tidsskrift for Matematik, 20(B), 33-39, 1909.
2. Palm C. “Intensitätsschwankungen im Fernsprechverkehr”. Ericsson Techniks, 44(39), 1-189, 1943.
3. Kendall DG. “Some problems in the theory of queues”. Journal of the Royal Statistical Society Series B (Methodological), 13(2), 151-185, 1960.
4. Gumbel M. “Waiting lines with heterogeneous servers”. Operations Research, 8(4), 219-225, 1960.
5. Takacs L. “On the generalization of Erlang’s formula”. Acta Mathematica Hungarica, 7(3), 419-433, 1956.
6. Yao DD. “Convexity properties of the overflow in an ordered-entry system with heterogeneous servers”. Operations Research Letters, 5(3), 145-147, 1986.
7. Neuts FM, Takahashi Y. “Asymptotic behavior of the stationary distributions in the GI/PH/c queue with heterogeneous servers”. Probability Theory and Related Fields, 57(4), 441-452, 1981.
8. Lin BW, Elsayed EA. “A general solution for multichannel queueing systems with ordered entry”. Computers & Operations Research, 5(4), 504-511, 1978.

9. Fakinos D. “The blocking system with heterogeneous servers”. The Journal of the Operational Research Society, 31(10), 388-394, 1980.
10. Nawijn MW. “On a two-server finite queuing system with ordered entry and deterministic arrivals”. European Journal of Operational Research, 18(3), 388-395, 1984.
11. Alpaslan F, Shahbazov A. “An analysis and optimization of stochastic service with heterogeneous channel and poisson arrival”. Pure and Applied Mathematika Sciences, 43(2), 15-20, 1996.
12. Kumar BK, Madheswari SP, Venkatakrishnan KS. “Transient solution of an M/M/2 queue with heterogeneous servers subject to catastrophes”. International Journal of Information and Management Sciences, 18(1), 63-80, 2007.
13. Alves FSQ, Yehia HC, Pedrosa LAC, Cruz FRB, Kerbache L. “Upper bounds on performance measures of heterogeneous M/M/c queues”. Mathematical Problems in Engineering, Vol: 2011 Article ID 702834, doi:10.1155/2011/702834, 2011.
14. Choudhury G, Deka M. “A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation”. Applied Mathematical Modelling, 36(12), 6050-6060, 2012.
15. Alpaslan F. “A queuing model with two heterogeneous servers and overflow”. Pure and Applied Mathematika Sciences, 55(1), 1-8, 2002.
16. Cooper, RB. “Queues with ordered servers that work at different rates”. Opsearch, 13(2), 69-78, 1976.
17. Matsui M, Fukuta J. “On a multichannel queueing system with ordered entry and heterogeneous servers”. AIIE Transactions, 9(2), 209-214, 1977.
18. Nath G, Enns E. “Optimal service rates in the multiserver loss system with heterogeneous servers”. Journal of Applied Probability, 18(3), 776-781, 1981.
19. Nawijn MW. “A note on many-server queueing systems with ordered entry, with an application to conveyor theory”. Journal of Applied Probability, 20(1), 144-152, 1983.
20. Pourbabai B, Sonderman D. “Service utilization factors in queueing loss systems with ordered entry and heterogeneous servers”. Journal of Applied Probability, 23(1), 236-242, 1986.
21. Yao DD. “The Arrangement of servers in an ordered-entry system”. Operations Research, 35(5), 759-763, 1987.
22. Saglam V, Shahbazov A. “Minimizing loss probability in queuing systems with heterogeneous servers”. Iranian Journal of Science and Technology, Transactions A: Science, 31(2), 199-206, 2007.
23. Isguder HO, Celikoglu CC. “Minimizing the loss probability in GI/M/3/0 queueing system with ordered entry”. Scientific Research and Essays, 7(8), 963-968, 2012.
24. Isguder HO, Kocer UK, Celikoglu CC. “Generalization of the Takacs' formula for GI/M/n/0 queuing system with heterogeneous servers”. Proceedings of the World Congress on Engineering I, London, UK, 6-8 July 2011.
25. Isguder HO, Kocer UU. “Analysis of GI/M/n/n queueing system with ordered entry and no waiting line”. Applied Mathematical Modelling, 38(3), 1024-1032, 2014.
26. Shahbazov A. Olasılık Teorisine Giriş. 1. Baskı. İstanbul, Türkiye, Birsen, 2005.
27. Isguder HO, Kocer UU. “Optimization of loss probability for GI/M/3/0 queuing system with heterogeneous servers”. Anadolu University Journal of Science and Technology: B-Theoretical Sciences, 1(1), 73-89, 2011.
28. Weber R. “On a conjecture about assigning jobs to processors of differing speeds”. IEEE Transactions on Automatic Control, 38(1), 166-170, 1993.
29. Gürcan M, Güral Y, Gokdere G. “Analysis of Repairable k-out-of-n System model using ınter-arrival failure times” CMES 2018 The Third International Conference on Computational Mathematics and Engineering Sciences, ITM Web of Conferences 22, 01054, 17 October 2018, https://doi.org/10.1051/itmconf/20182201054.