# A Roots Method in GI/PH/1 Queueing Model and Its Application

• Accepted : 2013.08.13
• Published : 2013.09.30
• 48 3

#### Abstract

In this paper, we introduce a roots method that uses the roots inside the unit circle of the associated characteristics equation to evaluate the steady-state system-length distribution at three epochs (pre-arrival, arbitrary, and post-departure) and sojourn-time distribution in GI/PH/1 queueing model. It is very important for an air base to inspect airplane oil because low-quality oil leads to drop or breakdown of an airplane. Since airplane oil inspection is composed of several inspection steps, it sometimes causes train congestion and delay of inventory replenishments. We analyzed interarrival time and inspection (service) time of oil supply from the actual data which is given from one of the ROKAF's (Republic of Korea Air Force) bases. We found that interarrival time of oil follows a normal distribution with a small deviation, and the service time follows phase-type distribution, which was first introduced by Neuts to deal with the shortfalls of exponential distributions. Finally, we applied the GI/PH/1 queueing model to the oil train congestion problem and analyzed the distributions of the number of customers (oil trains) in the queue and their mean sojourn-time using the roots method suggested by Chaudhry for the model GI/C-MSP/1.

#### Keywords

Roots Method;Phase-Type Distribution;Airplane Oil Inspection;Train Congestion Problem;GI/PH/1;Queueing Theory

#### References

1. Asmussen, S. (1992), Phase-type representations in random walk and queueing problems, Annals of Probability, 20(2), 772-789. https://doi.org/10.1214/aop/1176989805
2. Asmussen, S., Nerman, O., and Olsson, M. (1996), Fitting phase-type distributions via the EM algorithm, Scandinavian Journal of Statistics, 23(4), 419-441.
3. Baum, D. and Breuer, L. (2006), Applying Foster's criteria to a GI/PH/1 queueing system, Cybernetics and Systems Analysis, 42(3), 433-439. https://doi.org/10.1007/s10559-006-0081-8
4. Chakravarthy, S. (1992), A finite capacity GI/PH/1 queue with group services, Naval Research Logistics, 39(3), 345-357. https://doi.org/10.1002/1520-6750(199204)39:3<345::AID-NAV3220390305>3.0.CO;2-V
5. Chaudhry, M. L. and Templeton, J. G. C. (1983), A First Course in Bulk Queues, John Wiley & Sons, New York, NY.
6. Chaudhry, M. L., Harris, C. M., and Marchal, W. G. (1990), Robustness of root finding in single-server queueing models, INFORMS Journal on Computing, 2(3), 273-286. https://doi.org/10.1287/ijoc.2.3.273
7. Chaudhry, M. L., Samanta, S. K., and Pacheco, A. (2012), Analytically explicit results for the GI/C-MSP/1/${\infty}$ queueing system using roots, Probability in the Engineering and Informational Sciences, 26(2), 221-244. https://doi.org/10.1017/S0269964811000349
8. Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977), Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society: Series B, 39(1), 1-38.
9. Dudin, A. N. and Klimenok, V. I. (2003), Optimal admission control in a queueing system with heterogeneous traffic, Operations Research Letters, 31(2), 108-118. https://doi.org/10.1016/S0167-6377(02)00218-3
10. Dudin, A. N., Kim, C. S., and Semyonova, O. V. (2004), An optimal multithreshold control for the input flow of the GI/PH/1 queueing system with a BMAP flow of negative customers, Avtomatika i Telemekhanika, 9, 71-84.
11. Grassmann, W. K. (1982), The GI/PH/1 queue: a method to find the transition matrix, INFOR, 20, 144-156.
12. Guo, M. M., Tian, N. S., and Liu, A. Y. (2007), Queue system GI/PH/1 with repairable service station, Operations Research and Management Science, 16(5), 69-74.
13. Kao, E. P. C. (1991), Using state reduction for computing steady state probabilities of queues of GI/PH/1 types, ORSA Journal on Computing, 3(3), 231-240. https://doi.org/10.1287/ijoc.3.3.231
14. Kao, E. P. C. (1996), A comparison of alternative approaches for numerical solutions of GI/PH/1 queues, INFORMS Journal on Computing, 8(1), 74-85. https://doi.org/10.1287/ijoc.8.1.74
15. Kim, J. S. (2006), Asymptotic analysis of the loss probability in the GI/PH/1/K queue, Journal of Applied Mathematics & Computing, 22(1), 273-283. https://doi.org/10.1007/BF02896477
16. Latouche, G. (1993), Algorithms for infinite Markov chains with repeating columns. In Meyer, C. D. and Plemmons, R. D. (eds.), Linear Algebra, Markov Chains and Queueing Models, Springer, New York, NY, 231-265.
17. Latouche, G. and Ramaswami, V. (1999), Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, Philadelphia, PA.
18. Neuts, M. F. (1981a), Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press, Baltimore, MD.
19. Neuts, M. F. (1981b), Stationary waiting-time distributions in the GI/PH/1 queue, Journal of Applications Probability, 18(4), 901-912. https://doi.org/10.2307/3213064
20. Ramaswami, V. and Latouche, G. (1989), An experimental evaluation of the matrix-geometric method for the GI/PH/1 queue, Communications in Statistics: Stochastic Models, 5(4), 629-667. https://doi.org/10.1080/15326348908807128
21. Ramaswami, V. and Lucantoni, D. M. (1988), Moments of the stationary waiting time in the GI/PH/1 queue, Journal of Applications Probability, 25(3), 636-641. https://doi.org/10.2307/3213992
22. Sengupta, B. (1989), Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue, Advances in Applied Probability, 21(1), 159-180. https://doi.org/10.2307/1427202
23. Sengupta, B. (1990), The semi-Markovian queue: theory and applications, Communications in Statistics: Stochastic Models, 6(3), 383-413. https://doi.org/10.1080/15326349908807154