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Performance Analysis of a Discrete-Time Queue with Working Breakdowns and Searching for the Optimum Service Rate in Working Breakdown Period
Shaojun LAN, Yinghui TANG
Journal of Systems Science and Information ›› 2017, Vol. 5 ›› Issue (2) : 176-192.
PDF(489 KB)
PDF(489 KB)
Performance Analysis of a Discrete-Time Queue with Working Breakdowns and Searching for the Optimum Service Rate in Working Breakdown Period
This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject to breakdowns at random when it is in operation. As soon as the server fails, a repair process immediately begins. During the repair period, the defective server still provides service for the waiting customers at a lower service rate rather than completely stopping service. We analyze the stability condition for the considered system. Using the probability generating function technique, we obtain the probability generating function of the steady-state queue size distribution. Also, various important performance measures are derived explicitly. Furthermore, some numerical results are provided to carry out the sensitivity analysis so as to illustrate the effect of different parameters on the system performance measures. Finally, an operating cost function is formulated to model a computer system and the parabolic method is employed to numerically find the optimum service rate in working breakdown period.
discrete-time queue / working breakdowns / different arrival rates / performance measures / optimum service rate {{custom_keyword}} /
[1] Hunter J J. Mathematical techniques of applied probability, Vol. 2, discrete time models: Techniques and applications. Academic Press, New York, 1983.
[2] Bruneel H, Kim B G. Discrete-time models for communication systems including ATM. Kluwer Academic Publishers, Boston, 1993.
[3] Takagi H. Queueing analysis — A foundation of performance evaluation, Vol. 3, discrete-time systems. North-Holland, New York, 1993.
[4] Woodward M E. Communication and computer networks: Modelling with discrete-time queues. IEEE Computer Society Press, California, 1994.
[5] Alfa A S. Queueing theory for telecommunications: Discrete time modelling of a single node system. Springer, New York, 2010.
[6] Atencia I, Moreno P. A discrete-time Geo/G/1 retrial queue with server breakdowns. Asia-Pacific Journal of Operational Research, 2006, 23(2): 247-271.
[7] Tang Y, Yun X, Huang S. Discrete-time GeoX/G/1 queue with unreliable server and multiple adaptive delayed vacations. Journal of Computational and Applied Mathematics, 2008, 220: 439-455.
[8] Wang J, Zhang P. A discrete-time retrial queue with negative customers and unreliable server. Computers & Industrial Engineering, 2009, 56(4): 1216-1222.
[9] Lin C H, Ke J C. On the discrete-time system with server breakdowns: Computational algorithm and optimization algorithm. Applied Mathematics and Computation, 2011, 218(7): 3624-3634.
[10] Wang T Y, Ke J C, Chang F M. Analysis of a discrete-time queue with server subject to vacations and breakdowns. Journal of Industrial and Production Engineering, 2013, 30(1): 54-66.
[11] Gao S, Liu Z. A repairable GeoX/G/1 retrial queue with Bernoulli feedback and impatient customers. Acta Mathematicae Applicatae Sinica, English Series, 2014, 30(1): 205-222.
[12] Wang T Y. An unreliable Geo/G/1 queue with startup and closedown times under randomized finite vacations. Applied Mathematical Modelling, 2015, 39(3): 1383-1399.
[13] Atencia I. A discrete-time queueing system with server breakdowns and changes in the repair times. Annals of Operations Research, 2015, 235(1): 37-49.
[14] Kalidass K, Kasturi R. A queue with working breakdowns. Computers & Industrial Engineering, 2012, 63(4): 779-783.
[15] Li L, Wang J, Zhang F. Equilibrium customer strategies in Markovian queues with partial breakdowns. Computers & Industrial Engineering, 2013, 66(4): 751-757.
[16] Kim B K, Lee D H. The M/G/1 queue with disasters and working breakdowns. Applied Mathematical Modelling, 2014, 38(5): 1788-1798.
[17] Liu Z, Song Y. The MX/M/1 queue with working breakdown. RAIRO-Operations Research, 2014, 48(3): 399-413.
[18] Yang D Y, Wu Y Y. Transient behavior analysis of a finite capacity queue with working breakdowns and server vacations. Proceedings of the International Multiconference of Engineers and Computer Scientists, 2014: 1151-1156.
[19] Liou C D. Markovian queue optimisation analysis with an unreliable server subject to working breakdowns and impatient customers. International Journal of Systems Science, 2015, 46(12): 2165-2182.
[20] Servi L D, Finn S G. M/M/1 queues with working vacations (M/M/1/WV). Performance Evaluation, 2002, 50: 41-52.
[21] Hur S, Paik S J. The effect of different arrival rates on the N-policy of M/G/1 with server setup. Applied Mathematical Modelling, 1999, 23(4): 289-299.
[22] Sun W, Tian N, Li S. The effect of different rates on Geomξ /G/1 queue with multiple adaptive vacations and server setup/closedown times. World Journal of Modelling and Simulation, 2007, 3(4): 262-274.
[23] Wei Y, Yu M, Tang Y, et al. Queue size distribution and capacity optimum design for N-policy Geo(λ1,λ2,λ3)/G/1 queue with setup time and variable input rate. Mathematical and Computer Modelling, 2013, 57: 1559-1571.
[24] Luo C, Tang Y, Yu K, et al. Optimal (r,N)-policy for discrete-time Geo/G/1 queue with different input rate and setup time. Applied Stochastic Models in Business and Industry, 2015, 31(4): 405-423.
[25] Neuts M. Matrix-geometric solution in stochastic models — An algorithmic approach. Johns Hopkins University Press, Baltimore, 1981.
[26] Ronald L R. Optimization in operations research. Prentice Hall, New Jersey, 1997.
Supported by the National Natural Science Foundation of China (71571127), the Training Fund Program of Excellent Paper of Sichuan Normal University ([2016]4-1)
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