The matrix-form solution for \(Geo^{X}/G/1/N\) working vacation queue and its application to state-dependent cost control. (English) Zbl 1349.90226
Summary: This paper studies a finite buffer size \(Geo^X / G / 1 / N\) queue with single working vacation and different input rates. By combining two classic methods of supplementary variable and embedded Markov chain, the matrix-form solutions of queue-length distributions at departure, arbitrary, pre-arrival and outside observer’s observation epochs are obtained. Based on these queue-length distributions, some performance measures are also derived and some numerical experiments are carried out. Then, an optimal control on state-dependent operating cost of a production to order is presented.
MSC:
| 90B22 | Queues and service in operations research |
| 60K25 | Queueing theory (aspects of probability theory) |
Keywords:
stochastic model; single working vacation; finite capacity; matrix-form solution; state-dependent costReferences:
| [1] | Takagi, H., Queueing analysis—a foundation of performance evaluation, Discrete-time systems, vol. 3 (1993), North-Holland: North-Holland New York |
| [2] | Tian, N.; Xu, X.; Ma, Z., Discrete time queue theory (2007), Science Press: Science Press Beijing |
| [3] | Alfa, A. S., Vacation models in discrete time, Queueing Syst, 44, 5-30 (2003) · Zbl 1023.90009 |
| [4] | Servi, L. D.; Finn, S. G., \(M / M / 1\) queue with working vacations \((M / M / 1 / WV\), Perform Eval, 50, 41-52 (2002) |
| [5] | Li, J.; Tian, N.; Liu, W., Discrete-time \(GI / Geo / 1\) queue with working vacations, Queueing Syst, 56, 53-63 (2007) · Zbl 1177.90095 |
| [6] | Li, J.; Tian, N., The discrete-time \(GI / Geo / 1\) queue with working vacations and vacation interruption, Appl Math Comput, 185, 1-10 (2007) · Zbl 1109.60076 |
| [7] | Yi, X. W.; Kim, J. D.; Choi, D. W.; Chae, K. C., The \(Geo / G / 1\) queue with disasters and multiple working vacations, Stoch Models, 23, 537-549 (2007) · Zbl 1153.60395 |
| [8] | Li, J.; Liu, W.; Tian, N., Steady-state analysis of a discrete-time batch arrival queue with working vacations, Perform Eval, 67, 897-912 (2010) |
| [9] | Gao, S.; Liu, Z., Performance analysis of a discrete-time \(Geo^X / G / 1\) queue with single working vacation, World Acad Sci Eng Technol, 56, 1333-1341 (2011) |
| [10] | Goswami, C.; Selvaraju, N., The discrete-time \(MAP / PH / 1\) queue with multiple working vacations, Appl Math Model, 34, 931-946 (2010) · Zbl 1185.90047 |
| [11] | Li, J., Analysis of the discrete-time \(Geo / G / 1\) working vacation queue and its application to network scheduling, Comput Ind Eng, 65, 594-C604 (2013) |
| [12] | Banik, A. D.; Gupta, U. C.; Pathak, S. S., On the \(GI / M / 1 / N\) queue with multiple working vacations analytic analysis and computation, Appl Math Model, 31, 1701-1710 (2007) · Zbl 1167.90441 |
| [13] | Yu, M. M.; Tang, Y. H., Steady state analysis and computation of the \(GI^{[x]} / M^b / 1 / L\) queue with multiple working vacations and partial batch rejection, Comput Ind Eng, 56, 1243-1253 (2009) |
| [14] | Goswami, V.; Mund, G. B., Analysis of a discrete-time \(GI / GEO / 1 / N\) queue with multiple working vacations, J Syst Sci Syst Eng, 19, 367-384 (2010) |
| [15] | Goswami, V.; Laxmi, P. V., Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection, J Ind Manag Optim, 6, 911-927 (2010) · Zbl 1205.60158 |
| [16] | Yu, M. M.; Tang, Y. H.; Fu, Y. H.; Pan, L. M., \(GI / Geom / 1 / N / MWV\) queue with changeover time and searching for the optimum service rate in working vacation period, J Comput Appl Math, 235, 2170-2184 (2011) · Zbl 1209.60052 |
| [17] | Zhang, M.; Hou, Z. T., Steady state analysis of the \(GI / M / 1 / N\) queue with a variant of multiple working vacations, Comput Ind Eng, 61, 1296-1301 (2011) |
| [18] | Vidhyacharan, B.; Patrick, L., Modeling a supply chain using a network of queues, Appl Math Model, 34, 2074-2088 (2010) · Zbl 1193.90071 |
| [19] | Sun, H.; Seung, J. P., The effect of different arrival rates on the N-policy of \(M / G / 1\) with server setup, Appl Math Model, 23, 289-299 (1999) · Zbl 0957.90031 |
| [20] | Sun, W.; Tian, N.; Li, S., The effect of different rates on \(Geom / G / 1\) queue with multiple adaptive vacations and server setup/closedown times, World J Model Simul, 3, 262-274 (2007) |
| [21] | Luo, C. Y.; Xiang, K. L.; Yu, M. M.; Tang, Y. H., Recursive solution of queue length distribution for \(Geo / G / 1\) queue with single server vacation and variable input rate, Comput Math Appl, 61, 2401-2411 (2011) · Zbl 1221.90040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
