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# Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy

• * Corresponding author: Gábor Horváth
The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors.
• The paper introduces a class of vacation queues where the arrival and service processes are modulated by the same Markov process, hence they can be dependent. The main result of the paper is the probability generating function for the number of jobs in the system. The analysis follows a matrix-analytic approach. A step of the analysis requires the evaluation of the busy period of a quasi birth death process with arbitrary initial level. This element can be useful in the analysis of other queueing models as well. We also discuss several special cases of the general model. We show that these special settings lead to simplification of the solution.

Mathematics Subject Classification: Primary: 90B22; Secondary: 68M20, 60K25.

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• Figure 1.  Subset relations of the considered special vacation queue models

Figure 2.  Cycles in the evolution of the queue

Figure 3.  The mean number of jobs in the system

Table 1.  Vector β as a function of the vacation distribution

 Uniform Exponential Weibull The general model (0.546, 0.109, 0.345) (0.543, 0.116, 0.341) (0.539, 0.142, 0.319) The MAP/MAP/1 vacation queue (0.214, 0.097, 0.091, … 0.04, 0.382, 0.176) (0.214, 0.097, 0.091, … 0.04, 0.382, 0.176) (0.214, 0.097, 0.091, … 0.04, 0.382, 0.176) QBD vac. queue (0.546, 0.109, 0.345) (0.543, 0.115, 0.342) (0.53, 0.14, 0.33) The indep. QBD vacation queue (0.21, 0.101, 0.09, … 0.041, 0.373, 0.185) (0.207, 0.104, 0.089, … 0.042, 0.367, 0.191) (0.197, 0.113, 0.086, … 0.046, 0.348, 0.21)
•  [1] A.-S. Alfa, A discrete MAP/PH/1 queue with vacations and exhaustive time-limited service, Oper. Res. Lett., 18 (1995), 31-40.  doi: 10.1016/0167-6377(95)00015-C. [2] A.-S. Alfa, Discrete time analysis of MAP/PH/1 vacation queue with gated time-limited service, Queueing Systems, 29 (1998), 35-54.  doi: 10.1023/A:1019123828374. [3] Y. Baba, Analysis of a GI/M/1 queue with multiple working vacations, Operation Research Letters, 33 (2005), 201-209.  doi: 10.1016/j.orl.2004.05.006. [4] S. Chang and T. Takine, Factorization and stochastic decomposition properties in bulk queues with generalized vacations, Queueing Systems, 50 (2005), 165-183.  doi: 10.1007/s11134-005-0510-9. [5] B. T. Doshi, Queueing systems with vacations -a survey, Queueing Systems, 1 (1986), 29-66.  doi: 10.1007/BF01149327. [6] N. S. C. Goswami, The discrete-time MAP/PH/1 queue with multiple working vacations, Applied Mathematical Modelling, 34 (2010), 931-946.  doi: 10.1016/j.apm.2009.07.021. [7] G. Horváth, B. Van Houdt and M. Telek, Commuting matrices in the queue length and sojourn time analysis of MAP/MAP/1 queues, Stochastic Models, 30 (2014), 554-575.  doi: 10.1080/15326349.2014.930669. [8] J.-C. Ke, C.-H. Wu and Z. G. Zhang, Recent developments in vacation queueing models: A short survey, International Journal of Operations Research, 7 (2010), 3-8. [9] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling ASA-SIAM Series on Statistics and Applied Probability, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; American Statistical Association, Alexandria, VA, 1999. doi: 10.1137/1.9780898719734. [10] J. H. Li, N. S. Tian and W. Liu, Discrete-time GI/Geom/1 queue with multiple working vacations, Queueing Systems, 56 (2007), 53-63.  doi: 10.1007/s11134-007-9030-0. [11] M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.1017/S0021900200033465. [12] Z. Saffer and M. Telek, Analysis of BMAP/G/1 vacation model of non-M/G/1-type, in EPEW, vol. 5261 of LNCS, Springer, Mallorca, Spain, 2008,212–226. doi: 10.1007/978-3-540-87412-6_16. [13] Z. Saffer and M. Telek, Closed form results for BMAP/G/1 vacation model with binomial type disciplines, Publ. Math. Debrecen, 76 (2010), 359-378. [14] Z. Saffer and W. Yue, M/M/c multiple synchronous vacation model with gated discipline, Journal of Industrial and Management Optimization (JIMO), 8 (2012), 939-968.  doi: 10.3934/jimo.2012.8.939. [15] W. -H. Steeb, Matrix Calculus and Kronecker Product with Applications and C++ Programs World Scientific, 1997. doi: 10.1142/3572. [16] H. Takagi, Queueing Analysis -A Foundation of Performance Evaluation, Vacation and Prority Systems, vol. 1 North-Holland, New York, 1991. [17] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications vol. 93, Springer Science & Business Media, 2006. [18] D. Wu and H. Takagi, M/G/1 queue with multiple working vacations, Performance Evaluation, 63 (2006), 654-681.  doi: 10.1016/j.peva.2005.05.005.

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