×

Spectral analysis of a discrete-time queueing model with \(N\)-policy on an accelerated service. (English) Zbl 1470.60243

Dudin, Alexander (ed.) et al., Information technologies and mathematical modelling. Queueing theory and applications. 19th international conference, ITMM 2020, named after A. F. Terpugov, Tomsk, Russia, December 2–5, 2020. Revised selected papers. Cham: Springer. Commun. Comput. Inf. Sci. 1391, 404-416 (2021).
Summary: This paper analyses a discrete-time queueing model with two modes of service and \(N\)-policy. The arrival of customers constitutes a Bernoulli process. There are two types of service; mode 1 and mode 2 of which service times are geometrically distributed with parameters \(q_1\) and \(q_2\) respectively, where \(q_1<q_2\), so that the mode 2 is an accelerated service mode. Initially, the service starts with mode 1 and when the number of customers reaches \(N\), the server tries to change the type of the service to mode 2 with probability \(\theta \). Once the type of the service is changed from mode 1 to mode 2, it will resume the reduced rate when either of two cases happens; i) the number of customers in the system is either less than \(N\) and ii) the number of customers is reduced to zero. These two cases are studied in Model I and Model II respectively. The spectral value (or eigenvalue) approach is used to analyze Model I and consequently obtain the rate matrix of the model. Using the rate matrix of Model I, we analyze Model II. On the basis of a suitable cost function, numerical experiments are conducted for the models and obtained the optimum value of \(N\).
For the entire collection see [Zbl 1466.68011].

MSC:

60K25 Queueing theory (aspects of probability theory)
68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
90B22 Queues and service in operations research
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alfa, A.S.: Applied Discrete-Time Queues. Springer, Heidelberg (2016). doi:10.1007/978-1-4939-3420-1 · Zbl 1337.90001
[2] Anilkumar, MP; Jose, KP, A discrete time \(Geo/Geo/1\) inventory system with modified \(n \)-policy, Malaya J. Matematik (MJM), 8, 3, 868-876 (2020) · doi:10.26637/MJM0803/0023
[3] Anilkumar, M.P., Jose, K.P.: Discrete time priority queue with self generated interruption. In: AIP Conference Proceedings, vol. 2261, p. 030031. AIP Publishing LLC. (2020) · Zbl 1456.90007
[4] Anilkumar, MP; Jose, KP, A Geo/Geo/1 inventory priority queue with self induced interruption, Int. J. Appl. Comput. Math., 6, 4, 1-14 (2020) · Zbl 1456.90007 · doi:10.1007/s40819-020-00857-8
[5] Chandrasekaran, V.; Indhira, K.; Saravanarajan, M.; Rajadurai, P., A survey on working vacation queueing models, Int. J. Pure Appl. Math., 106, 6, 33-41 (2016)
[6] Drekic, S.; Grassmann, WK, An eigenvalue approach to analyzing a finite source priority queueing model, Ann. Oper. Res., 112, 1-4, 139-152 (2002) · Zbl 1013.90025 · doi:10.1023/A:1020933122382
[7] Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104) (1982). https://books.google.co.in/books?id=KwEItnMvwbgC · Zbl 0482.15001
[8] Goswami, V., A discrete-time queue with balking, reneging, and working vacations, Int. J. Stochastic Anal., 2014, 1-8 (2014) · Zbl 1322.60191
[9] Grassmann, W.; Tavakoli, J., The distribution of the line length in a discrete time GI/G/1 queue, Performance Eval., 131, 43-53 (2019) · doi:10.1016/j.peva.2019.03.001
[10] Grassmann, WK; Drekic, S., An analytical solution for a tandem queue with blocking, Queueing Syst., 36, 1-3, 221-235 (2000) · Zbl 0966.60090 · doi:10.1023/A:1019139405059
[11] Grassmann, WK; Tavakoli, J., A tandem queue with a movable server: an eigenvalue approach, SIAM J. Matrix Anal. Appl., 24, 2, 465-474 (2002) · Zbl 1018.60086 · doi:10.1137/S0895479801394088
[12] Haverkort, B.R., Ost, A.: Steady-state analysis of infinite stochastic petri nets: comparing the spectral expansion and the matrix-geometric method. In: Proceedings of the Seventh International Workshop on Petri Nets and Performance Models, pp. 36-45. IEEE (1997)
[13] Heyman, DP, Optimal operating policies for M/G/1 queuing systems, Oper. Res., 16, 2, 362-382 (1968) · Zbl 0164.47704 · doi:10.1287/opre.16.2.362
[14] Hunter, J.: Mathematical Techniques of Applied Probability, vol. ii (1983) · Zbl 0539.60065
[15] Krishnamoorthy, A.; Deepak, TG, Modified N-policy for M/G/1 queues, Comput. Oper. Res., 29, 12, 1611-1620 (2002) · Zbl 1259.90026 · doi:10.1016/S0305-0548(00)00108-8
[16] Lan, S.; Tang, Y., An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback, RAIRO-Oper. Res., 53, 2, 367-387 (2019) · Zbl 1423.60141 · doi:10.1051/ro/2017027
[17] Li, JH; Tian, NS, Analysis of the discrete time Geo/Geo/1 queue with single working vacation, Qual. Technol. Quant. Manage., 5, 1, 77-89 (2008) · doi:10.1080/16843703.2008.11673177
[18] Ma, Z.; Wang, P.; Yue, W., Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations, J. Ind. Manage. Optim., 13, 1467-1481 (2017)
[19] Meisling, T., Discrete-time queuing theory, Oper. Res., 6, 1, 96-105 (1958) · Zbl 1414.90112 · doi:10.1287/opre.6.1.96
[20] Mitrani, I.; Chakka, R., Spectral expansion solution for a class of Markov models: application and comparison with the matrix-geometric method, Performance Eval., 23, 3, 241-260 (1995) · Zbl 0875.68103 · doi:10.1016/0166-5316(94)00025-F
[21] Morse, P.M.: Queues, Inventories, and Maintenance (1958)
[22] Neuts, M.F.: Matrix-geometric solutions in stochastic models: an algorithmic approach. Courier Corporation (1994)
[23] Moreno, P., A discrete-time single-server queue with a modified N-policy, Int. J. Syst. Sci., 38, 6, 483-492 (2007) · Zbl 1149.90041 · doi:10.1080/00207720701353405
[24] Sun, G.Y., Zhu, Y.J.: The Geo/Geo/1 queue model with N-policy, negative customer, feedback and multiple vacation. J. Shandong Univ. (Nat. Sci.) 2 (2010) · Zbl 1224.90039
[25] Tian, N.; Ma, Z.; Liu, M., The discrete time Geom/Geom/1 queue with multiple working vacations, Appl. Math. Model., 32, 12, 2941-2953 (2008) · Zbl 1167.90461 · doi:10.1016/j.apm.2007.10.005
[26] Wang, T-Y; Ke, J-C, The randomized threshold for the discrete-time Geo/G/1 queue, Appl. Math. Model., 33, 7, 3178-3185 (2009) · Zbl 1205.90095 · doi:10.1016/j.apm.2008.10.010
[27] Yadin, M.; Naor, P., Queueing systems with a removable service station, J. Oper. Res. Soc., 14, 4, 393-405 (1963) · doi:10.1057/jors.1963.63
[28] Zhang, Z.J., Xu, X.l.: Analysis for the M/M/1 queue with multiple working vacations and N-policy. Int. J. Inf. Manage. Sci. 19(3), 495-506 (2008) · Zbl 1211.90054
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.