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http://dx.doi.org/10.14317/jami.2020.351

A RETRIAL QUEUEING MODEL WITH THRESHOLDS AND PHASE TYPE RETRIAL TIMES  

CHAKRAVARTHY, SRINIVAS R. (Departments of Industrial and Manufacturing Engineering & Mathematics, Kettering University)
Publication Information
Journal of applied mathematics & informatics / v.38, no.3_4, 2020 , pp. 351-373 More about this Journal
Abstract
There is an extensive literature on retrial queueing models. While a majority of the literature on retrial queueing models focuses on the retrial times to be exponentially distributed (so as to keep the state space to be of a reasonable size), a few papers deal with nonexponential retrial times but with some additional restrictions such as constant retrial rate, only the customer at the head of the retrial queue will attempt to capture a free server, 2-state phase type distribution, and finite retrial orbit. Generally, the retrial queueing models are analyzed as level-dependent queues and hence one has to use some type of a truncation method in performing the analysis of the model. In this paper we study a retrial queueing model with threshold-type policy for orbiting customers in the context of nonexponential retrial times. Using matrix-analytic methods we analyze the model and compare with the classical retrial queueing model through a few illustrative numerical examples. We also compare numerically our threshold retrial queueing model with a previously published retrial queueing model that uses a truncation method.
Keywords
Retrial; queueing; phase type distribution; thresholds;
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1 J.R. Artalejo, Accessible Bibliography on Retrial Queues, Mathematical and Computer Modelling, 30 (1999), 1-6.   DOI
2 J.R. Artalejo, M.J. Lopez-Herrero, On the Busy Period of the M/G/1 Retrial Queue, Naval Research Logistics 47 (2000), 115-127.   DOI
3 J.R. Artalejo, A. Gomez-Corral, M.F. Neuts, Analysis of multiserver queues with constant retrial rate, Euro. J. Oper. Res. 135 (2002), 569-581.   DOI
4 J.R. Artalejo, S.R. Chakravarthy, Algorithmic Analysis of a MAP/PH/1 Retrial Queue, TOP 14 (2006), 293-332.   DOI
5 J.R. Artalejo, A. Economou, M.J. Lopez-Herrero, Algorithmic approximations for the busy period distribution of the M/M/c retrial queue, Euro. J. Oper. Res. 176 (2007), 1687-1702.   DOI
6 J.R. Artalejo, A. Gomez-Corral, Modelling communication systems with phase type service and retrial times, IEEE Communications Letters 11 (2007), 955-957.   DOI
7 J.R. Artalejo, S.R. Chakravarthy, M.J. Lopez-Herrero, The busy period and the waiting time analysis of a MAP/M/c queue with finite retrial group, Stochastic Analysis and Applications 25 (2007), 445-469.   DOI
8 J.R. Artalejo, A. Gomez-Corral, Retrial Queueing Systems: A Computational Approach, Springer-Verlag, Berlin, Heidelberg, 2008.
9 K. Avrachenko, U. Yechiali, Retrail networks with finite buffers and their application to internet data traffc, Probability in the Engineering and Informational Sciences 22 (2008), 519-536.   DOI
10 K. Avrachenko, U. Yechiali, On tandem blocking queues with a common retrial queue, Computers and Operations Research 37 (2010), 1174-1180.   DOI
11 N. Baer, R.J. Boucherie, J-K. Ommeren, The PH/PH/1 multi-threshold queues, In: B. Sericola, M. Telek, and G. Horvath (Eds.): ASMTA 2014, LNCS 8499, 95-109. Springer International Publishing Switzerland, 2014.
12 S.R. Chakravarthy, A. Krishnamoorthy and V.C. Joshua, Analysis of a Multi-server Retrial Queue with Search of Customers from the Orbit, Performance Evaluation 63 (2006), 776-798.   DOI
13 S.R. Chakravarthy, A Multi-server Queueing Model with Markovian Arrivals and Multiple Thresholds, Asia-Pacific Journal of Operational Research 24 (2007), 223-243.   DOI
14 S.R. Chakravarthy, A multi-server synchronous vacation model with thresholds and a probabilistic decision rule, European Journal of Operational Research 182 (2007), 305-320.   DOI
15 S.R. Chakravarthy, Analysis of MAP/PH/c retrial queue with phase type retrials - Simulation approach, Communications in Computer and Information Science 356 (2013), 37-49.   DOI
16 B.D. Choi, Y.W. Shin, W.C. Ahn, Retrial queues with collision arising from unslotted CSMA/CD protocol, Queueing Systems 11 (1992), 335-356.   DOI
17 D.I. Choi, T.S. Kim, S. Lee, Analysis of an MMPP/G/1/K Queue with Queue Length Dependent Arrival Rates, and its Application to Preventive Congestion Control in Telecommunication Networks, European Journal of Operational Research 187 (2008), 652-659.   DOI
18 J.E. Diamond, A.S. Alfa, Approximation method for M/PH/1 retrial queues with phase type inter-retrial times, European Journal of Operational Research 113 (1999), 620-631.   DOI
19 D.V. Efrosinin, Controlled Queueing Systems with Heterogeneous Servers, Ph.D. Dissertation, Trier University, Germany, 2004.
20 D. Efrosinin, L. Breuer, Threshold policies for controlled retrial queues with heterogeneous servers, Ann Oper Res 141 (2006), 139-162.   DOI
21 G.I. Falin, J.G.C. Templeton, Retrial Queues, Chapman and Hall, London, 1997.
22 A. Graham, Kronecker Products and Matrix Calculus with Applications, Ellis Horwood, Chichester, UK, 1981.
23 Q.M. He, H. Li, Y.Q. Zhao, Ergodicity of the BMAP/PH/s/s + K retrial queue with PH-retrial times, Queueing Systems 35 (2000), 323-347.   DOI
24 O.C. Ibe, J. Keilson, Multi-server threshold queues with hysteresis, Performance Evaluation 21 (1995), 185-2135.   DOI
25 S.B. Khodadadi, F. Jolai, A fuzzy based threshold policy for a single server retrial queue with vacations, Central European Journal of Operations Research 20 (2012), 281-297.   DOI
26 J. Kim, B. Kim, A survey of retrial queueing systems, Ann. Oper. Res. 247 (2016), 3-36.   DOI
27 G. Latouche, V. Ramaswami, Introduction to matrix analytic methods in stochastic modeling, SIAM, 1999.
28 W. Lin, P.R. Kumar, Optimal Control of a Queueing System with two Heterogeneous Servers IEEE Trans. on Autom. Control 29 (1984), 696-703.   DOI
29 H. Luh, I. Viniotis, Threshold control policies for heterogeneous server systems, Mathematical Methods of OR 55 (2002), 121-142.   DOI
30 M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, MA, 1964.
31 M.F. Neuts, Matrix-geometric solutions in stochastic models: An algorithmic approach, The Johns Hopkins University Press, Baltimore, MD. [1994 version is Dover Edition], 1981.
32 R. Nobel, H.C. Tijms, Optimal Control of a Queueing System with Heterogeneous Servers, IEEE Transactions on Autom. Control 45 (2002), 780-784.
33 V. Ponomarov, E. Lebedev, Finite Source Retrial Queues with State-Dependent Service Rate, Communications in Computer and Information Science 356 (2013), 140146.
34 V. Ponomarov, E. Lebedev, Optimal Control of Retrial Queues with Finite Population and State-Dependent Service Rate, Advances in Intelligent Systems and Computing 754 (2018), 359-369.   DOI
35 V.V. Rykov, D.V. Efrosinin, Numerical Analysis of Optimal Control Polices for Queueing Systems with Heterogeneous Servers, Information Processes 2 (2002), 252-256.
36 M. Senthilkumar, K. Sohraby, K. Kim, On a multiserver retrial queue with phase type retrial time, Mathematical and Computational Models. Eds: R. Nadarajan et al., 65-78, 2012, Narosa Publishing House, New Delhi, India.
37 Y.W. Shin, Algorithmic solutions for M/M/c retrial queue with PH2 retrial time, Journal of Applied Mathematics and Informatics 29 (2011), 803-811.   DOI
38 Y.W. Shin, D.H. Moon, Approximation of M/M/c retrial queue with PH-retrial times, European Journal of Operational Research 213 (2011), 205-209.   DOI
39 W-H. Steeb and Y. Hardy, Matrix Calculus and Kronecker Product, World Scientific Publishing, Singapore, 2011.
40 T. Yang, M.J.M. Posner, J.G.C. Templeton, H. Li, An approximation method for the M/G/1 retrial queues with general retrial times, European Journal of Operational Research 76 (1994), 552-562.   DOI