Performance Analysis of M^X / (G1, G2) / 1 Retrial Queueing Model with Second Phase Optional Service and Bernoulli Vacation Schedule Using PGF Approach^[22]

Vishwa Nath Maurya^1,

¹Department of Mathematics, School of Science & Technology, University of Fiji, Saweni / Suva, Fiji Islands

Cite this paper:
Vishwa Nath Maurya. Performance Analysis of M^X / (G1, G2) / 1 Retrial Queueing Model with Second Phase Optional Service and Bernoulli Vacation Schedule Using PGF Approach^[22]. American Journal of Modeling and Optimization. 2014; 2(1):1-7. doi: 10.12691/ajmo-2-1-1

Abstract

Present paper describes with the bulk arrival retrial queueing M^X / G₁, G₂, / 1 model with two phase service and Bernoulli vacation schedule wherein first phase service is essential and the next second phase service is optional. If the second phase service is not demanded by arriving customer then the single server takes a vacation period according to Bernoulli vacation schedule in order to utilize it to complete some supplementary work and such type of vacation is assumed working vacation. In the queueing model taken into present consideration, the concepts of Bernoulli vacation schedule and next optional service have been incorporated along with realistic provision that the server has an option to avail a vacation with probability p (q) or may continue to serve the next customer, if any with complementary probability just after the completion of first phase essential service i.e. before the commencement of second phase optional service. In the present paper, our central goal is to investigate the steady state behavior of the bulk arrival retrial queueing M^X / G₁, G₂, / 1 model with two phase service and Bernoulli vacation schedule. By introducing supplementary variables, Chapman Kolmogorov equations are established and then the probability generating functions (PGFs) for first phase essential service (FPES), next phase optional service (NPOS) and working vacation and for the number of the customers in the orbit at an arbitrary epoch are investigated successfully. Besides investigating PGF for different states of the queueing system taken into consideration, performance measures such as the long run probabilities for which the server is in idle state, FPES state, NPOS state and in working vacation state are also explored in order to focus the application aspect of the investigated PGFs. By the end of the present paper, some numerical illustrations of the investigated results for M^X / E_k / 1 model as special case of M^X / G₁, G₂, / 1 have been presented in Table 1 and Table 2 for varying parameters.

Keywords:
bulk arrival retrial queue state dependent rate supplementary variables chapman-kolmogorov equations probability generating function approach Bernoulli vacation schedule principle of maximum entropy performance measures

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References:

[1]	Amandor, J. and Artalejo, J.R., “The M/G/1 retrial queue: New descriptors of the customer’s behavior,” Computers and Applied Mathematics, Vol. 223, No. 1, 2009, pp. 15-26.
[2]	Atencia and Moreno, P., “A single server retrial queue with general retrial times and Bernoulli schedule,” Applied Mathematics and Computation, Vol. 190, No. 2, 2005, pp. 1612-1626.
[3]	Boualem, M. Djellab, N. and Aissani, D., “Stochastic inequalities for M/G/1 retrial queues with vacations and constant retrial policy,” Mathematical and Computer Modelling, Vol. 50, No. 1-2, 2009, pp. 207-212.
[4]	Chakravarthy, S.R. and Dudin, A., “Analysis of retrial queueing model with MAP arrivals and two type of customers,” Mathematical and Computer Modelling, Vol. 37, No. 3-4, 2003, pp. 343-363.
[5]	Choudhury, G. and Deka, K., “An M/G/1 retrial queueing system with two phases of service subject to the server breakdown and repair,” Performance Evaluation, Vol. 65, No. 10, 2008, pp. 714-724.
[6]	Choudhury, G. and Deka, K., “An MX/G/1 unreliable retrial queue with two phase of service and Bernoulli admission mechanism,” Applied Mathematical Modelling, Vol. 215, No. 3, 2009, pp. 936-949.
[7]	Choudhury, G. and Madan, K.C., “A two phase batch arrival queueing system with vacation time under Bernoulli schedule,” Applied Mathematics and Computation, Vol. 149, No. 2, 2004, pp. 337-349.
[8]	Choudhury, G. and Paul, M., “A two phase queueing system with Bernoulli vacation schedule under multiple vacation policy,” Statistical Methods, Vol. 3, 2006, pp. 174-185.
[9]	Choudhury, G., Tadj., L. and Paul., M., “Steady state analysis of an MX/G/1 queue with two phase service and Bernoulli vacation schedule under multiple vacation policy,” Applied Mathematical Modelling, Vol. 31, No. 6, 2007, pp. 1079-1091.
[10]	Doshi, B.T., “Queueing systems with vacations: A survey,” Queueing Systems, Vol. 1, 1986, pp. 29-66.
[11]	Doshi, B.T., “Single server with vacations, in: H. Takagi (Ed.), Stochastic Analysis of Computer and Communication System,” North Holland, Amsterdam, 1990, pp. 217-265.
[12]	Jain, M. and Dhakad, M.R., “Queueing distribution MX/G/1 model using principle of maximum entropy,” Journal of Decision and Mathematical Sciences, Vol. 8, No. 1-3, 2003, pp. 37-44.
[13]	Jain, M., Sharma, G.C. and Sharma, R., “Unreliable server M/G/1 queue with multi-optional services and multi-optional vacations,” International Journal of Mathematical Operational Research, Vol. 5, No. 2, 2013 pp. 145-169.
[14]	Ke, J.C. and Chang, F.M., “MX/(G1,G2)/1 retrial queue under Bernoulli vacation schedules with general repeated attempts and starting failures,” Applied Mathematical Modelling, Vol. 33, No. 7, 2009, pp. 3186-3196.
[15]	Ke, J.C. and Lin, C.H., “Maximum entropy approach for batch arrival queue with an unreliable server and delaying vacations,” Computers and Applied Mathematics, Vol. 183, No. 2, 2006, pp. 1328-1340.
[16]	Kumar, M.S. and Arumuganathan, R., “On the single server batch arrival retrial queue with general vacation time under Bernoulli schedule and two phases of heterogeneous service,” Quality Technology and Quantitative Management, Vol. 5, No. 2, 2008, pp. 145-160.
[17]	Madan, K.C. and Choudhury, G., “A single server queue with two phase of heterogeneous service under Bernoulli schedule and a generalized vacation time,” International Journal of Information and Management Sciences, Vol. 16, No. 2, 2005, pp. 1-16.
[18]	Maurya V.N., “Investigation of probability generating function in an interdependent M/M/1: (∞; GD) queueing model with controllable arrival rates using Rouche’s theorem”, Open Journal of Optimization, Scientific Research Publishing, Irvine, California, USA, Vol.1, Issue 2, 2012, pp. 34-38.
[19]	Maurya V.N., “Sensitivity analysis on significant performance measures of bulk arrival retrial queueing MX/(G1,G2)/1 model with second phase optional service and Bernoulli vacation schedule”, International Open Journal of Operations Research, Academic and Scientific Publishing, New York, USA, Vol. 1, No. 1, 2013, pp. 1-15.
[20]	Maurya V.N., “Maximum entropy analysis of MX/(G1,G2)/1 retrial queueing model with second phase optional service and Bernoulli vacation schedule”, American Journal of Operational Research, Scientific and Academic Publishing, Rosemead, California, USA, Vol. 3, No. 1, 2013, pp. 1-12.
[21]	Maurya V.N., Arora D.K. and Maurya A.K., “Elements of Advanced Probability Theory and Statistical Techniques”, Scholar’s Press Publishing Co., Saarbrucken, Germany, 2013.
[22]	Maurya V.N., “Performance analysis of MX/(G1,G2)/1 retrial queueing model with second phase optional service and Bernoulli vacation schedule using PGF approach”, International Conference on Statistical Sciences (ICSS), Dubai, UAE; World Academy of Science, Engineering & Technology (WASET), Italy, 2014 (Communicated).
[23]	Sherman, N.P. and Kharoufeh, J.P., “An M/M/1 retrial queue with unreliable server,” Operational Research Letters, Vol. 34, No. 6, 2006, pp. 697-705.
[24]	Tian, N. and Zhang, Z.G., “Vacations queueing models-Theory and Applications,” Springer-Verlag, New York, 2006.
[25]	Wang, J., “An M/G/1 queue with second optional service and server breakdowns,” Computers and Mathematical Application, Vol. 47, No. 10-11, 2004, pp. 1713-1723.
[26]	Wang, J., Liu, B. and Li, J., “Transient analysis of an M/G/1 retrial queue subject to disasters and server failures,” Operational Research, Vol. 189, No. 3, 2008, pp. 1118-1132.
[27]	Wu, X., Brill, P., Hlynka, M. and Wang, J., “An M/G/1 retrial queue with balking and retrials during service,” International Journal of Operations Research, Vol. 1, 2005, pp. 30-51.