A stochastic version analysis of an M/G/1 retrial queue with Bernoulli‎ ‎schedule‎

Document Type : Research Paper

Authors

1 Research Unit LaMOS (Modeling and Optimization of Systems)‎, ‎Faculty of Technology‎, ‎University of Bejaia‎, ‎06000 Bejaia‎, ‎Algeria.

2 Department of Mathematics‎, ‎University of Biskra‎, ‎Biskra 07000‎, ‎Algeria‎ ‎Research Unit LaMOS (Modeling and Optimization of Systems)‎, ‎University of Bejaia‎, ‎06000 Bejaia‎, ‎Algeria.

3 Laboratory LaPS‎, ‎Department of athematics‎, ‎University of Badji Mokhtar‎, ‎Annaba 23000‎, ‎Algeria.

4 Research Unit LaMOS (Modeling and Optimization of Systems)‎, ‎Faculty of Exact Sciences‎, ‎University of Bejaia‎, ‎06000 Bejaia‎, ‎Algeria.

Abstract

‎In this work‎, ‎we derive insensitive bounds for various performance measures of a single-server‎ ‎retrial queue with generally distributed inter-retrial times and Bernoulli schedule‎, ‎under the special‎ ‎assumption that only the customer at the head of the orbit queue (i.e.‎, ‎a‎ ‎FCFS discipline governing the flow from the orbit to the server) is allowed‎ ‎to occupy the server‎. ‎The methodology is strongly based on stochastic comparison techniques‎. ‎Instead of studying a performance measure in a quantitative fashion‎, ‎this approach attempts to reveal the relationship between the performance measures and the parameters of the system‎. ‎We prove the monotonicity of the transition operator of the embedded Markov chain relative to strong stochastic ordering and increasing convex ordering‎. ‎We obtain comparability conditions for the distribution of the number of customers in the system‎. ‎Bounds are derived for the stationary distribution and‎ ‎some simple bounds for the mean characteristics of the system‎. ‎The proofs of these results are based on the validation of some inequalities for some cumulative probabilities associated with every state $(m‎, ‎n)$ of the system‎. ‎Finally‎, ‎the effects of various parameters on the performance of the system have been examined numerically.

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References

A.S. Alfa and W. Li, PCS networks with correlated arrival process and retrial phenomenon, IEEE Trans. Wireless Commun. 1 (2000) 630--637.
J.R. Artalejo and A. Gomez-Corral, Modelling communication systems with phase type service and retrial times, IEEE Commun. Lett. 11 (2007) 955--957.
J.R. Artalejo and A. Gomez-Corral, Retrial Queueing System: A Computational Approach, Springer-Verlag, Berlin, Heidelberg, 2008.
I. Atencia and P. Moreno, A single-server retrial queue with general retrial times and Bernoulli schedule, Appl. Math. Comput. 162 (2005) 855--880.
B.G. Bhaskaran, Almost sure comparison of birth and death processeswith application to M/M/s queueing systems, Queueing Syst. 1 (1986) 103--127.
M. Boualem, Insensitive bounds for the stationary distribution of a single server retrial queue with server subject to active breakdowns, Adv. Oper. Res. (2014), Article ID 985453, 12 pages.
M. Boualem, M. Cherfaoui and D. Assani, Monotonicity properties for a single server queue with classical retrial policy and service interruptions, Proc. Jangjeon Math. Soc. 19 (2016), no. 2, 225--236.
M. Boualem, N. Djellab and D. Assani, Stochastic inequalities for M=G=1 retrial queues with vacations and constant retrial policy, Math. Comput. Modelling 50 (2009), no. 1-2, 207--212.
M. Boualem, N. Djellab and D. Assani, Stochastic approximations and monotonicity of a single server feedback retrial queue, Math. Probl. Eng. (2012), Article ID 536982, 13 pages
M. Boualem, N. Djellab and D. Assani, Stochastic bounds for a single server queue with general retrial times, Bull. Iranian Math. Soc. 40 (2014), no. 1, 183--198.
H. Castel-Taleb and N. Pekergin, Stochastic monotonicity in queueing networks, in: Computer Performance Engineering, pp. 116--130, Lecture Notes in Comput. Sci. 5652, Springer, Berlin, Heidelberg, 2009.
B.D. Choi and Y. Chang, Single server retrial queues with priority calls, Math. Comput. Modelling 30 (1999), no. 3-4, 7--32.
B.D. Choi and K.K. Park, The M/G/1 retrial queue with Bernoulli schedule, Queueing Syst. 7 (1990), no. 2, 219--227.
G.I. Falin, Single line repeated orders queueing systems, Optimization 17 (1986), no. 5, 649--667.
G.I. Falin, J.R. Artalejo and M. Martin, On the single server retrial queue with priority customers, Queueing Syst. 14 (1993), no. 3-4, 439--455.
G.I. Falin, and J.G.C. Templeton, Retrial Queues, Chapman and Hall, London, 1997.
A. Gomez-Corral, Stochastic analysis of a single server retrial queue with general retrial times, Naval Res. Logist. 46 (1999), no. 5, 561--581.
Q.M. He, H. Li and Y.Q. Zhao, Ergodicity of the BMAP=PH=s=s + K retrial queue with PH-retrial times, Queueing Syst. 35 (2000), no. 1-4, 323--347.
H.M. Liang, Service station factors in monotonicity of retrial queues, Math. Comput Modelling 30 (1999), no. 3-4, 189--196.
H.M. Liang and V.G. Kulkarni, Monotonicity properties of single server retrial queues, Stoch. Models 9 (1993), no. 3, 373--400.
V.A. Kapyrin, A study of stationary distributions of a queueing system with recurring demands, Cybernatics 13 (1977) 548--590.
Z. Khalil and G. Falin, Stochastic inequalities for M/G/1 retrial queues, Oper. Res. Lett. 16 (1994) 285--290.
W.A. Massey, Stochastic orderings for Markov processes on partially ordered spaces, Math. Oper. Res. 12 (1987), no. 2, 350--367.
A. Muller and D. Stoyan, Comparison Methods for Stochastic Models and Risks, John Wiley & Sons, LTD, 2002.
M. Shaked and J.G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, San Diego, 1994.
M. Shaked and J.G. Shanthikumar, Stochastic Orders, Springer-Verlag, New York, 2007.
D. Stoyan, Comparison Methods for Queues and Other Stochastic Models, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Chichester, 1983.
R. Szekli, Stochastic Ordering and Dependence in Applied Probability, Lecture Notes in Statist. 97, Springer-Verlag, New York, 1995.
I.M. Verloop, U. Ayesta and S. Borst, Monotonicity properties for multi-class queueing systems, Discrete Event Dyn. Syst. 20 (2010), no. 4, 473--509.
T. Yang, M.J.M. Posner, J.G.C. Templeton and H. Li, An approximation method for the M/G/1 retrial queue with general retrial times, European J. Oper. Res. 76 (1994) 552--562.
T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Syst. 2 (1997) 201--233.
Bulletin of the Iranian Mathematical Society
Volume 43, Issue 5
October 2017
Pages 1377-1397

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History

  • Receive Date: 02 November 2015
  • Revise Date: 22 June 2016
  • Accept Date: 22 June 2016

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