# American Institute of Mathematical Sciences

January  2014, 10(1): 131-149. doi: 10.3934/jimo.2014.10.131

## The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times

 1 SMACS Research Group, TELIN Department, Ghent University, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium, Belgium, Belgium 2 Supply Networks and Logistics Research Center, Department of Industrial Management, Ghent University, Technologiepark 903, B-9052 Zwijnaarde, Belgium

Received  September 2012 Revised  June 2013 Published  October 2013

In this paper, we analyse the behaviour of a discrete-time single-server queueing system with general service times, equipped with the $NT$-policy. This is a threshold policy designed to reduce the number of service unit activation/deactivation cycles, whilst ensuring an acceptable delay trade-off. Once the server is deactivated, reactivation will be postponed until either $N$ customers have accumulated in the queue or the first customer has been in the queue for $T$ slots, whichever happens first. Due to this modus operandi, the system circulates between three phases: empty, accumulating and serving.
We assume a Bernoulli arrival process of customers and independent and identically distributed service times. Using a probability generating functions approach, we obtain expressions for the steady-state distributions of the phase sojourn times, the cycle length, the system content and the customer delay. The influence of the threshold parameters $N$ and $T$ on the mean sojourn times and the expected delay is discussed by means of numerical examples.
Citation: Bart Feyaerts, Stijn De Vuyst, Herwig Bruneel, Sabine Wittevrongel. The impact of the $NT$-policy on the behaviour of a discrete-time queue with general service times. Journal of Industrial & Management Optimization, 2014, 10 (1) : 131-149. doi: 10.3934/jimo.2014.10.131
##### References:
 [1] A. S. Alfa and W. Li, Optimal ($N$,$T$)-policy for M/G/1 system with cost structures,, Performance Evaluation, 42 (2000), 265.  doi: 10.1016/S0166-5316(00)00015-8.  Google Scholar [2] W. Böhm and S. G. Mohanty, On discrete-time Markovian $N$-policy queues involving batches,, Sankhya: The Indian Journal of Statistics, 56 (1994), 144.   Google Scholar [3] O. J. Boxma and W. P. Groenendijk, Waiting times in discrete-time cyclic-service systems,, IEEE Transactions on Communications, 36 (1988), 164.  doi: 10.1109/26.2746.  Google Scholar [4] H. Bruneel and B. G. Kim, "Discrete-Time Models for Communication Systems Including ATM,", The Springer International Series In Engineering And Computer Science, 205 (1993).  doi: 10.1007/978-1-4615-3130-2.  Google Scholar [5] B. Feyaerts, S. De Vuyst, S. Wittevrongel and H. Bruneel, Analysis of a discrete-time queueing system with an $NT$-policy,, ASMTA '10: 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications, 6148 (2010), 29.  doi: 10.1007/978-3-642-13568-2_3.  Google Scholar [6] P. Flajolet and R. Sedgewick, "Analytic Combinatorics,", Cambridge University Press, (2009).  doi: 10.1017/CBO9780511801655.  Google Scholar [7] A. G. Hernández-Díaz and P. Moreno, Analysis and optimal control of a discrete-time queueing system under the $(m,N)$-policy,, Valuetools '06: Proceedings of the 1st International Conference on Performance Evaluation Methodologies and Tools, (2006).   Google Scholar [8] D. P. Heyman, The T-policy for the M/G/1 queue,, Management Science, 23 (1977), 775.  doi: 10.1287/mnsc.23.7.775.  Google Scholar [9] J.-C. Ke, Optimal $NT$ policies for M/G/1 system with a startup and unreliable server,, Computers & Industrial Engineering, 50 (2006), 248.  doi: 10.1016/j.cie.2006.04.004.  Google Scholar [10] J.-C. Ke, H.-I Huang and Y.-K. Chu, Batch arrival queue with $N$-policy and at most $J$ vacations,, Applied Mathematical Modelling, 34 (2010), 451.  doi: 10.1016/j.apm.2009.06.003.  Google Scholar [11] H. W. Lee and W. J. Seo, The performance of the M/G/1 queue under the dyadic Min($N,D$)-policy and its cost optimization,, Performance Evaluation, 65 (2008), 742.   Google Scholar [12] S. S. Lee, H. W. Lee and K. C. Chae, Batch arrival queue with $N$-policy and single vacation,, Computers & Operations Research, 22 (1995), 173.   Google Scholar [13] P. Moreno, A discrete-time single-server queue with a modified $N$-policy,, International Journal of Systems Science, 38 (2007), 483.  doi: 10.1080/00207720701353405.  Google Scholar [14] H. Takagi, "Queueing Analysis, A Foundation of Performance Evaluation, Volume 3: Discrete-Time Systems,", North-Holland, (1993).   Google Scholar [15] K.-H. Wang, T.-Y. Wang and W. L. Pearn, Optimal control of the $N$-policy M/G/1 queueing system with server breakdowns and general startup times,, Applied Mathematical Modelling, 31 (2007), 2199.  doi: 10.1016/j.apm.2006.08.016.  Google Scholar [16] T.-Y. Wang, K.-H. Wang and W. L. Pearn, Optimization of the $T$ policy M/G/1 queue with server breakdowns and general startup times,, Journal of Computational and Applied Mathematics, 228 (2009), 270.  doi: 10.1016/j.cam.2008.09.021.  Google Scholar [17] M. Yadin and P. Naor, Queueing systems with a removable service station,, Operational Research Quarterly, 14 (1963), 393.   Google Scholar

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##### References:
 [1] A. S. Alfa and W. Li, Optimal ($N$,$T$)-policy for M/G/1 system with cost structures,, Performance Evaluation, 42 (2000), 265.  doi: 10.1016/S0166-5316(00)00015-8.  Google Scholar [2] W. Böhm and S. G. Mohanty, On discrete-time Markovian $N$-policy queues involving batches,, Sankhya: The Indian Journal of Statistics, 56 (1994), 144.   Google Scholar [3] O. J. Boxma and W. P. Groenendijk, Waiting times in discrete-time cyclic-service systems,, IEEE Transactions on Communications, 36 (1988), 164.  doi: 10.1109/26.2746.  Google Scholar [4] H. Bruneel and B. G. Kim, "Discrete-Time Models for Communication Systems Including ATM,", The Springer International Series In Engineering And Computer Science, 205 (1993).  doi: 10.1007/978-1-4615-3130-2.  Google Scholar [5] B. Feyaerts, S. De Vuyst, S. Wittevrongel and H. Bruneel, Analysis of a discrete-time queueing system with an $NT$-policy,, ASMTA '10: 17th International Conference on Analytical and Stochastic Modeling Techniques and Applications, 6148 (2010), 29.  doi: 10.1007/978-3-642-13568-2_3.  Google Scholar [6] P. Flajolet and R. Sedgewick, "Analytic Combinatorics,", Cambridge University Press, (2009).  doi: 10.1017/CBO9780511801655.  Google Scholar [7] A. G. Hernández-Díaz and P. Moreno, Analysis and optimal control of a discrete-time queueing system under the $(m,N)$-policy,, Valuetools '06: Proceedings of the 1st International Conference on Performance Evaluation Methodologies and Tools, (2006).   Google Scholar [8] D. P. Heyman, The T-policy for the M/G/1 queue,, Management Science, 23 (1977), 775.  doi: 10.1287/mnsc.23.7.775.  Google Scholar [9] J.-C. Ke, Optimal $NT$ policies for M/G/1 system with a startup and unreliable server,, Computers & Industrial Engineering, 50 (2006), 248.  doi: 10.1016/j.cie.2006.04.004.  Google Scholar [10] J.-C. Ke, H.-I Huang and Y.-K. Chu, Batch arrival queue with $N$-policy and at most $J$ vacations,, Applied Mathematical Modelling, 34 (2010), 451.  doi: 10.1016/j.apm.2009.06.003.  Google Scholar [11] H. W. Lee and W. J. Seo, The performance of the M/G/1 queue under the dyadic Min($N,D$)-policy and its cost optimization,, Performance Evaluation, 65 (2008), 742.   Google Scholar [12] S. S. Lee, H. W. Lee and K. C. Chae, Batch arrival queue with $N$-policy and single vacation,, Computers & Operations Research, 22 (1995), 173.   Google Scholar [13] P. Moreno, A discrete-time single-server queue with a modified $N$-policy,, International Journal of Systems Science, 38 (2007), 483.  doi: 10.1080/00207720701353405.  Google Scholar [14] H. Takagi, "Queueing Analysis, A Foundation of Performance Evaluation, Volume 3: Discrete-Time Systems,", North-Holland, (1993).   Google Scholar [15] K.-H. Wang, T.-Y. Wang and W. L. Pearn, Optimal control of the $N$-policy M/G/1 queueing system with server breakdowns and general startup times,, Applied Mathematical Modelling, 31 (2007), 2199.  doi: 10.1016/j.apm.2006.08.016.  Google Scholar [16] T.-Y. Wang, K.-H. Wang and W. L. Pearn, Optimization of the $T$ policy M/G/1 queue with server breakdowns and general startup times,, Journal of Computational and Applied Mathematics, 228 (2009), 270.  doi: 10.1016/j.cam.2008.09.021.  Google Scholar [17] M. Yadin and P. Naor, Queueing systems with a removable service station,, Operational Research Quarterly, 14 (1963), 393.   Google Scholar
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