# American Institute of Mathematical Sciences

July  2017, 13(3): 1237-1254. doi: 10.3934/jimo.2016070

## Stability of a queue with discriminatory random order service discipline and heterogeneous servers

 1 Department of Mathematics Education, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju, Chungbuk, 28644, Korea 2 Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Korea

* Corresponding author: Bara Kim

The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors.

Received  September 2015 Published  October 2016

We consider a queueing system with two classes of customers, two heterogeneous servers, and discriminatory random order service (DROS) discipline. The two servers may have either the same or different DROS weights for each class. Customers of each class arrive according to a Poisson process and the service times of each class of customers are assumed to be exponentially distributed with service rate depending on both the customer's class and the servers. We provide stability and instability conditions for this two-class two-server queue with DROS discipline.

Citation: Jeongsim Kim, Bara Kim. Stability of a queue with discriminatory random order service discipline and heterogeneous servers. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1237-1254. doi: 10.3934/jimo.2016070
##### References:
 [1] W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-3038-0.  Google Scholar [2] U. Ayesta, A. Izagirre and I. M. Verloop, Heavy-traffic analysis of the discriminatory random-order-service discipline, Performance Evaluation Review -Special Issue on IFIP Performance 2011 -29th International Symposium on Computer Performance, Modeling, Measurement and Evaluation, 39 (2011), 41-43.   Google Scholar [3] G. Fayolle, V. A. Malyshev and M. V. Menshikov, Topics in the Constructive Theory of Countable Markov Chains, Cambridge University Press, 1995. doi: 10.1017/CBO9780511984020.  Google Scholar [4] H. R. Gail, S. L. Hantler and B. A. Taylor, Analysis of a non-preemptive priority multiserver queue, Advances in Applied Probability, 20 (1988), 852-879.  doi: 10.2307/1427364.  Google Scholar [5] T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, Journal of Applied Probability, 24 (1987), 486-494.  doi: 10.1017/S0021900200031120.  Google Scholar [6] T. Hanschke, A matrix continued fraction algorithm for the multiserver repeated order queue, Mathematical and Computer Modelling, 30 (1999), 159-170.  doi: 10.1016/S0895-7177(99)00139-9.  Google Scholar [7] M. Haviv and J. van der Wal, Equilibrium strategies for processor sharing and queues with relative priorities, Probability in the Engineering and Informational Sciences, 11 (1997), 403-412.  doi: 10.1017/S0269964800004940.  Google Scholar [8] Q.-M. He, H. Li and Y. Q. Zhao, Ergodicity of the $BMAP/PH/s/s + K$ retrial queue with PH-retrial times, Queueing Systems, 35 (2000), 323-347.  doi: 10.1023/A:1019110631467.  Google Scholar [9] A. Izagirre, U. Ayesta and I. M. Verloop, Heavy-traffic analysis of a non-preemptive multi-class queue with relative priorities, Probability in the Engineering and Informational Sciences, 29 (2015), 153-180.  doi: 10.1017/S0269964814000278.  Google Scholar [10] B. Kim and J. Kim, Stability of a two-class two-server retrial queueing system, Performance Evaluation, 88/89 (2015), 1-17.  doi: 10.1016/j.peva.2015.02.002.  Google Scholar [11] B. Kim and I. Lee, Tests for nonergodicity of denumerable continuous time Markov processes, Computers and Mathematics with Applications, 55 (2008), 1310-1321.  doi: 10.1016/j.camwa.2007.07.003.  Google Scholar [12] J. Kim, J. Kim and B. Kim, Analysis of the M/G/1 queue with discriminatory random order service policy, Performance Evaluation, 68 (2011), 256-270.  doi: 10.1016/j.peva.2010.12.001.  Google Scholar [13] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Advances in Applied Probability, 25 (1993), 518-548.  doi: 10.2307/1427522.  Google Scholar [14] Y. W. Shin and D. H. Moon, M/M/c retrial queue with multiclass of customers, Methodology and Computing in Applied Probability, 16 (2014), 931-949.  doi: 10.1007/s11009-013-9340-0.  Google Scholar [15] R. L. Tweedie, Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, Journal of Applied Probability, 18 (1981), 122-130.  doi: 10.1017/S0021900200097667.  Google Scholar

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##### References:
 [1] W. J. Anderson, Continuous-Time Markov Chains: An Applications-Oriented Approach, Springer-Verlag, 1991. doi: 10.1007/978-1-4612-3038-0.  Google Scholar [2] U. Ayesta, A. Izagirre and I. M. Verloop, Heavy-traffic analysis of the discriminatory random-order-service discipline, Performance Evaluation Review -Special Issue on IFIP Performance 2011 -29th International Symposium on Computer Performance, Modeling, Measurement and Evaluation, 39 (2011), 41-43.   Google Scholar [3] G. Fayolle, V. A. Malyshev and M. V. Menshikov, Topics in the Constructive Theory of Countable Markov Chains, Cambridge University Press, 1995. doi: 10.1017/CBO9780511984020.  Google Scholar [4] H. R. Gail, S. L. Hantler and B. A. Taylor, Analysis of a non-preemptive priority multiserver queue, Advances in Applied Probability, 20 (1988), 852-879.  doi: 10.2307/1427364.  Google Scholar [5] T. Hanschke, Explicit formulas for the characteristics of the M/M/2/2 queue with repeated attempts, Journal of Applied Probability, 24 (1987), 486-494.  doi: 10.1017/S0021900200031120.  Google Scholar [6] T. Hanschke, A matrix continued fraction algorithm for the multiserver repeated order queue, Mathematical and Computer Modelling, 30 (1999), 159-170.  doi: 10.1016/S0895-7177(99)00139-9.  Google Scholar [7] M. Haviv and J. van der Wal, Equilibrium strategies for processor sharing and queues with relative priorities, Probability in the Engineering and Informational Sciences, 11 (1997), 403-412.  doi: 10.1017/S0269964800004940.  Google Scholar [8] Q.-M. He, H. Li and Y. Q. Zhao, Ergodicity of the $BMAP/PH/s/s + K$ retrial queue with PH-retrial times, Queueing Systems, 35 (2000), 323-347.  doi: 10.1023/A:1019110631467.  Google Scholar [9] A. Izagirre, U. Ayesta and I. M. Verloop, Heavy-traffic analysis of a non-preemptive multi-class queue with relative priorities, Probability in the Engineering and Informational Sciences, 29 (2015), 153-180.  doi: 10.1017/S0269964814000278.  Google Scholar [10] B. Kim and J. Kim, Stability of a two-class two-server retrial queueing system, Performance Evaluation, 88/89 (2015), 1-17.  doi: 10.1016/j.peva.2015.02.002.  Google Scholar [11] B. Kim and I. Lee, Tests for nonergodicity of denumerable continuous time Markov processes, Computers and Mathematics with Applications, 55 (2008), 1310-1321.  doi: 10.1016/j.camwa.2007.07.003.  Google Scholar [12] J. Kim, J. Kim and B. Kim, Analysis of the M/G/1 queue with discriminatory random order service policy, Performance Evaluation, 68 (2011), 256-270.  doi: 10.1016/j.peva.2010.12.001.  Google Scholar [13] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Advances in Applied Probability, 25 (1993), 518-548.  doi: 10.2307/1427522.  Google Scholar [14] Y. W. Shin and D. H. Moon, M/M/c retrial queue with multiclass of customers, Methodology and Computing in Applied Probability, 16 (2014), 931-949.  doi: 10.1007/s11009-013-9340-0.  Google Scholar [15] R. L. Tweedie, Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes, Journal of Applied Probability, 18 (1981), 122-130.  doi: 10.1017/S0021900200097667.  Google Scholar
The curve dividing the two regions
The stable and unstable regions of Example 1 (when $\mu_{11}=0.9$, $\mu_{12}=0.1$, and $\mu_{21}=\mu_{22}=0.5)$
Simulation results for $N_1(t)$ and $N_2(t)$ in Example 1 (when $\mu_{11}=0.9$, $\mu_{12}=0.1$, $\mu_{21}=\mu_{22}=0.5$, and $\lambda_1=\lambda_2=0.5$)
The curve dividing the stable and unstable regions
The stable and unstable regions of Example 2 (when $\mu_{11}=0.9$ and $\mu_{12}=0.1$)
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