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July  2011, 7(3): 735-751. doi: 10.3934/jimo.2011.7.735

## Partially shared buffers with full or mixed priority

 1 Department of Telecommunications and Information Processing, Ghent University, St-Pietersnieuwstraat 41, 9000 Gent

Received  September 2010 Revised  January 2011 Published  June 2011

This paper studies a finite-sized discrete-time two-class priority queue. Packets of both classes arrive according to a two-class discrete batch Markovian arrival process (2-DBMAP), taking into account the correlated nature of arrivals in heterogeneous telecommunication networks. The model incorporates time and space priority to provide different types of service to each class. One of both classes receives absolute time priority in order to minimize its delay. Space priority is implemented by the partial buffer sharing acceptance policy and can be provided to the class receiving time priority or to the other class. This choice gives rise to two different queueing models and this paper analyses both these models in a unified manner. Furthermore, the buffer finiteness and the use of space priority raise some issues on the order of arrivals in a slot. This paper does not assume that all arrivals from one class enter the queue before those of the other class. Instead, a string representation for sequences of arriving packets and a probability measure on the set of such strings are introduced. This naturally gives rise to the notion of intra-slot space priority. Performance of these queueing systems is then determined using matrix-analytic techniques. The numerical examples explore the range of service differentiation covered by both models.
Citation: Thomas Demoor, Dieter Fiems, Joris Walraevens, Herwig Bruneel. Partially shared buffers with full or mixed priority. Journal of Industrial & Management Optimization, 2011, 7 (3) : 735-751. doi: 10.3934/jimo.2011.7.735
##### References:
 [1] C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach,, Performance Evaluation, 16 (1992), 5.  doi: 10.1016/0166-5316(92)90064-N.  Google Scholar [2] T. Demoor, J. Walraevens, D. Fiems, S. De Vuyst and H. Bruneel, Influence of real-time queue capacity on system contents in DiffServ's expedited forwarding per-hop-behavior,, Journal of Industrial and Management Optimization, 6 (2010), 587.  doi: 10.3934/jimo.2010.6.587.  Google Scholar [3] T. Demoor, D. Fiems, J. Walraevens and H. Bruneel, Time and space priority in a partially shared priority queue,, in, (2010).   Google Scholar [4] D. Fiems and H. Bruneel, A note on the discretization of Little's result,, Operations Research Letters, 30 (2002), 17.  doi: 10.1016/S0167-6377(01)00112-2.  Google Scholar [5] D. Fiems, J. Walraevens and H. Bruneel, Performance of a partially shared priority buffer with correlated arrivals,, Lecture Notes in Computer Science, 4516 (2007), 582.  doi: 10.1007/978-3-540-72990-7_52.  Google Scholar [6] G. Hwang and B. Choi, Performance analysis of the $DAR(1)$/$D$/$c$ priority queue under partial buffer sharing policy,, Computers & Operations Research, 31 (2004), 2231.  doi: 10.1016/S0305-0548(03)00175-8.  Google Scholar [7] H. Kröner, G. Hébuterne, P. Boyer and A. Gravey, Priority management in ATM switching nodes,, IEEE Journal on Selected Areas in Communications, 9 (1991), 418.  doi: 10.1109/49.76641.  Google Scholar [8] K. Laevens and H. Bruneel, Discrete-time multiserver queues with priorities,, Performance Evaluation, 33 (1998), 249.  doi: 10.1016/S0166-5316(98)00024-8.  Google Scholar [9] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modeling,", Series on Statistics and Applied Probability. ASA-SIAM, (1999).   Google Scholar [10] M. Mehmet Ali and X. Song, A performance analysis of a discrete-time priority queueing system with correlated arrivals,, Performance Evaluation, 57 (2004), 307.  doi: 10.1016/j.peva.2004.01.001.  Google Scholar [11] H. Radha, Y.Chen, K. Parthasarathy and R. Cohen, Scalable internet video using MPEG-4,, Signal Processing: Image Communication, 15 (1999), 95.  doi: 10.1016/S0923-5965(99)00026-0.  Google Scholar [12] K. Spaey, "Superposition of Markovian Traffic Sources and Frame Aware Buffer Acceptance,", Ph.D. thesis, (2002).   Google Scholar [13] T. Takine, B. Sengupta and T. Hasegawa, An analysis of a discrete-time queue for broadband ISDN with priorities among traffic classes,, IEEE Transactions on Communications, 42 (1994), 1837.  doi: 10.1109/TCOMM.1994.582893.  Google Scholar [14] J. Van Velthoven, B. Van Houdt and C. Blondia, The impact of buffer finiteness on the loss rate in a priority queueing system,, Lecture Notes in Computer Science, 4054 (2006), 211.  doi: 10.1007/11777830_15.  Google Scholar [15] J. Walraevens, B. Steyaert and H. Bruneel, Performance analysis of a single-server ATM queue with a priority scheduling,, Computers & Operations Research, 30 (2003), 1807.  doi: 10.1016/S0305-0548(02)00108-9.  Google Scholar [16] J. Walraevens, S. Wittevrongel and H. Bruneel, A discrete-time priority queue with train arrivals,, Stochastic Models, 23 (2007), 489.  doi: 10.1080/15326340701471158.  Google Scholar [17] Y. Wang, C. Liu and C. Lu, Loss behavior in space priority queue with batch Markovian arrival process - discrete-time case,, Performance Evaluation, 41 (2000), 269.  doi: 10.1016/S0166-5316(99)00079-6.  Google Scholar [18] Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of Discrete-Time Space Priority Queue with Fuzzy Threshold,, Journal of Industrial and Management Optimization, 5 (2009), 467.  doi: 10.3934/jimo.2009.5.467.  Google Scholar [19] J. Zhao, B. Li, X. Cao and I. Ahmad, A matrix-analytic solution for the $DBMAP$/$PH$/$1$ priority queue,, Queueing Systems, 53 (2006), 127.  doi: 10.1007/s11134-006-8306-0.  Google Scholar

show all references

##### References:
 [1] C. Blondia and O. Casals, Statistical multiplexing of VBR sources: A matrix-analytic approach,, Performance Evaluation, 16 (1992), 5.  doi: 10.1016/0166-5316(92)90064-N.  Google Scholar [2] T. Demoor, J. Walraevens, D. Fiems, S. De Vuyst and H. Bruneel, Influence of real-time queue capacity on system contents in DiffServ's expedited forwarding per-hop-behavior,, Journal of Industrial and Management Optimization, 6 (2010), 587.  doi: 10.3934/jimo.2010.6.587.  Google Scholar [3] T. Demoor, D. Fiems, J. Walraevens and H. Bruneel, Time and space priority in a partially shared priority queue,, in, (2010).   Google Scholar [4] D. Fiems and H. Bruneel, A note on the discretization of Little's result,, Operations Research Letters, 30 (2002), 17.  doi: 10.1016/S0167-6377(01)00112-2.  Google Scholar [5] D. Fiems, J. Walraevens and H. Bruneel, Performance of a partially shared priority buffer with correlated arrivals,, Lecture Notes in Computer Science, 4516 (2007), 582.  doi: 10.1007/978-3-540-72990-7_52.  Google Scholar [6] G. Hwang and B. Choi, Performance analysis of the $DAR(1)$/$D$/$c$ priority queue under partial buffer sharing policy,, Computers & Operations Research, 31 (2004), 2231.  doi: 10.1016/S0305-0548(03)00175-8.  Google Scholar [7] H. Kröner, G. Hébuterne, P. Boyer and A. Gravey, Priority management in ATM switching nodes,, IEEE Journal on Selected Areas in Communications, 9 (1991), 418.  doi: 10.1109/49.76641.  Google Scholar [8] K. Laevens and H. Bruneel, Discrete-time multiserver queues with priorities,, Performance Evaluation, 33 (1998), 249.  doi: 10.1016/S0166-5316(98)00024-8.  Google Scholar [9] G. Latouche and V. Ramaswami, "Introduction to Matrix Analytic Methods in Stochastic Modeling,", Series on Statistics and Applied Probability. ASA-SIAM, (1999).   Google Scholar [10] M. Mehmet Ali and X. Song, A performance analysis of a discrete-time priority queueing system with correlated arrivals,, Performance Evaluation, 57 (2004), 307.  doi: 10.1016/j.peva.2004.01.001.  Google Scholar [11] H. Radha, Y.Chen, K. Parthasarathy and R. Cohen, Scalable internet video using MPEG-4,, Signal Processing: Image Communication, 15 (1999), 95.  doi: 10.1016/S0923-5965(99)00026-0.  Google Scholar [12] K. Spaey, "Superposition of Markovian Traffic Sources and Frame Aware Buffer Acceptance,", Ph.D. thesis, (2002).   Google Scholar [13] T. Takine, B. Sengupta and T. Hasegawa, An analysis of a discrete-time queue for broadband ISDN with priorities among traffic classes,, IEEE Transactions on Communications, 42 (1994), 1837.  doi: 10.1109/TCOMM.1994.582893.  Google Scholar [14] J. Van Velthoven, B. Van Houdt and C. Blondia, The impact of buffer finiteness on the loss rate in a priority queueing system,, Lecture Notes in Computer Science, 4054 (2006), 211.  doi: 10.1007/11777830_15.  Google Scholar [15] J. Walraevens, B. Steyaert and H. Bruneel, Performance analysis of a single-server ATM queue with a priority scheduling,, Computers & Operations Research, 30 (2003), 1807.  doi: 10.1016/S0305-0548(02)00108-9.  Google Scholar [16] J. Walraevens, S. Wittevrongel and H. Bruneel, A discrete-time priority queue with train arrivals,, Stochastic Models, 23 (2007), 489.  doi: 10.1080/15326340701471158.  Google Scholar [17] Y. Wang, C. Liu and C. Lu, Loss behavior in space priority queue with batch Markovian arrival process - discrete-time case,, Performance Evaluation, 41 (2000), 269.  doi: 10.1016/S0166-5316(99)00079-6.  Google Scholar [18] Y. C. Wang, J. S. Wang and F. H. Tsai, Analysis of Discrete-Time Space Priority Queue with Fuzzy Threshold,, Journal of Industrial and Management Optimization, 5 (2009), 467.  doi: 10.3934/jimo.2009.5.467.  Google Scholar [19] J. Zhao, B. Li, X. Cao and I. Ahmad, A matrix-analytic solution for the $DBMAP$/$PH$/$1$ priority queue,, Queueing Systems, 53 (2006), 127.  doi: 10.1007/s11134-006-8306-0.  Google Scholar
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