A continuous-time queueing model with class clustering and global FCFS service discipline

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January 2014, 10(1): 193-206. doi: 10.3934/jimo.2014.10.193

A continuous-time queueing model with class clustering and global FCFS service discipline

Willem Mélange ^1, , Herwig Bruneel ^2, , Bart Steyaert ^1, , Dieter Claeys ^1, and Joris Walraevens ^2,

1.	Department of Telecommunications and Information Processing, Ghent University - UGent, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium, Belgium, Belgium
2.	Department of Telecommunications and Information Processing, Ghent University, St-Pietersnieuwstraat 41, 9000 Gent

Received September 2012 Revised June 2013 Published October 2013

Abstract
Related Papers

In this paper the focus is on ``class clustering" in a continuous-time queueing model with two classes and dedicated servers. ``Class clustering" means that customers of any given type may (or may not) have a tendency to ``arrive back-to-back". We believe this is a concept that is often neglected in literature and we want to show that it can have a considerable impact on multiclass queueing systems, especially on the system considered in this paper. This system adopts a ``global FCFS" service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their types. The major aim of our paper is to quantify the intuitively expected (due to the service discipline) negative impact of ``class clustering" on the performance measures of our system. The motivation of our work are systems where this kind of inherent blocking is encountered, such as input-queueing network switches, road splits or security checks at airports.

Keywords: Queueing theory, class clustering, continuous time., global FCFS.

Mathematics Subject Classification: Primary: 60K25, 90B20; Secondary: 60J2.

Citation: Willem Mélange, Herwig Bruneel, Bart Steyaert, Dieter Claeys, Joris Walraevens. A continuous-time queueing model with class clustering and global FCFS service discipline. Journal of Industrial and Management Optimization, 2014, 10 (1) : 193-206. doi: 10.3934/jimo.2014.10.193

References:

[1]	I. Adan, T. de Kok and J. Resing, A multi-server queueing model with locking, EJOR, 116 (2000), 16-26.
[2]	I. J. B. F. Adan, J. Wessels and W. H. M. Zijm, A compensation approach for two-dimensional markov processes, Advances in Applied Probability, 25 (1993), 783-817. doi: 10.2307/1427792.
[3]	P. Beekhuizen and J. Resing, Performance analysis of small non-uniform packet switches, Performance Evaluation, 66 (2009), 640-659.
[4]	Z. Berdowski, F. van den Broek-Serlé, J. Jetten, Y. Kawabata, J. Schoemaker and R. Versteegh, Survey on standard weights of passengers and baggage, Survey. EASA 2008.C.06/30800/R20090095/30800000/FBR/RLO, (2009).
[5]	D. Bertsimas, An exact fcfs waiting time analysis for a general class of G/G/s queueing systems, Queueing Systems Theory Appl., 3 (1988), 305-320. doi: 10.1007/BF01157853.
[6]	D. Bertsimas, An analytic approach to a general class of G/G/s queueing systems, Operations Research, 38 (1990), 139-155. doi: 10.1287/opre.38.1.139.
[7]	P. P. Bocharov and C. D'Apice, "Queueing Theory," Walter de Gruyter, 2004.
[8]	W. Grassmann, Real eigenvalues of certain tridiagonal matrix polynomials, with queueing applications, Linear Algebra and its Applications, 342 (2002), 93-106. doi: 10.1016/S0024-3795(01)00462-1.
[9]	M. Karol, M. Hluchyj and S. Morgan, Input versus output queueing on a space-division packet switch, IEEE Transactions on Communications, 35 (1987), 1347-1356.
[10]	K. Laevens, A processor-sharing model for input-buffered ATM-switches in a correlated traffic environment, Microprocessors and Microsystems, 22 (1999), 589-596.
[11]	S. Liew, Performance of various input-buffered and output-buffered ATM switch design principles under bursty traffic: Simulation study, IEEE Transactions on Communications, 42 (1994), 1371-1379.
[12]	W. Mélange, H. Bruneel, B. Steyaert and J. Walraevens, A two-class continuous-time queueing model with dedicated servers and global fcfs service discipline, In "Analytical and Stochastic Modeling Techniques and Applications," Lecture Notes in Computer Science, Springer Berlin / Heidelberg, 6751 (2011), 14-27.
[13]	M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach," Corrected reprint of the 1981 original. Dover Publications, Inc., New York, 1994.
[14]	D. Ngoduy, Derivation of continuum traffic model for weaving sections on freeways, Transportmetrica, 2 (2006), 199-222.
[15]	R. Nishi, H. Miki, A. Tomoeda and K. Nishinari, Achievement of alternative configurations of vehicles on multiple lanes, Physical Review E, 79 (2009), 066119.
[16]	A. Stolyar, MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic, Annals of Applied Probability, 14 (2004), 1-53. doi: 10.1214/aoap/1075828046.
[17]	T. Van Woensel and N. Vandaele, Empirical validation of a queueing approach to uninterrupted traffic flows, 4OR, A Quarterly Journal of Operations Research, 4 (2006), 59-72.
[18]	T. Van Woensel and N. Vandaele, Modeling traffic flows with queueing models: A review, Asia-Pacific Journal of Operational Research, 24 (2007), 435-461.

show all references

References:

[1]	I. Adan, T. de Kok and J. Resing, A multi-server queueing model with locking, EJOR, 116 (2000), 16-26.
[2]	I. J. B. F. Adan, J. Wessels and W. H. M. Zijm, A compensation approach for two-dimensional markov processes, Advances in Applied Probability, 25 (1993), 783-817. doi: 10.2307/1427792.
[3]	P. Beekhuizen and J. Resing, Performance analysis of small non-uniform packet switches, Performance Evaluation, 66 (2009), 640-659.
[4]	Z. Berdowski, F. van den Broek-Serlé, J. Jetten, Y. Kawabata, J. Schoemaker and R. Versteegh, Survey on standard weights of passengers and baggage, Survey. EASA 2008.C.06/30800/R20090095/30800000/FBR/RLO, (2009).
[5]	D. Bertsimas, An exact fcfs waiting time analysis for a general class of G/G/s queueing systems, Queueing Systems Theory Appl., 3 (1988), 305-320. doi: 10.1007/BF01157853.
[6]	D. Bertsimas, An analytic approach to a general class of G/G/s queueing systems, Operations Research, 38 (1990), 139-155. doi: 10.1287/opre.38.1.139.
[7]	P. P. Bocharov and C. D'Apice, "Queueing Theory," Walter de Gruyter, 2004.
[8]	W. Grassmann, Real eigenvalues of certain tridiagonal matrix polynomials, with queueing applications, Linear Algebra and its Applications, 342 (2002), 93-106. doi: 10.1016/S0024-3795(01)00462-1.
[9]	M. Karol, M. Hluchyj and S. Morgan, Input versus output queueing on a space-division packet switch, IEEE Transactions on Communications, 35 (1987), 1347-1356.
[10]	K. Laevens, A processor-sharing model for input-buffered ATM-switches in a correlated traffic environment, Microprocessors and Microsystems, 22 (1999), 589-596.
[11]	S. Liew, Performance of various input-buffered and output-buffered ATM switch design principles under bursty traffic: Simulation study, IEEE Transactions on Communications, 42 (1994), 1371-1379.
[12]	W. Mélange, H. Bruneel, B. Steyaert and J. Walraevens, A two-class continuous-time queueing model with dedicated servers and global fcfs service discipline, In "Analytical and Stochastic Modeling Techniques and Applications," Lecture Notes in Computer Science, Springer Berlin / Heidelberg, 6751 (2011), 14-27.
[13]	M. F. Neuts, "Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach," Corrected reprint of the 1981 original. Dover Publications, Inc., New York, 1994.
[14]	D. Ngoduy, Derivation of continuum traffic model for weaving sections on freeways, Transportmetrica, 2 (2006), 199-222.
[15]	R. Nishi, H. Miki, A. Tomoeda and K. Nishinari, Achievement of alternative configurations of vehicles on multiple lanes, Physical Review E, 79 (2009), 066119.
[16]	A. Stolyar, MaxWeight scheduling in a generalized switch: State space collapse and workload minimization in heavy traffic, Annals of Applied Probability, 14 (2004), 1-53. doi: 10.1214/aoap/1075828046.
[17]	T. Van Woensel and N. Vandaele, Empirical validation of a queueing approach to uninterrupted traffic flows, 4OR, A Quarterly Journal of Operations Research, 4 (2006), 59-72.
[18]	T. Van Woensel and N. Vandaele, Modeling traffic flows with queueing models: A review, Asia-Pacific Journal of Operational Research, 24 (2007), 435-461.

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