Abstract
The MM CPP/GE/c/L G-Queue is a Markov modulated queue with compound Poisson arrivals of both positive and negative customers and generalised exponential service times at c parallel servers. The system considered has either finite or infinite (L = ∞) capacity and customers in service cannot be killed by a negative arrival (immune servicing). The equilibrium queue length probabilities are derived as well as the Laplace transform of the response time probability density function of successful customers. This model can form the basis of a building block for networks with bursty, correlated traffic and with load balancing and unreliable servers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
R. Chakka, Performance and Reliability Modelling of Computing Systems Using Spectral Expansion, Ph.D. Thesis, University of Newcastle upon Tyne, UK: 1995.
R. Chakka and P.G. Harrison, Analysis of MMPP/M/c/L queues, Proceedings of the Twelfth UK Computer and Telecommunications Performance Engineering Workshop, Edinburgh (1996) 117–128.
J.M. Fourneau and M. Hernandez, Modelling defective parts in a flow system using G-networks, Second International Workshop on Performability Modelling of Computer Communication Systems, Le Mont Saint-Michel, June 1993.
J.M. Fourneau, E. Gelenbe and R. Suros, G-networks with multiple classes of positive and negative customers, Theoretical Computer Science, Vol. 155, pp. 141–156, 1996.
R. Fretwell and D. Kouvatsos, ATM traffic burst lengths are geometrically bounded, Proceedings of the 7 th IFIP Workshop on Performance Modelling and Evaluation of ATM & IP Networks, Antwerp, Belgium (1999).
E. Gelenbe. Random neural networks with positive and negative signals and product form solution, Neural Computation, Vol. 1, No. 4, pp 502–510, 1989.
P.G. Harrison and A. de C. Pinto, An approximate analysis of a synchronous, packet-switched banyan networks with blocking, Performance Evaluation 19 (1994) 223–258.
P.G. Harrison and E. Pitel. Sojourn times in single server queues with negative customers. Journal of Applied Probability 30, 943–963, 1993.
P.G. Harrison, E. Pitel. Response time distributions in tandem G-networks. Journal of Applied Probability 32, 224–246, 1995.
P.G. Harrison and E. Pitel. The M/G/1 queue with negative customers. Advances in Applied Probability 28, 540–566, 1996.
D.D. Kouvatsos, A maximum entropy analysis of the G/G/1 Queue at Equilibrium, Journal of Operations Research Society, 39, 2 (1988) 183–200.
D. Kouvatsos, Entropy maximisation and queueing network models, Annals of Operations Research, 48 (1994), 63–126.
D. Kouvatsos, J. Wilkinson, P.G. Harrison and M.K. Bhabuta, Performance Analysis of Buffered Banyan ATM Switch Architectures, in Performance Modelling and Evaluation of ATM Networks, Vol. II, D. Kouvatsos, ed., Chapman- Hall (1996).
B.B. Mandelbrot and J.W.V. Ness, Fractional Brownian Motions, fractional noises and applications, SIAM Review, 10,4 (1968) 422–437.
I. Mitrani, Probabilistic Modelling, Cambridge University Press (1998).
I. Mitrani and R. Chakka, Spectral expansion solution for a class of Markov models: Application and comparison with the matrix-geometric method, Performance Evaluation 23 (1995) 241–260.
W.J. Stewart, Introduction to Numerical Solution of Markov Chains, Princeton University Press, 1994
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag London Limited
About this paper
Cite this paper
Harrison, P.G., Chakka, R. (2001). The MM CPP/GE/c/L G-Queue at equilibrium. In: Goto, K., Hasegawa, T., Takagi, H., Takahashi, Y. (eds) Performance and QoS of Next Generation Networking. Springer, London. https://doi.org/10.1007/978-1-4471-0705-7_19
Download citation
DOI: https://doi.org/10.1007/978-1-4471-0705-7_19
Publisher Name: Springer, London
Print ISBN: 978-1-4471-1183-2
Online ISBN: 978-1-4471-0705-7
eBook Packages: Springer Book Archive