Abstract
Loss networks have long been used to model various types of telecommunication network, including circuit-switched networks. Such networks often use admission controls, such as trunk reservation, to optimize revenue or stabilize the behaviour of the network. Unfortunately, an exact analysis of such networks is not usually possible, and reduced-load approximations such as the Erlang Fixed Point (EFP) approximation have been widely used. The performance of these approximations is typically very good for networks without controls, under several regimes. There is evidence, however, that in networks with controls, these approximations will in general perform less well. We propose an extension to the EFP approximation that gives marked improvement for a simple ring-shaped network with trunk reservation. It is based on the idea of considering pairs of links together, thus making greater allowance for dependencies between neighbouring links than does the EFP approximation, which only considers links in isolation.
Similar content being viewed by others
References
N.H. Antoniu, Optimal admission policies for small star networks, Ph.D. thesis, University of Durham (1995).
R.A. Barry and P.A. Humblet, Models of blocking probabilities in all-optical networks with and without wavelength changers, IEEE Journal on Selected Areas in Communications 14 (1996) 858-867.
M.S. Bebbington, P.K. Pollett and I. Ziedins, Improved fixed point methods for loss networks with linear structure, in: Proc. of the 4th Internat. Conf. on Telecommunications, Vol. 3, ed. W.J. Lavery, 1997, pp. 1411-1416.
M.S. Bebbington, P.K. Pollett and I. Ziedins, Two-link approximation schemes for linear loss networks without controls, Journal of the Korean Mathematical Society 35 (1998) 539-557.
A. Birman, Computing approximate blocking probabilities for a class of all-optical networks, IEEE Journal on Selected Areas in Communications 14 (1996) 852-857.
D.Y. Burman, J.P. Lehoczky and Y. Lim, Insensitivity of blocking probabilities in a circuit-switching network, Journal of Applied Probability 21 (1984) 850-859.
G. Ciardo and K.S. Trivedi, Solution of large GSPN models, in: Numerical Solution of Markov Chains, ed. W.J. Stewart (Marcel Dekker, New York, 1991) pp. 565-595.
G. Ciardo and K.S. Trivedi, A decomposition approach for stochastic Petri net models, in: Proc. of the 4th Internat. Workshop on Petri Nets and Performance Models (IEEE Computer Soc. Press, Silver Spring, MD, 1991) pp. 74-83.
A.J. Coyle, W. Henderson and P.G. Taylor, Reduced load approximations for loss networks, Telecom-munication Systems 2 (1993) 21-50.
A.J. Coyle, W. Henderson and P.G. Taylor, Decomposition methods for loss networks with circuit reservation, in: Proc. of the 7th Australian Teletraffic Res. Seminar, ed. W. Henderson, Teletraffic Research Centre, University of Adelaide, Adelaide, pp. 229-241.
O. Gerstel, R. Ramaswami and G.H. Sasaki, Fault tolerant multiwavelength optical rings with limited wavelength conversion, IEEE Journal on Selected Areas in Communications 16 (1998) 1166-1177.
R.J. Gibbens, P.J. Hunt and F.P. Kelly, Bistability in communications networks, in: Disorder in Physical Systems, eds. G.R. Grimmett and D.J.A. Welsh (Oxford Univ. Press, Oxford, 1990) pp. 113-127.
J.Y. Hui, Switching and Traffic Theory for Integrated Broadband Networks (Kluwer, Boston, 1990).
P.J. Hunt, Loss networks under diverse routing: the symmetric star network, Advances in Applied Probability 25 (1995) 255-272.
P.J. Hunt and T.G. Kurtz, Large loss networks, Stochastic Processes and their Applications 53 (1994) 363-378.
P.J. Hunt and C.N. Laws, Asymptotically optimal loss network control, Mathematics of Operations Research 18 (1991) 880-900.
P.J. Hunt and C.N. Laws, Optimization via trunk reservation in single resource loss systems under heavy traffic, The Annals of Applied Probability 7 (1997) 1058-1079.
F.P. Kelly, Stochastic models for computer communication systems, Journal of the Royal Statistical Society. Series B 47 (1985) 379-395.
F.P. Kelly, Blocking probabilities in large circuit-switched networks, Advances in Applied Probability 18 (1986) 473-505.
F.P. Kelly, Routing and capacity allocation in networks with trunk reservation, Mathematics of Operations Research 15 (1990) 771-793.
F.P. Kelly, Loss networks, Annals of Applied Probability 1 (1991) 319-378.
P.B. Key, Optimal control and trunk reservation in loss networks, Probability in the Engineering and Informational Sciences 4 (1990) 203-242.
J.C. Lagarias, A.M. Odlyzko and D.B. Zagier, Realizable traffic patterns and capacity of disjointly shared networks, Comput. Networks 10 (1985) 275-285.
S.A. Lippman, Applying a new device in the optimization of exponential queueing systems, Operations Research 23 (1975) 687-710.
G. Louth, M. Mitzenmacher and F. Kelly, Computational complexity of loss networks, Theoretical Computer Science 125 (1994) 45-59.
I.M. MacPhee and I. Ziedins, Admission controls for loss networks with diverse routing, in: Stochastic Networks: Theory and Applications, eds. F.P. Kelly, S. Zachary and I. Ziedins, Royal Statistical Society Lecture Note Series (Oxford Univ. Press, Oxford, 1996) pp. 205-214.
D. Mitra and P.J. Weinberger, Probabilistic models of database locking: solutions, computational algorithms and asymptotics, Advances in Applied Probability 19 (1984) 219-239.
D. Mitra and I. Ziedins, Virtual partitioning by dynamic priorities: fair and efficient resource-sharing by several services, in: Broadband Communications: Networks, Services, Applications, Future Directions, ed. B. Plattner, Lecture Notes in Computer Science, Vol. 1044 (Springer, New York, 1996) pp. 173-185.
B. Moretta and I. Ziedins, Admission controls for Erlang's loss system with service times distributed as a finite sum of exponential random variables, Journal of Applied Mathematics and Decision Science 2 (1998) 119-132.
D.L. Pallant, A reduced load approximation for cellular mobile networks including handovers, Australian Telecommunications Research 26 (1992) 21-29.
D. Pallant and P.G. Taylor, Approximation of performance measures in cellular mobile networks with dynamic channel allocation, Telecommunication Systems 3 (1994) 129-163.
E. Pinsky and Y. Yemeni, A statistical mechanics of some interconnection networks, in: Performance' 84, ed. E. Gelenbe (Elsevier/North-Holland, Amsterdam, 1984) pp. 147-158.
K.W. Ross, Multiservice Loss Models for Broadband Telecommunication Networks (Springer, Berlin, 1994).
D. Singer, A guide to RINGNET, Mathematics Department, The University of Queensland (1993).
W. Whitt, Blocking when service is required from several facilities simultaneously, AT&T Technical Journal 64 (1985) 1807-1856.
S. Zachary, Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice, Stochastic Processes and their Applications 20 (1985) 247-256.
S. Zachary and I. Ziedins, Loss networks and Markov random fields, Journal of Applied Probability 36 (1999) 403-414.
I. Ziedins, Optimal admission controls for Erlang's loss system with phase-type arrivals, Probability in the Engineering and Informational Sciences 10 (1996) 397-414.
I.B. Ziedins and F.P. Kelly, Limit theorems for loss networks with diverse routing, Advances in Applied Probability 21 (1989) 804-830.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bebbington, M., Pollett, P. & Ziedins, I. Two-Link Approximation Schemes for Loss Networks with Linear Structure and Trunk Reservation. Telecommunication Systems 19, 187–207 (2002). https://doi.org/10.1023/A:1013394009996
Issue Date:
DOI: https://doi.org/10.1023/A:1013394009996