Abstract
Simple and computationally attractive lower and upper bounds are presented for the call congestion such as those representing multi-server loss or delay stations. Numerical computations indicate a potential usefulness of the bounds for quick engineering purposes. The bounds correspond to product-form modifications and are intuitively appealing. A formal proof of the bounds and related monotonicity results will be presented. The technique of this proof, which is based on Markov reward theory, is of interest in itself and seems promising for further application. The extension to the non-exponential case is discussed. For multiserver loss stations the bounds are argued to be insensitive.
Similar content being viewed by others
References
I.J.B.F Adan and J. van der Wal, Monotonicity of the throughput of a closed queueing network in the number of jobs, Memorandum 87-03, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1987. To appear: Opns. Res.
I.J.B.F. Adan and J. van der Wal, Monotonicity of the throughput in single server production and assembly networks with respect to the buffer sizes. To appear: Proc. 1th Int. workshop on queuing systems with blocking.
U.N. Bhat, Finite capacity assembly-like queues, Queueing Systems 1 (1986) 85–101.
N.M. van Dijk, Simple bounds for queueing systems with breakdowns, Perf. Evaluation 8 (1988) 117–128.
N.M. van Dijk, A formal proof for the insensitivity of simple bounds for finite multi-server non-exponential tandem queues, Stochastic Processes 27 (1988) 261–277.
N.M. van Dijk and B.F. Lamond, Bounds for the call congestion of finite single-server exponential tandem queues, Opns. Res. 36 (1988) 470–477.
N.M. van Dijk, P. Tsoucas and J. Walrand, Simple bounds and monotonicity of the call congestion of finite multiserver delay systems, Probability in the Engineering and Informational Sciences 2 (1988), 129–138.
A. Hordijk and N. van Dijk, Networks of queues with blocking, Performance '81 (North-Holland, 1981) 51–65.
A. Hordijk and N. van Dijk, Adjoint process, job-local-balance and insensitivity for stochastic networks Bull. 44-th Session Int.Stat.Inst., 50 (1983) 776–788.
A. Hordijk and A. Ridder, Stochastic inequalities for an overflow model, J. Appl. Probability 24 (1987) 696–708.
E.H. Lipper and B. Sengupta, Assembly-like queues with finite capacity: Bounds, asymptotics and approximations, Queueing Systems 1 (1986) 67–83.
T.S. Robertazzi and A.A. Lazar. On the modelling and optimal flow control of the Jacksonian network Perf. Evaluation 5 (1985) 29–43.
J.G. Shanthikumar and D.D. Yao, Stochastic monotonicity of the queue lengths in closed queueing networks, Research Report, University of California, Berkeley, Opns. Res. 35 (1987), 583–588.
J.G. Shanthikumar and D.D. Yao, General queueing networks: Representation and stochastic monotonicity, Proc. of 26th IEEE Conference on Decision and Control, (1987) 1084–1087.
D. Stoyan,Comparison Method for Queues and other Stochastic models (Wiley, New York, 1983).
R. Suri, A concept of monotonicity and its characterization for closed queueing networks, Opns. Res. 33 (1985) 606–624.
P. Tsoucas and J. Walrand, Monotonicity of throughput in non-Markovian networks, To appear: J. Appl. Probability.
J. van der Wal. Monotonicity of the throughput of a closed exponential network in the number of jobs, Research report COSOR 85-21, Eindhoven University of Technology, 1985.
W. Whitt, Comparing counting processes and queues, Adv. Appl. Probability 13 (1981) 207–220.
R.W. Wolff, Poisson arrivals see time averages, Opns. Res. 30 (1982) 223–231.
D.D. Yao, Some properties of the throughput function of closed networks of queues, Oper. Res. Letters 3 (1985) 313–317.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van Dijk, N.M., van der Wal, J. Simple bounds and monotonicity results for finite multi-server exponential tandem queues. Queueing Syst 4, 1–15 (1989). https://doi.org/10.1007/BF01150852
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01150852