Abstract
With the aim of studying a multi-station feedforward fluid network with Markov-modulated input and output rates we first study a two-dimensional parallel network having the property that the second station cannot be empty unless the first station is. A method for computing the steady state characteristics of such a process is given and it is shown that this can be used to determine the steady state characteristics of two-dimensional tandem fluid networks and more general networks. Finally a multi-station feedforward network is considered. Under appropriate conditions, it is explained how to determine the joint steady state Laplace–Stieltjes transform (LST) and it turns out that in order to compute conditional means and the covariance structure (given the state of the underlying Markov chain) all that is needed is the methods developed for the two-dimensional parallel model together with some trivial linear algebra.
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References
D. Anick, D. Mitra and M.M. Sondhi, Stochastic theory of a data handling system with multiple sources, Bell System Tech. J. 61 (1982) 1871–1894.
S. Asmussen, Stationary distributions for fluid flow models with or without brownian noise, Stochastic Models 11 (1995) 21–49.
S. Asmussen and O. Kella, A multi-dimensional martingale for Markov additive processes and its applications, Adv. in Appl. Probab. 32 (2000) 376–393.
M.T. Barlow, L.C.G. Rogers and D. Williams, Wiener-Hopf factorization for matrices, in: Séminaire de Probabilités XIV, Lecture Notes in Mathematics, Vol. 784 (Springer, Berlin, 1980) pp. 324–331.
A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences (SIAM, Philadelphia, 1994).
H. Chen and A. Mandelbaum, Leontief systems, RBV's and RBM's, in: Proc. Imperial College Workshop on Applied Stochastic Processes, eds. M.H.A. Davis and R.J. Elliott (Gordon and Breach, New York, 1991).
H. Glickman, Stochastic fluid networks, M.Sc. thesis, Dept. of Statistics, The Hebrew University of Jerusalem (1993).
J.M. Harrison and M.I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab. 9 (1981) 302–308.
O. Kella, Parallel and tandem fluid networks with dependent Lévy inputs, Ann. Appl. Probab. 3 (1993) 682–695.
O. Kella, Stability and nonproduct form of stochastic fluid networks with Lévy inputs, Ann. Appl. Probab. 6 (1996) 186–199.
O. Kella and W. Whitt, A storage model with a two-state random environment, Oper. Res. 40 (1992) 257–262.
O. Kella and W. Whitt, A tandem fluid network with Lévy input, in: Queueing and Related Models, eds. U.N. Bhat and I.V. Basawa (Oxford University Press, 1992) pp. 112–128.
O. Kella and W. Whitt, Stability and structural properties of stochastic storage networks, J. Appl. Probab. 33 (1996) 1169–1180.
D. Kroese and W. Scheinhardt, Joint distributions for interacting fluid queues, Queueing Systems (2000), to appear.
A. Pacheco and N.U. Prabhu, A Markovian storage model, Ann. Appl. Probab. 6 (1996) 76–91.
L.C.G. Rogers, Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains, Ann. Appl. Probab. 4 (1994) 390–413.
W. Scheinhardt, Markov-modulated and feedback fluid queues, Ph.D. dissertation, Faculty of Mathematical Sciences, University of Twente (1998).
T.E. Stern and A.I. Elwalid, Analysis of separable Markov-modulated rate models for informationhandling systems, Adv. in Appl. Probab. 23 (1991) 105–139.
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Kella, O. Markov-modulated Feedforward Fluid Networks. Queueing Systems 37, 141–161 (2001). https://doi.org/10.1023/A:1011096201765
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DOI: https://doi.org/10.1023/A:1011096201765