Abstract
We consider Jackson networks with state-dependent arrival and service rates which show product form or nearly product form steady-states and come up with examples of load-dependent admission control. For these networks we prove an arrival theorem for external as well as for internal arrivals. In case of open tandem systems with state-independent service rates we compute the joint distribution of the sojourn times of a customer in the nodes and the distribution of the customer’s end-to-end-delay.
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References
F. Baccelli and P. Brémaud, Elements of Queueing Theory (Springer, Berlin, 1994).
F. Baskett, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. Assoc. Comput. Mach. 22 (1975) 248–260.
G. Bolch, S. Greiner, H. de Meer and K.S. Trivedi, Queueing Networks and Markov Chains (Wiley, New York, 1998).
O.J. Boxma and H. Daduna, Sojourn times in queueing networks, in: Stochastic Analysis of Computer and Communication Systems, ed. H. Takagi (North-Holland, Amsterdam, 1990) pp. 401–450.
O.J. Boxma, F.P. Kelly and A.G. Konheim, The product form for sojourn time distributions in cyclic exponential queues, J. Assoc. Comput. Mach. 31 (1984) 128–133.
P.J. Burke, The output process of a stationary M=M=s queueing system, Ann. Math. Statist. 39 (1968) 1144–1152.
H. Daduna, Discrete time analysis of a state dependent tandem with different customer types, in: Foundations of Computer Science, Potential-Theory-Cognition, eds. C. Freksa, M. Jantzen, and R. Valk, Lecture Notes in Computer Science, Vol. 1337 (Springer, Berlin, 1997) pp. 287–296.
H. Daduna, Sojourn time distributions in non-product-form queueing networks, in: Frontiers in Queueing: Models and Applications in Science and Engineering, ed. J.H. Dshalalow (CRC Press, Boca Raton, FL, 1997) pp. 194–224.
H. Daduna, Some results for steady-state and sojourn time distributions in open and closed linear networks of Bernoulli servers with state-dependent service and arrival rates, Performance Evaluation 30 (1997) 3–18.
P.G. Harrison, Laplace transform inversion and passage-time distributions in Markov processes, J. Appl. Probab. 27 (1990) 74–87.
F.P. Kelly, Reversibility and Stochastic Networks (Wiley, New York, 1979).
K.H. Kook and R.F. Serfozo, Travel and sojourn times in stochastic networks, Ann. Appl. Probab. 3 (1993) 228–252.
S.S. Lavenberg and M. Reiser, Stationary state probabilities at arrival instants for closed queueing networks with multiple types of customers, J. Appl. Probab. 17 (1980) 1048–1061.
E. Reich, Waiting times when queues are in tandem, Ann. Math. Statist. 28 (1957) 768–773.
E. Reich, Note on queues in tandem, Ann. Math. Statist. 34 (1963) 338–341.
R. Schassberger and H. Daduna, Sojourn times in queuing networks with multiserver nodes, J. Appl. Probab. 24 (1987) 511–521.
K.C. Sevcik and I. Mitrani, The distribution of queueing network states at input and output instants, J. Assoc. Comput. Mach. 28 (1981) 358–371.
H. Takagi, ed., Stochastic Analysis of Computer and Communication Systems (North-Holland, Amsterdam, 1990).
D.D. Yao, ed., Stochastic Modeling and Analysis of Manufacturing Systems, Springer Series in Operations Research (Springer, New York, 1994).
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Daduna, H., Meyer, S. Individual customer’s behaviour in networks with state-dependent arrival rates. Queueing Systems 32, 351–362 (1999). https://doi.org/10.1023/A:1019103507843
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DOI: https://doi.org/10.1023/A:1019103507843