Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-16T08:24:30.365Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear stochastic fluid networks

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  14 July 2016

Offer Kella  and
Ward Whitt
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Ward Whitt*
Affiliation:
AT&T Labs
*
∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: mskella@olive.mscc.huji.ac.il
∗∗Postal address: Room A117, AT&T Labs, Shannon Laboratory, 180 Park Avenue, Florham Park, NJ 07932–0971, USA. Email address: wow@research.att.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce open stochastic fluid networks that can be regarded as continuous analogues or fluid limits of open networks of infinite-server queues. Random exogenous input may come to any of the queues. At each queue, a c.d.f.-valued stochastic process governs the proportion of the input processed by a given time after arrival. The routeing may be deterministic (a specified sequence of successive queue visits) or proportional, i.e. a stochastic transition matrix may govern the proportion of the output routed from one queue to another. This stochastic fluid network with deterministic c.d.f.s governing processing at the queues arises as the limit of normalized networks of infinite-server queues with batch arrival processes where the batch sizes grow. In this limit, one can think of each particle having an evolution through the network, depending on its time and place of arrival, but otherwise independent of all other particles. A key property associated with this independence is the linearity: the workload associated with a superposition of inputs, each possibly having its own pattern of flow through the network, is simply the sum of the component workloads. As with infinite-server queueing models, the tractability makes the linear stochastic fluid network a natural candidate for approximations.

Keywords

Fluid modelsstochastic fluid networksstorage networksLévy processesinfinite-server queuesqueueing networksshot noise

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B05: Inventory, storage, reservoirs 90B15: Network models, stochastic
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 244 - 260
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work was partially supported by Israel Science Foundation Grant No. 794/97.

References

Apostol, T. M. (1957). Mathematical Analysis. Addison-Wesley, Reading, MA.Google Scholar
Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York.Google Scholar
Asmussen, S., and Kella, O. (1996). Rate modulation in dams and ruin problems. J. Appl. Prob. 33, 523–535.CrossRefGoogle Scholar
Berman, A., and Plemmons, R. J. (1979). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York.Google Scholar
Brandt, A. (1986). The stochastic equation Y n+1 = An Yn + Bn with stationary coefficients. Adv. Appl. Prob. 18, 211–220.Google Scholar
Chaudhry, M. L., and Templeton, J. G. C. (1983). A First Course in Bulk Queues. Wiley, New York.Google Scholar
Daley, D. J. (1971). The definition of a multi-dimensional generalization of shot noise. J. Appl. Prob. 8, 128–135.CrossRefGoogle Scholar
Daley, D. J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Denardo, E. V., and Lee, T. Y. S. (1966). Managing uncertainty in a serial production line. Operat. Res. 44, 382–392.CrossRefGoogle Scholar
Denardo, E. V., and Tang, C. S. (1992). Linear control of a Markov production system. Operat. Res. 40, 259–278.CrossRefGoogle Scholar
Duffield, N. G., and Whitt, W. (1997). Control and recovery from rare congestion events in a large multi-server system. Queueing Systems 26, 69–104.CrossRefGoogle Scholar
Eick, S. G., Massey, W. A., and Whitt, W. (1993a). The physics of the M t /G/∞ queue. Operat. Res. 41, 731–742.CrossRefGoogle Scholar
Eick, S. G., Massey, W. A., and Whitt, W. (1993b). M t/G/∞ queues with sinusoidal arrival rates. Management Sci. 39, 241–252.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II, 2nd edn. Wiley, New York.Google Scholar
Glasserman, P., and Yao, D. D. (1994). Monotone Structure in Discrete-Event Systems. Wiley, New York.Google Scholar
Glynn, P. W., and Whitt, W. (1991). A new view of the heavy-traffic limit theorem for infinite-server queues. Adv. Appl. Prob. 23, 188–209.CrossRefGoogle Scholar
Grier, N., Massey, W. A., McKoy, T., and Whitt, W. (1997). The time-dependent Erlang loss model with retrials. Telecommunication Systems 7, 253–265.CrossRefGoogle Scholar
Holman, D. F., Chaudhry, M. L., and Kashyap, B. R. K. (1982). On the number in the system GI X /M/∞. Sankhyā A 44, 294–297.Google Scholar
Jacobsen, M. (1993). A brief account of the theory of homogeneous Gaussian diffusions in finite dimensions. In Frontiers in Pure and Applied Probability, Proc. 3rd Finnish/Soviet Symposium on Probability Theory and Mathematical Statistics, eds. Niemi, H., Hognas, G., Shiryaev, A. N., Melinkov, A. V., VSP Utrecht, TVP Moscow, pp. 86–94.Google Scholar
Jennings, O. B., Mandelbaum, A., Massey, W. A., and Whitt, W. (1996). Server staffing to meet time-varying demand. Management Sci. 42, 1383–1394.CrossRefGoogle Scholar
Jennings, O. B., and Massey, W. A. (1997). A modified-offered-load approximation for time-dependent circuit-switched networks. Telecommunication Systems 7, 229–251.CrossRefGoogle Scholar
Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682–695.CrossRefGoogle Scholar
Kella, O. (1996). Stability and non-product form of stochastic fluid networks with Lévy inputs. Ann. Appl. Prob. 6, 186–199.CrossRefGoogle Scholar
Kella, O. (1997). Stochastic storage networks: stationarity and the feedforward case. J. Appl. Prob. 34, 498–507.CrossRefGoogle Scholar
Kella, O., and Whitt, W. (1992). A tandem fluid network with Lévy input. In Queues and Related Models, eds. Basawa, I. and Bhat, U. Oxford University Press, Oxford, pp. 112–128.Google Scholar
Kella, O., and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 1169–1180.CrossRefGoogle Scholar
Klüppelberg, C., and Mikosch, T. (1995). Explosive Poisson shot noise processes with application to risk reserves. Bernoulli 1, 125–147.CrossRefGoogle Scholar
Kurtz, T. (1996). Limit theorems for workload input processes. In Stochastic Networks, eds. Kelly, F. P., Zachary, S. and Ziedins, I. Clarendon, Oxford, pp. 119–139.Google Scholar
Liu, L., Kashyap, B. R. K., and Templeton, J. G. C. (1990). On the GI X/G/∞ system. J. Appl. Prob. 27, 671–683.CrossRefGoogle Scholar
Liu, L., and Templeton, J. G. C. (1991). The GR X n /G n /∞ system: system size. Queueing Systems 8, 323–356.CrossRefGoogle Scholar
Massey, W. A., and Whitt, W. (1993). Network of infinite server queues with nonstationary Poisson input. Queueing Systems 13, 183–250.CrossRefGoogle Scholar
Massey, W. A., and Whitt, W. (1994). An analysis of the modified offered-load approximation for the nonstationary Erlang loss model. Ann. Appl. Prob. 4, 1145–1160.CrossRefGoogle Scholar
Melamed, B., and Whitt, W. (1990). On arrivals that see time averages. Operat. Res. 38, 156–172.CrossRefGoogle Scholar
Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.CrossRefGoogle Scholar
Ross, S. M. (1983). Stochastic Processes. Wiley, New York.Google Scholar
Shandbhag, D. N. (1966). On the infinite-server queue with batch arrivals. J. Appl. Prob. 3, 274–279.CrossRefGoogle Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.CrossRefGoogle Scholar
Whitt, W. (1990). Queues with service times and interarrival times depending linearly and randomly upon waiting times. Queueing Systems 6, 335–352.CrossRefGoogle Scholar