Abstract
This paper investigates mathematical methods methods which may be used to study state dependent routing networks. In these networks, the nature of network management, network design, and network servicing begins to coalesce in some regards; but in other regards each function begins to take on new responsibilities. One example of this is how a network manager might utilize information of less than perfect accuracy. Updates to status maps take place at discrete intervals, and between updates the quality of information ages.
We now summarize the most important points that will be shown in the body of this paper. Using an expansion of the Chapman-Kolmogorov equation we find a Fokker-Planck approximation to the evolution of the network in a state dependent routing (SDR) system. Applying methods of stochastic integrals we find solutions to the Fokker-Planck equation. In an SDR system in which one routes to the least occupied resource (either a queue with a single server or a trunk group) with a high probability, say 95% of the time, then the probability density as a function of the difference in resource occupancy decays exponentially proportional to the square of the magnitude in that difference. The proportionality constant is given by the mean arrival rate and divided by the total variability in service time and arrival. When the proportion of customers misrouted is greater than about 5%, the decay becomes an exponential function of the first power of the magnitude in this difference.
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References
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© 1990 Plenum Press, New York
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Schwartz, S.D. (1990). Stochastic Integrals and the Calculation of Performance in Dynamic Control/State Dependent Routing Networks. In: Kershenbaum, A., Malek, M., Wall, M. (eds) Network Management and Control. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1471-4_32
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DOI: https://doi.org/10.1007/978-1-4613-1471-4_32
Publisher Name: Springer, Boston, MA
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