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Basic Routing Problem

Source-Destination Routing Optimal Strategies Eric Chi EE228a, Fall 2002 Dept. of EECS, U.C. Berkeley. Basic Routing Problem. Network with links of finite capacity Connection requests for various node-pairs arrive one by one A decision is made to either deny the request or

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Basic Routing Problem

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  1. Source-Destination RoutingOptimal StrategiesEric ChiEE228a, Fall 2002Dept. of EECS, U.C. Berkeley

  2. Basic Routing Problem • Network with links of finite capacity • Connection requests for various node-pairs arrive one by one • A decision is made to either • deny the request or • admit the connection along a given route • An admitted call simultaneously holds some capacity along all links along the route for some amount of time before departing • Objective: Make decisions that minimize blocking probability

  3. Approaches • Suboptimal: Greedy algorithms • Always admit if there is space. • Choose good heuristics for where to place calls. • Maximize spare capacity • Minimize “Interference” • Optimal: Dynamic programming • Balances • Immediate gains • Long term opportunity costs

  4. Markov Decision Process • State specified by a Markov Chain • Request arrivals are Poisson • Calls holding times are exponentially distributed • Rewards (Costs) associated with • Residing in a state • Making a transition • Transition probabilities depend on policies for a given state.

  5. Discrete Time MDP

  6. Bellman Principle of Optimality • Given an optimal control for n steps to go, the last n-1 steps provide optimal control with n-1 steps to go. • Example: Dijstkra’s Shortest Path Algorithm

  7. Solving MDPs: Value Iteration • Solve the fixed point equation. Then

  8. Solving MDPs: Policy Iteration

  9. Optimal Policy: Route to least loaded Example: Symmetric l X/C l’ Y/C l

  10. Proof (Sketch) • Prove that load balancing is optimal for any finite time to go n. (Monotone convergence allows us to take the limit.) • Prove inductively that for all n, b, a

  11. Example: Unbalanced l1 X/C l2 Y/C l3

  12. Optimal Policy: Route to lower link until full. If full route to top link. Example: Unbalanced l X/C l’ Y/C

  13. Comparison

  14. Example: Alternate Routing l1 • Policy A: Route up 1st, Route down 2nd • Policy B: Route down 1st, Route up 2nd X/C l2 Y/C

  15. Comparison • Two policies

  16. Literature • K. R. Krishnan and T. J. Ott, "State-dependent routing for telephone traffic: theory and results," in 25th IEEE Control and Decision Conf., Athens, Greece, Dec. 1986, pp. 2124-2128. • A. Ephremides, P. Varaiya, and J. Walrand. A simple dynamic routing problem. IEEE Transactions on Automatic Control, 25(4):690-693, August 1980. • R.J. Gibbon and F.P. Kelly. Dynamic routing in fully connected networks. IMA journal of Mathematical Control and Information, 7:77--111, 1990. • Marbach, P., Mihatsch, M., Tsitsiklis, J.N., "Call admission control and routing in integrated service networks using neuro-dynamic programming," IEEE J. Selected Areas in Comm., v. 18, n. 2, pp. 197--208, Feb. 2000. • K. Kar, M. Kodialam, and T.V. Lakshman, “Minimum Interference Routing of Bandwidth Guaranteed Tunnels with Applications to MPLS Traffic Engineering,” IEEE JSAC, 1995, Special Issue on Advances in the Fundamentals of Networking, pp. 1128-36.

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