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Source-Destination Routing Optimal Strategies Eric Chi EE228a, Fall 2002 Dept. of EECS, U.C. Berkeley
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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
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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
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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.
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Discrete Time MDP
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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
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Solving MDPs: Value Iteration Solve the fixed point equation. Then
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Solving MDPs: Policy Iteration
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Example: Symmetric Y/C X/C ’ Optimal Policy: Route to least loaded
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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, , a
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Example: Unbalanced Y/C X/C 2
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Example: Unbalanced Y/C X/C ’ Optimal Policy: Route to lower link until full. If full route to top link.
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Comparison
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Example: Alternate Routing Policy A: Route up 1 st, Route down 2 nd Policy B: Route down 1 st, Route up 2 nd Y/C X/C 2
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Comparison Two policies
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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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