Basic Routing Problem
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Transcript of Basic Routing Problem
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Source-Destination RoutingOptimal Strategies
Eric ChiEE228a, 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 1st, Route down 2nd
• Policy B: Route down 1st, Route up 2nd
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.