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Optimal Strategies for Network Routing: Minimizing Blocking Probability in Connection Requests

This document explores the fundamental problem of routing in networks with finite capacity links, addressing the challenges of admitting connection requests for various node pairs that arrive sequentially. The goal is to make decisions that minimize the blocking probability of connections through two main approaches: suboptimal greedy algorithms and optimal dynamic programming techniques. Utilizing Markov Decision Processes, it examines how to balance immediate gains with long-term opportunity costs, employing value and policy iteration methods to derive optimal routing policies.

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Optimal Strategies for Network Routing: Minimizing Blocking Probability in Connection Requests

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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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