Flows and Networks Plan for today (lecture 6):

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Presentation transcript:

Flows and Networks Plan for today (lecture 6): Last time / Questions? Jackon network Kelly / Whittle network Interpretation traffic equations Optimal design of a Kelly / Whittle network

Jackson network : Mean sojourn time Simple queues, FCFS, Transition rates Traffic equations Open network Partial balance equations: Equilibrium distribution Sojourn time in each queue: Sojourn time on path i,j,k:

Kelly / Whittle network Transition rates for some functions :S[0,),  :S(0,) Traffic equations Open network Partial balance equations: Theorem: Assume then satisfies partial balance, and is equilibrium distribution Kelly / Whittle network

Flows and Networks Plan for today (lecture 6): Last time / Questions? Jackon network Kelly / Whittle network Interpretation traffic equations Optimal design of a Kelly / Whittle network

Interpretation traffic equations Transition rates for some functions :S(0,), Traffic equations Open network Theorem: Suppose that the equilibrium distribution is then and rate jk PROOF

Flows and Networks Plan for today (lecture 6): Last time / Questions? Jackon network Kelly / Whittle network Interpretation traffic equations Optimal design of a Kelly / Whittle network

Optimal design of Kelly / Whittle network (1) Transition rates for some functions :S[0,),  :S(0,) Routing rules for open network to clear input traffic as efficiently as possible Cost per time unit in state n : a(n) Cost for routing jk : Design : b_j0=+ : cannot leave from j; sequence of queues Expected cost rate PROOF

Optimal design of Kelly / Whittle network (2) Transition rates Given: input traffic Maximal service rate Optimization problem : minimize costs Under constraints

Optimal design of Kelly / Whittle network (3) Lagrangian (without nonnegativity constraints) Interpretation Lagrange multipliers Theorem : (i) the marginal costs of input satisfy with equality for those nodes j which are used in the optimal design, where (ii) If the routing jkis used in the optimal design the equality holds in (i) and the minimum in the rhs is attained at given k. (iii) If node j is not used in the optimal design then αj =0. If it is used but at less that full capacity then cj =0.

Optimal design of Kelly / Whittle network (4) PROOF: Kuhn-Tucker conditioins Dynamic programming equation: for nodes that are used in the optimal design: For networks with fixed routing or fixed service: Optimize lambda or p

Optimal design of Kelly / Whittle network (5) Some structure is known: Corollary: Suppose Then optimal network has no cycles Dynamic programming equation: for nodes that are used in the optimal design: Notice that cj is also determined by αj This is bridge to deterministic network flow problems.

Next: Multiple job-types General queues Exercises