1. 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

2. 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:

3. 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

4. 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

5. 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

6. 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

7. 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

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

9. 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.

10. 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

11. 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.

12. Next:
Multiple job-types
General queues
Exercises