1. Flows and Networks Plan for today (lecture 5): Last time / Questions? Blocking of transitions Kelly / Whittle network Optimal design of a Kelly / Whittle network: optimisation problem Intermezzo: mathematical programming Optimal design of a Kelly / Whittle network: Lagrangian and interpretation Optimal design of a Kelly / Whittle network: Solution optimisation problem Optimal design of a Kelly / Whittle network: network structure Summary Exercises Questions

2. Flows and Networks Plan for today (lecture 5): Last time / Questions? Blocking of transitions Kelly / Whittle network Optimal design of a Kelly / Whittle network: optimisation problem Intermezzo: mathematical programming Optimal design of a Kelly / Whittle network: Lagrangian and interpretation Optimal design of a Kelly / Whittle network: Solution optimisation problem Optimal design of a Kelly / Whittle network: network structure Summary Exercises Questions

3. Blocking in tandem networks of simple queues (1) Simple queues, exponential service queue j, j=1,…,J state move depart arrive Transition rates Traffic equations Solution

4. Blocking in tandem networks of simple queues (2) Simple queues, exponential service queue j, j=1,…,J Transition rates Traffic equations Solution Equilibrium distribution Partial balance PICTURE J=2

5. Blocking in tandem networks of simple queues (3) Simple queues, exponential service queue j, j=1,…,J Suppose queue 2 has capacity constraint: n2<N2 Transition rates Partial balance? PICTURE J=2 Stop protocol, repeat protocol, jump-over protocol

6. Exercises Exercise Blocking Consider a tandem network of two simple queues. Let the arrival rate to queue 1 be Poisson , and let the service rate at each queue be exponential  i, i=1,2. Let queue 1 have capacity N1. Queue 2 is a standard simple queue. For N1= , give the equilibrium distribution. For N1<  formulate three distinct blocking protocols that preserve product form, indicate graphically what the implication of these protocols is on the transition diagram, and proof (by partial balance) that the equilbrium distribution is of product form.

7. Flows and Networks Plan for today (lecture 5): Last time / Questions? Blocking of transitions Kelly / Whittle network Optimal design of a Kelly / Whittle network: optimisation problem Intermezzo: mathematical programming Optimal design of a Kelly / Whittle network: Lagrangian and interpretation Optimal design of a Kelly / Whittle network: Solution optimisation problem Optimal design of a Kelly / Whittle network: network structure Summary Exercises Questions

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

9. Examples Independent service, Poisson arrivals Alternative

10. Examples Simple queue s-server queue Infinite server queue Each station may have different service type

11. 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

12. Flows and Networks Plan for today (lecture 5): Last time / Questions? Blocking of transitions Kelly / Whittle network Optimal design of a Kelly / Whittle network: optimisation problem Intermezzo: mathematical programming Optimal design of a Kelly / Whittle network: Lagrangian and interpretation Optimal design of a Kelly / Whittle network: Solution optimisation problem Optimal design of a Kelly / Whittle network: network structure Summary Exercises Questions

13. Source How to route jobs, and how to allocate capacity over the nodes? sink

14. Optimal design of Kelly / Whittle network (1) Transition rates for some functions  :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

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

16. Intermezzo: mathematical programming Optimisation problem Lagrangian Lagrangian optimization problem Theorem : Under regularity conditions: any point that satisfies Lagrangian optimization problem yields optimal solution of Optimisation problem

17. Intermezzo: mathematical programming (2) Optimisation problem Introduce slack variables Kuhn-Tucker conditions: Theorem : Under regularity conditions: any point that satisfies Lagrangian optimization problem yields optimal solution of Optimisation problem Interpretation multipliers: shadow price for constraint. If RHS constraint increased by , then optimal objective value increases by  i 

18. Optimal design of Kelly / Whittle network (3) Optimisation problem Lagrangian form Interpretation Lagrange multipliers :

19. Optimal design of Kelly / Whittle network (4) KT-conditions Computing derivatives:

20. Optimal design of Kelly / Whittle network (5) Theorem : (i) the marginal costs of input satisfy with equality for those nodes j which are used in the optimal design. (ii) If the routing j  k is 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 α j =0. If it is used but at less that full capacity then c j =0. Dynamic programming equations for nodes that are used

21. Optimal design of Kelly / Whittle network (6) PROOF: Kuhn-Tucker conditions :

22. Flows and Networks Plan for today (lecture 5): Last time / Questions? Blocking of transitions Kelly / Whittle network Optimal design of a Kelly / Whittle network: optimisation problem Intermezzo: mathematical programming Optimal design of a Kelly / Whittle network: Lagrangian and interpretation Optimal design of a Kelly / Whittle network: Solution optimisation problem Optimal design of a Kelly / Whittle network: network structure Summary Exercises Questions

23. Exercise: Optimal design of Jackson network (1) Consider an open Jackson network with transition rates Assume the service rates and arrival rates are given. Let the costs per time unit for a job residing at queue j be.Let the costs for routing a job from station j to station k be (i) Formulate the design problem (allocation of routing probabilities) as an optimisation problem. (ii) Consider the case of parallel simple queues, i.e. a fresh job routes to station j with probability and leaves the network upon completion at that station. Provide the solution to the optimization problem for the case for all j,k

24. Exercise: Optimal design of Jackson network (2) Consider an open Jackson network with transition rates Assume that the routing probabilities and arrival rates are given Let the costs per time unit for a job residing at queue j be Let the costs for routing a job from station i to station j be Let the total service rate that can be distributed over the queues be, i.e., (i) Formulate the design problem (allocation of service rates) as an optimisation problem. (ii) Now consider the case of a tandem network, and provide the solution to the optimisation problem