1. Networks (3TU) Plan for today (lecture 5): Last time / Questions? Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

2. Jackson network : Definition Simple queues, exponential service queue j, j=1,…,J state move depart arrive Transition rates Traffic equations Irreducible, unique solution, interpretation, exercise Jackson network: open Gordon Newell network: closed

3. Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

4. Jackson network : Equilibrium distribution Simple queues, Transition rates Traffic equations Closed network Open network Global balance equations: Closed network: Open network:

5. closed network : equilibrium distribution Transition rates Traffic equations Closed network Global balance equations: Theorem: The equilibrium distribution for the closed Jackson network containing N jobs is Proof

6. Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

7. Partial balance Global balance verified via partial balance Theorem: If distribution satisfies partial balance, then it is the equilibrium distribution. Interpretation partial balance

8. Jackson network : Equilibrium distribution Transition rates Traffic equations Open network Global balance equations: Theorem: The equilibrium distribution for the open Jackson network containing N jobs is, provided α j <1, j=1,…,J, Proof

9. Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

10. Kelly / Whittle network Transition rates for some function  :S  (0,  ∞ ) Traffic equations Open network Partial balance equations: Theorem: Assume that then satisfies partial balance, and is equilibrium distribution Kelly / Whittle network

11. Examples Independent service, Poisson arrivals Alternative

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

13. Flows and Networks Plan for today (lecture 4): Last time / Questions? Output simple queue Tandem network Jackson network: definition Jackson network: equilibrium distribution Partial balance Kelly/Whittle network Summary / Next Exercises

14. Summary / next: Equilibrium distributions Reversibility Output reversible Markov process Tandem network Jackson network Partial balance Kelly-Whittle network NEXT: Optimization (W sec 9.7) Quasi-reversibility (R+SN Ch 3)

15. Exercises [R+SN] 2.1.1, 2.1.2, 2.3.1, 2.3.4, 2.3.5, 2.3.6, 2.4.1, 2.4.2, 2.4.6, 2.4.7