1. Flows and Networks Plan for today (lecture 6): Last time / Questions? 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? Waiting time simple queue Little Sojourn time tandem network Jackson network: mean sojourn time Product form preserving blocking Summary / Next Exercises

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. 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

7. 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

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

9. 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

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

11. 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

12. 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 

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

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

15. 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

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

17. Customer types : routes Customer type identified route Poisson arrival rate per type Type i: arrival rate  (i), i=1,…,I Route r(i,1), r(i,2),…,r(i,S(i)) Type i at stage s in queue r(i,s) Fixed number of visits; cannot use Markov routing 1, 2. or 3 visits to queue: use 3 types

18. Customer types : queue discipline Customers ordered at queue Consider queue j, containing n j jobs Queue j contains jobs in positions 1,…, n j Operation of the queue j: (i) Each job requires exponential(1) amount of service. (ii) Total service effort supplied at rate  j (n j )  (iii) Proportion  j (k,n j ) of this effort directed to job in position k, k=1,…, n j ; when this job leaves, his service is completed, jobs in positions k+1,…, n j move to positions k,…, n j -1. (iv) When a job arrives at queue j he moves into position k with probability  j (k,n j + 1), k=1,…, n j +1; jobs previously in positions k,…, n j move to positions k+1,…, n j +1.

19. Customer types : equilibrium distribution Transition rates type i job arrival (note that queue which job arrives is determined by type) type i job completion (job must be on last stage of route through the network) type i job towards next stage of its route Notice that each route behaves as tandem network, where each stage is queue in tandem Thus: arrival rate of type i to stage s :  (i) Let State of the network: Equilibrium distribution

20. Symmetric queues; insensitivity Operation of the queue j: (i) Each job requires exponential(1) amount of service. (ii) Total service effort supplied at rate  j (n j )  (iii) Proportion  j (k,n j ) of this effort directed to job in position k, k=1,…, n j ; when this job leaves, his service is completed, jobs in positions k+1,…, n j move to positions k,…, n j -1. (iv) When a job arrives at queue j he moves into position k with probability  j (k,n j + 1), k=1,…, n j +1; jobs previously in positions k,…, n j move to positions k+1,…, n j +1. Symmetric queue is insensitive

21. Flows and network: summary stochastic networks Contents 1.Introduction; Markov chains 2.Birth-death processes; Poisson process, simple queue; reversibility; detailed balance 3.Output of simple queue; Tandem network; equilibrium distribution 4.Jackson networks; Partial balance 5.Sojourn time simple queue and tandem network 6.Performance measures for Jackson networks: throughput, mean sojourn time, blocking 7.Application: service rate allocation for throughput optimisation Application: optimal routing further reading[R+SN] chapter 3: customer types; chapter 4: examples

22. Exercises [R+SN] 3.1.2, 3.2.3, 3.1.4.

23. Exercise: Optimal design of Jackson network (1) Consider an open Jackson network with transition rates Assume that 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 i to station j be (i) Formulate the design problem (allocation of routing probabilities) as an optimisation problem. (ii) Provide the solution to this problem

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) Provide the solution to this problem (iii) Now consider the case of a tandem network, and provide the solution to the optimisation problem for the case for all j,k