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 Product form preserving blocking

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. Examples Independent service, Poisson arrivals Alternative

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

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. 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

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

12. 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

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

14. 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

15. 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 

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

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

18. 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

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

20. 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

21. 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.

22. 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

23. Quasi-reversibility Multi class queueing network, class c  C A queue is quasi-reversible if its state x(t) is a stationary Markov process with the property that the state of the queue at time t0, x(t0), is independent of (i) arrival times of class c customers subsequent to time t0 (ii) departure times of class c custmers prior to time t0. Theorem If a queue is QR then (i) arrival times of class c customers form independent Poisson processes (ii) departure times of class c customers form independent Poisson processes.

24. Quasi-reversibility Multi class queueing network, class c  C S(c,x) set of states queue contains one more class c than in state x Arrival rate class c customer Departure rate class c customer Characterise QR, combine QR vs R: stronger condition on arrival process, but weaker form of balance

25. Quasi-reversibility: network Multi class queueing network, class c  C J queues Customer type identifies route Poisson arrival rate per type  (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) State X(t)=(x 1 (t),…,x J (t)) Construct a network by multiplying the rates for the individual queues Arrival of type i causes queue k=r(i,1) to change at Departure type i from queue j = r(i,S(i)) Routing

26. Quasi-reversibility: network Theorem : For an open network of QR queues (i) the states of individual queues are independent (ii) an arriving customer sees the equilibrium distri bution (ii’) the equibrium distribution for a queue is as it would be in isolation with arrivals forming Poisson process. (iii) time-reversal: another open network of QR queues (iv) system is QR, so departures form Poisson process Proof of part (i)

27. 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

28. Quasi-reversiblity vs Partial balance QR: fairly general queues, service disciplines, Markov routing, product form equilibrium distribution factorizes over queues. PB: fairly general relation between service rate at queues, state-dependent routing (blocking), product form equilibrium distribution factorizes over service and routing parts. Identical for single type queueing network with Markov routing

29. Exercises [R+SN] 3.1.2, 3.2.3, 3.1.4, 3.1.3, 3.1.6, 3.3.2

30. 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

31. 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