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 αj =0. If it is used but at less that full capacity then cj =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 nj jobs
Queue j contains jobs in positions 1,…, nj
Operation of the queue j: (i) Each job requires exponential(1) amount of service. (ii) Total service effort supplied at rate j(nj) (iii) Proportion  j(k,nj) of this effort directed to job in position k, k=1,…, nj ; when this job leaves, his service is completed, jobs in positions k+1,…, nj move to positions k,…, nj -1. (iv) When a job arrives at queue j he moves into position k with probability j(k,nj + 1), k=1,…, nj +1; jobs previously in positions k,…, nj move to positions k+1,…, nj +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)=(x1(t),…,xJ(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(nj) (iii) Proportion  j(k,nj) of this effort directed to job in position k, k=1,…, nj ; when this job leaves, his service is completed, jobs in positions k+1,…, nj move to positions k,…, nj -1. (iv) When a job arrives at queue j he moves into position k with probability j(k,nj + 1), k=1,…, nj +1; jobs previously in positions k,…, nj move to positions k+1,…, nj +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