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Transcript of Flows and Networks Plan for today (lecture 5): Last time / Questions? Blocking of transitions Kelly...
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
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
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
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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
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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
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Exercises
• Exercise BlockingConsider 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.
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
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
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• Independent service, Poisson arrivals
• Alternative
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Interpretation traffic equations
• Transition rates
for some functions
:S(0,),
• Traffic equations
• Open network
• Theorem: Suppose that the equilibrium distribution is
then
and rate jk
• PROOF
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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
• Source
• How to route jobs, and • how to allocate capacity over the nodes?
• sink
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 jk :
• Design : b_j0=+ : cannot leave from j; sequence of queues
• Expected cost rate
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Optimal design of Kelly / Whittle network (2)
• Transition rates
• Given: input traffic
• Maximal service rate
• Optimization problem :
minimize costs
• Under constraints
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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
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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
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Optimal design of Kelly / Whittle network (3)
• Optimisation problem
• Lagrangian form
• Interpretation Lagrange multipliers :
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Optimal design of Kelly / Whittle network (4)
• KT-conditions
• Computing derivatives:
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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 jk 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
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Optimal design of Kelly / Whittle network (6)
• PROOF: Kuhn-Tucker conditions :
0 if 0 and
(**) 0
0 if 0 and
(*) 0
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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
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
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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
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