Download Optimal design of Kelly / Whittle network

Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
Blocking in tandem networks of simple queues (1)
• Simple queues, exponential service queue j, j=1,…,J
• state
n  (n1 ,..., nJ )
move
T j , j 1 (n)  (n1 ,..., n j  1, n j 1  1,..., nJ )
depart
T j 0 (n)  (n1 ,..., n j  1,..., nJ )
arrive
T0 k (n)  (n1 ,..., nk  1,..., nJ )
• Transition rates
q (n, T j , j 1 (n))   j
q (n, TJ 0 (n))   J
q (n, T01 (n))  
• Traffic equations
 j  j   j 1 j 1 , j  2,..., J
11  
• Solution

j 
j
, j  1,..., J
Blocking in tandem networks of simple queues (2)
•
Simple queues, exponential service queue j, j=1,…,J
•
Transition rates
q (n, T j , j 1 (n))   j
q (n, TJ 0 (n))   J
q (n, T01 (n))  
•
Traffic equations
 j  j   j 1 j 1 , j  2,..., J
11  

j 
j
•
Solution
•
Equilibrium distribution
J
 (n)   (1   j ) j n
j
, j  1,..., J
n  S  {n : n  0}
j 1
•
Partial balance
J
J
k 0
k 0
 (n) q(n, T jk (n))    (T jk (n)) q(T jk (n), n)
J
J


 (n) j    jk    j (T j 0 (n))    (T jk (n))kj
k 1
k 1


•
PICTURE J=2
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
q (n, T j , j 1 (n))   j , j  2,..., J
q (n, T1, 2 (n))  11(n2  N 2 )
q (n, TJ 0 (n))   J
q (n, T01 (n))  
•
Partial balance?
•
PICTURE J=2
•
Stop protocol, repeat protocol, jump-over protocol
J
 (n)   (1   j ) j n
n  S  {n : n  0}
j
j 1
J
J
k 0
k 0
 (n) q(n, T jk (n))    (T jk (n)) q(T jk (n), n)
J
J


 (n) j    jk    j (T j 0 (n))    (T jk (n))kj
k 1
k 1


Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
Kelly / Whittle network
•
Transition rates
for some functions
ψ:S(0,∞),
•
Traffic equations
 (T j 0 (n))
q (n, T jk (n))   j
p jk
 ( n)
 (T j 0 (n))
q (n, T j 0 (n))   j
p j0
 ( n)
 ( n)
q (n, T0 k (n))  0
p0 k
 ( n)
 j ( j  p jk )  0 p0 k    k k pkj
k 0
k 1
 j ( j )   0 p0 k    k pkj
•
Open network
•
Partial balance equations:
k 1
J
J
k 0
k 0
 (n) q(n, T jk (n))    (Tkj (n)) q(Tkj (n), n)
•
Theorem: Assume B
1
 1


 j
J
   (n)
nS
then
J
 (n)  B (n)
j 1
satisfies partial balance,
j 1
 1


 j




nj
J





nj
 jn
J

 jn  
j
j 1
j
nS
j 1
and is equilibrium distribution Kelly / Whittle network
Interpretation traffic equations
• Transition rates
for some functions
ψ:S(0,∞),
 (T j 0 (n))
q (n, T jk (n))   j
p jk
 ( n)
 (T j 0 (n))
q (n, T j 0 (n))   j
p j0
 ( n)
 ( n)
q (n, T0 k (n))  0
p0 k
 ( n)
• Traffic equations
• Open network
 j ( j  p jk )  0 p0 k    k k pkj
k 0
k 1
• Theorem: Suppose that the equilibrium distribution is
J
 (n)  B (n)
j 1
then




J

j 1
j
 (T j 0 (n)) 
E
 j 
j
  ( n) 
and rate jk
• PROOF
 1


 j
nj
 j  jk
 jn
j
nS
Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
• Source
• How to route jobs, and
• how to allocate capacity over the nodes?
•
sink
Optimal design of Kelly / Whittle network (1)
•
 (T j 0 (n))
q (n, T jk (n))   j
p jk
 ( n)
 (T j 0 (n))
q (n, T j 0 (n))   j
p j0
 ( n)
 ( n)
q (n, T0 k (n))  0
p0 k
 ( n)
Transition rates
for some functions
ψ:S(0,∞),
•
Routing rules for open network to clear input traffic
0 p0 k
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
b jk
C  A( )   b jk  jk j
j ,k
A( ) 

nS
J
a (n) (n)  j

nS
j 1
J
 (n)  j n
j 1
j
nj
Optimal design of Kelly / Whittle network (2)
• Transition rates
 (T j 0 (n))
q(n, T jk (n))   j
p jk , j, k  0,..., J
 (n)
• Given: input traffic
0 p0 k
• Maximal service rate
 j    jk    j p jk   j
k 0
k 0
• Optimization problem :
C  A( )   b jk  jk j
minimize costs
j ,k
A( ) 
k 0
j
jk
nS
a (n) (n)  j
  (n) 
j 1
   k kj , j  1,..., J
k 0
 j   j , j  1,..., J
 j  0, j  1,..., J
0  1
 jk  0, j  1,..., J , k  0,..., J
0 k fixed
j 1
J
nS
• Under constraints


J
nj
j
nj
Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
Intermezzo: mathematical programming
• Optimisation problem
min f ( x1 ,..., xn )
s.t. g i ( x1 ,..., xn )  bi
i  1,..., m
• Lagrangian L  f ( x ,..., x ) 
1
n
m

i 1
i (bi  g i ( x1 ,..., xn ))
L  b  g ( x ,..., x )  0
i
i
1
n

i
m
g i
L  f 


i
x
x
x
j
j i 1
j
• Lagrangian optimization problem
min L( x1 ,..., xn , 1 ,..., m )
• Theorem : Under regularity conditions: any point
( x1 ,..., xn , 1 ,..., m ) that satisfies Lagrangian
optimization problem yields optimal solution ( x1 ,..., xn )
of Optimisation problem
Intermezzo: mathematical programming (2)
• Optimisation problem
min f ( x1 ,..., xn )
s.t. g i ( x1 ,..., xn )  bi
i  1,..., m
• Introduce slack variables
• Kuhn-Tucker conditions:
m
g i
f
  i
 0, j  1,..., n
x

x
j i 1
j
 i (bi  g i ( x1 ,..., xn ))  0, i  1,..., m
 i  0, i  1,..., m
• Theorem : Under regularity conditions: any point
( x1 ,..., xn )
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
Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
Optimal design of Kelly / Whittle network (3)
• Optimisation problem
min C ({ j ,  jk })  A( )   b jk  jk j
j ,k

k 0
j
jk
   k kj  0, j  1,..., J
k 0
 j   j , j  1,..., J
 j  0, j  1,..., J
0  1
 jk  0, j  1,..., J , k  0,..., J
0 k fixed
• Lagrangian form
L C
j 0

k 0
 j ( k kj   j  jk )    j (  jk   j )
   j j 
j 0
j 0

j ,k 0
jk
k 0
 jk
 0   0   0  00  0
• Interpretation Lagrange multipliers :
j 
L
 ( )
0j
 j   L
 j
Optimal design of Kelly / Whittle network (4)
• KT-conditions
L  0, j  1,..., J

j
L  0, j , k  0,..., J

jk

k 0
 j ( k kj   j  jk )  0, j  1,..., J
 j (  jk   j )  0, j  1,..., J
k 0
 j j  0, j  1,..., J
 jk  jk  0, j , k  0,..., J
 j , j ,  j , jk  0
• Computing derivatives:
L  c  
b jk  jk   j j    k  jk   j

j j

k
k
j
L  b           
jk j
j j
k j
j
jk
 jk
1 A( )
cj 
 j  j
Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
Optimal design of Kelly / Whittle network (5)
• Theorem : (i) the marginal costs of input satisfy
 j  c j  min k (b jk   k ), j  1,..., J
0  0
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
 j  c j  min k (b jk   k )
0  0
Optimal design of Kelly / Whittle network (6)
• PROOF: Kuhn-Tucker conditions :
c j  j   b jk  jk   j j    k  jk  0
k
(*)
k
and  0 if  j  0
b jk j   j j   k j   j  0
and  0 if  jk  0
(**)
Networks of Queues
Plan for today (lecture 6):
•
•
•
•
•
•
•
•
•
•
•
Last time / Questions?
Product form preserving blocking
Interpretation traffic equations
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
Exercises
Questions??
Exercise 1: insensitivity and
Norton
Consider a network consisting of subnetworks.
Each subnetwork is quasi reversible. The
subnetworks are linked via Markov routing
(as in the transparancy in class, but now
state independent routing).
1. Formulate the transition rates at local and
global level
2. Formulate and prove the decomposition
theorem
3. Suppose that we want to aggregate a
component into a single queue. Give the
service rates for this queue.
4. Now assume that all queues in the network
are infinite server queues. Give the service
rate for the aggregated components. Show
the relation between Norton’s theorem and
the insensitivity property.
5. Investigate the relation between Norton’s
theorem and insensitivity
Exercise 2: Optimal design of Jackson network
(1)
•
Consider an open Jackson network
q (n, T jk (n))   j p jk
with transition rates
q (n, T j 0 (n))   j p j 0
q (n, T0 k (n))  0 p0 k
•
Assume that the service rates
j
and arrival rates
0
are given
•
Let the costs per time unit for a job residing at queue j be
aj
•
Let the costs for routing a job from station i to station j be
b jk
•
(i) Formulate the design problem (allocation of routing
probabilities) as an optimisation problem.
•
(ii) Provide the solution to this problem
Exercise 3: Optimal design of Jackson network
(2)
•
Consider an open Jackson network
q (n, T jk (n))   j p jk
with transition rates
q (n, T j 0 (n))   j p j 0
q (n, T0 k (n))  0 p0 k
•
Assume that the routing probabilities
p jk and arrival rates 0
are given
•
Let the costs per time unit for a job residing at queue j be
aj
•
Let the costs for routing a job from station i to station j be
b jk
•
Let the total service rate that can be distributed over the
queues be
 , i.e.,

j  
j
•
(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 b
for all j,k
jk
0
Exercise 4 : routes vs Markovian routing
1. Show that an open Jackson network
with transition rates
q (n, T jk (n))   j p jk
q (n, T j 0 (n))   j p j 0
q (n, T0 k (n))  0 p0 k
Can equivalently be represented as a network with fixed routing
(i.e., the route is determined by the customer type).
2. Now consider an open Jackson network with queues with finite
capacity, i.e. ni<Ni, i=1,…,J. Consider the network under a
product form preserving blocking protocol. Investigate whether
or not the network can equivalently be represented as a
network with fixed routing as in 1.
Exercise 5
Consider an open tandem network of 3 single server queues
(M/M/1 queues).
1. Formulate the transitions rates.
2. Give the equilibrium distribution, and show that this
distribution satisfies partial balance
3. Show that a customer arriving to queue j, j=1,2,3
observes the equilibrium distribution
4. Now assume that the total number of customers in the
network is restricted not to exceed N. Formulate the
transition rates. Give the equilibrium distribution, and
the distribution of the state observed by a customer
arriving to queue 1
5. Now assume that each queue has a capacity restriction,
say the number in queue i is restricted not to exceed Ni.
A queue that has reached its capacity constraint is
blocked, which means that no extra customer may enter
the queue. This means that a customer cannot leave its
predecessor, i.e., when queue 3 is blocked a customer
cannot leave queues 2 (the server at queue 2 is
stopped). Formulate the transition rates, and show that
the equilibrium distribution cannot satisfy partial
balance.
6. Again consider the tandem network of part 5. Consider
the stop protocol, that is the following modification of
the transition rates. The arrival process, and each nonblocked queue is stopped until a service completion has
occurred at the blocked queue. Give the state space, and
formulate the transition rates. Show that the
equilibrium distribution has a product form and
satisfies partial balance. Give the distribution of the
state observed by a customer arriving to (and entering)
queue 2.
7. Again consider the tandem network of part 5. Assume
that a customer arriving to a queue that has reached its
capacity restriction, jumps over the queue and moves to
the next queue in the tandem. Formulate the transition
rates. Give the equilibrium distribution (proof!).
Exercise 6
Consider a tandem network consisting of 4 queues. Queue 1
is a FCFS queue, queue 2 a PS queue, queue 3 a LCFS
queue, and queue 4 an Infinite server queue. Let
customers of type 1 and 2 arrive to queue 1 according to
a Poisson process with rates λ1, and λ2.
1. Let the service requirements at each of the queues be
exponentially distributed. Formulation the network as
BCMP network. Also give the transition rates.
2. Show that the queues are quasi reversible (you may do
this for the queues in isolation, i.e., for each queue
consider only that queue [thus part of the network] with
Poisson arrival process)
3. Give the equilibrium distribution for the tandem
network.
4. Show that this distribution satisfies partial balance.
5. Prove that the equilibrium distribution of queue 2, 3
and 4 is insensitive to the distribution of the service
time except for its mean (you may do this for the queue
in isolation).
6. Prove that the equilibrium distribution of the tandem
network is insensitive to the distribution of the service
time at each of the queues 2,3,4 except for their means.
7. Indicate why the distribution cannot be insensitive to
the service time distribution in an FCFS queue.
Exercises: rules
In all exercises:
give proof or derivation of results
(you are not allowed to state something like: proof as
given in class, or proof as in book…)
motivate the steps in derivation
hand in exercises week before oral exam, that is based on
these exercises
All oral exams (30 mins) for this part on the same day,
you may suggest the date…
Hand in 4 exercises: 1 and $ and select 2 or 3 and 5 or 6
Each group 2 or 3 persons: hand in a single set of answers
G1: 1,2,4,5
G2: 1,2,4,6
G3: 1,3,4,5
G4: 1,3,4,6