STUDENTS PROBABILITY DAY Weizmann Institute of Science March 28, 2007 Yoni Nazarathy (Supervisor: Prof. Gideon Weiss) University of Haifa 0 10 20 30 40 0 5 10 15 20 Queueing Networks with Infinite Virtual Queues An Example, An Application and a Fundamental Question
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STUDENTS PROBABILITY DAY Weizmann Institute of Science March 28, 2007 Yoni Nazarathy (Supervisor: Prof. Gideon Weiss) University of Haifa Yoni Nazarathy.
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STUDENTS PROBABILITY DAYWeizmann Institute of Science
March 28, 2007
STUDENTS PROBABILITY DAYWeizmann Institute of Science
March 28, 2007
Yoni Nazarathy(Supervisor: Prof. Gideon Weiss)
University of Haifa
Yoni Nazarathy(Supervisor: Prof. Gideon Weiss)
University of Haifa
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Queueing Networks withInfinite Virtual Queues
An Example, An Application and a Fundamental Question
Queueing Networks withInfinite Virtual Queues
An Example, An Application and a Fundamental Question
Yoni Nazarathy, University of Haifa, 2007 2
Multi-Class Queueing Networks (Harrison 1988, Dai 1995,…)Multi-Class Queueing Networks (Harrison 1988, Dai 1995,…)
Near Optimal Control over a Finite Time HorizonNear Optimal Control over a Finite Time Horizon
Approximation Approach:1) Approximate the problem using a fluid system.2) Solve the fluid system (SCLP).3) Track the fluid solution on-line (Using MCQN+IVQs).4) Under proper scaling, the approach is asymptotically optimal.
Approximation Approach:1) Approximate the problem using a fluid system.2) Solve the fluid system (SCLP).3) Track the fluid solution on-line (Using MCQN+IVQs).4) Under proper scaling, the approach is asymptotically optimal.
Solution is intractable3
10
( )T
kk
Min Q t dt
Finite Horizon Control of MCQN
Weiss, Nazarathy 2007
Yoni Nazarathy, University of Haifa, 2007 11
Fluid formulationFluid formulation
1 2 3
0
1 1 1
0
2 2 1 2
0 0
3 3 2 3
0 0
1 31 3
22
min ( ) ( ) ( )
( ) (0) ( )
( ) (0) ( ) ( )
( ) (0) ( ) ( )
1 1( ) ( ) 1
1( ) 1
, 0
T
t
t t
t t
q t q t q t dt
q t q u s ds
q t q u s ds u s ds
q t q u s ds u s ds
u t u t
u t
u q
(0, )t T
s.t.
This is a Separated Continuous Linear Program (SCLP)
Server 1Server 2
1
23
Yoni Nazarathy, University of Haifa, 2007 12
Fluid solutionFluid solution
•SCLP – Bellman, Anderson, Pullan, Weiss.•Piecewise linear solution. •Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss).
The Optimal Solution:
•SCLP – Bellman, Anderson, Pullan, Weiss.•Piecewise linear solution. •Simplex based algorithm, finds the optimal solution in a finite number of steps (Weiss).
The Optimal Solution:
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3 3
2 2
1 1
1 3
2
(0) (0) 15
(0) (0) 1
(0) (0) 8
1.0
0.25
40
Q q
Q q
Q q
T
3( )q t
2 ( )q t
1( )q t
Yoni Nazarathy, University of Haifa, 2007 13
4 Time Intervals4 Time Intervals
For each time interval, set a MCQN with Infinite Virtual Queues.
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3
1
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0 {} {} {2} {2,3}nK
31 1 10 0 1 0 14 4 4 4
{1,2,3}{1,2,3} {1,3} {1}nK
0 { | ( ) 0, }nk nk q t t
{ | ( ) 0, }nk nk q t t
Yoni Nazarathy, University of Haifa, 2007 14
Maximum Pressure (Dai, Lin) is such a policy, even when
ρ=1Maximum Pressure (Dai, Lin) is such a policy, even when
ρ=1
Now Control the MCQN+IVQ Using a Rate Stable Policy
Yoni Nazarathy, University of Haifa, 2007 15
Example realizations, N={1,10,100}Example realizations, N={1,10,100}