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On the Variance ofOutput Counts of Some
Queueing Systems
Yoni NazarathyGideon Weiss
SE Club, TU/eApril 20, 2008
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Haifa
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Overview
1. Introduction and background2. Results for M/M/1/K3. Results for Re-entrant lines4. Possible Future Work
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A Bit On Queueing Output Processes
Buffer Server
0 1 2 3 4 5 6 …State:
A Single Server Queue:
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The Classic Theorem on M/M/1 Outputs:
Burkes Theorem (50’s):Output process of stationary version is Poisson ( ).
Buffer Server
0 1 2 3 4 5 6 …State:
( )D t
t
OutputProcess:
•Poisson Arrivals: M/M/1 Queue:
•Exponential Service times: •State Process is a birth-death CTMC
A Bit On Queueing Output Processes
A Single Server Queue:
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PLANTOUTPUT
Problem Domain: Analysis of Output Processes
Desired:
1. High Throughput
2. Low Variability
Model as a Queueing System
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Example 1: Stationary stable M/M/1, D(t) is PoissonProcess:( )
Example 2: Stationary M/M/1/1 with . D(t) is RenewalProcess(Erlang(2, )):
Variability of OutputsVariability of Outputs
(1)Vt B o Asymptotic
Variance Rate of Outputs
t
1( , )D t
3( , )D t
t1( , )X t
3( , )X t 2( , )X t
2( , )D t
Var( ( ))D t
V
21 1 1Var( ( ))
4 8 8tD t t e
Var( ( ))D t t
2
3V
m
For Renewal Processes:
Plant
8Taken from Baris Tan, ANOR, 2000.
Previous Work: NumericalPrevious Work: Numerical
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Summary of our Results
Queueing System Without Losses Finite Capacity Birth Death Queue
Push Pull Queueing Network Infinite Supply Re-Entrant Line
1*
0
K
ii
V v
stable
? critical
instable
arrivals
service
V
V
V
1 2
Explicit Expressions
for , V V
1
1
2
3
kk C
kk C
V
m
V
Diffusion LimitsDiffusion Limits
Matrix Analytic MethodsSimple
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Overview
1. Introduction and background2. Results for M/M/1/K3. Results for Re-entrant lines4. Possible Future Work
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The M/M/1/K Queue
K
* (1 )K
1
11
1
11
1
iK
i
K
KFiniteBuffer
0,...,i K
NOTE: output process D(t) is non-renewal.
1 1 1
Stationary Distribution:
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What values do we expect for ?V
?
( )V
Keep and fixed.K
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?
( )V
K / / 1( )M M
What values do we expect for ?VKeep and fixed.K
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?
( )V 40K
?* (1 )KV
Similar to Poisson:
What values do we expect for ?VKeep and fixed.K
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?
( )V
40K
What values do we expect for ?VKeep and fixed.K
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( )V
40K
2
3
Balancing
Reduces
Asymptotic
Variance of
Outputs
What values do we expect for ?VKeep and fixed.K
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**
* *
VV
V V
BRAVO Effect
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1*
0
K
ii
V v
2
2 ii i
i
Mv M
d
*1i i iM D P
1
i
i jj
P
0
i
i jj
D d
Theorem
i i id
Part (i)
Part (ii)
0iv
1 2 ... K
0 1 1... K
*1
V
0 0
1 1 1 1
1 1 1 1
( )
( )K K K K
K K
*1KD
Scope: Finite, irreducible, stationary,birth-death CTMC that represents a queue.
0 10
1
ii
i
0 1
0 0 1
1iK
j
i j i
and
If
Then
Calculation of iv
(Asymptotic Variance Rate of Output Process)
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Explicit Formula in case of M/M/1/K2
2
1 2 1
1 3
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3 6 3
(1 )(1 (1 2 ) (1 ) )1
(1 )
K K K
K
K K
K KV
K
2lim
3KV
20
0 1 KK-1
Some (partial) intuition for M/M/1/K
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Overview
1. Introduction and background2. Results for M/M/1/K3. Results for Re-entrant lines4. Possible Future Work
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Infinite Supply Re-entrant Line
4
2
1C
1 3
56
78
10 9
( )D t
2C 3C
4C
1
1Infinite QueuesSupply
1 i
1
2 21
1
1 {2,..., } ... ,
1 , C
Means: ,...,
Variances: ,...,
1, i=2,...,Ii
Operations Servers
I
j
k
k
kk C
i kk C
K C C
C C
m m
m
m
1
Control:
1) Premptive - Non-Idling.
2) In give lowest priority to infinity supply (operation 1).
example: Last Buffer First Serve (Priority)
C
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Stability Result for Re-entrant Line (Guo, Zhang, 2008 – Pre-print)
( ) Q(t), U(t)X t Queues Residuals
is Markov with state space ( )X t
Theorem (Guo Zhang): X(t) is positive (Harris) recurrent.
Proof follows framework of Jim Dai (1995)
2 Things to Prove:
1. Stability of fluid limit model
2. Compact sets are petite
Positive Harris Recurrence: There exists,
1K K
Note: We have similar result for Push-Pull Network.
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: For any stable (PHR) policy (e.g. LBFS):
2
1 3
1
Theorem
kk C
V
mk
k C
for Re-entrant linesV
( )D nt nt
nt
Remember for renewal Process:
Proof Method: Find diffusion limit of:
2
3V
m
It is Brownian Motion (0, )V
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“Renewal Like”
4
2
1C
1 3
56
78
10 9
2C 3C
4C1
1
2
3
kk C
kk C
V
m
1C
1
6
8
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Renewal Output1 1 1 1 2 2 2 2 3 3 3 31 6 8 10 1 6 8 10 1 6 8 10
Job 1 Job 2 Job 3
, , , , , , , , , , , ,....x x x x x x x x x x x x
1 1 1 1 2 2 2 3 3 3 31 6 8 10 6 8 1 1 6 8 10
201, , , , , , , , , , , , ,...x x x x x x x x x x xx
NON-Renewal Output
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Overview
1. Introduction and background2. Results for M/M/1/K3. Results for Re-entrant lines4. Possible Future Work
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t
1( , )D t
3( , )D t 2( , )D t
( )Var D TV
T
( )(1)
Var D T BV o
T T
T
Naive Estimation of:
There is bias due to intercept:
V
( ( )) (1)Var D T VT B o Remember:
( )R t
( )D t
tBusy Cycle Duration
Number Customers
Served
Use “Regenerative Simulation:”
Alternative:
(estimated moments)D RV V f
12 121 2( ( ), ( )) (1)DCov D t D t C t B o Future Work:
Smith (50’s), Brown Solomon (1975)
12 12 12 (estimated moments)D RC C f
???
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Thank You