On the Variance of Output Counts of Some Queueing Systems

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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

13

?

( )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

15

?

( )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

21

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