1

On the variance curve of outputs for some queues and

networks

Yoni NazarathyGideon WeissYoav Kerner

QPA Seminar,EURANDOM January 8, 2009

2

Queueing System

Output Counts

t

1( , )D t

3( , )D t 2( , )D t

Var( ( ))D t

t

( )D t

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, )):

21 1 1

Var( ( ))4 8 8

tD t t e

Var( ( ))D t t

(1)Vt B o

B

V

Asymptotic Variance RateV

B Y-intercept

3

Outline of the Talk: Models and Methods

Outline of the Talk: Models and Methods

Models:• Finite Capacity Birth Death Queues (M/M/1/K)

• General Lossless Queues

• M/G/1 Queue

• Push-Pull (infinite supply) Network

• Infinite supply re-entrant line

Methods:• Markovian Arrival Process (MAP)

• Embedding in Renewal Reward

• Regenerative Simulation (Renewal Reward)

• Diffusion Limits

4

Finite CapacityBirth-Death Queues

5

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)

6

**

V V

BRAVO Effect (M/M/1/K)

2

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

7

C D

* * 2 * 2 3 2Var ( ) 2( ) 2 2( ) 2 ( )1 1 r bt

BV

D t D D t D O t e

0 0

1 1 1 1

1 1 1 1

( )

( )K K K K

K K

1

1

0 0

0 0

0

0K

K

* 1D *E[ ( )]D t t

0 0

1 1 1

1 1 1

0 ( )

0 ( )

0K K K

K

Generator Transitions without events Transitions with events

1( )1

, 0r b

Method: Markovian Arrival Process

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General LosslessQueues

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Stable Lossless Queues Preserve Asymptotic Variance

stable

BRAVO (?) critical

instable

arrivals

service

V

V

V

Cov ( ), ( )1

Var ( ) Var ( )

A t Q t

A t Q t Cov ( ), ( )A t Q t O t

( ) ( ) ( )D t A t Q t

0

Var ( ) Var ( ) +Var ( ) 2Cov ( ), ( )

Cov ( ), ( ) 0 2 limD A

t

D t A t Q t A t Q t

A t Q tV V

t

Proof for stable case:

10

M/G/1 Queue

11

M/G/1 Linear AsymptoteTheorem:

4 4 2 4 2 2 2 3 3 2 4 3 3

2

2

3 6 18 18 10 18 4 (1 )Stationary

12(1 )

Starting Empty(1 )

c c c c c

B

( ) (1)Var D t t B o

1, 2 (Exponential like) B 0

1, 2 B 0

1, 2 B 0

c

c

c

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Shape of Variance Curve (?)

0 B

0 B

0t

Var ( )D t

1, 2 B 0

1, 2 B 0

c

c

Pas Op: Possible non-sense ahead

!!!

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( )R t

Derivation Method:Embedding in Renewal Reward

( )D t

t

V

V

a

ar

r ( )

( ) (1

(1)

)

R RR t V t

D t V

o

t o

B

B

2

11

1

R

R

R

nV m

m

V

B

V

B

0t

0 0( ) ( )D t R t

Busy Cycle Duration

Number Customers

Served

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Linear Asymptote of Renewal Reward is Known

Linear Asymptote of Renewal Reward is Known

Var ( ) (1)R RD t V t B o

22 1 11 1 23 2

1 1 1

1 2 1 2 111 2 3 1 2 11 12 21

1 1 1 = 2

m

long expression( , , , , , , , , , , , , )

R

R

V m n k n nm m

B m m m n n k k k m m n n k

Brown, Solomon 1975:

2

11

1

R

R R

V V

nB B V m

m

15

Using in Regenerative Simulation

16

t

1( , )D t

3( , )D t 2( , )D t ( )Var D T

VT

( )D t

( )(1)

Var D T BV o

T T T

Naive Estimation of Asymptotic Variance:

There is bias due to intercept:

Regenerative Estimation of Asymptotic Variance:

22 1 11 1 23 2

1 1 1

1 1 1 = 2

mRV m n k n nm m

Estimate moments of busy cycle and number served…. Plug in…

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0 .2 0 .4 0 .6 0 .8 1 .0 1 .2

0 .2

0 .4

0 .6

0 .8

Example

M/M/1/K “like” systems (D. Perry, Boxma, et. al.)

V

Customers that have to wait more than 5 time units will not enter the queue.

18

A Push-Pull Queueing Network

19 2 ( )Q t

4 ( )Q t

1S

2S

• 2 job streams, 4 steps

• Queues at 2 and 4

• Infinite job supply at 1 and 3

• 2 servers

The Push-Pull Network

1 2

34

1S 2S

2 4( ), ( )Q t Q t• Control choice based on

• No idling, FULL UTILIZATION

• Preemptive resume

Push

Push

Pull

Pull

Push

Push

Pull

Pull

2Q

4Q

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Policies

1 2

4 3

Inherently stable

Inherently unstable

Policy: Pull priority (LBFS)

Policy: Linear thresholds

1 2

4 3

1 2

34

TypicalBehavior:

2 ( )Q t

4 ( )Q t

2,4

1S 2S

3

4

2 1

1,3

TypicalBehavior:

5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0

5

1 0

2 2 4Q Q

4 1 2Q Q

Server: “don’t let opposite queue go below threshold”

1S

2S

Push

Pull

Pull

Push

1,3

21

KSRS

1 2

34

22

M/G/. Pull Priority

M G

MG

2 2 2 2 2 21 11 1 2 1 2 2 1 1 1 2 1 1 2 2 23

1 2 1 2

(1 )( ) ( )( )( )

V c c

Using the Renewal Reward Method :

1

2

1 1,c

2 2,c

Number served of type 1, during a cycle is 0 w.p. . 2

1 2

23

Using DiffusionLimits

Now assume general processing times with finite second moment.

24

Network View of the Model

1

1 4 2 3

k

k

1

Dynamics

( ) sup{ : }

(0) 0, ( )

( ) ( ) , ( ) ( )

D ( ) ( ( ))

(0) 0, Q (t) 0

( ) (0) ( ) ( )

nj

k kj

k k

k k

k

k k k k

S t n t

T T t

T t T t t T t T t t

t S T t

Q

Q t Q D t D t

4 1 2 1

0 0

Pull priority policy

( ) ( ) 0 ( ) ( ) 0t t

Q s dT s Q s dT s 4 1 2 1 2 2 4 3

0 0

2 4 4 4 2 21 20 0

Linear thresholds policy

{0 ( ) ( )} ( ) 0 {0 ( ) ( )} ( ) 0

1 1

{ ( ) ( )} ( ) 0 { ( ) ( )} ( ) 0

1 1

1 1

t t

t t

Q s Q s dT s Q s Q s dT s

Q s Q s dT s Q s Q s dT s

2 4 1 2 3 4

Network process

( ) ( ), ( ), ( ), ( ), ( ), ( )Y t Q t Q t T t T t T t T t

or

1 2

34

1S 2S

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Stability Result

( ) Q(t), U(t)X t

1 2

34

Queue Residual

is strong Markov with state space

( )X t

Theorem: 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:

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Diffusion Scaling

Now find a limiting process, such that . ( )D t ( ) ( )wn

nD t D t

Var (1)V D

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Diffusion Limit

Theorem: When network is PHR and follows rates,

With .

( ) ( ) ( ) B(0, )n wn n

D t T t Q t

10 dimensional Brownian motion

( ) 0wn

Q t

Proof Outline: Use positive Harris recurrence to show, , simple calculations along with functional CLT for renewal processes yields the result.

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Infinite SupplyRe-entrant Lines

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Infinite Supply Re-entrant Line

4

2

1C

1 3

56

78

10 9

( )D t

2C 3C

4C

2

13

1

: For any stable policy (e.g. LBFS): .k

k C

mkk C

Thm V

1

1Infinite QueuesSupply

1

1

2 21

1

1 {2,..., } ... ,

1 .

Means: ,...,

Variances: ,...,

1, i=2,...,Ii

I

k

k

kk C

i kk C

K C C

C

m m

m

m

30

“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

10

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

31

Thank You

32

Extensions

33

• Inherently stable network

• Inherently unstable network

• Unbalanced network

• Completely balanced network

Configuration 1 2

34

1 2

4 3

1 2

4 3

1 2 1 2

4 3 4 3

or

1 2

4 3

34

Calculation of Rates

1

2

2 3 41 2 1

1 3 2 4

4 1 23 4 3

1 3 2 4

( )

( )

1 4 3 21 , 1

1 2

4 3

1 1 2 2

4 4 3 3

1 2

34

Corollary: Under assumption (A1), w.p. 1,

every fluid limit satisfies: .

k - Time proportion server works on k

k -Rate of inflow, outflow through k

Full utilization:

Stability:

( ) , ( )k kk kT t t D t t

35

Memoryless Processing(Kopzon et. al.)

1 2

4 3

Inherently stable

Inherently unstable

Policy: Pull priority

Policy: Generalized thresholds

1 2

4 3

1 2

34

1S 2S

Alternating M/M/1 Busy Periods

Results:Explicit steady state:

Stability (Foster – Lyapounov)

- Diagonal thresholds

2 ( )Q t

4 ( )Q t

- Fixed thresholds

36

37

38

39

40

41

Proof Outline

1*

0

K

ii

V v

Whitt: Book: 2001 - Stochastic Process Limits,.

Paper: 1992 - Asymptotic Formulas for Markov Processes…

1) Lemma: Look at M(t) instead of D(t).

2) Proposition: The “Fully Counting” MAP of M(t) has associated MMPP with same variance.

3) Results of Ward Whitt: An explicit expression of asymptotic variance rate of birth-death MMPP.

42

t

3 2 1

2 4 2

1 1 2

a b c

a

b

c

MMPP (Markov Modulated Poisson Process)

Example:

0 ( )N t

tabc

( )Q t

rate 4Poisson Process

rate 2

rate 3

rate 4

rate 2

rate 4

rate 3

rate 2

rate 3

rate 4

rate 2

1 0

1 0

E[ ( )] E[ ( )]

Var( ( )) Var( ( ))

N t N t

N t N t

Proposition

3 0 0 0 2 1

0 4 0 2 0 2

0 0 2 1 1 0

6 2 1 3 0 0

2 8 2 0 4 0

1 1 4 0 0 2

Transitions without events Transitions with events

1( )N tFully Counting MAP

1( ),N t

( )Q t

0 ( )N t

43

0 1 KK – 1

Some intuition for M/M/1/K-BRAVO

…

44

V

c

c

M/M/40/40

M/M/10/10

M/M/1/40

1

K=20K=30

c=30

c=20

45

V

1

MAP used for PH/PH/1/40 with Erlang and Hyper-Exp distributions

1

2

46

The “2/3 property”The “2/3 property”

K

• GI/G/1/K

• SCV of arrival = SCV of service

• 1 V2 42 33

3 21

2 3

6 2 455 3

1 2 132 3

47

,

2 3

,

1

11

lim Corr( ( ), ( )) 15 5 3

4 12 2 4

1

K

t

K

R

KD t L t

K K K

R

1 12

,1 2 1 2 2 1

(1 )(1 3 ) (1 )(3 )

(1 )(1 (2 1)(1 ) )((1 )(1 ) 4( 1)(1 ) )

K K K K

KK K K K K

KR

K

0.139772 1

1lim Corr( ( ), ( )) 1

41

12

tD t L t

For Large K

( )D t

( )L t

48

Proposition: For ,4 3 2

2

7 28 37 18

180 360 180D

K K K KB

K K

M/M/1/K1

* * 2 * 2 3 2Var ( ) 2( ) 2 2( ) 2 ( )

D

r bt

BV

D t D De t De O t e , 0r b

1