On the Variance of Output Counts of Some Queueing Systems

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On the Variance of Output Counts of Some Queueing Systems

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On the Variance of Output Counts of Some Queueing Systems

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On the Variance ofOutput Counts of SomeQueueing Systems

Yoni Nazarathy

Gideon Weiss

SE Club, TU/e

April 20, 2008

Haifa

- Introduction and background
- Results for M/M/1/K
- Results for Re-entrant lines
- Possible Future Work

A Single Server Queue:

Server

Buffer

…

State:

2

3

4

5

0

1

6

The Classic Theorem on M/M/1 Outputs:Burkes Theorem (50’s):

Output process of stationary version is Poisson ( ).

A Single Server Queue:

Server

Buffer

…

State:

2

3

4

5

0

1

6

M/M/1 Queue:

- Poisson Arrivals:

- Exponential Service times:

- State Process is a birth-death CTMC

OutputProcess:

PLANT

OUTPUT

- Desired:
- High Throughput
- Low Variability

Model as a Queueing System

Asymptotic Variance Rate of Outputs

For Renewal Processes:

Plant

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, )):

Taken from Baris Tan, ANOR, 2000.

Queueing System Without Losses

Finite Capacity Birth Death Queue

Push Pull Queueing Network

Infinite Supply Re-Entrant Line

- Introduction and background
- Results for M/M/1/K
- Results for Re-entrant lines
- Possible Future Work

m

FiniteBuffer

NOTE: output process D(t) is non-renewal.

Stationary Distribution:

What values do we expect for ?

Keep and fixed.

What values do we expect for ?

Keep and fixed.

What values do we expect for ?

Keep and fixed.

Similar to Poisson:

What values do we expect for ?

Keep and fixed.

What values do we expect for ?

Keep and fixed.

Balancing

Reduces

Asymptotic

Variance of

Outputs

BRAVO Effect

Theorem

Scope: Finite, irreducible, stationary,birth-death CTMC that represents a queue.

(Asymptotic Variance Rate of Output Process)

Part (i)

Part (ii)

Calculation of

If

and

Then

K-1

K

0

1

- Introduction and background
- Results for M/M/1/K
- Results for Re-entrant lines
- Possible Future Work

1

2

3

5

4

6

8

7

9

10

Queues

Residuals

is Markov with state space

Theorem (Guo Zhang):X(t) is positive (Harris) recurrent.

- Proof follows framework of Jim Dai (1995)
- 2 Things to Prove:
- Stability of fluid limit model
- Compact sets are petite

Note: We have similar result for Push-Pull Network.

Positive Harris Recurrence: There exists,

for Re-entrant lines

Proof Method: Find diffusion limit of:

It is Brownian Motion

Remember for renewal Process:

1

1

2

3

6

5

4

6

8

8

7

10

9

10

- Introduction and background
- Results for M/M/1/K
- Results for Re-entrant lines
- Possible Future Work

Naive Estimation of :

Remember:

There is bias due to intercept:

Alternative:

Smith (50’s), Brown Solomon (1975)

Use “Regenerative Simulation”:

Future Work:

Number Customers Served

Busy Cycle Duration

???

Thank You