On the variance of output counts of some queueing systems
This presentation is the property of its rightful owner.
Sponsored Links
1 / 28

On the Variance of Output Counts of Some Queueing Systems PowerPoint PPT Presentation


  • 48 Views
  • Uploaded on
  • Presentation posted in: General

On the Variance of Output Counts of Some Queueing Systems. Yoni Nazarathy Gideon Weiss. SE Club, TU/e April 20, 2008. Haifa. Overview. Introduction and background Results for M/M/1/K Results for Re-entrant lines Possible Future Work. A Bit On Queueing Output Processes.

Download Presentation

On the Variance of Output Counts of Some Queueing Systems

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


On the variance of output counts of some queueing systems

On the Variance ofOutput Counts of SomeQueueing Systems

Yoni Nazarathy

Gideon Weiss

SE Club, TU/e

April 20, 2008


On the variance of output counts of some queueing systems

Haifa


Overview

Overview

  • Introduction and background

  • Results for M/M/1/K

  • Results for Re-entrant lines

  • Possible Future Work


A bit on queueing output processes

A Bit On Queueing Output Processes

A Single Server Queue:

Server

Buffer

State:

2

3

4

5

0

1

6


A bit on queueing output processes1

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

Output process of stationary version is Poisson ( ).

A Bit On Queueing Output Processes

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:


Problem domain analysis of output processes

Problem Domain: Analysis of Output Processes

PLANT

OUTPUT

  • Desired:

  • High Throughput

  • Low Variability

Model as a Queueing System


Variability of outputs

Variability of Outputs

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, )):


Previous work numerical

Previous Work: Numerical

Taken from Baris Tan, ANOR, 2000.


Summary of our results

Summary of our Results

Queueing System Without Losses

Finite Capacity Birth Death Queue

Push Pull Queueing Network

Infinite Supply Re-Entrant Line


Overview1

Overview

  • Introduction and background

  • Results for M/M/1/K

  • Results for Re-entrant lines

  • Possible Future Work


The m m 1 k queue

The M/M/1/K Queue

m

FiniteBuffer

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

Stationary Distribution:


On the variance of output counts of some queueing systems

What values do we expect for ?

Keep and fixed.


On the variance of output counts of some queueing systems

What values do we expect for ?

Keep and fixed.


On the variance of output counts of some queueing systems

What values do we expect for ?

Keep and fixed.

Similar to Poisson:


On the variance of output counts of some queueing systems

What values do we expect for ?

Keep and fixed.


On the variance of output counts of some queueing systems

What values do we expect for ?

Keep and fixed.

Balancing

Reduces

Asymptotic

Variance of

Outputs


On the variance of output counts of some queueing systems

BRAVO Effect


On the variance of output counts of some queueing systems

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


Explicit formula in case of m m 1 k

Explicit Formula in case of M/M/1/K


Some partial intuition for m m 1 k

K-1

K

0

1

Some (partial) intuition for M/M/1/K


Overview2

Overview

  • Introduction and background

  • Results for M/M/1/K

  • Results for Re-entrant lines

  • Possible Future Work


Infinite supply re entrant line

Infinite Supply Re-entrant Line

1

2

3

5

4

6

8

7

9

10


Stability result for re entrant line guo zhang 2008 pre print

Stability Resultfor Re-entrant Line (Guo, Zhang, 2008 – Pre-print)

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,


On the variance of output counts of some queueing systems

for Re-entrant lines

Proof Method: Find diffusion limit of:

It is Brownian Motion

Remember for renewal Process:


Renewal like

“Renewal Like”

1

1

2

3

6

5

4

6

8

8

7

10

9

10


Overview3

Overview

  • Introduction and background

  • Results for M/M/1/K

  • Results for Re-entrant lines

  • Possible Future Work


On the variance of output counts of some queueing systems

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

???


On the variance of output counts of some queueing systems

Thank You


  • Login