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

1 / 28

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

On the Variance of Output Counts of Some Queueing Systems

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 - - - - - - - - - - - - - - - - - - - - - - - - - -

## On the Variance ofOutput Counts of SomeQueueing 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

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 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

PLANT

OUTPUT

• Desired:

• High Throughput

• Low Variability

Model as a Queueing System

### 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

Taken from Baris Tan, ANOR, 2000.

### Summary of our Results

Queueing System Without Losses

Finite Capacity Birth Death Queue

Push Pull Queueing Network

Infinite Supply Re-Entrant Line

### Overview

• Introduction and background

• Results for M/M/1/K

• Results for Re-entrant lines

• Possible Future Work

### The M/M/1/K Queue

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

### Overview

• 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

### 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,

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

### Overview

• 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