On the variance of output counts of some queueing systems
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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.

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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


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


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


K-1

K

0

1

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


Overview

  • Introduction and background

  • Results for M/M/1/K

  • Results for Re-entrant lines

  • Possible Future Work


Infinite Supply Re-entrant Line

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:


“Renewal Like”

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


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