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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

On the Variance ofOutput Counts of SomeQueueing Systems

Yoni Nazarathy

Gideon Weiss

SE Club, TU/e

April 20, 2008



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:




What values do we expect for ?

Keep and fixed.

Similar to Poisson:



What values do we expect for ?

Keep and fixed.

Balancing

Reduces

Asymptotic

Variance of

Outputs



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



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,


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


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

???



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