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Stability or Stabilizability? Seidman’s FCFS example revisited. José A.A. Moreira Agilent Technologies Germany. Carlos F.G. Bispo Instituto de Sistemas e Robótica Portugal. Outline. Motivation Proposed Solution Active Idleness Time Window Controller Simulation Results Conclusions.

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Stability or stabilizability seidman s fcfs example revisited l.jpg

Stability or Stabilizability?Seidman’s FCFS example revisited

José A.A. Moreira

Agilent Technologies

Germany

Carlos F.G. Bispo

Instituto de Sistemas e Robótica

Portugal


Outline l.jpg
Outline

  • Motivation

  • Proposed Solution

    • Active Idleness

    • Time Window Controller

  • Simulation Results

  • Conclusions


Motivation the system l.jpg
Motivation – The system

  • Multi-class, Non-Acyclic Queuing network

    • Random service times

    • Random external inter-arrival times

    • Diferent types of customers

      • Each type has a deterministic routing

      • Same type may visit a server more than once

      • Each service a different class

      • Each class a different service distribution

        • Not a Jackson network


Motivation the control policies l.jpg
Motivation – The control policies

  • Open networks

    • No adimission policy

    • Scheduling policy

  • Scheduling policy

    • Distributed: buffer priority; ESPT; FCFS; etc.

    • Non-idling or work conserving

    • No preemption


Motivation the stability condition l.jpg
Motivation – The stability condition

  • Assume all classes are uniquely numbered

    • k = 1, 2, ..., K

    • Let mk be the first moment of the service for class k

  • Each server operates over a subset of all classes

  • Each class has an associated type of customer for wich an external arrival rate is defined

    • Let lk be the first moment for the arrival rate of class k

  • Then the traffic intensity condition is

    • Sk c(i)lkmk < 1, for all i = 1, 2, ..., S


Motivation the problem l.jpg
Motivation – The problem

  • Is the traffic intensity condition sufficient or simply a necessary condition for stability?

    • It is sufficient for Jackson networks

      • Service distribution associated with the server, not the customer

      • FCFS as the scheduling policy

    • It seems sufficient for acyclic networks

    • But, some examples of unstable non-acyclic networks

      • Lu-Kumar example (’91); Seidman’s example (’94); Dai’s example (’95)


Motivation seidman s example i l.jpg
Motivation – Seidman’s example I

  • FCFS as the scheduling policy

  • Originally presented with deterministic processing times and inter-arrival intervals


Motivation seidman s example ii l.jpg

Server #1

Server #2

Server #3

Server #4

Sum of customers at each server

X-axis goes up to 40,000 periods

Y-axis goes up to 20,000 customers

Motivation – Seidman’s example II

  • Our simulation results in a stochastic setting


Motivation consequences l.jpg
Motivation – Consequences

  • After these examples, the answer seems to be

    • The traffic intensity condition is NOT a sufficient stability condition for general queuing networks.

  • However,

    • Most authors focused on non-idling policies

    • From the static and deterministic scheduling theory we know that their equivalent to non-idling policies may not contain the optimal solution

    • Clear-a-Fraction policies with Backoff resorts to idling policies to establish stability (Kumar & Seidman, ‘90)


Proposed solution active idleness i l.jpg
Proposed solution – Active Idleness I

  • Why determine if a network is stable under all non-idling policies?

  • Or, why determine regions for which some topologies are stable for all non-idling policies?

  • Why not asking if a network is stabilizable?

    • That is, can a given policy be changed to make the network stable?

    • Is this property intrinsic to the pair network/policy or just a property of the network?


Proposed solution active idleness ii l.jpg
Proposed solution – Active Idleness II

  • By using non-idling policies we are forcing idleness due to lack of customers

    • Burstiness in the arrival and services times is allowed to freely spread trough the network

  • Actively resort to idleness

    • That is, allow a server to stay idle in the presence of customers

    • Take the server’s past history to provide a measure of global state of the network


Proposed solution tw controller i l.jpg
Proposed solution – TW Controller I

  • The Time Window Controller is an implementation of the Active Idleness concept

    • Define a finite size window of time looking into the past history of each class

      • Tk [0, [

    • Define a maximum fraction of time each server operates over each class during that window

      • fkmax [0, 1]

    • Compute the fraction actually used through exponential smoothing

      • fk(t), with ak [0, 1]

    • Use original policy only on classes not exceeding their fraction


Proposed solution tw controller ii l.jpg
Proposed solution – TW Controller II

  • Classes exceeding their maximum fraction are blocked

    • If all costumers waiting belong to blocked classes, the server will remain idle

    • Idleness is kept until a new customer from a non blocked class arrives or until one of the blocked classes present drops below its maximum time fraction

  • Controller filters burstiness on individual classes

  • The filtering procedure is local


Proposed solution tw controller iii l.jpg
Proposed solution – TW Controller III

  • What is good for an individual server is not necessarily good for the network

    • Idleness is bad for a single server when customers are present

    • Local scheduling policies are based on what is good for a single server

      • Getting rid of waiting customers

    • Active Idleness hurts single servers to preserve the network

      • Past history of a single server is a measure of load to remaining servers


Simulation results seidman s example l.jpg
Simulation results – Seidman’s example

  • Choice of parameters for the Controller

    • All fractions add up to 1 at each server

    • Each fraction is sligthly above the long term needs


Simulation results buffer trajectories l.jpg

Server #1

Server #2

Server #3

Server #4

Sum of customers at each server

X-axis goes up to 40,000 periods

Y-axis goes up to 1,000 customers

Simulation results – Buffer trajectories

  • Red line – the original trajectories

  • Blue line – the modified trajectories


Simulation results active idleness l.jpg
Simulation results – Active Idleness

  • There is no Active Idleness on the original system, but Passive Idleness accounts for a huge capacity waste

  • The modified system has a significant reduction of Passive Idleness at the expense of a very small amount of Active Idleness


Conclusions i l.jpg
Conclusions I

  • Consequences

    • The traffic intensity condition is sufficient to ensure stabilizability, if processing times have upper bounds and original policy is non-idling

    • Stabilizability is intrinsic to the network’s topology

    • Optimal controller is stable

  • Limitations

    • We can construct a provably stabilizing controller if all services have an upper bound

      • Leaves out Markovian systems, but not critical for real life systems


Conclusions ii l.jpg
Conclusions II

  • Features

    • The maximum time fractions can add up to more than one

    • Performance gains even when the original is already stable

  • Future

    • Characterize the performance measures as functions of the parameters – convex?; unimodal?; etc.

    • Design an optimization package to tune the TW Controller


Stability or stabilizability seidman s fcfs example revisited20 l.jpg

Stability or Stabilizability?Seidman’s FCFS example revisited

José A.A. Moreira

[email protected]

Carlos F.G. Bispo

[email protected]

http://www.isr.ist.utl.pt


Dai s example l.jpg

Dai’s network

Performance

Idleness

Parameters

Dai’s example


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