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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

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
Outline
  • Motivation
  • Proposed Solution
    • Active Idleness
    • Time Window Controller
  • Simulation Results
  • Conclusions
motivation the system
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
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
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
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
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

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
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
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
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
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
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
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
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

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
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
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
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

Stability or Stabilizability?Seidman’s FCFS example revisited

José A.A. Moreira

jose_moreira@agilent.com

Carlos F.G. Bispo

cfb@isr.ist.utl.pt

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

dai s example

Dai’s network

Performance

Idleness

Parameters

Dai’s example
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