Stability or Stabilizability? Seidman

1
Stability or Stabilizability?Seidmans FCFS
example revisited

• José A.A. Moreira
• Agilent Technologies
• Germany

Carlos F.G. Bispo Instituto de Sistemas e
Robótica Portugal
2
Outline

• Motivation
• Proposed Solution
• Active Idleness
• Time Window Controller
• Simulation Results
• Conclusions

3
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

4
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

5
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) lk mk lt 1, for all i 1, 2, ..., S

6
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) Seidmans example (94)
  Dais example (95)

7
Motivation Seidmans example I

• FCFS as the scheduling policy
• Originally presented with deterministic
  processing times and inter-arrival intervals

8
Motivation Seidmans example II

• Our simulation results in a stochastic setting

9
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)

10
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?

11
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 servers past history to provide a
  measure of global state of the network

12
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

13
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

14
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

15
Simulation results Seidmans 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

16
Simulation results Buffer trajectories

• Red line the original trajectories
• Blue line the modified trajectories

17
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

18
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 networks
  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

19
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

20
Stability or Stabilizability?Seidmans FCFS
example revisited

• José A.A. Moreira
• jose_moreira_at_agilent.com

Carlos F.G. Bispo cfb_at_isr.ist.utl.pt http//www.is
r.ist.utl.pt
21
Dais example