Stability or Stabilizability?Seidman’s FCFS example revisited

José A.A. MoreiraAgilent Technologies

Germany

Carlos F.G. BispoInstituto de Sistemas e Robótica

Portugal

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Outline

• Motivation

• Proposed Solution– Active Idleness

– Time Window Controller

• Simulation Results

• Conclusions

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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

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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

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Motivation – The stability condition

• Assume all classes are uniquely numbered– k = 1, 2, ..., K– Let k 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 k be the first moment for the arrival rate of class k

• Then the traffic intensity condition is k c(i) k k < 1, for all i = 1, 2, ..., S

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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)

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Motivation – Seidman’s example I

• FCFS as the scheduling policy

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

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Motivation – Seidman’s example II

• Our simulation results in a stochastic setting

Server #1Server #2Server #3Server #4Sum of customers at each server

X-axis goes up to 40,000 periods

Y-axis goes up to 20,000 customers

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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)

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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?

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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

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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

• fktwithk [0, 1]– Use original policy only on classes not exceeding their fraction

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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

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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

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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

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Simulation results – Buffer trajectories

• Red line – the original trajectories

• Blue line – the modified trajectories

Server #1Server #2Server #3Server #4Sum of customers at each server

X-axis goes up to 40,000 periods

Y-axis goes up to

1,000 customers

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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

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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

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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 revisited

José A.A. Moreirajose_moreira@agilent.com

Carlos F.G. Bispocfb@isr.ist.utl.pt

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

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Dai’s example

Acrobat DocumentDai’s network Acrobat DocumentPerformance

Acrobat DocumentIdlenessAcrobat DocumentParameters