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

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Stability or Stabilizability? Seidman’s FCFS example revisited

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  1. 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

  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)lkmk < 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); Seidman’s example (’94); Dai’s example (’95)

  7. Motivation – Seidman’s example I • FCFS as the scheduling policy • Originally presented with deterministic processing times and inter-arrival intervals

  8. 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

  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 server’s 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 – 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

  16. 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

  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 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

  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?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

  21. Dai’s network Performance Idleness Parameters Dai’s example

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