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Stability or Stabilizability? Seidman’s FCFS example revisited

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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

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

- 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

- 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

- 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

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

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

- 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

- 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

- 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

- 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

- 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

- 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

- Choice of parameters for the Controller
- All fractions add up to 1 at each server
- Each fraction is sligthly above the long term needs

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

- 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

- 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

- 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

José A.A. Moreira

Carlos F.G. Bispo

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

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