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

MED Lisbon, Portugal1 Outline Motivation Proposed Solution –Active Idleness –Time Window Controller Simulation Results Conclusions

MED Lisbon, Portugal2 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

MED Lisbon, Portugal3 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

MED Lisbon, Portugal4 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

MED Lisbon, Portugal5 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)

MED Lisbon, Portugal6 Motivation – Seidman’s example I FCFS as the scheduling policy Originally presented with deterministic processing times and inter-arrival intervals

MED Lisbon, Portugal7 Motivation – Seidman’s example II Our simulation results in a stochastic setting Server #1Server #2Server #3Server #4 Sum of customers at each server X-axis goes up to 40,000 periods Y-axis goes up to 20,000 customers

MED Lisbon, Portugal8 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)

MED Lisbon, Portugal9 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?

MED Lisbon, Portugal10 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

MED Lisbon, Portugal11 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 T k  [0,  [ –Define a maximum fraction of time each server operates over each class during that window f k max  [0, 1] –Compute the fraction actually used through exponential smoothing f k  t  with  k  [0, 1] –Use original policy only on classes not exceeding their fraction

MED Lisbon, Portugal12 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

MED Lisbon, Portugal13 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

MED Lisbon, Portugal14 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

MED Lisbon, Portugal15 Simulation results – Buffer trajectories Red line – the original trajectories Blue line – the modified trajectories Server #1Server #2Server #3Server #4 Sum of customers at each server X-axis goes up to 40,000 periods Y-axis goes up to 1,000 customers

MED Lisbon, Portugal16 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

MED Lisbon, Portugal17 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

MED Lisbon, Portugal18 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. Moreira Carlos F.G. Bispo

MED Lisbon, Portugal20 Dai’s example Dai’s network PerformanceIdleness Parameters