Distributional fixed points and attractors in queueing theory - - PowerPoint PPT Presentation

Distributional fixed points and attractors in queueing theory

Sergio L´

• pez

Universidad Nacional Aut´

• noma de M´

exico (Joint work with Pablo Ferrari, Universidad de Buenos Aires)

June 11 PIMS Probability Summer School 2012

Sergio L´

• pez (UNAM)

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Discrete queues

The M \ M \ 1 queue.

S A D

t t t

At arrival process. St service process. For the M \ M \ 1, At ∼ Poisson(λ), St ∼ Poisson(µ), independent. Stability (recurrence): λ < µ. Dt effective departure process. Qt queue length.

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• pez (UNAM)

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Discrete queues

Regulator mapping

f : [0, ∞) → R, f(0) ≥ 0, g : R → R s.t. limt→−∞ gt = ∞ and limt→∞ gt = −∞. Reflective mapping: R(f)t = ft − inf0≤s≤t{fs ∧ 0}, R(g)t = gt − infs≤t{gs ∧ 0} = sups≤t[gt − gs]+.

Reflection of f Function f

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• pez (UNAM)

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Discrete queues

Construction using the regulator mapping

Q t A S A - S

t t t t Sergio L´

• pez (UNAM)

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Discrete queues

Burke’s theorem

[Burke 56’] For a stationary stable M/M/1 queue, with arrival Poisson(λ), service Poisson(µ), the departure process is a Poisson(λ) process.

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• pez (UNAM)

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Discrete queues

Proof of Burke’s theorem:

T T Q Q

t s * t s *

D A ,

Reverse process Q∗

t ≡ Q(T − t−).

{A∗

s : 0 ≤ s ≤ T} d ={At : 0 ≤ t ≤ T} (reversibility)

{A∗

s} a.s. = {DT − DT−s} (identification) d ={Dt : 0 ≤ t ≤ T} (reversibility).

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• pez (UNAM)

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Discrete queues

Tandem queues

Under stationary stable Poisson(µ) servers and independence, if the initial arrival process has Poisson(λ) law, so the other arrival processes.

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• pez (UNAM)

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Discrete queues

Tandem queues

Under stationary stable Poisson(µ) servers and independence, if the initial arrival process has Poisson(λ) law, so the other arrival processes.

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• pez (UNAM)

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Discrete queues

Attractiveness

[Mountford-Prabhakar 95’] Let A an ergodic stationary point process with λ < 1 such that lim

t→∞

A(t) t

= λ P − a.s. {Nn}∞

n=1 ind. Poisson(1) process. Let

A1 = A, An = Q(An−1, Nn−1)

∀n ≥ 2

where Q(A, S) is the departure process of the queue operator with arrival A and service S. Then An converges to a Poisson(λ) process.

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• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

Proof’s idea: Let P an independient Poisson(λ) process. Let it pass over the tandem queues P1 = A, Pn = Q(Pn−1, Nn−1)

∀n ≥ 2

using the same services {Nn}∞

n=1.

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• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Discrete queues

Mountford-Prabhakar’s theorem

A t S t A t S t A t P

t

P

t

P

t , , , 1 2 3 1 1 2 2 3

• All clients in An and Pn eventually couple.
• Pn is Poisson(λ), ∀n.
• Sergio L´
• pez (UNAM)

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Continuous queues

Continuous valued queues

Let At the arrival process, St the service process, continuous process on

R+ (or R).

Define Q = R(A − S). Stability: At − St has negative drift (descends from infinity). Q[s,t] = A[s,t] − D[s,t], then D[s,t] ≡ A[s,t] − Q[s,t].

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• pez (UNAM)

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Continuous queues

Continuous valued queues

Let At the arrival process, St the service process, continuous process on

R+ (or R).

Define Q = R(A − S). Stability: At − St has negative drift (descends from infinity). Q[s,t] = A[s,t] − D[s,t], then D[s,t] ≡ A[s,t] − Q[s,t].

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• pez (UNAM)

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Continuous queues

Brownian queue

Let Bt ⊥ Wt be standard B.M. At = Bt, St = Wt + c t, c > 0. x ≥ 0. Qt = R(x + At − St), is a regulated B.M. with drift −c < 0.

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• pez (UNAM)

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Continuous queues

Brownian Burke’s analogue

Theorem [Harrison 91’, O’Conell-Yor 01’] Let At be a B.M., let St + c t an ind. B.M., where A0 − S0 is exp(c). Let Dt the B.M. driven by St with barrier At. Then Dt is a B.M.

A S

D

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• pez (UNAM)

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Continuous queues

Burke’s analogue

Proof’s idea:

• Scale a discrete valued queue, using heavy traffic limit (λn ∼ 1 −

c √n ),

then use continuity of R, [Whitt 02’]. Calculate limits using and not using Burke’s theorem and conclude by weak limit unicity.

• Or use properties of Brownian Motion (2M − X Pitman’s

representation theorem).

• Sergio L´
• pez (UNAM)

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Continuous queues

A Brownian analogue to Mountford-Prabhakar’s theorem

[Ferrari, L.] Let A be a non-explosive continuous process. Let {Sn}n∈N be B.M.’s with drift c > 0 and {En}n∈N exp(c) r.v. the initial workload of each

• queue. Assume independence. Define

A1 = A An = Q(An−1, Sn−1), then An converges to a B.M.

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• pez (UNAM)

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Continuous queues

M-P’s analogue

Proof’s idea: Define an independent B.M. and apply the queueing operator B1 = B Bn = Q(Bn−1, Sn−1), with the same services. [Non-increasing distance in synchrony] Let f, g, h : R+ → R be c` adl` ag functions with f(0), g(0) > h(0). Define f∗ (g∗) the function driven by h and reflected on the barrier f (g). Then,

||f∗ − g∗||[0,T] ≤ ||f − g||[0,T].

Sergio L´

• pez (UNAM)

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Continuous queues

M-P’s analogue

Proof’s idea: Define an independent B.M. and apply the queueing operator B1 = B Bn = Q(Bn−1, Sn−1), with the same services. [Non-increasing distance in synchrony] Let f, g, h : R+ → R be c` adl` ag functions with f(0), g(0) > h(0). Define f∗ (g∗) the function driven by h and reflected on the barrier f (g). Then,

||f∗ − g∗||[0,T] ≤ ||f − g||[0,T].

Sergio L´

• pez (UNAM)

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Continuous queues

M-P’s analogue

If Sn ≤ An ∧ Bn on [0, T], the paths couple (and it persists). Define On ≡ {Sn ≤ An ∧ Bn on [0, T]}.

• Use monotonicity (derived by the lemma) and properties of B.M.
• Conclude by (a markovian version of) Borel-Cantelli lemma.
• Sergio L´
• pez (UNAM)

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Continuous queues

M-P’s analogue

If Sn ≤ An ∧ Bn on [0, T], the paths couple (and it persists). Define On ≡ {Sn ≤ An ∧ Bn on [0, T]}.

• Use monotonicity (derived by the lemma) and properties of B.M.
• Conclude by (a markovian version of) Borel-Cantelli lemma.
• Sergio L´
• pez (UNAM)

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Work in progress

Work in progress: Towards a stationary analogue to M-P

Prove the convergence with stationary distribution in each queue instead

• f exp r.v.’s as initial workloads.

Obtain a stationary tandem system.

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• pez (UNAM)

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Work in progress

Towards a stationary analogue to M-P

[Ferrari, L.] Graphical construction of reflections of simple random walks.

+ F=S-A

• +

D S A D

Sergio L´

• pez (UNAM)

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Work in progress

Towards a stationary analogue to M-P

[Ferrari, L.] Graphical construction of reflections of simple random walks.

+ F=S-A

• +

D S A D

Sergio L´

• pez (UNAM)

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Work in progress

Towards a stationary analogue to M-P

[Ferrari, L.] Graphical construction of reflections of simple random walks.

+ F=S-A

• +

D S A D

Sergio L´

• pez (UNAM)

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Work in progress

Towards a stationary analogue to M-P

[Ferrari, L.] Graphical construction of reflections of simple random walks.

+ F=S-A

• +

D S A D

Sergio L´

• pez (UNAM)

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Work in progress

Towards a stationary analogue to M-P

[Ferrari, L.] Non-existence of a SRW as a fixed point, i.e. reflection of a SRW over another SRW is not a SRW in any case. Conjecture: For SRW servers, there exists an attractive fixed point for the queueing operation which converges to BM under scaling.

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• pez (UNAM)

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Work in progress

References 1 K. Burdzy and Z. Chen. Coalescence of synchronous couplings. Probability Theory and Related Fields,(’02). 2 P . Burke. The output of a queuieng system. Operations Research, (’56). 3 J. Harrison, R. Williams. On the quasireversibility of a multiclass brownian station. Annals of Probability, (’90). 4 P . Lieshout, M. Mandjes. Transient analysis of Brownian queues. Probability, Networks and Algorithms, (’07). 5 N. O’Connell , M. Yor. Brownian analogues of Burkes theorem. Stochastic Processes and their Applications, (’01).

Sergio L´

• pez (UNAM)

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Work in progress

6 J. Mairesse, B. Prabhakar. The existence of fixed points for the ·/GI/1

• queue. Annals of Probability, (’03).

7 J. Martin, B. Prabhakar. Fixed points for multiclass queues. Arxive: 1003.3024v1, (’10) 8 T. Mountford and B. Prabhakar. On the Weak Convergence of Departures from an Infinite Series of ·/M/1 Queues. Annals of Applied Probability, (’95). 9 I. Norros, P . Salminen. On busy periods of the unbounded Brownian

• storage. Queueing Systems, (’01).

10 B. Prabhakar. The attractiveness of the fixed points of a ·/GI/1 queue. Annals of Probability, (’03). 11 W. Whitt. Stochastic Process Limits. Springer-Verlag, (’02).

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• pez (UNAM)

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