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

Distributional fixed points and attractors in queueing theory Sergio L´ opez Universidad Nacional Aut´ onoma de M´ exico (Joint work with Pablo Ferrari, Universidad de Buenos Aires) June 11 PIMS Probability Summer School 2012 Sergio L´ opez (UNAM) 1 / 33

Discrete queues The M \ M \ 1 queue. D S A t t t A t arrival process. S t service process. For the M \ M \ 1, A t ∼ Poisson ( λ ) , S t ∼ Poisson ( µ ) , independent. Stability (recurrence): λ < µ . D t effective departure process. Q t queue length. Sergio L´ opez (UNAM) 2 / 33

Discrete queues Regulator mapping f : [ 0 , ∞ ) → R , f ( 0 ) ≥ 0, g : R → R s.t. lim t →−∞ g t = ∞ and lim t →∞ g t = −∞ . Reflective mapping: R ( f ) t = f t − inf 0 ≤ s ≤ t { f s ∧ 0 } , R ( g ) t = g t − inf s ≤ t { g s ∧ 0 } = sup s ≤ t [ g t − g s ] + . Function f Reflection of f Sergio L´ opez (UNAM) 3 / 33

Discrete queues Construction using the regulator mapping A t S t A - S t t Q t Sergio L´ opez (UNAM) 4 / 33

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. Sergio L´ opez (UNAM) 5 / 33

Discrete queues Proof of Burke’s theorem: Q * T 0 s * D A , s t 0 Q T t Reverse process Q ∗ t ≡ Q ( T − t − ) . s : 0 ≤ s ≤ T } d { A ∗ = { A t : 0 ≤ t ≤ T } (reversibility) s } a . s . { A ∗ = { D T − D T − s } (identification) d = { D t : 0 ≤ t ≤ T } (reversibility). � Sergio L´ opez (UNAM) 6 / 33

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. Sergio L´ opez (UNAM) 7 / 33

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. Sergio L´ opez (UNAM) 7 / 33

Discrete queues Attractiveness [Mountford-Prabhakar 95’] Let A an ergodic stationary point process with λ < 1 such that A ( t ) lim = λ P − a . s . t t →∞ { N n } ∞ n = 1 ind. Poisson ( 1 ) process. Let A 1 = A , A n = Q ( A n − 1 , N n − 1 ) ∀ n ≥ 2 where Q ( A , S ) is the departure process of the queue operator with arrival A and service S. Then A n converges to a Poisson ( λ ) process. Sergio L´ opez (UNAM) 8 / 33

Discrete queues Mountford-Prabhakar’s theorem Proof’s idea: Let P an independient Poisson ( λ ) process. Let it pass over the tandem queues P 1 = A , P n = Q ( P n − 1 , N n − 1 ) ∀ n ≥ 2 using the same services { N n } ∞ n = 1 . Sergio L´ opez (UNAM) 9 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 10 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 11 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 12 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 13 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 14 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 15 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 16 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 17 / 33

Discrete queues Mountford-Prabhakar’s theorem 1 1 A t P , t 1 S t 2 2 A t P , t S t 2 3 3 A t P , t • All clients in A n and P n eventually couple. • P n is Poisson ( λ ) , ∀ n . � Sergio L´ opez (UNAM) 18 / 33

Continuous queues Continuous valued queues Let A t the arrival process, S t the service process, continuous process on R + (or R ). Define Q = R ( A − S ) . Stability: A t − S t 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 ] . Sergio L´ opez (UNAM) 19 / 33

Continuous queues Continuous valued queues Let A t the arrival process, S t the service process, continuous process on R + (or R ). Define Q = R ( A − S ) . Stability: A t − S t 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 ] . Sergio L´ opez (UNAM) 19 / 33

Continuous queues Brownian queue Let B t ⊥ W t be standard B.M. A t = B t , S t = W t + c t , c > 0. x ≥ 0. Q t = R ( x + A t − S t ) , is a regulated B.M. with drift − c < 0. Sergio L´ opez (UNAM) 20 / 33

Continuous queues Brownian Burke’s analogue Theorem [Harrison 91’, O’Conell-Yor 01’] Let A t be a B.M., let S t + c t an ind. B.M., where A 0 − S 0 is exp ( c ) . Let D t the B.M. driven by S t with barrier A t . Then D t is a B.M. A S D Sergio L´ opez (UNAM) 21 / 33

Continuous queues Burke’s analogue Proof’s idea: c • Scale a discrete valued queue, using heavy traffic limit ( λ n ∼ 1 − √ 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 (2 M − X Pitman’s representation theorem). � Sergio L´ opez (UNAM) 22 / 33

Continuous queues A Brownian analogue to Mountford-Prabhakar’s theorem [Ferrari, L.] Let A be a non-explosive continuous process. Let { S n } n ∈ N be B.M.’s with drift c > 0 and { E n } n ∈ N exp ( c ) r.v. the initial workload of each queue. Assume independence. Define A 1 = A A n = Q ( A n − 1 , S n − 1 ) , then A n converges to a B.M. Sergio L´ opez (UNAM) 23 / 33

Continuous queues M-P’s analogue Proof’s idea: Define an independent B.M. and apply the queueing operator B 1 = B B n = Q ( B n − 1 , S n − 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´ opez (UNAM) 24 / 33

Continuous queues M-P’s analogue Proof’s idea: Define an independent B.M. and apply the queueing operator B 1 = B B n = Q ( B n − 1 , S n − 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´ opez (UNAM) 24 / 33

Continuous queues M-P’s analogue If S n ≤ A n ∧ B n on [ 0 , T ] , the paths couple (and it persists). Define O n ≡ { S n ≤ A n ∧ B n 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´ opez (UNAM) 25 / 33

Continuous queues M-P’s analogue If S n ≤ A n ∧ B n on [ 0 , T ] , the paths couple (and it persists). Define O n ≡ { S n ≤ A n ∧ B n 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´ opez (UNAM) 25 / 33

Work in progress Work in progress: Towards a stationary analogue to M-P Prove the convergence with stationary distribution in each queue instead of exp r.v.’s as initial workloads. Obtain a stationary tandem system. Sergio L´ opez (UNAM) 26 / 33

Work in progress Towards a stationary analogue to M-P [Ferrari, L.] Graphical construction of reflections of simple random walks. D A S + F=S-A - + D - Sergio L´ opez (UNAM) 27 / 33

Work in progress Towards a stationary analogue to M-P [Ferrari, L.] Graphical construction of reflections of simple random walks. D A S + F=S-A - + D - Sergio L´ opez (UNAM) 28 / 33