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Queuing Networks Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Queuing Networks Florence Perronnin Polytech'Grenoble - UGA March 23, 2017 F. Perronnin (UGA) Queuing Networks March 23, 2017 1 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Outline 1 Introduction to Queuing Networks 2 Refresher: M/M/1 queue 3 Reversibility 4 Open Queueing Networks 5 Closed queueing networks 6 Multiclass networks 7 Other product-form networks F. Perronnin (UGA) Queuing Networks March 23, 2017 2 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Introduction to Queuing Networks Single queues have simple results They are quite robust to slight model variations We may have multiple contention resources to model: I servers I communication links I databases with various routing structures. Queuing networks are direct results for interaction of classical single queues with probabilistic or static routing. F. Perronnin (UGA) Queuing Networks March 23, 2017 3 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Outline 1 Introduction to Queuing Networks 2 Refresher: M/M/1 queue 3 Reversibility 4 Open Queueing Networks 5 Closed queueing networks 6 Multiclass networks 7 Other product-form networks F. Perronnin (UGA) Queuing Networks March 23, 2017 4 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Refresher: M/M/1 Infinite capacity λµ Poisson(λ) arrivals Exp(µ) service times M/M/1 queue FIFO discipline Definition λ ρ = µ is the traffic intensity of the queueing system. λ λ λ λ λλ i−1 i+1 . .. 0 1 ... i µ µ µ µµ µ Number of clients X (t) in the system follows a birth and death process. F. Perronnin (UGA) Queuing Networks March 23, 2017 5 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Results for M/M/1 queue 1 Stable if and only if ρ < 1 i 2 Clients follow a geometric distribution 8i 2 N; πi = (1 − ρ)ρ ρ 3 E Mean number of clients [X ] = (1−ρ) 4 E 1 Average response time [T ] = µ−λ 1000 response time M/M/1 900 800 700 600 500 400 response time 300 200 100 0 0 0.2 0.4 0.6 0.8 1 traffic load F. Perronnin (UGA) Queuing Networks March 23, 2017 6 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks M/M/1/K In reality, buffers are finite: M/M/1/K is a queueing system with blocking. λµ K λ λ λ λ K−1 0 1 ... K µ µ µ µ Results for M/M/1/K queue Geometric distribution with finite state space (1 − ρ)ρi π(i) = 1 − ρK+1 F. Perronnin (UGA) Queuing Networks March 23, 2017 7 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks M/M/c 1 µ λ λ λ λ λ 2 µ i c c . c . iµ cµ cµ cµ c µ Results for M/M/c queue Stability condition λ < cµ. λ where ρ = µ and with ( ρi Cρ,c i! if i ≤ c 1 π(i) = i 1 ρ Cρ,c = i c Cρ,c c! c if i > c Pc−1 ρ ρ 1 i=0 i! + c! 1−ρ/c F. Perronnin (UGA) Queuing Networks March 23, 2017 8 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks F. Perronnin (UGA) Queuing Networks March 23, 2017 9 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Outline 1 Introduction to Queuing Networks 2 Refresher: M/M/1 queue 3 Reversibility 4 Open Queueing Networks 5 Closed queueing networks 6 Multiclass networks 7 Other product-form networks F. Perronnin (UGA) Queuing Networks March 23, 2017 10 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Reversibility Definition A continuous-time Markov chain X (t) is time-reversible if and only if there 2 exist a probability distribution π such that 8(i; j) 2 E ; πi qij = πj qji Theorem the π distribution is also the stationary distribution. Proof Check the balance equations. F. Perronnin (UGA) Queuing Networks March 23, 2017 11 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Reversibility Proposition An ergodic birth and death process is time-reversible. Proof λ λi−1 λi λi+1 λ0 λ1 i−2 i−1 i+1 . .. 0 1 ... i µ µ µ µi+2 µ1 µ2 i−1 i i+1 By induction: 1 π0λ0 = π1µ1 2 Suppose πi−1λi−1 = πi µi . Then πi (λi + µi ) = πi+1µi+1 + πi−1λi−1 Which gives πi λi = πi+1µi+1. F. Perronnin (UGA) Queuing Networks March 23, 2017 12 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Burke's theorem Theorem The output process of an M/M/s queue is a Poisson process that is independent of the number of customers in the queue. Sketch of Proof. λ i i+1 µ X (t) increases by 1 at rate λπi (Poisson process λ). Reverse process increases by 1 at rate µπi+1= λπi by reversibility. F. Perronnin (UGA) Queuing Networks March 23, 2017 13 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Outline 1 Introduction to Queuing Networks 2 Refresher: M/M/1 queue 3 Reversibility 4 Open Queueing Networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues 5 Closed queueing networks 6 Multiclass networks 7 OtherF. Perronnin product-form (UGA) networks Queuing Networks March 23, 2017 14 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Tandem queues λ µ µ 1 2 Let X1 and X2 denote the number of clients in queues 1 and 2 respectively. Lemma X1 and X2 are independent rv's. Proof Arrival process at queue 1 is Poisson(λ) so future arrivals are independent of X1(t). By time reversibility X1(t) is independent of past departures. Since these departures are the arrival process of queue 2, X1(t) and X2(t) are independent. F. Perronnin (UGA) Queuing Networks March 23, 2017 15 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Tandem queues Theorem The number of clients at server 1 and 2 are independent and λ n1 λ λ n2 λ P(n1; n2) = 1 − 1 − µ1 µ1 µ2 µ1 Proof By independence of X1 and X2 the joint probability is the product of M/M/1 distributions. This result is called a product-form result for the tandem queue. This product form also appears in more general networks of queues. F. Perronnin (UGA) Queuing Networks March 23, 2017 16 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Acyclic networks Example of a feed-forward network: 0 0 λi λk pik µi µk p pk0 il pi0 0 0 jk λl λj p µl µj p jl pl0 pj0 Exponential service times Routing matrix: output of i is routed to j with 0 1 0 pij pik pil probability pij B pji 0 pjk pjl C external traffic arrives at i with rate λ0 R = B C i @ pki pkj 0 pjl A packets exiting queue i leave the system pli plj plk 0 with probability pi0. F. Perronnin (UGA) Queuing Networks March 23, 2017 17 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Decomposition of a Poisson Process Problem N(t) Poisson process with rate λ Z(n) sequence of iid rv's ∼ Bernoulli(p) independent of N. Suppose the nth trial is performed at the nth arrival of the Poisson process. Result Resulting process M(t) Poisson(λ) p of successes is Bernoulli Poisson(λp) Poisson(λp). 1−p Process of failures L(t) is Poissonλ(1 − p) and Poisson(λ(1 − p)) is independent of M(t). F. Perronnin (UGA) Queuing Networks March 23, 2017 18 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Acyclic networks 0 0 λi λk pik µi µk Total arrival rate at node i: λ p pk0 i il pi0 (1 ≤ i ≤ K). λ0 λ0 pjk l j µ µj l p jl pl0 pj0 No feedback: using Burke theorem, all internal flows are Poisson! Thus we can consider K independent M/M/1 queues with Poisson arrivals with rate λi , where K 0 X 0 λi = λi + λj pji i.e. ~Λ = ~Λ + ~ΛR in matrix notation. j=0 Stability condition All queues must be stable independently: λi < µi , 8i = 1; 2;:::; K: F. Perronnin (UGA) Queuing Networks March 23, 2017 19 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Backfeeding Example Switch transmitting frames with random errors. A NACK is sent instantaneously if the frame is incorrect. Frame success probability is p. Arrivals ∼ Poisson(λ0) Frame transmission times ∼ Exp (µ) 0 p λ µ 1−p F. Perronnin (UGA) Queuing Networks March 23, 2017 20 / 46 Intro Refresher Reversibility Open networks Closed networks Multiclass networks Other networks Tandem queues Acyclic networks Backfeeding Jackson networks Open networks of M/M/c queues Backfeeding 0 λ µ Remark Arrivals are not Poisson anymore! p Result The departure process is still Poisson with rate λp.
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