Poisson processes, Markov chains and M/M/1 queues

Transcription

1 Poisson processes, Markov chains and M/M/1 queues Naveen Arulselvan Advanced Communication Networks Lecture 3

2 Little s law λ Queue Server T Avg. no. in system N = λ T Arrival rate Avg. delay in system N : Time average / Statistical average

3 Little s law applications Single server Queue Notation Entire system: N = λt Just Queue: N Q = λw Just server: ρ = λ X N, N Q = number of customers W, T = average delay λ = arrival rate X = service time ρ = utility

4 Single server Queue contd. System equations Total number is system: N = N Q + ρ Total delay: T = W + X N = λt N Q = λw ρ = λ X

5 Single server Queue contd. System equations Total number is system: N = N Q + ρ Total delay: T = W + X N = λt N Q = λw ρ = λ X Fix λ, X 5 equations, 5 unknowns (N, N Q, W, T, ρ) Not independent, Need more info!

6 Stochastic Modeling: Arrival times Easy yet interesting model! {A(t); t 0} is the arrival process A(t) is Number of arrivals in [0, t] Basic Model for A(t): Poisson Process also called Poisson counting process Non-decreasing, integer valued sample paths

7 Poisson process A(t)is Poisson process with rate λ t > s and τ = t s A(t) A(s): Number of arrivals in (s, t]

8 Poisson process A(t)is Poisson process with rate λ t > s and τ = t s A(t) A(s): Number of arrivals in (s, t] Definition Definition 1 A(t) A(s) is a Poisson R.V with parameter λτ (P1) Pr(A(t) A(s) = n) = e λτ (λτ) n n! n = 0, 1, 2, No. of arrivals in any 2 disjoint intervals are independent (P2) Pr(A(t) A(s) = n, A(t ) A(s ) = n ) = Pr(A(t) A(s) = n) Pr(A(t ) A(s ) = n )

9 Sanity Check A(t) is Poisson t 1 < t 2 < t 3 : 3 time instants B(t i t j ) denotes A(t i ) A(t j ) Pr(B(t 3 t 1 ) = 1) = e λ(t 3 t 1 ) λ(t 3 t 1 ) (P1) Alternately, { B(t 3 t 1 ) = 1 B(t 3 t 2 )=1 & B(t 2 t 1 )=0 E 1 B(t 3 t 2 )=0 & B(t 2 t 1 )=1 E 2 E 1 and E 2 disjoint Pr(B(t 3 t 1 ) = 1) = Pr(E 1 ) + Pr(E 2 )

10 Sanity check Contd.. Pr(E 1 ) = Pr(B(t 3 t 2 ) = 1, B(t 2 t 1 ) = 0) = Pr(B(t 3 t 2 ) = 1)Pr(B(t 2 t 1 ) = 0) ind. events, (P2) = e λ(t 3 t 2 ) λ(t 3 t 2 )e λ(t 2 t 1 ) Pr(E 2 ) = e λ(t 3 t 2 ) e λ(t 2 t 1 ) λ(t 2 t 1 ) Pr(E 1 ) + Pr(E 2 ) = e λ(t 3 t 1 ) λ(t 3 t 1 ) (P1), (P2) consistent for any n

11 Poisson: Alternate Viewpoint Not a counting process? t n = time of nth arrival A(t n ) = n, A(t) < n for all t < t n τ n = t n+1 t n be the nth interarrival time A(t) determined by sequence of R.Vs τ 1, τ 2, A(t) τ 1 τ 2 τ 3 t 1 t 2 t 3 t 4 time

12 Poisson Processes Definition Definition 2 A(t) is a Poisson process with rate λ if τ 1, τ 2, are an i.i.d sequence of exponential R.Vs with mean 1 λ

13 Exponential R.Vs τ n is exponential with mean 1 λ CDF: Pr(τ n s) = 1 e λs, s 0 0, s < 0 Claim Pr(τ n > s) = e λs PDF: f τn (s) = λe λs Var(τ n ) = 1 λ 2 Defn 1 and Defn 2 are equivalent Intuition: Consider zero arrivals in an interval [t n, t n + s) Pr(A(t n + s) A(t n ) = 0) = exp( λs) Pr(τ n > s) = exp( λs) (Exponential r.v) (Poisson p.m.f)

14 Memoryless Property For an exponential R.V τ n, Pr(τ n > t + s τ n > t) = Pr(τ n > s) Pr(τ n > t + s τ n > t) = Pr(τ n > t + s and τ n > t) Pr(τ n > t) = Pr(τ n > t + s) Pr(τ n > t) = e λs = e λ(t+s) e λt Exponentials only continuous R.V with this property

15 Example: Bus Arrivals Pr(Bus at No Bus in ) = 10λexp( 10λ) Pr(Bus at 11.10) = 10λexp( 10λ)

16 Other Useful properties A 1 (t), A 2 (t),, A k (t) are independent Poisson processes with rates λ 1, λ 2,, λ k. A TOT (t) counts total number of arrivals from all processes A TOT (t) = A 1 (t) + A 2 (t) + + A k (t) Eg: In network, several input links with Poisson streams combine into one output 1. Adding Independent Poisson processes A TOT (t) is also Poisson with rate λ 1 + λ λ k

17 2. Splitting Randomly split process A(t), Poisson process, rate λ Poisson, λ p Poisson, λ(1 p) Splitting must be independent of arrival times

18 A 1 (t), A 2 (t),, A N (t) are independent counting processes ith process has i.i.d interarrival times with mean 1 µ i and variance σi Sum of means: N i=1 1 µ i = 1 λ 2. Sum of variances is finite: N i=1 σ2 i < M (constant) 3. Limiting Property As N N A i (t) Poisson i=1 A combination of large number of independent arrivals streams can be modeled as Poisson

19 Suitability to Network Traffic Lots of independent streams in internet Limiting property - Poisson models reasonable model for network traffic Caveat: Sum of variances need to be finite Finite variance might not be true

20 Service Statistics Service time of a packet is Packet size/link Rate = L C Assume variable length packets Service time is exponential with parameter µ Pr(X n x) = 1 e µs E(X n ) = 1 µ = E(Ln) C = X Assume interarrival times and service times are independent M/M/1 Queue FCFS single server system, infinite buffer with Poisson arrivals and exponential service times Wait till next class to learn more!!