Photo: US National Archives

Transcription

1 ESD.86. Markov Processes and their Application to Queueing II Richard C. Larson March 7, 2007 Photo: US National Archives

2 Outline Little s Law, one more time PASTA treat Markov Birth and Death Queueing Systems

3 Queueing System Arriving Customers SERVICE FACILITY Queue of Waiting Customers Departing Customers Figure by MIT OCW.

4 Cumulative # of Arrivals FCFS=First Come, First Served SJF=Shortest Job First FCFS SJF L(t) L SJF (t) What about LJF, Longest Job 1st? 0 t = time

5 System is General L = λw Our results apply to entire queue system, queue plus service facility But they could apply to queue only! S.F. L q = λw q Or to service facility only! L SF = λw SF = λ /μ 1/μ = mean service time

6 All of this means, You buy one, you get the other 3 for free! W = 1 μ +W q L = L q + L SF = L q + λ μ L = λw

7 Markov Queues Markov here means, No Memory

8 Source: Larson and Odoni, Urban Operations Research

9 Balance of Flow Equations λ P = μ (λ + μ )P = λ P + μ P for n =1,2,3,... n n n n 1 n 1 n +1 n +1 Another way to balance the flow: Source: Larson and Odoni, Urban Operations Research λ n P n = μ n +1 P n +1 n = 0,1,2,...

10 λ 0 = μ 1 P 1 P 1 = μ 2 P 2... λ n P n = μ n +1 P n +1 P 1 =(λ 0 /μ 1 ) P 2 =( /μ 2 )P 1 =(λ 0 /μ 1 )( /μ 2 ) =(λ 0 /[μ 1 μ 2 ]) P n +1 = (λ n /μ n +1 )P n = (λ 0...λ n /[μ 1 μ 2...μ n +1 ]) Telescoping! Source: Larson and Odoni, Urban Operations Research λ n P n = μ n +1 P n +1 n = 0,1,2,...

11 λ 0 = μ 1 P 1 P 1 = μ 2 P 2... λ n P n = μ n +1 P n +1 P 1 =(λ 0 /μ 1 ) P 2 =( /μ 2 )P 1 =(λ 0 /μ 1 )( /μ 2 ) =(λ 0 /[μ 1 μ 2 ]) P n +1 = (λ n /μ n +1 )P n = (λ 0...λ n /[μ 1 μ 2...μ n +1 ]) Telescoping! + P 1 + P = P P n =1 n= 0 +(λ 0 /μ 1 ) + (λ 0 /[μ 1 μ 2 ]) (λ 0...λ n /[μ 1 μ 2...μ n +1 ]) +... =1 {1+(λ 0 /μ 1 ) + (λ 0 /[μ 1 μ 2 ]) (λ 0...λ n /[μ 1 μ 2...μ n +1 ]) +...} =1 Now, you easily solve for and then for All other P n s.

12 PASTA: Poisson Arrivals See Time Averages

13 Time to Buckle your Seatbelts!

14 The M/M/1 Queue Source: Larson and Odoni, Urban Operations Research +(λ 0 /μ 1 ) + (λ 0 /[μ 1 μ 2 ]) (λ 0...λ n /[μ 1 μ 2...μ n +1 ]) +...=1 {1+(λ 0 /μ 1 ) + (λ 0 /[μ 1 μ 2 ]) (λ 0...λ n /[μ 1 μ 2...μ n +1 ]) +...} =1 {1 +(λ /μ) + (λ 2 /μ 2 ) (λ n +1 /μ n +1 ) +...} =1 {1+(λ /μ) + (λ 2 /μ 2 ) (λ n +1 /μ n +1 ) +...} =1/[1 (λ /μ)] For λ/μ < 1.

15 The M/M/1 Queue =1 λ /μ for λ /μ <1. Source: Larson and Odoni, Urban Operations Research P n = (λ /μ) n =(λ /μ) n (1 λ /μ) for n =1,2,3,... {1 +(λ /μ) + (λ 2 /μ 2 ) (λ n +1 /μ n +1 ) +...} =1 {1+(λ /μ) + (λ 2 /μ 2 ) (λ n +1 /μ n +1 ) +...} =1/[1 (λ /μ)] For λ/μ < 1.

16 P T (z) d dz PT (z) The M/M/1 Queue P n z n = n= 0 n= 0 z=1 n= 0 np n =1 λ /μ for λ /μ <1. (λ /μ) n (1 λ /μ) z n = 1 ρ 1 ρz = L = (1 ρ)( ρ) (1 ρz) 2 = ρ 1 ρ for ρ <1 P n = (λ /μ) n =(λ /μ) n (1 λ /μ) for n =1,2,3,... L = λw = ρ /(1 ρ) implies W = (1/ λ)ρ /(1 ρ) = (1/μ)/(1 ρ) L q = λw q etc.

17 Mean Wait vs. Rho Note the Elbow! Series Rho

18 More on M/M/1 Queue Let w(t) = pdf for time in the system (including queue and service) Assume First-Come, First-Served (FCFS) Queue Discipline w(t) = w(t k)p = k k= 0 w(t) = μe μt (1 ρ) μ k +1 t k e μt ρ k (1 ρ) k! k= 0 Exercise: Do the same for Time in queue k= 0 (μtρ) k k! w(t) = μ(1 ρ)e μ(1 ρ )t t 0 = μ(1 ρ)e μt e μρt

19 Blackboard Modeling 3 server zero line capacity 3 server capacity for 4 in queue Same as above, but 50% of queuers balk due to having to wait in queue Single server who slows down to half service rate when nobody is in queue More??.

20 About the cut between states to write the balance of flow equations 1,0 1,1 0,0 0,1

21 Optional Exercise: Is it better to enter a single server queue with service rate μ or a 2-server queue each with rate μ /2? Can someone draw one or both of the state-rate-transition diagrams? Then what do you do?

22 Final Example: Single Server, Discouraged Arrivals λ /2 λ /3 λ /4 λ /5 State-Rate-Transition Diagram, Discouraged Arrivals P k = 1 k! (λ μ )k = [1 + ( λ μ ) + 1 2! (λ μ ) ! (λ μ ) k! (λ μ )k +...] 1 = (e λ / μ ) 1 = e λ / μ

23 = (e λ / μ ) 1 = e λ / μ > 0 ρ = utilization factor =1 =1 e λ / μ <1. P k = (λ /μ)k k! e λ / μ, k = 0,1,2,... Poisson Distribution! L = time - average number in system = λ/μ How? L = λ A W Little's Law, where λ A average rate of accepted arrivals into system

24 Apply Little s Law to Service Facility ρ = λ (average service time) A ρ = average number in service facility = λ /μ A λ = μρ = μ(1 e λ / μ ) A W = L λ A = λ /μ μ(1 e λ / μ ) = λ μ 2 (1 e λ / μ )