Richard C. Larson. March 5, Photo courtesy of Johnathan Boeke.

Transcription

1 ESD.86 Markov Processes and their Application to Queueing Richard C. Larson March 5, 2007 Photo courtesy of Johnathan Boeke.

2 Outline Spatial Poisson Processes, one more time Introduction to Queueing Systems Little s Law Markov Processes

3 Spatial Poisson Processes Courtesy of Andy Long. Used with permission.

4 Spatial Poisson Processes Entities distributed in space (Examples?) Follow postulates of the (time) Poisson process λdt = Probability of a Poisson event in dt History not relevant What happens in disjoint time intervals is independent, one from the other The probability of a two or more Possion events in dt is second order in dt and can be ignored Let s fill in the spatial analogue..

5 Set S has area A(S). Poisson intensity is γ Poisson entities/(unit area). X(S) is a random variable X(S) = number of Poisson entities in S S P{X(S) = k} = (γa(s))k k! e γa(s ), k = 0,1,2,...

6 Nearest Neighbors: Euclidean Define D 2 = distance from a random point to nearest Poisson entity Want to derive f D 2 (r). Happiness: F D2 (r) P{D 2 r} =1 P{D 2 > r} F D2 (r) =1 Prob{no Poisson entities in circle of radius r} F D2 (r) =1 e γπr 2 r 0 r f D2 (r) = d dr F D 2 (r) = 2rγπe γπr 2 r 0 Rayleigh pdf with parameter 2γπ Random Point

7 Nearest Neighbors: Euclidean Define D 2 = distance from a random point to nearest Poisson entity Want to derive f D 2 (r). E[D 2 ] = (1/2) 2 σ D2 1 γ 1 = (2 π /2) 2πγ "Square Root Law" r f D2 (r) = d dr F D 2 (r) = 2rγπe γπr 2 r 0 Rayleigh pdf with parameter 2γπ Random Point

8 Nearest Neighbor: Taxi Metric F D1 (r) P{D 1 r} F D1 (r) =1 Pr{no Poisson entities in diamond} 2 γ 2r F D1 (r) =1 e f D1 (r) = d dr F D 1 (r) = 4rγe 2γr 2 r

9 How Might you Derive the PDF fo the k th Nearest Neighbor? Blackboard exercise!

10 To Queue or Not to Queue, That May be a Question!

11 Queueing System Arriving Customers Service Facility Queue of Waiting Customers Departing Customers Figure by MIT OCW.

12 Finite or Infinite? Finite or Infinite? Servers: Statistical Clones? Queue Discipline: How queuers Are selected for service Source: Larson and Odoni, Urban Operations Research

13 What Kinds of Queues Occur in Systems of Interest to ESD? ESD Queues? Photos courtesy, from top left, clockwise: U.S. FAA: Flickr user *keng Luke Hoersten

14 Little s Law for Queues a(t) L(t) d(t) Source: Larson and Odoni, Urban Operations Research a(t) = cumulative # arrivals to system in (0,t] d(t) = cumulative # departures from system in (0,t] L(t) = a(t) d(t) L(t) = number of customers in the system (in queue and in service) at time t

15 Little s Law for Queues γ(t) = t [a(τ) d(τ)]dτ = L(τ)dτ 0 t 0 γ(t) = total number of customer minutes spent in the system a(t) = cumulative # arrivals to system in (0,t] d(t) = cumulative # departures from system in (0,t] L(t) = a(t) d(t) L(t) = number of customers in the system (in queue and in service) at time t

16 Let s Get an expression for Each of 3 Quantities λ t average customer arrival rate = a(t)/t W t average time that an arrived customer has spent in the system W t = γ(t)/a(t) L t = time average # customers in system during (0,t] L t = 1 t t L(τ)dτ = γ(t)/t 0 L = γ(t) = a(t) γ(t) t t t a(t) = λ t W t In the limit, L = λw, Little's Law

17 Key Issues L = λw L in a time-average. Explain λ is average of arrival rate of customers who actually enter the system W is average time in system (in queue and in service) for actual customers who enter the system

18 More Issues L = λw Little s Law is general. It does not depend on Arrival process Service process # servers Queue discipline Renewal assumptions, etc. It just requires that the 3 limits exist.

19 Still More Issues L = λw What about balking? Reneging? Finite capacity? Do we need iid service times? Iid interarrival times? Do we need each busy period to behave statistically identically? Look at role of γ(t). Can change queue statistics by changing queue discipline.

20 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

21 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

22 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

23 Utilization Factor ρ ={ 1 if server is busy Single Server. Set Y 0 if server is idle E[Y] =1*P{server is busy} + 0*P{server is idle} E[Y] =1*ρ + 0 = ρ = E[# customers in SF] =? E[Y] is time-average number of customers in the SF Buy Little s Law, ρ = λ /μ <1

24 Utilization Factor ρ Similar logic for N identical parallel servers gives ρ = ( λ N ) 1 μ = λ Nμ <1 Here, λ/μ corresponds to the timeaverage number of servers busy

25 Markov Queues Markov here means, No Memory

26 Source: Larson and Odoni, Urban Operations Research

27 Balance of Flow Equations λ 0 P 0 = μ 1 P 1 (λ n + μ n )P n = λ n 1 P n 1 + μ n +1 P n +1 for n =1,2,3,... To be continued..