Základy teorie front II

Transcription

1 Základy teorie front II Aplikace Poissonova procesu v teorii front Mgr. Rudolf B. Blažek, Ph.D. prof. RNDr. Roman Kotecký, DrSc. Katedra počítačových systémů Katedra teoretické informatiky Fakulta informačních technologií České vysoké učení technické v Praze Rudolf Blažek & Roman Kotecký, 2011 Statistika pro informatiku MI-SPI, ZS 2011/12, Přednáška 17 Evropský sociální fond Praha & EU: Investujeme do vaší budoucnos@

2 Introduction to Queueing Theory II The Poisson Process in Queueing Theory Mgr. Rudolf B. Blažek, Ph.D. prof. RNDr. Roman Kotecký, DrSc. Department of Computer Systems Department of Theoretical Informatics Faculty of Information Technologies Czech Technical University in Prague Rudolf Blažek & Roman Kotecký, 2011 Statistics for Informatics MI-SPI, ZS 2011/12, Lecture 17 The European Social Fund Prague & EU: We Invest in Your Future

3 Queueing Theory Review Queueing System Diagram Fronta Waiting Queue Obsluha Service Facility Customer Population Arriving Customers Vstupní tok požadavků Departing Customers Výstupní tok požadavků Prvky systému hromadné obsluhy 3

4 Queueing Theory Review Kendall Notation A / B / X / Y / Z A:( Customer arrival pattern ( ( (Interarrival time distribution) B:( Service pattern (Service time distribution) X:( Number of parallel servers Y: ( System capacity Z:( Queueing discipline Default values:( Y =, Z = FCFS Example: ( M / M / 3 = M / M / 3 / / FCFS ( ( ( ( (Poisson arrivals, Exp. service times, 3 servers) 4

5 Queueing Theory Review Characteristics of the Input Process Arriving patterns M: ( Markovian or Memoryless Poisson Process ( ( (I.e. exponential & independent interarrival times) D:( Ek:( G:( GI:( Deterministic, constant interarrival times Erlang distribution of order k of interarrival times General probability distribution of interarrival times General & Independent distribution of interarrival times Default Assumption: Poisson Process Charakteristiky vstupního toku požadavků 5

6 Queueing Theory Review Characteristics of the Output Process Service time distribution M: ( Markovian or Memoryless exponential service times D:( Deterministic, constant service times Ek:( Erlang distribution of order k of service times G:( General probability distribution of service times Default Assumption: Exponential service times Charakteristiky výstupního toku požadavků 6

7 Queueing Theory Review Queueing System M/M/m Infinite FCFS Queue m Parallel Servers w/ Exp(μ) service time Infinite Customer Population Arriving Customers Poisson Process w/ rate λ Departing Customers 7

8 Queueing Theory Review Web and Database Servers Example Pool of m application servers (e.g. Tomcat) submits a job to a central database server Application server 1 Application server 2 Central Database Server Application server m 8

9 Queueing Theory Review Web and Database Servers Example We assume Poisson arrival process (λ) for the requests. Scenario 1: Application servers can submit multiple requests We have m application servers Then we obtain a Poisson arrival process with the rate ( ( ( ( ( ( ( ( ( μ = m λ We will see later why... 9

10 Queueing Theory Review Web and Database Servers Example Case 1: Application servers can submit multiple requests Application server 1 Rate λ Application server 2 Rate λ Rate mλ Central Database Server Application server m Rate λ 10

11 Queueing Theory Review Web and Database Servers Example We assume Poisson arrival process (λ) for the requests. Scenario 2: Application servers must wait for their request to finish State k: If k servers are waiting for their requests to finish, then only (m-k) servers can submit requests Then we obtain a state-dependent Poisson arrival process with the rate (k) = (m k) k < m 0 k m 11

12 Queueing Theory Review Plan of Study We will focus on M/M/m systems We must therefore study The Exponential Distribution (interarrival & service times) The Poisson Process (interarrival times are Exponential) Birth & Death Markov chains with continuous time (the number of customers in the system) We will also look at a M/G/ system Poisson arrivals, General service time, many servers 12

13 The Poisson Process 13

14 Defining the Poisson Process by Number of arrivals N(t) = number of arrivals during (0,t) N(t)... Poisson Process with rate λ t3 ~ Exp(λ) t4 ~ Exp(λ) t1 ~ Exp(λ) t2 ~ Exp(λ) t5 ~ Exp(λ) t5 > t -T4 N(t) = 4 0 T1 T2 T3 T4 t Time t Tk(...( arrival time of the k th customer tk ~ Exp(λ)(...( independent exponential interarrival times 14

15 Exponential Distribution Definition!!!!!!!!!!!!!! T ~ Exp(λ) A random variable T has an exponential distribution with rate λ if its density is e t for t 0 f T (t) = 0 for t < 0. We can also write the definition in terms of the distribution function F T (t) =P(T apple t) =1 e t, 8t 0 or in terms of the survival function P(T > t) =e t, 8t 0. 15

16 Main Properties of the Exponential Distribution Properties of T ~ Exp(λ) If T ~ Exp(λ) then ET =1/ ET 2 =2/ 2 Var T = ET 2 (ET ) 2 =1/ 2 Example: Average arrival rate: λ = 10 arrivals per minute Average wait time between arrivals: 1/10 = 0.1 minutes 16

17 Exponential Distribution Lack of Memory Property Lemma Assume waiting time T ~ Exp(λ). Given that we already waited s units of time, the remaining waiting time T s has the same distribution as if we did not wait at all, i.e. Exp(λ). Abbreviated notation: T ~ Exp(λ) (T s T>s) ~ Exp(λ) We waited s, but no arrival Conditional remaining waiting time We say: Exponential distribution is Memoryless 17

18 Exponential Distribution Lack of Memory Property Lack of memory: Conditional remaining waiting time: (T-s T > s) ~ Exp(λ) Remaining waiting time: T-s Original waiting time: T ~ Exp(λ) 0 s Time t Fix time s and observe that there was no arrival... we know T > s 18

19 Lack of Memory & the Poisson Process Number of arrivals N(t) = number of arrivals during (0,t) N(t)... Poisson Process with rate λ t3 ~ Exp(λ) t4 ~ Exp(λ) t1 ~ Exp(λ) t2 ~ Exp(λ) t5 ~ Exp(λ) t5 > t -T4 N(t) = 4 0 T1 T2 T3 T4 Different notation: Tk( ( ( (...( arrival time of the k th customer t Time t tk ~ Exp(λ)(...( independent exponential interarrival times 19

20 Lack of Memory & the Poisson Process Number of arrivals Conditional remaining waiting time: (t5-s t5 > s)~ Exp(λ) Original waiting time: t5 ~ Exp(λ) Given: t5 > t -T4 Given: t5 > s s = t -T4 0 T4 T1 T2 T3 t Time t Tk( ( ( (...( arrival time of the k th customer tk ~ Exp(λ)(...( independent exponential interarrival times 20

21 Proof of the Lack of Memory Property We want to prove: T ~ Exp(λ) (T s T>s) ~ Exp(λ) It s easier to use the survival function: T Exp( ) if and only if P(T > t) =e t, 8t 0. P(T s > t T > s) 8s, t 0 = = P(T > t + s, T > s) P(T > s) P(T > t + s) = e (t+s) P(T > s) e s = e t... Exp( ). 21

22 Exponential Races Lemma Let S ~ Exp(λ) and T ~ Exp(μ) be independent. Then min(s, T ) Exp( + µ). Proof: P(min(S, T ) > t) =P(S > t, T > t) =P(S > t)p(t > t) = e t e µt = e ( +µ)t... Exp( + µ). 22

23 Exponential Races & Queueing Systems Line 1 Independent waiting times Waiting time for arrival S ~ Exp(λ) Waiting time for arrival V = min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 T ~ Exp(μ) Waiting time for arrival 23

24 Exponential Races & Queueing Systems E.g. average rate of arrivals λ = 3/min Line 1 Waiting S ~ Exp(λ) Independent waiting times Line 2 Waiting T ~ Exp(μ) E.g. average rate of arrivals μ = 5/min Waiting min(s, T ) ~ Exp(λ+μ) Then the average rate of arrivals is λ + μ = 3+5 = 8 / min 24

25 Exponential Races & Queueing Systems Line 1 Waiting S ~ Exp(λ) Independent waiting times Waiting min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 Waiting T ~ Exp(μ) OK, the first arrival has exponential waiting time But how about the next arrivals? Is it a Poisson Process? Are all interarrival times exponential? Are all interarrival times independent? 25

26 Exponential Races & Lack of Memory Waiting for arrival on line 1 or 2 Line 1 s1 ~ Exp(λ) s2 ~ Exp(λ)... independent Line 2 Merged Line t1 ~ Exp(μ) v1 =min(s1, t1) v1 ~ Exp(λ+ μ) 0 v t 1 = (t1-v t1 > v) ~ Exp(μ)... independent v2 = min(s2, t 1) ~ Exp(λ+ μ) Time t Independent interarrival times ~ Exp(λ+ μ) Poisson Process (λ+ μ) Arrival observed at time v. Assume it was from line 1. 26

27 Example Web and Database Servers Example Case 1: Application servers can submit multiple requests Application server 1 Rate λ Application server 2 Rate λ Rate mλ Central Database Server Application server m Rate λ If the Poisson Processes are independent then the we obtain a Poisson Process on the merged line 27

28 The Winner of an Exponential Race Lemma Let S ~ Exp(λ) and T ~ Exp(μ) be independent. Then the probability that S arrives first is P(S < T )= + µ P(T < S) = µ + µ 28

29 The Winner of an Exponential Race & Queueing Systems Line 1 Waiting S ~ Exp(λ) Independent waiting times Waiting min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 Waiting T ~ Exp(μ) An arrival is observed on the merged line. Where from? µ P(From Line 1) = + µ P(From Line 2) = + µ 29

30 The Winner of an Exponential Race & Queueing Systems Line 1 S ~ Exp(λ), λ = 3/min Independent waiting times λ + μ = 3+5 = 8 / min min(s, T ) ~ Exp(λ+μ) Merged Line Line 2 T ~ Exp(μ), μ = 5/min An arrival is observed on the merged line. Where from? µ P(Line 1) = + µ = 3 8 P(Line 2) = + µ =

31 Exponential Races for n Variables Corollary Let Ti ~ Exp(λi) be independent, i = 1, 2,...,n. Then min(t 1,..., T n ) Exp( n ). Proof is very similar as for two variables 31

32 The Winner of an Exponential Race Corollary Let Ti ~ Exp(λi) be independent, i = 1, 2,...,n. Then the probability that Tk arrives first is P(T k = min(t 1,..., T n )) = k n 32

33 The Poisson Process Definition!!!!!! Poissonův Process s intenzitou λ Let ti ~ Exp(λ) be independent random variables, i = 1, 2,... Let Tn = t1 + t tn with T0 = 0, and define N(s) = max {n: Tn s} for all s 0. Then N(s) is called the Poisson Process with rate λ. ti ~ Exp(λ)(...( independent exponential interarrival times Tn( ( (...( arrival time of the n th customer N(s)( (...( number of arrivals during time interval (0,s) 33

34 Definition and Basic Properties Defining the Poisson Process by Number of arrivals N(s) = number of arrivals during (0,s) N(s)... Poisson Process with rate λ t1 ~ Exp(λ) t2 ~ Exp(λ) t3 ~ Exp(λ) t4 ~ Exp(λ) t5 ~ Exp(λ) t5 > s -T4 N(s) = max {n: Tn s} N(s) = max {1,2,3,4} N(s) = 4 0 T2 T4 T1 =t1 T3 =t1+t2+t3 (T2 =t1+t2) (T4 =t1+t2+t3+t4) s Time t ti ~ Exp(λ)(...( independent exponential interarrival times Tn( ( ( (...( arrival time of the n th customer 34

35 Poisson Processes Definition and Basic Properties The Poisson Distribution Lemma N(s) has a Poisson distribution with mean λs. Definition!!!!!!!!!!!! X ~ Poisson(μ) A random variable X has a Poisson distribution with mean μ if P(X = n) =e µ µn for n = 0, 1,... n! 35