Multi Heterogeneous Queueing Server System. General Exam Oral Examination Fall 2012 prepared by Husnu Saner Narman

Transcription

1 Multi Heterogeneous Queueing Server System General Exam Oral Examination Fall 2012 prepared by Husnu Saner Narman

2 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 2

3 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 3

4 Queueing System Service rate λ μ Arrival rate 4

5 Queueing System Problems 3) If queue is finite, what is drop rate? 1) How much time does a customer wait? 1) Average waiting time 2) Average queue length 3) Drop rate 2) How many customers are In queue? served 5

6 Heterogeneous Multi Server Queueing System μ=1 μ=2 1) Average waiting time 2) Average queue length 3) Drop rate μ=3 served 6

7 Heterogeneous Multi Server Allocation Policies 1) SSF μ=1 μ=2 2) FSF 1) SSF = Slowest Server First 2) FSF = Fastest Server First 3) RCS = Randomly Chosen Server μ=3 7

8 Different class costumer Heterogeneous Multi Server Multi Class μ=1 μ=2 μ=3 1) Priority = Which type of customer is served first? 2) Flexibility = Which type of customer will be served by which server? 8

9 Heterogeneous Multi Server Queueing System Performance metrics: Average waiting time Average queue length Drop rate (for finite queue) Allocation Policies FSF, SSF, RCS... Priority in Multi Class Multi Server System Flexibility in Multi Class Multi Server System 9

10 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 10

11 Contribution of Works Upper and Lower bound average waiting time for Multi Heterogeneous Single Queue System Upper and Lower bound average queue length for Multi Heterogeneous Single Queue System Lower bound queue drop rate for Multi Heterogeneous Single Finite Queue System Developed performance approximations are better than homogeneous approximation 11

12 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 12

13 Multi Heterogeneous System First Model (M/Mi/c) μ 1 λ... μ 2 μ c 13

14 Assumption for First Model μ 1 μ 2... μ c Poisson distribution with rate λ, Exponential distribution with rate μ i, c number of servers, Slowest Server First (SSF) allocation Because of SSF, upper bound performance metrics 14

15 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 15

16 Notations λ μ i p i u E n u E T μ ti ρ Job arrival rate, Service rate of i th server, State probability of i th state, Upper bound average queue length of the system, Upper bound average waiting time of the system, Total service rates until i th server, or Utilization of the system, or ρ=λ /μ tc c μ ti = i=1 μ i 16

17 Probability of States λ p 0 =μ t1 p 1 p 1 = p 0 λ μt1 λ p 1 =μ t2 p 2 p 2 = p 0 λ 2 μ t1 μ t2 λ p c 1 =μ tc p c p c = p 0 λ c μ t1 μ t2...μ tc λ i λ p i 1 =μ tc p i p i = p 0 μ t1 μ t2...μ tc μ tc i=0 p i =1 and ρ<1 i c = i c μ tc λ i c j=1 μ tj where i>c 17

18 Probability of States p 0 = c 1 1+ ( j=1 1 λ j j μ ti) + ( λ c c (1 ρ) i=1 i=1 μ ti) ρ= λ μ tc <1 where ρ=μ tc / λ 18

19 Average Queue Length and Waiting Time General formula of average queue length for single server E n = i=1 ip i Modified formula of average queue length for multi servers E u n = (i c) p i i=c+1 E u n = p 0 i=c+1 (i c)λ i p i = λ because i c μ tj μ tc i c μ tc c j=1 i c j=1 μ tj 19

20 Average Queue Length and Waiting Time Upper bound average queue length c μ tc = i=1 μ i E n u = p 0 c μ tc c i=1 μ ti ρ c+1 (1 ρ) 2 ρ= λ μ tc <1 From Little's Law Upper bound average waiting time E u T = E u n λ =p 0 Arrival rate c μ tc c i=1 μ ti ρ c+1 (1 ρ) 2 1 λ 20

21 Result for First Model Arrival rate Number of servers Heterogeneity level of server rates Gini Index: μ 1 =1,μ 2 =2,μ 3 =3 (0,0) (1/3,1/6) (2/3,3/6) (3/3,6/6) (0,0) (1/ 3,1/ 6) (2/3,1/2) (1,1) 21

22 Waiting Time for First Model Increase Better performance FSF heterogeneity allocation than homogeneous decreases better service rates performance 22

23 Error Comparison Error Rate Formula Which M/M/c or developed formula is better error= x simulation x calculation x simulation 23

24 Error Figures of Different Allocations for First Model Hom. Prop. approx. better with high low heterogeneity and servers 24

25 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 25

26 Multi Heterogeneous System Second Model (M/Mi/3/N) μ 1 N λ μ 2 μ 3 26

27 Assumption for Second Model μ 1 μ 2 μ 3 Poisson distribution with rate λ, Exponential distribution with rate μ i, 3 number of servers, Fastest Server First (FSF) allocation Because of FSF, lower bound performance metrics 27

28 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 28

29 Notations λ μ 1, μ 2, μ 3 π i, j, k π n,3 μ t Job arrival rate, Service rate of 1 st, 2 nd, and 3 rd servers, State probability of servers, idle:0, busy:1 State probability of n th state when servers are busy Total service rates Server state transition diagram Jobs state transition diagram 29

30 Probability of States λ π 0,3 +μ t π 0,3 =λ π 0,1,1 +λ π 1,0,1 +λ π 1,1,0 +μ t π 1,3 λ π n,3 =μ t π n+1,3 where 0 n< N π n,3 =π 0,3 ( μ λ ) n or π n,3 =π N,3 ( λ where 0 n N t μ ) n N t 30

31 Probability of States All derived formulas are invalid 31

32 Different Approach Hard to develop approximations after correction Similar methodology with first model Only Job states (no server states) Can be expendable n number of servers 32

33 Notations λ μ 1, μ 2, μ 3 p i E n l E T l P B l γ u ρ μ t1 =μ 1 Job arrival rate, Service rate of 1 st, 2 nd, and 3 rd servers, State probability of jobs in queue Lower bound average queue length of the system, Lower bound average waiting time of the system, Lower bound queue drop rate Upper bound throughput of queue Utilization of the system, or μ t2 =μ 1 +μ 2 μ t3 =μ 1 +μ 2 +μ 3 ρ=λ /μ t3 33

34 Probability of States λ p 0 =μ t1 p 1 p 1 = p 0 λ μt1 λ p 1 =μ t2 p 2 p 2 = p λ 2 0 μ t1 μ t2 λ p i 1 =μ t3 p i p i = p λ i 0 μ t1 μ t2 μ = p μ 2 t3 ρ i i 2 0 μ t1 μ t2 N +3 t3 p i =1 ={ and ρ>0 i= μ λ + λ 2 p t1 μ t1 μ + μ 2 t3 t2 μ t1 μ ( ρ3 ρ N +4 t2 1 ρ ) μ λ t1 + λ 2 μ t1 μ t2 + μ 2 t3 μ t1 μ t2 ( N +1) where 3 i N +3 ρ 1 ρ=1} ρ=λ/μ t3 34

35 Queue Drop Rate and Throughput Lower bound because of FSF allocation μ 2 P l t3 ρ N +3 B = p N +3 = p 0 μ t1 μ t2 p i = p 0 μ t3 ρ i μ t1 μ t2 Upper bound because of FSF allocation γ u =λ(1 P B l ) Arrival rate Not blocking probability 35

36 Average Queue Length and Waiting Time General formula of average queue length for single server N E n = i=1 ip i Modified formula of average queue length for multi servers N +3 E l n = (i 3) p i i=4 E n l = p 0 2 μ t3 μ t1 μ t2 i=4 N+3 (i 3)ρ i because p i = p 0 μ t3 ρ i μ t1 μ t2 36

37 Average Queue Length and Waiting Time Lower bound average queue length E n l ={p 0 2 μ t3 μ t1 μ ρ 4 ( 1 ( N +1)ρN + N ρ N +1 ) ρ 1 t2 (1 ρ) 2 2 μ p t3 N ( N +1) 0 μ t1 μ t2 ρ=1} 2 From Little's Law Lower bound average waiting time E l T = E l l n γ = E n u λ(1 P l B ) Throughput or effective arrival rate 37

38 Result for Second Model Arrival rates Average queue length Average queue delay Drop Probability Throughput Buffer Size 38

39 Queue length and Waiting time (Delay) Queue length comparison between analytical and simulation Waiting time comparison between analytical and simulation Good approximation 39

40 Drop rate and Throughput Drop rate comparison between analytical and simulation Throughput comparison between analytical and simulation Very good approximation 40

41 Drop Rate and Queue Length λ = 32, μ 1 = 1, μ 2 = 5, μ 3 = 20, N= 20, μ 1 = 1, μ 2 = 5, μ 3 = 20, Effect of buffer size to drop rate Effect of utilization to queue length After N=30, buffer size has no effect 41

42 Content Motivation Contribution Multi Heterogeneous System First Model Analysis of First Model Multi Heterogeneous System Second Model Analysis of Second Model Conclusion References 42

43 Conclusion Approximation formulas developed Queue length Waiting time Drop and Throughput for finite queue Verified by simulation Tested by different allocations, FSF, SSF, RCS 43

44 Future Work Different allocation can be used LUF : Low utilization first Allocation policies behaviors in in finite queue Developing Probabilistic allocation approximation Considering multi class system with different class and flexibility level 44

45 Questions 45

46 References [1] T. Heath, B. Diniz, E. V. Carrera, W. M. Jr., and R. Bianchini, Energy conservation in heterogeneous server clusters, in Principles and Practice of Parallel Programming, Chicago, Illinois, June 2005, pp [2] S. Gurumurthi and S. Benjaafar, Modeling and analysis of flexible queueing systems, Naval Research Logistics, vol. 51, pp , June [3] F. S. Q. Alves, H. C. Yehia, L. A. C. Pedrosa, F. R. B. Cruz, and L. Kerbache, Upper bounds on performance measures of heterogeneous M/M/c queues, Mathematical Problems in Engineering, vol. 2011, p. 18, May [4] C. Misra and P. K. Swain, Performance analysis of finite buffer queueing system with multiple heterogeneous servers, in 6 th international conference on Distributed Computing and Internet Technology, ser. ICDCIT 10, Bhubaneswar, India, Feb 2010, pp [5] G. Appenzeller, I. Keslassy, and N. McKeown, Sizing router buffers, Computer Communication Review, vol. 34, pp , Oct

47 Probability of States of First Model 47

48 Probability of States of Second Model if if 48