1 / 39

Chapter 4

Chapter 4. λ. 2λ. 0. 1. 2. μ. 2μ. Non-nearest neighbor MC’s. Example. μ. Erlang Distribution. r = 1. 2μ. 2μ. Erlang Distribution. r = 2. rμ. rμ. rμ. 1. 2. r. r-stage Erlang Distribution. b(x). r = 1. 2. x. b(x). r = ∞. r = 1. μe -μ. 2. x. λ. departures. rμ.

gareth-giles
Download Presentation

Chapter 4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 4

  2. λ 2λ 0 1 2 μ 2μ Non-nearest neighbor MC’s • Example

  3. μ Erlang Distribution r = 1

  4. 2μ 2μ Erlang Distribution r = 2

  5. rμ rμ rμ 1 2 r r-stage Erlang Distribution

  6. b(x) r = 1 2 x b(x) r = ∞ r = 1 μe-μ 2 x

  7. λ departures rμ rμ rμ 1 2 r Queue λ λ λ 0, 0 1, 1 2, 1 3, 1 γμ γμ λ 1, 2 2, 2 γμ γμ γμ γμ γμ λ 1, r 2, r 3, r M/Er/1 • (k, i) stage of service customer is in # in system

  8. k-1, i γμ k-1, i λ k-1, i λ λ λ k-1, i λ λ λ λ λ 0 1 2 r-1 r r+1 r+2 γμ k-1, i rμ rμ rμ rμ rμ rμ rμ rμ λ λ λ λ λ λ j-r j-1 j j+1 j+r rμ rμ rμ rμ rμ rμ rμ rμ

  9. departures μ rλ rλ rλ 1 2 r Queue Arrival box rλ rλ rλ rλ rλ rλ rλ 0 1 2 r-1 r r+1 rλ rλ rλ rλ rλ rλ rλ rλ μ μ μ μ μ μ j-r j-1 j j+1 j+r μ μ μ μ μ μ μ μ Er/M/1 二維 or 一維: j = stages of arrival completed by all customers in system plus 〃 all customers in arrival box

  10. (r+1) roots 中有一個根 at Z=1 Im Z=1 Re

  11. λ λ λ λ λ λ λ 0 1 2 r-1 r r+1 r+2 μ μ μ μ μ μ μ μ Bulk Arrival System • M/M/1 Bulk Arrival • Bulk size = r • 與 M/Er/1 比較, 把 M/Er/1 中的 • rμ 改成 μ • ρ 改成

  12. λ λ λ λ λ λ λ 0 1 2 r-1 r r+1 μ μ μ μ μ μ 註: Bulk Service • M/M/1 Bulk Service • Bulk size = r • 與 Er/M/1 比較, 把 Er/M/1 中的 • rλ 改成 λ • ρ 改成

  13. rgk rgk-1 rg2 rg2 rgk-2 rg1 rg1 rg1 rg1 0 1 2 k-2 k-1 k k+1 k+1 μ μ μ μ μ μ μ μ μ Bulk Arrival System • M/M/1 Bulk Arrival • Bulk size = random • gk = P[Bulk size = k] • 與 M/M/1 Bulk Arrival, bulk size = k 比較

  14. λ λ λ λ λ λ λ 0 1 2 r-1 r r+1 μ μ μ μ μ μ Bulk Service System • M/M/1 Bulk Service • Bulk size = r • 與 Er/M/1 比較, 把 Er/M/1 中的 • rλ 改成 λ • ρ 改成

  15. λ λ λ λ λ λ λ 0 1 2 r-1 r r+1 μ μ μ μ μ μ μ μ • Suppose less than r customers can be served immediately • (no need to wait until full bulk = r)

  16. b(x) r = 1 rμ μr rμ μ1 rμ μ2 2 1 1 2 2 r r x r-stage Erlang dist (Er)

  17. μ1 α1 μ2 α2 αR μR 2 2 Hyperexponential Dist (HR)

  18. r1μ1 r1μ1 r1μ1 r1 α1 1 2 αk rkμk rkμk rkμk rk αR 1 2 rRμR rRμR rRμR service facilityone customer at one time rR 1 2 Series-Parallel Stage-Type device

  19. β1 βr β2 μ1 μr μ2 1-β1 1-βr 1-β2 Coxian Stage-Type Device

  20. Poisson ? Poisson Networks of Queues • Paul Burke(1954) • Burke’s Theorem: • The only(FCFS) queuing systems which give Poisson out for Poisson in is M/M/1/n

  21. Cn-1 Cn Cn+1 Cn+2 λ Xn+1 departures Xn+2 Xn idle μ Server Queue Cn Cn+1 Cn+2 Wn Wn+1 Wn+2=0 Queue Cn Cn+1 Cn+2 • M/M/1:

  22. Case 1: Case 2:

  23. pλ p pλ λ λ λ 2 (1-p)λ 1-p (1-p)λ 1 4 3 Feed forward ??M/M/mi

  24. j i rij • J.R.Jackson(1957) • = External arrival rate to node i (Poisson) • mi = Number of parallel servers in node i (Exponential) with mean service (1/μi) sec. • rij = P[node j next after node i] • P[leave network after service in node i] = • λi = Total traffic handled by node i (sum of external + internal arrivals)

  25. p λ μ Poisson Poisson λ 1-p NOT Poisson

  26. (1963)force the system to always have K customers

  27. j i rij • Gordon and Newell(1967) • mi = # servers in node i (Exponential) with mean service (1/μi) sec. • rij = P[node j next after node i].

  28. Bottlenecknode Closed Queuing Networks

  29. terminal A K A B A B 1 1 0 0 1 SWAPPINGDEVICE 1 CPU B 不同 class of job have different probability to go

More Related