1. 1 Chapter 4

2. 2 Non-nearest neighbor MC’s  Example 012 λ2λ μ 2μ

3. 3 Erlang Distribution μ r = 1

4. 4 Erlang Distribution r = 2 2μ

5. 5 r-stage Erlang Distribution rμ 12r

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

7. 7 M/Er/1  (k, i) λdepartures Queue 12r rμ # in system stage of service customer is in 0, 0 1, 1 1, 2 1, r 2, 1 2, 2 2, r λ λ γμ 3, 1 λ λ λ 3, r γμ

8. 8 k-1, i γμ λλ j-r rμ j-1jj+1 λ rμ λλλ j+r rμ λλ 012 r-1rr+1 λ rμ λλ r+2 λλλλ rμ

9. 9

10. 10

11. 11

12. 12 Er/M/1 二維 or 一維 : j = stages of arrival completed by all customers in system plus 〃 all customers in arrival box 012 r-1rr+1 rλ μμμμμμ j-r j-1jj+1 rλ μμμμ j+r rλ μμμ μ departures Queue μ 12r rλ Arrival box

13. 13

14. 14 (r+1) roots 中有一個根 at Z=1 ImIm ReRe Z=1

15. 15

16. 16 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 012 μ μ μ r-1rr+1 λ μ λλ μ μ μ r+2 λλλλ μ

17. 17 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 012 r-1rr+1 λλλλλλ λ μμμμμμ

18. 18 Bulk Arrival System  M/M/1 Bulk Arrival Bulk size = random g k = P[Bulk size = k] 與 M/M/1 Bulk Arrival, bulk size = k 比較  012 μ μ μ k-2k-1k rg k-1 μ μ μ μ k+1 μ rg k-2 rg 2 rg 1 rg k k+1 μ rg 1 rg 2

19. 19

20. 20

21. 21 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 012 r-1rr+1 λλλλλλ λ μμμμμμ

22. 22  Suppose less than r customers can be served immediately (no need to wait until full bulk = r) 012 r-1rr+1 λλλλλλ λ μμμμμμ μ μ

23. 23

24. 24

25. 25 r-stage Erlang dist (Er) r = 1 2 b(x) x rμ 12r μ2μ2 μ1μ1 μrμr 12r

26. 26 Hyperexponential Dist (H R) μ2μ2 μ1μ1 μRμR α1α1 α2α2 αRαR 2 2

27. 27

28. 28 Series-Parallel Stage-Type device α1α1 αkαk αRαR r1μ1r1μ1 r1μ1r1μ1 r1μ1r1μ1 12 r1r1 rkμkrkμk rkμkrkμk rkμkrkμk 12 rkrk rRμRrRμR rRμRrRμR rRμRrRμR 12 rRrR service facility one customer at one time

29. 29 Coxian Stage-Type Device β1β1 1-β 1 μ1μ1 β2β2 1-β 2 μ2μ2 βrβr 1-β r μrμr

30. 30 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 Poisson ?

31. 31  M/M/1: λ μ departures Queue Server Queue CnCn C n+1 C n+2 CnCn C n+1 C n+2 C n-1 CnCn C n+1 C n+2 WnWn W n+1 W n+2 =0 XnXn X n+1 X n+2 idle

32. 32 Case 1: Case 2:

33. 33 Feed forward λ p 1-p pλ (1-p)λ 1 2 3 λλ pλ (1-p)λ 4 ??M/M/m i

34. 34  J.R.Jackson(1957) = External arrival rate to node i (Poisson) m i = Number of parallel servers in node i (Exponential) with mean service (1/μ i ) sec. r ij = P[node j next after node i] P[leave network after service in node i] = λ i = Total traffic handled by node i (sum of external + internal arrivals) r ij i j

35. 35 μ Poisson1-p p λ λ NOT Poisson Poisson

36. 36 (1963) force the system to always have K customers

37. 37  Gordon and Newell(1967) m i = # servers in node i (Exponential) with mean service (1/μ i ) sec. r ij = P[node j next after node i]. r ij i j

38. 38 Closed Queuing Networks Bottleneck node

39. 39 K SWAPPING DEVICE CPU A B A 0 B 1 A 1 B 0 1 1 不同 class of job have different probability to go terminal