1. MAT 4830 Mathematical Modeling07
Servicing Requests II

2. HW
Individual HW

3. HW
HW 4 – well done
One of you earned bonus points for your next exam.

4. HW HW 5 – I hope your MCM paper is explained better than this HW.
Last night, I started this HW at 8pm and hoped that I would be home for dinner by 9:30pm.
Describing how the codes works is not the same as explaining the idea of the solutions.

5. Preview Look at Server-Customer Systems with Queues Maple:
The function nature of proc
Use of user-defined proc
Strings Handling

6. Server-Customer System with Queues
Suppose requests cannot be rejects
Requests will be assigned to the (first) server with the shortest wait
One queue, multi-server (Post Office)
We want to know the average waiting time of customers

7. Server-Customer System with Queues

8. Example 1:
Suppose we have 1 server to handle requests.

9. Idea Generate 𝑛 customers
If the server is busy, record the waiting time of the customer
Compute the average waiting time

10. Tool: Clocks System Clock t: Indicate the current time
Server Clock s: Indicate the completion time for the last customer being served

11. Tool: Clocks System Clock t: Indicate the current time
Server Clock s: Indicate the completion time for the last customer being served
if s<t, the server is idle

12. Tool: Clocks System Clock t: Indicate the current time
Server Clock s: Indicate the completion time for the last customer being served
if s≥t, the server is busy

13. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

14. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
First customer comes in at t=7
s<t? Idle?

15. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
The server will be busy until t=7+12=19

16. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
Next customer comes in 11 min. after the first customer.
s<t? Idle?

17. Idea n T t S s W 1 7 12 19 2 11 18 15 33 3 8 26 40 4 41 48 5 6 47 10 58 9 57 66
Update the Wait counter with W=0+1

18. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
The server will be busy until t=19+14=33

19. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
Next customer comes in 8 min. after the last customer.
s<t? Idle?

20. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
Update the Wait counter with W=1+(33-26)=8

21. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
The server will be busy until t=33+7=40

22. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
Next customer comes in 15 min. after the last customer.
s<t? Idle?

23. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66
The server will be busy until t=41+7=48

24. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

25. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

26. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

27. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

28. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

29. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

30. Idea n T t S s W 1 7 12 19 2 11 18 14 33 3 8 26 40 4 15 41 48 5 6 47 10 58 9 57 66

31. Example 2: Suppose we have 2 servers to handle requests
Requests will be assigned to the server with the shortest wait

32. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

33. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

34. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

35. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

36. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

37. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

38. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

39. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

40. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

41. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

42. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

43. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

44. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

45. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

46. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

47. Idea n T t S s1 s2 W 1 7 12 19 2 11 18 14 32 3 8 26 33 4 29 39 5 6 35 10 45 36 47

48. Example 3: Suppose we have 𝑚 servers to handle requests
Requests will be assigned to the server with the shortest wait

49. Example 3: 𝒎 Servers 𝑠[1], 𝑠[2],…, 𝑠[𝑚] are the server clocks
Find the (first) server with the minimum value
Find 𝑘 such that

50. Maple
Let us switch to Maple

51. The Function nature of proc
Maple returns the result of the last executable statement in a procedure.
We can also use return() to return the output.
return() can be placed anywhere in the program.
The output can be assigned to a variable
Useful when called within another procedure

52. Example 4a Write a program to compute the function value of
Of course, there are easier ways to do this. We only want to demo the point.
Example 4b below does not represent the best programming practices

53. Example 4a

54. Example 4b

55. Example 4c Write a function(procedure) to
Generate a normally generated random number 𝑥 with 𝑚𝑒𝑎𝑛=0 and 𝑠𝑡𝑑=1.
Compute the square root of the cut off of 𝑥.
Output this as the function value for the procedure.
Call the cut off function from example 4b.

56. Example 4c

57. Maple: Strings Handling

58. Maple: Strings Handling
Assignments

59. Maple: Strings Handling
Subscripting

60. Maple: Strings Handling
Subscripting

61. Maple: Strings Handling
Subscripting

62. Maple: Strings Handling
Counting