Simulation

1
Simulation

• A Queuing Simulation

2
Example

• The arrival pattern to a bank is not Poisson
• There are three clerks with different service
  rates
• A customer must choose which idle server to go to
• These conditions do not meet the restrictions of
  queuing models developed earlier

3
TIME BETWEEN ARRIVALS

• MINUTES PROB RN
• 1 .40 00-39
• 2 .30 40-69
• 3 .20 70-89
• 4 .10 90-99

4
SERVICE TIME FOR ANN

• MINUTES PROB RN
• 3 .10 00-09
• 4 .20 10-29
• 5 .35 30-64
• 6 .15 65-79
• 7 .10 80-89
• 8 .05 90-94
• 9 .05 95-99

5
SERVICE TIME FOR BOB

• MINUTES PROB RN
• 2 .05 00-04
• 3 .10 05-14
• 4 .15 15-29
• 5 .20 30-49
• 6 .20 50-69
• 7 .15 70-84
• 8 .10 85-94
• 9 .05 95-99

6
SERVICE TIME FOR CARL

• MINUTES PROB RN
• 6 .25 00-24
• 7 .50 25-74
• 8 .25 75-99

7
CHOICE OF SERVER

• ALL THREE SERVERS IDLE
• CHOICE PROB RN
• ANN 1/3 0000-3332
• BOB 1/3 3333-6665
• CARL 1/3 6666-9999
• ( Carls prob. is .0001 more than 1/3)
• TWO SERVERS IDLE (A/B), (A/C), (B,C)
• CHOICE A/B A/C B/C PROB RN
• Ann Ann Bob 1/2 0-4
• Bob Carl Carl 1/2 5-9

8
ARBITRARY CHOICE OFCOLUMNS FOR SIMULATION

• EVENT COLUMN
• ARRIVALS 10
• CHOICE OF SERVER 15
• ANNS SERVICE 1
• BOBS SERVICE 2
• CARLS SERVICE 3

9
DESIRED QUANTITIES

• WQ -- the average waiting time in queue
• W -- the average waiting time in system
• LQ -- the average customers in the queue
• L -- the average customers in the
  system
• If we get estimates for Wq and W, then from
  Littles Laws we can estimate
• LQ ? WQ
• L ? W

10
WILL WE REACH STEADY STATE?

• Average time between arrivals 1/?
• .4(1) .3(2) .2(3) .1(4) 2.0 minutes
• ? 60/2 30/hr.
• Anns average service time 1/?A
• .1(3) .2(4) .05(9) 5.3 minutes
• ?A 60/5.3 11.32/hr.

11
WILL WE REACH STEADY STATE?

• Bobs average service time 1/?B
• .05(2) .1(3) .05(9) 5.5 minutes
• ?B 60/5.5 10.91/hr.
• Carls average service time 1/?C
• .25(6) .50(7) .25(8) 7 minutes
• ?C 60/7 8.57/hr.
• ? 30/hr.
• ?A ?B ?C 11.32 10.91 8.57 30.8/hr.
• ? lt ?A ?B ?C gt Will reach Steady State!

12
THE SIMULATION

• RN IAT AT WQ RN SERV SB RN ST SE W
• 1 36 1 801 0 4231 B 801 33
  5 806 5
• 2 52 2 803 0 7 C 803 98
  8 811 8
• 3 99 4 807 0 9 B 807 26
  4 811 4
• 4 54 2 809 0 ------ A 809 88
  7 816 7
• 5 96 4 813 0 8 C 813 00
  6 819 6
• 6 20 1 814 0 ------ B 814 48
  5 819 5
• 7 41 2 816 0 ------ A 816 11
  4 820 4
• 8 31 1 817 2 6 C 819 61
  7 826 9
• 9 33 1 818 1 ------ B 819 96
  9 828 10

13
SIMULATION (CONTD)

• RN IAT AT WQ RN SERV SB RN ST SE
  W
• 10 07 1 819 1 ------ A 820 62
  5 825 6
• 11 21 1 820 5 ------ A 825 54
  5 830 10
• 12 01 1 821 5 ------ C 826 49
  7 833 12
• 13 20 1 822 6 ------ B 828 84
  7 835 13
• 14 18 1 823 7 ------ A 830 69
  6 836 13
• 15 92 4 827 6 ------ C 833 95
  8 841 14
• 16 10 1 828 7 ------ B 835 63
  6 841 13
• 17 90 4 832 4 ------ A 836 31
  5 841 9
• 18 66 2 834 7 3711 B 841 05
  3 844 10

14
CALCULATING THE STEADY STATE QUANTITIES

• The quantities we want are steady state
  quantities --
• The system must be allowed to settle down to
  steady state
• Throw out the results from the first n customers
• Here we use n 8
• Average the results of the rest
• Here we average the results of customers 9 -18

15
CALCULATIONS FOR W, Wq

• Total Wait in the queue of the last 10 customers
  (1155676747) 49 min.
• WQ ? 49/10 4.9 min.
• Total Wait in the queue of the last 10 customers
  (106101213131413910) 90 min.
• W? 90/10 9.0 min.

16
CALCULATIONS FOR L, Lq

• Littles Laws LQ ?WQ and L ?W
• ? and W and Wq must be in the same time units
• ? 30/hr. .5/min.
• LQ ?WQ ? (.5)(4.9) 2.45
• L ?W ? (.5)(9.0) 4.50
• ? est. of system utilization ? 4.50 -2.45
  2.05
• Est. of Average number of idle workers ? 3- 2.05
  0.95

17
Review

• Simulation of Queuing Models to Determine System
  Parameters
• Check to See if Steady State Will Be Reached
• Determine random number mappings
• Use of pseudorandom numbers to estimate WQ and W
• Ignore the results from the first few arrivals
• Use Littles Laws to get L, LQ
• Average Number of Busy Workers L - LQ