Robert Griffiths
HW. # 9 11/9/99
Promodel #1
Develop a model of the single teller M/M/1 queue. Using an arrival rate of 1.5, a service rate of 2 per
hour and 25 replications- obtain averages and standard deviations of performance parameters. I will
use the model developed for hw#6:
********************************************************************************
*
Formatted Listing of Model:
*
*
C:\My Documents\grad_school\sma\hw9.MOD
*
********************************************************************************
Time Units:
Minutes
Distance Units:
Feet
********************************************************************************
*
Locations
*
********************************************************************************
Name
Cap
Units Stats
Rules
Cost
------------- -------- ----- ----------- -------------- -----------waiting_lines INFINITE 1 Time Series Oldest, FIFO,
Teller
1
1 Time Series Oldest, ,
********************************************************************************
*
Entities
*
********************************************************************************
Name
Speed (fpm) Stats
Cost
---------- ------------ ----------- -----------customer 150
Basic
********************************************************************************
*
Processing
*
********************************************************************************
Process
Routing
Entity Location
Operation
Blk Output Destination Rule Move Logic
-------- ------------- ------------------ ---- -------- ----------- ------- -----------customer waiting_lines move
1 customer Teller
FIRST 1
customer Teller
WAIT E(30)
1 customer EXIT
FIRST 1
********************************************************************************
*
Arrivals
*
********************************************************************************
Entity Location
Qty each First Time Occurrences Frequency Logic
-------- ------------- ---------- ---------- ----------- ---------- -----------customer waiting_lines 1
0
infinite E(40)
Given the data in the problem statement, arrival rate averages 40 minutes, service rate averages 30
minutes. The model is updated and ran:
M/M/1 simulation: Run time = 8, Replications = 25
1.33
L
w
57.18, 2=44.7
Lq
wq
0.84, 2=1.05
30.14, 2=35.71
Evaluate Steady-state parameters:
Analytical
value
8hr run
50hr
500hr
5000hr
1.62
25.24
0.193
6.62
0.66
68.14
0.98
38.17
1.46
95.2
1.62
66.5
1.52
112.8
2.09
82.8
(given by Table 7.4)
L (avg # of customers in system)
w (avg time spent in system per customer)
Lq (avg # of customers in queue)
wq (avg time spent in queue per customer)
3
120mins
2.25
90mins
So, after 5000hrs, parameters have reached within 90% of their steady state values.
(except L, haven’t
figured that out yet…)
Utilizing a “warmup” period” results in the following:
L (avg # of customers in system)
w (avg time spent in system per customer)
Lq (avg # of customers in queue)
wq (avg time spent in queue per customer)
8hr warmup,
50hr run
50hr warmup,
50hr run
100hr warmup,
500hr run
500hr warmup,
500hr run
1.56
76.9
1.2
47.4
1.54
57.29
0.82
32.47
1.45
99.5
1.7
70.67
1.5
126.4
2.4
96.3
Don’t really understand the “effect” of the warmup option. Increasing the warmup time with a fixed
run-time doesn’t seem to advance parameters to steady-state for the 50 hour runs. However, this is not
the case with the 500 hour runs. Increasing the warmup time does advance the simulation to steady
state. Further questions are raised as to why the parameters have exceeded their steady state values.
Promodel #2
Develop a model to simulate the system in Example 11.2 of DESS. Assume an arrival rate of 45
customers/hr and a service time normally distributed with mean 1.1 minutes and standard deviation of
0.2 minutes. Assume also that the measured average time in system of customers is 4.3 minutes.
Perform 25 replications of a 2 hour-long run to obtain averages and standard deviations of performance
parameters.
********************************************************************************
*
Formatted Listing of Model:
*
*
C:\My Documents\grad_school\sma\hw9.MOD
*
********************************************************************************
Time Units:
Minutes
Distance Units:
Feet
********************************************************************************
*
Locations
*
********************************************************************************
Name
Cap Units Stats
Rules
Cost
------------- --- ----- ----------- -------------- -----------waiting_lines infinite 1 Time Series Oldest, FIFO,
Teller
1 1 Time Series Oldest, ,
********************************************************************************
*
Entities
*
********************************************************************************
Name
Speed (fpm) Stats
Cost
---------- ------------ ----------- -----------customer 150
Basic
********************************************************************************
*
Processing
*
********************************************************************************
Process
Routing
Entity Location
Operation
Blk Output Destination Rule Move Logic
-------- ------------- ------------------ ---- -------- ----------- ------- -----------customer waiting_lines move
1 customer Teller
FIRST 1
customer Teller
WAIT N(1.1,0.2) 1 customer EXIT
FIRST 1
********************************************************************************
*
Arrivals
*
********************************************************************************
Entity Location
Qty each First Time Occurrences Frequency Logic
-------- ------------- ---------- ---------- ----------- ---------- -----------customer waiting_lines 1
0
infinite P(1.333)
After running the model the results are: Average Minutes in System = 3.127, with a S.D. of 0.98.
Focusing on the performance parameter average time of customers in system, test the null hypothesis
that the expected value of this parameter (based on your simulation results) is in agreement with the
observed value.
First we determine a critical value for the test statistic ‘t’. For the two-sided test, I went to Table A.5.
Using =0.05 and n=25, tcrit,0.025,24=2.06.
Secondly, the test statistic is computed per eqn 11.2:
Y 2   0 3.127  4.3
t0 

 5.924
S/ n
0.98 / 25
Finally, the critical value is compared to the test statistic. Since |-5.924| > 2.06, reject the hypothesis
and conclude the model is inadequate.