IOE 416 Group Project - Winter 2024
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Industrial Engineering
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Apr 3, 2024
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IOE 416 - Queueing Systems – Winter 2024
GROUP PROJECT
This project is a practical assignment. Its purpose is to observe a queuing system for 1 hour and obtain quantitative results about the queue's performance from your observations. Please read carefully, there are several areas asking for comments on results (even if brief) that should appear in the report.
1.
Form a group of 3-4 students (see group selection assignment). Based on the group selection assignment assignments will be modified so all are in a group.
2.
Pick a queueing system to observe and a 1-hour period for your observations. For simplicity, choose a single-server queue, however a multiple-server queue is acceptable. Pick a queue that will likely see 30 or more arrivals in an hour (but not 100s).
Do NOT select a queue that puts you in vehicle traffic for safety and to avoid police interest. Do NOT select the security line at the airport. Choose a queue at any location (North Campus, Central Campus, downtown Ann Arbor, etc.) and a time when it is reasonably busy (when some queueing occurs most of the time). If another group in the class is observing your selected queue, please go at a different time or find a different queue. Assign data collection responsibilities to ensure all data collected and no one is overwhelmed with data collection.
Observations
3.
Record location of queue, date, observation start/end times, and server count.
Briefly note peculiarities (if any) of your queueing system.
4.
Record the time each arrival joins the queue. 5.
Record start time for each service. 6.
Record, for each service in step 5, its departure time.
7.
Record the number of customers in the system at:
-
the start of observations,
-
1-minute (or finer) intervals during the hour,
-
the end of observations.
Data Analysis
8.
From your arrival observations in step 4,
a)
Obtain the interarrival times b
i
for each arrival i
=
1
,
2
,…
b)
Calculate estimates of
-
average interarrival time v
and ¿
1
/
v
,
-
standard deviation of interarrival times ❑
b
.
c)
Plot a histogram (probability density) of observed interarrival times using appropriate time bins (somewhere between 5 and 20 bins as seems reasonable). Overlay an exponential distribution with rate on the same plot. 1
Times recorded in 4, 5 and 6 should be to nearest 10 seconds or (or less).
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d)
Calculate the standard deviation of the average interarrival time ❑
b
.
e)
Use ❑
b
to develop a (Normal) 95% confidence interval for the average
interarrival time. Do the same 95% confidence interval assuming the interarrival time standard deviation equals the average interarrival
time (exponential). Comment on their comparison.
f)
Use average v
and standard deviation ❑
b
to estimate how much longer you would have to observe the system to get a 95% confidence interval for average
interarrival time with limits within 5% of the true average interarrival time.
g)
Perform a Chi-Squared goodness of fit test of the exponential distribution to the histogram (Excel function). Report its p-value and explain what it means, in simple language, for an “executive”.
Comment briefly on the fit, or lack of fit, of the exponential distribution to your empirical distribution of inter-arrival times.
See Lecture Notes 1A for help. Consider using Excel to do calculations and plots, however, it’s OK to do the work and plots by hand.
9.
Repeat step 8 for service times, i.e., from observations in steps 5 and 6,
a)
Obtain the service time s
i
for each observed service i
=
1
,
2
…
(ok if i
was not observed as an arrival).
b)
Calculate estimates of
-
average service time τ
and service rate μ
=
1
/
τ ,
-
standard deviation of service times ❑
s
.
c)
Plot a histogram of service times and overlay an exponential distribution with rate μ
using appropriate bin sizes (might be the same as done for interarrivals).
d)
Calculate the standard deviation of the average service time ❑
s
.
e)
Use ❑
s
to develop a 95% confidence interval for the average
service time. Do the same 95% confidence interval assuming the service time standard deviation equals the average service time. Comment on their
comparison.
f)
Use average τ
and standard deviation ❑
s
to estimate how much longer (hours) you would have to observe the system to get a 95% confidence interval with limits within 5% of the true average service time.
g)
Perform a Chi-Squared goodness of fit test of the exponential distribution to the histogram and report its p-value. Explain what it means, in simple language, for an “executive”.
Comment briefly on the fit, or lack of fit, of the exponential distribution to your empirical distribution of service
times. If poor, what might be a better distribution to use?
10.
From observations in step 7
, obtain estimates of: (a) average number in system L
(b) standard deviation of number in system ❑
L
11. Using the above estimates from data, include the following table in the report. Only use 3 significant digits for numbers 2
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