IEGR 367 - Report for service design
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Morgan State University *
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367
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Industrial Engineering
Date
Jan 9, 2024
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docx
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3
Uploaded by PresidentStorkMaster683
Jabriea Johnson
IEGR 367
Report for Service design chapter
Example 4 (5.3) The Minute Stop Market has one pump for gasoline, which can service 10 customers per hour (Poisson distributed). Cars arrive at the pump at a rate of 5 per hour (Poisson distributed).
A. Determine the average queue length, the average time a car is in the system, and the average time a car must wait. Given, μ =10 and λ =5 Average queue length
L = λ / (μ - λ) = 5 / (10 - 5) = 1 car average in the queuing system Average time a car is in the system. W = 1 / (μ – λ) = 1 / (10 - 5) = 0.2 hours (12 minutes) average time in the system Average time a car must wait. W
q
= λ / μ (μ – λ)
= 5 / 10(10 - 5) = 0.1 hours (6 minutes) average time waiting
B. If, during the period from 4:00 P.M. to 5:00 P.M., the arrival rate increases to 12 cars per hour, what will be the effect on the average queue length? Given, μ =10 and λ=12. L = λ / (μ – λ) = 12 / (10 - 12) = -6 cars average in the queuing system This means the arrival rate is more than the service rate and as a result the average number of cars in the queue would increase.
Example 5 (5.8) During registration at Tech every quarter, students in the Department of
Management must have their courses approved by the departmental advisor. It takes the advisor an average of five minutes (exponentially distributed) to approve each schedule, and students arrive at the advisor’s office at the rate of 10 per hour (Poisson distributed). Compute L, Lq, W, Wq, and ρ. What do you think about this system? How would you change it?
Average number of students in the system (L) = λ / (μ - λ)
L = 10 / (12 - 10) = 5 students
Average number of students in the queue (Lq) = λ
2
/ (μ * (μ - λ))
Lq = 10
2
/ (12 * (12 - 10)) = 8.33 students
Average time a student spends in the system (W) = 1 / (μ - λ)
W = 1 / (12 - 10) = 0.5 hours (30 minutes)
Average time a student waits in the queue (Wq) = λ / (μ * (μ - λ))
Wq = 10 / (12 * (12 - 10)) = 0.83 hours (50 minutes)
Probability that the advisor is busy (p) = λ / μ
p = 10 / 12 = 0.83
The system seems to experience a moderate level of congestion, as there are students waiting in the queue and the average waiting times are not negligible. The advisor utilization is relatively high, indicating that the advisor is often busy. To improve the system, you could consider implementing appointment scheduling, employing additional advisors during peak times, or introducing an online approval system to streamline the process and reduce waiting times. Each approach has its pros and cons, so it's important to consider the specific needs and constraints of your department. Example 6 (5.11) The head of the Management Department at Tech is considering the addition of
a second advisor in the college advising office to serve students waiting to have their schedules approved. This new advisor could serve the same number of students per hour as the present advisor. Determine L, Lq, W, and Wq for this altered advising system. As a student, would you recommend adding the advisor?
Average number of students in the system (L): L = λ / (μ_total - λ)
L = 10 / (24 - 10) = 0.714 students
Average number of students in the queue (Lg): Lg = λ
2
/ (μ_total * (μ_total - λ))
Lg = 10
2
/ (24 * (24 - 10)) = 0.208 students
Average time a student spends in the system (W): W = 1 / (μ_total - λ)
W = 1 / (24 - 10) = 0.0714 hours (4.29 minutes)
Average time a student spends being advised (Wd): Wd = 1 / μ_total
Wd = 1 / 24 = 0.0417 hours (2.5 minutes)
In this case, since the new advisor can serve the same number of students per hour as the present advisor, the service rate (μ) remains the same, but the arrival rate (λ) will be doubled due to the addition of a second advisor. Once you calculate these metrics, you can analyze whether adding the advisor is a good idea. If the average waiting times (W and Wd) are significantly reduced and the average number of students in the system and queue (L and Lg) are also lower, it might be a beneficial change. However, if the costs associated with the second advisor outweigh the improvements in these metrics, a more detailed cost-benefit analysis might be necessary. The advisor depends on your personal experience and the impact you believe it would have on the advising process. If you
frequently encounter long wait times and delays in schedule approvals, you might see value in the addition of a second advisor.
When customers arrive at Gilley’s Ice Cream Shop, they take a number and wait to be called to purchase ice cream from one of the counter servers. From experience in past summers,
the store’s staff knows that customers arrive at the rate of 35 per hour (Poisson distributed)
on summer days between 3:00 p.m. and 10:00 p.m. and a server can serve 15 customers per
hour on average (Poisson distributed). Gilley’s wants to make sure that customers wait no
longer than 5 minutes for service.
Gilley’s is contemplating keeping three servers behind the ice-cream counter during the peak
summer hours. Will this number be adequate to meet the waiting time policy?
Using the M/M/1 queuing model for this analysis, where:
"M" stands for Poisson arrivals (customers arriving at a rate of 35 per hour).
"M" stands for Poisson service times (servers can serve customers at a rate of 15 per hour).
"1" represents a single server.
λ (arrival rate) = 35 customers per hour
μ (service rate per server) = 15 customers per hour
For one server: ρ = 35 / 15 = 2.33
If ρ is less than 1, it means the server is not fully utilized, and there is some spare capacity.
If ρ is greater than 1, it means the server is overloaded and cannot keep up with the arrivals.
In this case, ρ is greater than 1 (2.33), which indicates that one server is not enough to meet the demand efficiently. Customers will experience longer wait times, and the waiting time policy of 5 minutes or less may not be met.
μ (combined service rate for three servers) = 3 * 15 = 45 customers per hour
Now, we can calculate the utilization factor for three servers:
ρ = 35 / 45 = 0.778
With three servers, ρ is less than 1 (0.778), which means that the servers are not fully utilized. This suggests that three servers should be adequate to meet the waiting time policy of customers waiting no longer than 5 minutes for service during the peak summer hours.
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