RSM 270 Introduction to Operations Management
This is the sample questions for lecture 4 for your preparation.
Problem 1 Our chap from last sample problems again injured himself, this time he slipped on an orange peel his friend drives him to the Emergency Room. Similar to the previous hospital they went to, they see 24 patients in the waiting room but in this hospital, unlike the other hospital, they also see patients leaving the ER before they are seen by a physician. The phenomenon they are observing is called Reneging. How will this affect their wait time as compared the hospital they went to last time(assuming they are seen by the exact same physician) and what is the utilization of the ER if there are always customers waiting?
Reneging makes their wait time decrease. Assuming there is always patient waiting to be served, the server is always busy, therefore utilization is 1.
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Problem 2 The average queue length in a system increases if …
the number of servers decreases, the number of customers arriving per minute increases, the
variability in customer arrivals increases, and the variability of service times increases.
Problem 3 Mount Sinai hospital is striving to meet an increase in demand after its popu- larity recently soared. Outline three ways in which it can do this.
Increase the number of beds they have in the hospital, Hire twice as many doctors as nurses as they currently do, Spend money on a state of the information system that tells you what medical test any patient in the hospital needs at any point in time)
Problem 4 Clearly explain the information–capacity tradeoff in the OM triangle using an example.
A family doctor who makes fixed appointments and who limits the time he spends with each patient reduces the variability significantly, i.e., has more more information than a walk-in clinic where patients arrive purely randomly. If the walk-in clinic wants to offer the same short wait times as the family doctor above, they need to have several doctors on duty, i.e., much higher capacity to meet the random demand
Problem 5 Assume eight RSM270 students are enjoying a day in an amusement park during the reading weak in sunny California! One of the attractions is a roller coaster that has only one car which can seat a group of four people. A ride will take exactly 1.5 minutes because it is computer controlled. The time for one group to get off and another group to get into the car is fixed to 1.5 minutes, so that everybody has sufficient time to get seated and an employee can check whether all passengers are restrained correctly. On average, a group of four park visitors arrives every four minutes to this attraction (consider the most natural distribution for arrivals).
(a) Given arrival rate, service rate, coefficient of variation of interarrival time and coffe- cient of variation of service time with respect to this attraction.
(b) On average, how long does a group of four people have to wait before they can take the ride?
(c) How much money would the park visitors who want to take the roller coaster above spend in the park per hour if 50% of those are using FastPass?
Assume that this park uses a similar virtual queuing system as Disney World’s Fast- Pass. Further suppose that guests using FastPass can walk right on to the attraction without waiting, and that they spend $1 every 5 minutes that they roam about in the park instead of waiting in line.
Assume eight people are already waiting to get on the attraction. Just in the moment when Megan and her seven friends arrive, the car comes back from its round-trip. Now people get off and the next group is boarding. To calm down her nerves, Yan would like to get a chocolate bar from a close-by vending machine. So she plans to leave the queue and is hoping to rejoin her friends just in time so that she does not have to queue up by herself at the end of the roller coaster line-up. Here is some information about the vending machine. Every customer takes 30 seconds (constant) to buy a snack while the average time between customer arrivals is 40 seconds. Assume that the standard deviation of the inter-arrival times is 2 minutes.
(d) Give arrival rate, service rate, Ca, and Cs for the vending machine.
(e) What is Yan’s expected time to get her chocolate bar?
(f) Using the result from the previous part of the question, can Yan still make it back on time to join her friends for the ride? (We assume that nobody complains that she is ‘jumping the line”.) Give detailed explanations.
(a) E[a] = 4 [min/groups], i.e., we have a “group” arriving every 4 minutes. So, arrival rate λ = 1 = 1/4 [min/group]= 15 group/hr.
Service time is 1.5 minutes for trip plus 1.5 minutes for getting on/off. Thus, E[s] = 3
minutes per group, and service rate μ = 1 = 1/3 [min/group] = 20 group/hr. E [s]
Ca = 1(We assume exponentially distributed, 4 min on average). Cs = 0 (deterministic: exactly, fixed).
(b) Using what we calculated in the first part, ρ = μλ = 15/20. Therefore we have ρ2 Ca2 + Cs2 (15/20)2 1 + 0
Iq =1−ρ× 2 =1−15/20×( 2 )=1.125 Therefore, Tq = Iq = 1.125/15 hours, or 1.125/15 × 60 = 4.5 minutes.
(c) Per hour 15 groups arrive, thus, 60 guests arrive. 50% of those have a FastPass, i.e., 30 guests per hour can spend 4.5 min in the park buying drinks/snacks/souvenirs rather than standing in line. Thus in total 30(4.5) = 135 minute-guests. At $1 every 5 minute-guests, this results in 135/5 = 27$ per hour.
(d) λ = 1 = 1/40 customer/second= 1/40 × 60 = 1.5 customers per minute. E [a]
μ = 1 = 1 customers/second= 1 × 60 = 2 customers per minute. E[s] 30 30
Ca = σa E [a]
= 2×60seconds =3. Cs =0. 40seconds
(e) ρ=λ/μ=1.5/2=3/4.
Iq = 1 − ρ × 2 = 1 − 3/4 × 2 = 8 .
ρ2 Ca2 + Cs2 (3/4)2 32 + 0 81
Tq = Iq = 81/8 = 27/4 minutes, or 6 minutes 45 seconds. Therefore, we have T =
Tq + Ts = 27/4 + 1/2 = 7 minutes 15 seconds.
(f) The eight people before Megan and her seven friends are back after 6 minutes. Also Among Megan’s group, indeed Megan and three friends have already boarded the car and are ready to go (that is included in the first 6 minutes). 75 seconds later (when Yan arrives) this group is already half through the trip. But the other three friends are still waiting and Yan can join them.
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