Vaccinations against Covid-19 highlighted an interesting problem. Because there was a shortage of vaccines, the government segmented the population into different categories essentially based on risk. Healthcare workers were in the first group to get the vaccine, those over 75 were in the second group, etc. People had to make appointments to get vaccinated. The problem with the system was that there were unused vaccines when people failed to show up for appointments. Once Covid-19 vaccines are thawed out, they have a short shelf life; if they are not used they must be thrown away. Initially, thousands of vaccines were thrown out since New York State regulations did not allow remaining doses of the vaccine to be used on lower-risk people who were not scheduled to get vaccinated.
Model the following simple solution:
Add a second “standby” line. Line one is for those with an appointment, the high-risk people. Line two is a standby line for anyone. For the purpose of this example, suppose that 10 per cent of all people with appointments do not show up for the vaccination. Unused vaccines are given to those on the standby line on a FCFS basis.
What additional assumptions will you need to model this problem (and possible solution)?
What outcome measures will be of interest in order to determine the effectiveness of the solution presented?
A bank manager is considering modernizing his bank to use a single waiting line rather than separate queues for each teller. He needs to be convinced that the customer will actually be better served as a result of the change. Simulate both systems, given the following information:
Customer inter-arrival times and service times are distributed according to an exponential distribution. (Lambda =20 per hour, µ=10 per hour).
The number of tellers (k=3)
The length of a run (1 day = 8 hours).
All tellers are identical. Upon arrival, a customer will go immediately to any available teller. If no teller is available, the customer will
(a) wait in the single queue until one becomes available
OR
(b) choose the shortest line and wait there until served.
The bank closes at a specified time but all customers in the bank are served before tellers leave.
Measures desired are:
per-server utilization
average and maximum queue length
average customer waiting time
Write a very brief report comparing the results of programs (a) and (b). Also, compare these results to the ones you obtained in the SimpleQ Assignment
Add features to the original simulation to model balking, reneging, and/or jockeying.
[LINK]
A cafeteria chain is designing a new food service concept for business lunch customers. In addition to the regular cafeteria line, a parallel self-service line with limited food items will be added as an option. Data was collected at a prototype operation. During the peak 45-minute period, customer interarrival times were found to be normally distributed with an average of 15 seconds and a standard deviation of 3 seconds. half of the customers used the self-service line. The regular line consists of the salad, entree, dessert, and beverage stations. Each station’s service time is exponentially distributed with averages of 15, 30, 10, and 20 seconds, respectively. The entree station has two servers; all others have one. The self-service line can accommodate up to six customers at a time. Its service time was normally distributed with an average of 45 seconds and a standard deviation of 5 seconds. After obtaining their food, customers pay the cashier. There are two cashiers at the cafeteria. The cashier service time is exponentially distributed with an average of 15 seconds. Assuming that customers stop at each stations of the regular line, w would like to simulate the system for 1000 customers served to determine the following:
The average waiting time and average queue length at the cashier
The station in the regular line that has the largest average queue length
Average system time for regular line customers and for self-service line customers
Current system. A DMV office currently has 3 clerks assisting customers to obtain license plates for their cars. Each clerk services only customers from one particular geographic region. Customers arrive and enter one of three lines based on their area of residence. You may assume that customers are distributed equally across the three regions. Each customer type is assigned a single separate clerk to process the application forms and accept payment with a separate queue for each. Service times are uniformly distributed anchored at (8,10) minutes for all customer types. After completion of this phase all customers are sent to a single supervisor who reviews the application paperwork and issues the plates. Thus all 3 customer types merge into a single FIFO queue for this phase. Service time for this activity is uniform (2.66, 3.33) minutes. Output MOEs: the average and max time in system for all customer types combined.
A) Model this arrival activity as three independent arrival streams. For each stream, customer interarrival times are exponentially distributed with mean 10 minutes. (Begin each stream with an arrival at time 00:00.) Develop a model of this current system and run for 5,000 minutes. You may consider first 500 min as a warm-up period. Animate your model.
B) Run A with 5 replications.
c) Show the results graphically for single replication and for 5 reps.
Alternate system. A consultant has recommended that the office not differentiate among customer types instead all customers will line up in a single queue for any of 3 identical clerks for the first phase. Run this alternative model - do parts A, B, and C again for this alternate planned system - and compare to the current model. Write up your conclusions.
A lecturer teaching a simulation class of 15 students gives a homework assignment every class meeting and these assignments are collected at the beginning of the next class. The number of students completing the day's assignment follows a discrete probability distribution with probabilities listed in the table below.
The lecturer grades the assignments during office hours, which are scheduled for 90 minutes after class each day. The time to review an assignment follows a uniform distribution with a minimum of 2 minutes and a maximum of 3 minutes. Students may also make appointments to see the lecturer during office hours; visits are given a higher priority than assignments but the lecturer finishes grading the current assignment before helping the student. The lecturer schedules appointments with three to seven students per day (with equal probability for each number in the range) at 15 minute intervals starting at the beginning of office hours. Of the students who schedule appointments, 10% typically do not show up. The time that the lecturer spends with each student is triangular distributed (5, 10, 20 mins.).
Simulate the operation of a days office hours, collecting statistics on the time required to complete the days tasks (grading papers, seeing students), the average time students wait to see the lecturer and the percentage of time spent by the lecturer on each of the two tasks.
[still under construction]
(Source: Adapted from S. Ross, Simulation 5th edition, Academic Press, 2013.)
Consider a shop that stocks a particular type of product that it sells for a price of r per unit. customers demanding this product appear in accordance with a Poisson process with rate lambda and the amount demanded by each is a random variable having distribution of G. in order to meet demand, the shopkeeper must keep an amount of product on hand and whenever the on-hand inventory becomes low, additional units are ordered from the distributor. The shopkeeper uses a so-called (s, S) ordering policy; namely, whenever the on-hand inventory becomes low, additional units are ordered.
Whenever the on-hand inventory is less than S and there is no presently outstanding order then an amount is ordered to bring it up to S, where x < S. that is if inventory is at x we order S - x. The cost of ordering y units is a specified function C(y), and it takes L units of time until the order is delivered, with the payments being made upon delivery. In addition, the shop pays an inventory holding cost of H per unit item per unit time. Suppose further that whenever a customer demands more of the product than is presently available then the amount on hand is sold and the remainder of the order is lost to the shop.
MOE: Use simulation to estimate the shop's expected profit up to some fixed time T.
variables:
System state variables (x,y) where x is the amount of inventory on hand and y is the amount on order.
Counter variables:
C total amount of ordering costs by t
H, total amount of inventory holding costs by t
R total amount of revenue earned by time t
t1 time at which the order being filled will be delivered
Events
a customer arrives with demand
an order arrives
Updating is accomplished by considering which of the event times is smaller.
A conveyor belt brings an item to a work station at the rate of one every 5 minutes. Service is performed on the item with service times exponentially distributed. Which of the following service rates is required if the average time in the waiting line is not to exceed 3 minutes?
(a) 30/hr (b) 20/hr (c) 15/hr (d) 12/hr
Base your analysis on 8 hours of simulated activity.
(Source:Watson 1981)
[still under construction]
(Source: Adapted from S. Ross, Simulation 5th edition, Academic Press, 2013.)
A system needs n working machines to be operational. To guard against machine breakdown, additional machines are kept available as spares. Whenever a machine breaks down it is immediately replaced by a spare and is itself sent to the repair facility which consists of a single repair person who repairs failed machines one at a time.
Once a failed machine has been repaired it becomes available as a spare to be used when the need arises.
==FIGURE==
all repair times are independent random variables having the common distribution function G. each time a machine is put into use the amount of time it functions before breaking down is a random variable independent of the past having distribution function F.
the system is said to "crash" when a machine fails and no spares are available . assume that there are initially n+s functional machines of which n are put into use and s are kept as spares.
MOE: we are interested in simulating this system so as to approximate E(T), where T is the time at which the system crashes.
variables
system state - the number of machines that are down at time t
Events: the system state variable will change either when a working machine breaks down or when a repair is completed
to know when the next event will occur, we need to keep track of the times at which the machines presently is use will fail and the time that the machine presently being repaired (if any) will complete its repair. it will be convenient to store the n failure times in an ordered list.