ISE 580 Fall 2025
Homework #3
Show all your work step by step. Your grade depends on the clarity of solutions and the accuracy of answers.
1. [60 points] Hungry’s Fine Fast Foods is interested in looking at their staffing for the lunch rush, running from 10 am to 2 pm (4 hours). People arrive as walk-ins, by car, or on a (roughly) scheduled bus, as follows:
• Walk-ins—one at a time, interarrivals are exponential with mean 3 minutes; the first walk-in occurs EXPO(3) minutes past 10 am.
• By car—with one, two, three, or four people to a car with respective probabilities 0.2, 0.3, 0.3, and 0.2 (DISC(0.2, 1, 0.5, 2, 0.8, 3, 1, 4)); interarrivals distributed as exponential with mean 5 minutes; the first car arrives EXPO(5) minutes past 10 am.
• A single bus arrives every day sometime between 11 am and 1 pm (arrival time distributed uniformly over this period). The number of people on the bus varies from day to day, but it appears to follow a Poisson distribution with a mean of 30 people.
Once people arrive, either alone or in a group from any source, they operate independently regardless of their source. The first stop is with one of the servers at the order/payment counter, where ordering takes TRIA(1, 2, 4) minutes and payment then takes TRIA(1, 2, 3) minutes; these two operations are sequential, first order-taking then payment, by the same server for a given customer. The next stop is to pick up the food ordered, which takes an amount of time distributed uniformly between 30 seconds and 2 minutes. Then each customer goes to the dining room, which has 30 seats (people are willing to sit anywhere, not necessarily with their group), and partakes of the sublime victuals, taking an enjoyable TRIA(11, 20, 31) minutes. After that, the customer walks fulfilled to the door and leaves. Queueing at each of the three “service” stations (order/ pay, pickup food, and dining room) is allowed, with FIFO discipline. There is a travel time of EXPO(30) seconds from each station to all but the exit door—entry to order/pay, order/pay to pickup food, and pickup food to dining. After eating, people move more slowly, so the travel time from the dining room to the exit is EXPO(1) minute.
The servers at both order/pay and pickup food have a single break that they “share” on a rotating basis. More specifically, at 10:50, 11:50, 12:50, and 1:50, one server from each station goes on a 10-minute break; if the person due to go on break at a station is busy at break time, he or she finishes serving the customer but still has to be back at the top of the hour (so the break could be a little shorter than 10 minutes). You can use the images below for the current staff schedule.
Order and Payment Servers Schedule
Pickup Food Servers Schedule
Staffing is the main issue facing Hungry’s. Currently, there are six servers at the order/pay station and two at the pickup food station throughout the 4-hour period. Since they know that the bus arrives sometime during the middle 2 hours, they’re considering a variable staffing plan that, for the first and last hour would have three at order/pay and one at pickup food, and for the middle 2 hours would have nine at order/pay and three at pickup food (note that the total number of person-hours on the payroll is the same, 32, under either the current staffing plan or the alternate plan, so the payroll cost is the same). What’s your advice? (You can use the images below for this staff schedule.)
Order and Payment Servers Schedule
Pickup Food Servers Schedule
In terms of output, observe the average and maximum length of each queue, the average and maximum time in each queue, and the total number of customers completely served and out the door. Make plots ofthe queues to get into order/pay, pickup food, and the dining room.
2. [40 points] Patients arrive to a 24-hour, 7-days-a-week outpatient clinic with interarrival times being distributed as exponential with mean 5.95 (all times are in minutes); the first patient arrives at time 0. The clinic has five different stations (like nodes in a network), where patients might be directed for service before leaving. All patients first sign in with a single receptionist; sign-in times have a triangular distribution with parameters 1, 4, and 8. From there, they might go to the nurses’ station (probability 0.9888) orto one of three exam rooms (probability 0.0112). The table below gives the five stations, service times at those stations, and transition probabilities out of each station into the next station for a patient (including out of the sign-in station, just described as an example):
All patients eventually go through the checkout station and go home. Note that it is possible that, after a visit to an exam room, a patient is directed to an exam room again (but may have to queue for it). After a patient checks in but is queueing for either the nurses’ station or an exam room, regard that patient as being in the waiting room (and those patients leaving an exam room but again directed to an exam room are also regarded as being in the waiting room). There are three identical exam rooms, but only one resource unit at each of the other four stations. Queues for each station are first-in, first-out, and we assume that movement time between stations is negligible. Run a simulation of 30 round-the-clock 24-hour days and observe the average total time in system of patients, the average number of patients present in the clinic, as well as the throughput of the clinic (number of patients who leave the clinic and go home over the 30 days); also make a plot of the number of patients present in the clinic. If you could afford to add resources, where is the need most pressing?