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University of Toronto

Department of Mechanical and Industrial Engineering

MIE263: Operations Research II Stochastic OR

(Winter 2022)

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Homework 6

Due: Tuesday April 12, 2022 at 5pm

Please scan and submit to Quercus Dropbox

Please post questions on Ed Discussion

Problem 1:

At Starbuck’s DriveThru with one cashier, customers arrive with an arrival rate of 10 per hour

and the cashier serves customers at a service rate of 15 customers per hour,

(a) What is the average number of customers in the system?

(b) What is the average cycle time in the system?

(c) What is the average queueing length?

(d) What is the average waiting time?

Problem 2: Consider a cinema with only one ticket booth. People arrive to purchase tickets

according to a Poisson process with rate of 6 people per minute, and the time it takes to purchase

a ticket is exponentially distributed with mean 7.5 seconds. After purchasing a ticket, an attendee

takes exactly 1.5 minutes to reach his assigned seat. If a person arrives 2 minutes before the picture

starts, can he expect to be seated for the start of the picture?

Problem 3: MacBurger’s is attempting to determine how many servers should be available

during the breakfast shift. During each hour, an average of 100 customers arrive at the restaurant.

Each line or server can handle an average of 50 customers per hour. A server costs $5 per hour

and the cost of waiting in line for 1 hour is $20. Assuming that an M/M/m model is applicable,

determine the number of servers that minimizes the sum of delay and service costs.

a) What is the minimum number of servers MacBurger’s must have? Give an expression of

the service rate if MacBurgers has m servers.

b) Construct a Markov chain for MacBurger’s breakfast shift if they have m servers, but

their line-ups can be infinitely long. Be sure to include the transition rates for births and

deaths

c) It is found that the optimal number of servers is 4, with an average queue length of Lq =

0.17. On average, how long does a customer spend waiting in line? In the whole system?

What is the optimal total hourly cost to MacBurger’s?

Problem 4: Pieces arrive to a machine in a factory according to a Poisson process with rate

20/min. The processing time of each piece follows an exponential distribution with rate 24/min.

The company incurs in a cost of $100 per minute for each piece waiting to be processed and of $

for every minute the machine operates, regardless of if it is processing a piece or not. The company

wants to minimize its costs, and is considering adding another identical machine to this process.

For which values of would you recommend to add the extra machine?

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