# Matching Supply with Demand

## Quiz 7 :Variability and Its Impact on Process Performance: Waiting Time Problems

(Online Retailer) Customers send e-mails to a help desk of an online retailer every 2 minutes, on average, and the standard deviation of the interarrival time is alsO2 minutes. The online retailer has three employees answering e-mails. It takes on average 4 minutes to write a response e-mail. The standard deviation of the processing times iS2 minutes. a. Estimate the average customer wait before being served. b. How many e-mails would there be, on average, that have been submitted to the online retailer but not yet answered
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Essay

Customers' emails arrive at in a single queue and then served by one of the three servers (employees answering emails). It is assumed that the service is delivered on a First Come First Serve (FCFS) basis. The arrival process has a defined empirical distribution with given mean and standard deviation of interarrival time. It is also assumed that the population in infinite and there is no restriction or constraint in the queue length.
Note the given data as follows:
Average interarrival time,
Standard deviation of interarrival time,
Average processing time,
Standard deviation of processing time,
Number of servers,
a.
To find the average waiting time
, first, compute the coefficients of variation for interarrival and processing time as follows:
Compute the utilization
as follows:
Use the following formula to compute the value of average waiting time
.
So, the average waiting time for a customer is
b.
Use the Little's law to compute the number of emails waiting to be processed
.
So, the total inventory in the system,
So, the number of emails that is delivered and yet to be processed is

(My-law.com) My-law.com is a recent start-up trying to cater to customers in search of legal services who are intimidated by the idea of talking to a lawyer or simply too lazy to enter a law office. Unlike traditional law firms, My-law.com allows for extensive interaction between lawyers and their customers via telephone and the Internet. This process is used in the upfront part of the customer interaction, largely consisting of answering some basic customer questions prior to entering a formal relationship. In order to allow customers to interact with the firm's lawyers, customers are encouraged to send e-mails to my-lawyer@My-law.com. From there, the incoming e-mails are distributed to the lawyer who is currently "on call." Given the broad skills of the lawyers, each lawyer can respond to each incoming request. E-mails arrive from 8 A.M. tO6 P.M. at a rate of 10 e-mails per hour (coefficient of variation for the arrivals is 1). At each moment in time, there is exactly one lawyer "on call," that is, sitting at his or her desk waiting for incoming e-mails. It takes the lawyer, on average, 5 minutes to write the response e-mail. The standard deviation of this is 4 minutes. a. What is the average time a customer has to wait for the response to his/her e-mail, ignoring any transmission times Note: This includes the time it takes the lawyer to start writing the e-mail and the actual writing time. b. How many e-mails will a lawyer have received at the end of a 10-hour day c. When not responding to e-mails, the lawyer on call is encouraged to actively pursue cases that potentially could lead to large settlements. How much time on a 10-hour day can a My-law.com lawyer dedicate to this activity (assume the lawyer can instantly switch between e-mails and work on a settlement) To increase the responsiveness of the firm, the board of My-law.com proposes a new operating policy. Under the new policy, the response would be highly standardized, reducing the standard deviation for writing the response e-mail to 0.5 minute. The average writing time would remain unchanged. d. How would the amount of time a lawyer can dedicate to the search for large settlement cases change with this new operating policy e. How would the average time a customer has to wait for the response to his/her e-mail change Note: This includes the time until the lawyer starts writing the e-mail and the actual writing time.
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Variabilities can affect the overall performance of a process. Any discrepancies in the system should be controlled so as to achieve maximum output.
The formula to calculate the length of queue in parallel servers is as shown below:
Consider the following data for the services provided by a legal services startup:
Arrival rate = 10 mails per hour
Coefficient of variation CV a = 1
Activity time, p = 5 minutes
Standard deviation of service time = 4 minutes
a.
Calculate the average response time in the queue as shown below:
Step 1: Calculate the inter-arrival time by using the formula as shown below:
Step 2: Calculate
to compute the utilization as shown below:
= 1 (given in the case)
Step 3: Calculate the utilization by using the formula as shown below:
Step 4: Calculate the time in queue or total waiting time by using the formula as shown below:
Step 5: Calculate the average response time in the queue as shown below:
Thus, the average response time is 25.49 minutes.
b.
Calculate the total number of emails received by the lawyer at the end of the day as shown below:
c.
Calculate the time that can be dedicated to side by side cases as shown below:
Note: The time that can be dedicated to side by side cases will equal to the idle time per day.
Calculate the idle time per day as shown below:
Thus, the time that can be dedicated to side by side cases is 102 minutes.
d.
Compute the impact of change in standard deviation of writing the response e-mails as shown below:
The idle time is calculated by using the formula,
The formula for utilization is mentioned below:
Where, a is the interarrival time of emails and p is the activity time respectively.
One can say that idle time is directly dependent on the utilization but utilization depends on interarrival time of emails and activity time. There is no role of standard deviation in calculating the utilization and idle time. So, there will be no effect of changing the standard deviation on the idle time.
e.
Calculate the change in average response time if the standard deviation is changed to 0.5 minutes as shown below:
Calculate the average response time in the queue as shown below:
Step 1: Calculate the inter-arrival time by using the formula as shown below:
Standard deviation = 0.5 minutes
Step 2: Calculate
to compute the utilization as shown below:
= 1 (given in the case)
Step 3: Calculate the utilization by using the formula as shown below:
Step 4: Calculate the time in queue or total waiting time by using the formula as shown below:
Step 5: Calculate the average response time in the queue as shown below:
Thus, the average response time will be 17.62 minutes.

(Car Rental Company) The airport branch of a car rental company maintains a fleet of 50 SUVs. The interarrival time between requests for an SUV is 2.4 hours, on average, with a standard deviation of 2.4 hours. There is no indication of a systematic arrival pattern over the course of a day. Assume that, if all SUVs are rented, customers are willing to wait until there is an SUV available. An SUV is rented, on average, for 3 days, with a standard deviation of 1 day. a. What is the average number of SUVs parked in the company's lot b. Through a marketing survey, the company has discovered that if it reduces its daily rental price of $80 by$25, the average demand would increase to 12 rental requests per day and the average rental duration will become 4 days. Is this price decrease warranted Provide an analysis! c. What is the average time a customer has to wait to rent an SUV Please use the initial parameters rather than the information in part (b). d. How would the waiting time change if the company decides to limit all SUV rentals to exactly 4 days Assume that if such a restriction is imposed, the average interarrival time will increase to 3 hours, with the standard deviation changing to 3 hours.
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The demand and supply may vary based on the capacity rate, flow time etc. This results in waiting lines. If the demand exceeds the supply, the waiting time increases. If the supply is more than demand, the utilization decreases. In such cases, the service provider may choose other alternatives to match demand and supply.
Following are some of the problems to calculate the utilization, coefficient of variation of inter arrival times, coefficient of variation of processing times and waiting time:
…… (1)
…… (2)
…… (3)
…… (4)
Consider the following information:
Fleet of cars, m is 50
Inter arrival time, a is 2.4 hours
Standard deviation is 2.4 hours
Average number of days rented, p is 3 days
Standard deviation is 1 day
The total number of cars in the parking lot are not in use or not utilized. So, first calculate the utilization rate of the cars using equation (1):
60% of the cars are utilized. Calculate the cars in parking lot as follows:
Therefore, there are 20 cars in the parking lot
The average demand will increase to 12 rentals per day if the price is decreased from $80 by$25. The following are the changes when there is an increase in demand:
Changes in inter arrival time:
Changes in processing time:
Changes in utilization rate:
Change in profit:
As there is an increase in the utilization rates and profit, the company should implement the changes in price.
Calculate the average wait time as follows:
Calculate the coefficient of variation of inter arrival times and coefficient of variation of processing times using equation (2) and equation (3):
Calculate the time in queue using equation (4):
Hence, the average waiting time is 0.019 hours.
The company decides to rent the cars only for 4 days which results in increase of inter arrival time by 3 hours and standard deviation changes to 3 hours.
Calculate the average wait time:
Calculate the coefficient of variation of inter arrival times and coefficient of variation of processing times as follows:
Calculate the time in queue as follows:
Hence, the average waiting time is 0.049 hours.

(Tom Opim) The following situation refers to Tom Opim, a first-year MBA student. In order to pay the rent, Tom decides to take a job in the computer department of a local department store. His only responsibility is to answer telephone calls to the department, most of which are inquiries about store hours and product availability. As Tom is the only person answering calls, the manager of the store is concerned about queuing problems. Currently, the computer department receives an average of one call every 3 minutes, with a standard deviation in this interarrival time of 3 minutes. Tom requires an average oF2 minutes to handle a call. The standard deviation in this processing time is 1 minute. The telephone company charges $5.00 per hour for the telephone lines whenever they are in use (either while a customer is in conversation with Tom or while waiting to be helped). Assume that there are no limits on the number of customers that can be on hold and that customers do not hang up even if forced to wait a long time. a. For one of his courses, Tom has to read a book ( The Pole, by E. Silvermouse). He can read 1 page per minute. Tom's boss has agreed that Tom could use his idle time for studying, as long as he drops the book as soon as a call comes in. How many pages can Tom read during an 8-hour shift b. How long does a customer have to wait, on average, before talking to Tom c. What is the average total cost of telephone lines over an 8-hour shift Note that the department store is billed whenever a line is in use, including when a line is used to put customers on hold. Essay Answer: (Atlantic Video) Atlantic Video, a small video rental store in Philadelphia, is open 24 hours a day, and-due to its proximity to a major business school-experiences customers arriving around the clock. A recent analysis done by the store manager indicates that there are 30 customers arriving every hour, with a standard deviation of interarrival times oF2 minutes. This arrival pattern is consistent and is independent of the time of day. The checkout is currently operated by one employee, who needs on average 1.7 minutes to check out a customer. The standard deviation of this check-out time is 3 minutes, primarily as a result of customers taking home different numbers of videos. a. If you assume that every customer rents at least one video (i.e., has to go to the checkout), what is the average time a customer has to wait in line before getting served by the checkout employee, not including the actual checkout time (within 1 minute) b. If there are no customers requiring checkout, the employee is sorting returned videos, of which there are always plenty waiting to be sorted. How many videos can the employee sort over an 8-hour shift (assume no breaks) if it takes exactly 1.5 minutes to sort a single video c. What is the average number of customers who are at the checkout desk, either waiting or currently being served (within 1 customer) d. Now assume for this question only that 10 percent of the customers do not rent a video at all and therefore do not have to go through checkout. What is the average time a customer has to wait in line before getting served by the checkout employee, not including the actual checkout time (within 1 minute) Assume that the coefficient of variation for the arrival process remains the same as before. e. As a special service, the store offers free popcorn and sodas for customers waiting in line at the checkout desk. ( Note: The person who is currently being served is too busy with paying to eat or drink.) The store owner estimates that every minute of customer waiting time costs the store 75 cents because of the consumed food. What is the optimal number of employees at checkout Assume an hourly wage rate of$10 per hour.
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(RentAPhone) RentAPhone is a new service company that provides European mobile phones to American visitors to Europe. The company currently has 80 phones available at Charles de Gaulle Airport in Paris. There are, on average, 25 customers per day requesting a phone. These requests arrive uniformly throughout the 24 hours the store is open. ( Note: This means customers arrive at a faster rate than 1 customer per hour.) The corresponding coefficient of variation is 1. Customers keep their phones on average 72 hours. The standard deviation of this time is 100 hours. Given that RentAPhone currently does not have a competitor in France providing equally good service, customers are willing to wait for the telephones. Yet, during the waiting period, customers are provided a free calling card. Based on prior experience, RentAPhone found that the company incurred a cost of $1 per hour per waiting customer, independent of day or night. a. What is the average number of telephones the company has in its store b. How long does a customer, on average, have to wait for the phone c. What are the total monthly (30 days) expenses for telephone cards d. Assume RentAPhone could buy additional phones at$1,000 per unit. Is it worth it to buy one additional phone Why e. How would waiting time change if the company decides to limit all rentals to exactly 72 hours Assume that if such a restriction is imposed, the number of customers requesting a phone would be reduced to 20 customers per day.
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