1. 18 Management of Waiting Lines
Homework; 1, 3, 6ab(hint: P0=.1111), 4abc(hint: P0=.4286), Add1

2. Add1 Homework Problem
Benny the Barber owns a one-chair shop. At barber college, they told Benny that his customers would exhibit a Poisson arrival distribution and that he would provide service via an exponential service distribution. His market survey data indicate that customers arrive at an average rate of two per hour. It will take Benny an average of 20 minutes to give a haircut. Based on this data, find the following:
The probability that at least two customers will arrive in an hour.
The probability that no more than one customer will arrive in an hour.
Once the customer is in the chair, what is the probability that his haircut will last between 10 and 20 minutes (i.e. he get a fairly quick haircut)?
Once the customer is in the chair, what is the probability that his haircut will last between 20 and 30 minutes (i.e. he get a longer haircut)?
If the haircut times followed a normal distribution (instead of an exponential distribution) with a standard deviation of 5 minutes, determine the probabilities in questions 3 and 4). It is important to understand the difference in your answers for questions 3, 4, and 5. Use the normal distribution for this question only.

3. Add1 Homework Problem
6. Use manual calculations and the Excel queuing worksheet to determine:
The average number of customers waiting.
The average time a customer waits.
The average time a customer is in the shop.
The average utilization of Benny’s time.
The probability that no customers are in the shop (either waiting or being serviced).
The probability that three customers are in the shop.

4. Add1 Homework Problem
Benny is considering adding a second chair (i.e. second server). Customers would be selected for a haircut on a first-come-first-served (FCFS) basis from those waiting. Benny has assumed that each barber takes an average of 20 minutes to give a haircut, and that business would remain unchanged with customers arriving at a rate of two per hour.
7. Find the following information to help Benny decide if a second chair should be added:
The average number of customers waiting.
The average time a customer waits.
The average time a customer is in the shop.

5. Probability Distributions
A table (or graph) of possible outcomes of an experiment (or study) and associated probabilities.
Discrete –
Continuous –

6. The Uniform Distribution

7. The Binomial Distribution

8. The Poisson Distribution
Used to model arrivals

9. The Poisson Distribution

10. Arrivals, Example
Sam’s wholesale club. Customers arrive at an average rate of 3 per hour and the rates follow a Poisson distribution.
P(X=3)=?
P(X<2)=?
P(X > 2)=?

11. The Negative Exponential Distribution
Used to model service times

12. Service Times, Example
Sam’s wholesale club. Service desk with a clerk. The clerk spends 4 minutes with each customer, on average. Assume service times follow a negative exponential distribution.
P(T<5)=?
P(2<T<4)=?
P(4<T<6)=?
Numbers are in minutes per customer.

13. M/M/1 Systems Poisson distributed arrival rates, l
Exponentially distributed service rates, m

14. M/M/1 Systems, Formulas

15. M/M/1 System, Example
Bank with one teller. One customer arrives every 15 minutes on average. Service for a customer takes 10 minutes on average. Assume Poisson arrivals and exponential servicing times.
Find r ?
Average time a customer spends waiting in line?
How long is the line on average?
The probability there will be at least 2 other customers in the system, either in line or being serviced?

16. M/M/2 Systems Poisson distributed arrival rates, l
Exponentially distributed service rates, m
Servers work at the same average rate

17. M/M/2 Systems, Formulas

18. M/M/2 System, Example
Bank with two tellers. One customer arrives every 15 minutes on average. Service for a customer takes 10 minutes on average. Assume Poisson arrivals and exponential servicing times.
Find r ?
Probability no customers in the bank?
How long is the line on average?
Average time a customer waits in the line?
Average total time a customer spends in the bank?