Operations and Supply Chain Management
Operations and Supply Chain Management
14th Edition
ISBN: 9780078024023
Author: F. Robert Jacobs
Publisher: MCG
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Question
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Chapter 10, Problem 10OQ

a)

Summary Introduction

To determine: The average number of customers at the coffee urn.

a)

Expert Solution
Check Mark

Explanation of Solution

Given information:

It is given that the arrival rate is 3 per minute and the serving time is 15 seconds, which is exponentially distributed.

Determine the average number of customers at the coffee urn (use model 1):

λ=3 per minute

μ=6015=4 per minute

Average number ofcustomers at the coffee urn(Ls)}=λμλ=343=3 customers

Hence, the average number of customers at the coffee urn is 3 customers.

b)

Summary Introduction

To determine: The time taken to get a cup of coffee.

b)

Expert Solution
Check Mark

Explanation of Solution

Given information:

It is given that the arrival rate is 3 per minute and the serving time is 15 seconds, which is exponentially distributed.

Determine the time taken to get a cup of coffee (use model 1):

λ=3 per minute

μ=6015=4 per minute

Time taken to get acup of coffee(Ws)}=Lsλ=33=1 minute

Hence, the time taken to get a cup of coffee is 1 minute.

c)

Summary Introduction

To determine: The percentage of time that the urn being used.

c)

Expert Solution
Check Mark

Explanation of Solution

Given information:

It is given that the arrival rate is 3 per minute and the serving time is 15 seconds, which is exponentially distributed.

Determine the percentage of time that the urn being used (use model 1):

λ=3 per minute

μ=6015=4 per minute

ρ=λμ=34=0.75 or 75%

Hence, the percentage of time that the urn being used is 75%.

d)

Summary Introduction

To determine: The probability of having three or more people in the cafeteria.

d)

Expert Solution
Check Mark

Explanation of Solution

Given information:

It is given that the arrival rate is 3 per minute and the serving time is 15 seconds, which is exponentially distributed.

Determine the probability of having three or more people in the cafeteria (use model 1):

λ=3 per minute

μ=6015=4 per minute

Probability=1(P0+P1+P2)=1{(134)(34)0+(134)(34)1+(134)(34)2}=1(0.2500+0.1875+0.1406)

  =10.5781=0.4219

Hence, the probability of having three or more people in the cafeteria is 0.4219.

e)

Summary Introduction

To determine: The changes in part (a) and part (b) is the cafeteria installs an automatic vendor that dispenses a cup of coffee at 15 seconds.

e)

Expert Solution
Check Mark

Explanation of Solution

Given information:

It is given that the arrival rate is 3 per minute and the serving time is 15 seconds, which is exponentially distributed.

Determine the average number of customers at the coffee urn (use model 2):

λ=3 per minute

μ=6015=4 per minute

Average number ofcustomers at the coffee urn(Ls)}=Lq+λμ=λ22μ(μλ)+λμ=322×4(43)+34

        =98+34=1.125+0.75=1.875

Hence, the average number of customers at the coffee urn is reduced from 3 to 1.875 customers.

Determine the time taken to get a cup of coffee (use model 2):

λ=3 per minute

μ=6015=4 per minute

Time taken to get acup of coffee(Ws)}=Lsλ=1.8753=0.625 minute

Hence, the time taken to get a cup of coffee is reduced from 1 minute to 0.625 minutes.

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