Chapter 10, Problem 17OQ

a)

Summary Introduction

To determine: The percentage utilization of the operator.

Explanation of Solution

Given information:

It is given that the arrival rate is 25 per hour and the service time is 2 minutes.

Determine the percentage utilization of the operator (use model 1):

$\lambda =25\text{ per hour}$

$\mu =\frac{60}{2\text{ minutes}}=30\text{ per hour}$

$\begin{array}{c}\rho =\frac{\lambda }{\mu }\\ =\frac{25}{30}\\ =0.833\text{ or 83}\text{.3\%}\end{array}$

Hence, the utilization of the clerk is 83.3%.

b)

Summary Introduction

To determine: The average number of documents in the system.

Explanation of Solution

Given information:

It is given that the arrival rate is 25 per hour and the service time is 2 minutes.

Determine the average number of documents in the system (use model 1):

$\lambda =25\text{ per hour}$

$\mu =\frac{60}{2\text{ minutes}}=30\text{ per hour}$

$\begin{array}{c}\begin{array}{l}\text{Average number of documents}\\ \text{in the system}\left(L_s\right)\end{array}\}=\frac{\lambda }{\mu -\lambda }\\ =\frac{25}{30-25}\\ =5\text{ document}\end{array}$

Hence, the average number of documents in the system is 5 documents.

c)

Summary Introduction

To determine: The average time in the system.

Explanation of Solution

Given information:

It is given that the arrival rate is 25 per hour and the service time is 2 minutes.

Determine the average time in the system (use model 1):

$\lambda =25\text{ per hour}$

$\mu =\frac{60}{2\text{ minutes}}=30\text{ per hour}$

$\begin{array}{c}{\text{W}}_{\text{s}}=\frac{L_s}{\lambda }\\ =\frac{5}{25}\\ =0.20\text{ hours or 12 minutes}\end{array}$

Hence, the average time in the system is 12 minutes.

d)

Summary Introduction

To determine: The probability that there will be four or more documents in the system.

Explanation of Solution

Given information:

It is given that the arrival rate is 25 per hour and the service time is 2 minutes.

Determine the probability that there will be four or more documents in the system (use model 1):

$\lambda =25\text{ per hour}$

$\mu =\frac{60}{2\text{ minutes}}=30\text{ per hour}$

$\begin{array}{c}\text{Probability}=1-\left(P_0+P_1+P_2+P_3\right)\\ =1-\left\{\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^0+\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^1+\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^2+\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^3\right\}\\ =1-\left\{\left(1-\frac{25}{30}\right){\left(\frac{25}{30}\right)}^0+\left(1-\frac{25}{30}\right){\left(\frac{25}{30}\right)}^1+\left(1-\frac{25}{30}\right){\left(\frac{25}{30}\right)}^2+\left(1-\frac{25}{30}\right){\left(\frac{25}{30}\right)}^3\right\}\end{array}$

  $\begin{array}{l}=1-\left(0.1667+0.1389+0.1157+0.0965\right)\\ =1-0.5178\\ =0.4822\end{array}$

Hence, the probability that there will be four or more documents in the system is 0.4822.

e)

Summary Introduction

To determine: The average number of documents in the system if another clerk were added.

Explanation of Solution

Given information:

It is given that the arrival rate is 25 per hour and the service time is 2 minutes.

Determine the average number of documents in the system if another clerk were added (use model 1):

$\lambda =25\text{ per hour}$

$\mu =\frac{60}{2\text{ minutes}}=30\text{ per hour}$

$\begin{array}{l}{L}_q=\frac{{\lambda }^2}{\mu \left(\mu -\lambda \right)}\\ =\frac{{30}^2}{30\left(30-30\right)}\\ =\infty \end{array}$