Chapter 10, Problem 18OQ

a)

Summary Introduction

To determine: The percentage utilization of the graduate student.

Explanation of Solution

Given information:

It is given that the arrival rate is 4 per hour and the service time is 10 minutes.

Determine the percentage utilization of the graduate student (use model 1):

$\lambda =4\text{ per hour}$

$\mu =\frac{60}{10\text{ minutes}}=6\text{ per hour}$

$\begin{array}{c}\rho =\frac{\lambda }{\mu }\\ =\frac{4}{6}\\ =0.667\text{ or 66}\text{.7\%}\end{array}$

Hence, the utilization of the graduate student is 66.7%.

b)

Summary Introduction

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

Explanation of Solution

Given information:

It is given that the arrival rate is 4 per hour and the service time is 10 minutes.

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

$\lambda =4\text{ per hour}$

$\mu =\frac{60}{10\text{ minutes}}=6\text{ per hour}$

$\begin{array}{c}\begin{array}{l}\text{Average number of students}\\ \text{in the system}\left(L_s\right)\end{array}\}=\frac{\lambda }{\mu -\lambda }\\ =\frac{4}{6-4}\\ =2\text{ students}\end{array}$

Hence, the average number of students in the system is 2 students.

c)

Summary Introduction

To determine: The average time in the system.

Explanation of Solution

Given information:

It is given that the arrival rate is 4 per hour and the service time is 10 minutes.

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

$\lambda =4\text{ per hour}$

$\mu =\frac{60}{10\text{ minutes}}=6\text{ per hour}$

$\begin{array}{c}{\text{W}}_{\text{s}}=\frac{L_s}{\lambda }\\ =\frac{2}{4}\\ =0.50\text{ hours or 30 minutes}\end{array}$

Hence, the average time in the system is 30 minutes.

d)

Summary Introduction

To determine: The probability that there will be four or more students in the line.

Explanation of Solution

Given information:

It is given that the arrival rate is 4 per hour and the service time is 10 minutes.

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

$\lambda =4\text{ per hour}$

$\mu =\frac{60}{10\text{ minutes}}=6\text{ per hour}$

$\begin{array}{c}\text{Probability}=1-\left(P_0+P_1+P_2+P_3\right)\\ =1-\left\{\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^0+\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^1+\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^2+\left(1-\frac{\lambda }{\mu }\right){\left(\frac{\lambda }{\mu }\right)}^3\right\}\\ =1-\left\{\left(1-\frac{4}{6}\right){\left(\frac{4}{6}\right)}^0+\left(1-\frac{4}{6}\right){\left(\frac{4}{6}\right)}^1+\left(1-\frac{4}{6}\right){\left(\frac{4}{6}\right)}^2+\left(1-\frac{4}{6}\right){\left(\frac{4}{6}\right)}^3\right\}\end{array}$

  $\begin{array}{l}=1-\left(0.3333+0.2222+0.1481+0.0988\right)\\ =1-0.8024\\ =0.1976\end{array}$

Hence, the probability that there will be four or more students in the line is 0.1976.

e)

Summary Introduction

To determine: The average number of students in the line if the arrival rate is increased to 6 per hour.

Explanation of Solution

Given information:

It is given that the arrival rate is 4 per hour and the service time is 10 minutes.

Determine the average number of students in the line if the arrival rate is increased to 6 per hour (use model 1):

$\lambda =4\text{ per hour}$

$\mu =\frac{60}{10\text{ minutes}}=6\text{ per hour}$

$\begin{array}{l}{L}_q=\frac{{\lambda }^2}{\mu \left(\mu -\lambda \right)}\\ =\frac{6^2}{6\left(6-6\right)}\\ =\infty \end{array}$