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Chapter 2: 11OQ (page 254)

An engineering firm retains a technical specialist to assist four design engineers working on a project. The help that the specialist gives engineers ranges widely in time consumption. The specialist has some answers available in memory, others require computation, and still, others require significant search time. On average, each request for assistance takes the specialist one hour. The engineers require help from the specialist on average of once each day. Because each assistance takes about an hour, each engineer can work for seven hours, on average, without assistance. One further point: Engineers needing help do not interrupt if the specialist is already involved with another problem. Treat this as the finite queuing problem and answer the following questions:

a. How many engineers, on average, are waiting for the technical specialist for help?

b. What is the average time that an engineer has to wait for the specialist?

c. What is the probability that an engineer will have to wait in line for the specialist?

Short Answer

Expert verified

Consider the information given for the engineering firm

Assume the population is finite.

Time is taken for assistance,T鈥=鈥1鈥hour

Average time engineer can work before assistance, U = 7 hr.

Number of Engineers, N = 4

Number of assistants, S = 1

Calculate the service factor with the help of the given formula:

X鈥=鈥TT+鈥U

Where,

X= Service factor

T= Time taken for assistance

U= Average time engineer can work before assistance

Thus,

Service Factor,

X鈥=鈥11+鈥7鈥X鈥=鈥0.125

Step by step solution

01

(a) Calculate the average number of engineers that are waiting for the help of a technical specialist with the help of the given formula

The average number of engineers waiting in the line

L= Average number of engineers waiting in the line

N = Number of Engineers

F= Efficiency factor

F for S = 1 and

X = 0.125

From the Finite Queuing Table given in the

Thus, Efficiency factor (F) = 0.945.

L鈥=鈥N脳鈥1-鈥F鈥鈥鈥鈥=鈥4脳1-鈥0.945鈥鈥鈥鈥=鈥0.22

Hence, the average number of engineers waiting in the line is 0.22

To get the average waiting time, compute the value of H.

H is the number of engineers to be helped.

02

(b) Calculate the average number of engineers being helped with the help of the given formula

The average number of engineers being helped,

H鈥=鈥N鈥脳F鈥脳X鈥鈥鈥鈥鈥=鈥4脳0.945脳0.125H鈥=鈥0.475

Calculate the average time an engineer has to wait with the help of the given formula:

Average time an engineer has to wait

W鈥=鈥LTH鈥鈥鈥鈥鈥鈥=鈥0.22脳10.475鈥鈥鈥鈥鈥鈥=0.463鈥Hour

Hence, the average time an engineer has to wait is 0.463 hours.

03

(c) Calculate the probability that an engineer will have to wait in line for the specialist with the help of the Finite Queuing Table given in the chapter, as follows

Service factor (X) = 0.125

Efficiency factor (F) = 0.945

Number of assistants (S) = 1

The value of D in the table is 0.362.

Therefore, the probability (D) that an engineer will have to wait in line for the specialist is D=0.362

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