Question

Repair calls are handled by one repairman at a photocopy shop. Repair time, including travel time, is exponentially distributed, with a mean of 2.7 hours per call. Requests for copier repairs come in at a mean rate of 1.8 per eight-hour day (assume Poisson)

a. Determine the average number of customers awaiting repairs. (Round your answer to 2 decimal places.) Number of customers

b. Determine system utilization. (Round your answer to 2 decimal places. Omit the "%" sign in your response.) System utilization %

c. Determine the amount of time during an eight-hour day that the repairman is not out on a call. (Round your answer to 2 decimal places.) Amount of time hours

d. Determine the probability of two or more customers in the system. (Do not round intermediate calculations. Round your answer to 4 decimal places.) Probability

Answer 1

Answer to Question 1:

Probability is important in case of making decisions under uncertainty. Under uncertainty, a decision can have multiple outcomes. But chances of each outcome happening are different and thus we can say that there will be multiple outcome with different probability of occurrence. Each outcome will have their respective expected value

To summarize , when one makes decision under uncertainty, there will be multiple outcome with their respective expected values and corresponding probabilities.

It will be prudent to choose the outcome which will have the highest value of “Probability x Expected outcome “ since that will give the true picture of the best outcome. Probability thus is an important determining factor for decision making under uncertain situation.

Answer to question 2 :

Undoubtedly , Never catching a fish and Catching a fish are two mutually exclusive events.

Two statements with one highlighting the act of catching a fish and another the act of never catching a fish can also be considered as two exhaustive items.

However never catching a fish and catching a fish longer than certain length are not two exhaustive events as there will be multiple possibilities of catching fishes with minimum length restrictions.

Therefore,

ANSWER : C ) A& B ARE MUTUALLY EXCLUSIVE BUT NOT COLLECTIVELY EXHAUSTIVE

Answer to question 3 :

Probability of flipping a head while tossing a coin = ½

Since, flipping a coin three times are three mutually independent events, probability of flipping 3 heads in a row

= Probability of flipping head in first flipping x Probability of flipping head in 2^nd tossing x Probability of head in third flipping

= ½ x ½ x ½

= 1/8

PROBABILITY OF FLIPPING 3 HEADS IN A ROW = 1/8

Given are following data :

Request arrival data = a = 1.8 per day

Repair time = r = 8/2.7 per day

1. Average number of customers awaiting repair

= a^2/r x ( r – a )

= 1.8 x 1.8 / ( 8/2.7 x ( 8/2.7 – 1 ) )

= 1.8 x 1.8 / 2.96 x ( 2.96 – 1 )

= 1.8 x 1.8 / 2.96 x 1.96

= 0.5584 ( 0.56 rounded to 2 decimal places )

AVERAGE NUMBER OF CUSTOMERS AWAITING REPAIR = 0.56

1. System utilization = a/r x 100 percent = 1.8 x 2.7/8 x 100 = 60.75 %

1. Amount of time in an 8 hour day the repairman is not on call

= ( 100 – system utilization) percent of 8 hour

= 39.25% of 8 hours

= 3.14 hours

d. Probability of ZERO customers in the system

= P0 = ( 1 – a/r) = 1 – 1.8 x 2.7/8 = 1 – 0.6075 = 0.3925

Probability of 1 customer in the system

= P1 = ( a/r) x Po = (1.8 x 2.7/8) x 0.3925 = 0.6075 x 0.3925 = 0.2384

Therefore , probability of either Zero or 1 customer in the system

= P0 + P1

= 0.3925 + 0.2384

= 0.6309

Probability of 2 or more customers in the system

= 1 – Probability of either zero or 1 customer in the system

= 1 – 0.6309

= 0.3691

PROBABILITY OF TWO OR MORE CUSTOMERS IN THE SYSTEM = 0.3691