Arrivals at telephone booth are considered to be Poisson with an average time of 10 minutes between are arrival and the next. The length of phone call is assumed to be distributed exponentially with mean 3 minutes.

i) What is the probability that a person arriving at the booth will have to wait?

ii) The telephone department will install a second booth when convinced that an arrival would expect waiting for at least 3 minutes for phone call. By how much should the flow of arrivals increases in order to justify a second booth?

iii) What is the average length of the queen that forms from time to time?

iv) What is the probability that it will take him or her more than 10 minutes altogether to wait for the phone and complete his or her call?

For the following Auto regressive model

X_t = 0.7X _{t -1} - 0.4 X_{t -2} + a t

i) Verify whether the series is Stationary

ii) Obtain p_k : k = 1, 2 ,3, 4 and 5

iii) Plot the Correlogram.

The annual sales revenue (in lakhs of Rs) of a product as a function of sales force (number of salesmen) and annual adverting expenditure (in Thousands of Rs) for the past 10 years are summarised in following table:

A department has five employees with five jobs to be performed. The time (in hours) each man will take to perform each job is given in the effectiveness matrix:

A company has factories at F₁ , F₂ and F₃ which supply to Ware houses at W₁ , W₂ and W₃ .Weekly factory capacities are 200, 160 and 90 units, respectively. Weekly warehouse requirements are 180, 120 and 150 units, respectively. Unit shipping costs (in Rs) are as follows:

State whether the following statements are true or false and also give the reason in support of your answer.             

i) The number of possible basic solutions of an LPP of order 2 × 3 is 5.

ii) If there are 3 jobs to be performed one at a time on each of the 2 machines, then the possible number of sequences would be 36.

iii) If the arrival rate is 6 per hour and service rate is 3 per hour then the probability of no customer in queue is 0.7

iv)  The estimated value of ‘b’ in the regression line Y = a + bX + e in terms of variancecovariance is

v) Given the trend equation, y = 108 + 2.88 with 1980 as origin and yearly data from 1980 to 1982, the estimated value for 1985 is 119.52.

A private company has two bottling plants located in city I and City II. Each plant produces three cold drinks Drink A, Drink B and Drink C. The number of bottles produced per day are as follows.