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?

The demand (in thousands of units) of refrigerators manufactured by a company are    summarised in the following table:

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:

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.

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 company makes two kinds of leather belts. Belt A is a high quality belt and Belt B is of lower quality. The respective profits are Rs 4 and Rs 3 per belt. The production of each of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 per day. The supply of leather is sufficient for only 800 belts per day (Both A and B combined). Belt A requires a fancy buckle and only 400 per day are available. There are only 700 buckles a day available for Belt B. What should be the daily production of each type of belt? Formulate this problem as an LP Model and solve it using Simplex Method.

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.