STAT S315F Supplementary Assignment PDF

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Page 1STAT S315F Supplementary AssignmentQuestion 1 (20 marks)(a) Arrivals at and departures from a tax enquiry office by members of the public may be modelled as a simple queue. (i) If on average eight people arrive every hour and if the average time the tax inspector spends with each person is fiv...

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STAT S315F Supplementary Assignment Question 1 (20 marks) (a)

Arrivals at and departures from a tax enquiry office by members of the public may be modelled as a simple queue. (i)

If on average eight people arrive every hour and if the average time the tax inspector spends with each person is five minutes, calculate the traffic intensity for this queue.

[2]

The pattern of arrivals at and departures from the tax enquiry office may for practical purposes be assumed to attain equilibrium shortly after the office opens each day. (ii)

At any time, what is the average number of people in the office (excluding the tax inspector)?

[2]

(iii) What proportion of people arriving at the office find there is no-one else there to make an enquiry, and so receive immediate attention?

[2]

(iv) Find the average total time that a person spends in the tax enquiry office. What proportion of people are in the office for more than half an hour?

[3]

(v)

(b)

What is the average length of idle periods for the tax inspector? For how long, on average, does he answer enquiries between idle periods?

[3]

Four cashiers are on duty in a post office where customers may be assumed to arrive independently and at random, at an average rate of 60 per hour. If a cashier is free, then an arriving customer receives immediate attention; otherwise a central queue is formed. The service time for each cashier may be assumed to be exponentially distributed with mean 3 minutes. (i)

Write down the specification of this queue, and calculate the traffic intensity .

[2]

Assume that the queue is in equilibrium. (ii)

Find the probability that at any particular time all four cashiers are idle. What proportion of the time are exactly two of the four cashiers busy?

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[6]

Question 2 (20 marks) The lifetime T (in years) of a heating element in an electric fire may be described in terms of its hazard function h( t) 

(a)

(b)

t , 4

t  0.

Find the survivor function and hence identify the distribution of the random variable T. What proportion of heating elements last for less than a year?

[6]

Find the mean lifetime and the median lifetime of these heating elements.

[4]

Whenever a heating element fails, it is immediately replaced. Mr. Jones has an electric fire containing one of these heating elements. Suppose that, at some time when the fire is quite old, Mr Jones gives it to his daughter to use.

(c)

(d)

(e)

Find the p.d.f. of the residual lifetime and the p.d.f. of the total lifetime of the heating element in the electric fire when Mr. Jones gives it to his daughter.

[5]

Find the mean total lifetime of the heating element in the electric fire when Mr Jones gives it to his daughter.

[4]

If she keeps the fire for four years, how many times should she expect to replace the heating element in that time?

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