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STAT0007 exercise sheet 7

A factory produces racing bikes. The first stage is to assemble the bike on the factory floor, which takes an exponentially distributed length of time with mean 10 hours. Once assembled, the bike is then immediately inspected.

• If the bike passes inspection (which happens at a rate of 1 per hour), then the bike is shipped to its new owner and does not return to the factory floor or for inspection.

• If it does not pass inspection (which happens at a rate of 0.5 per hour) then it is sent back to the factory floor to be reassembled.

The distribution of time it takes to reassemble is the same as for the original assembly. This transfer between assembly and inspection will continue indefinitely until the bike passes inspection. Let Xt be the status of a bike at time t, and assume that {Xt;t ≥ 0} is a continuous time Markov chain with state space, S, given by:

S = {’being assembled or reassembled’, ‘being inspected’, ‘shipped’}.

a. Draw a state-space diagram for the process {Xt;t ≥ 0}.

b. Relabel the states for notational convenience: 1 is ‘being assembled or reassembled’, 2 is ‘being inspected’, and 3 is ‘shipped’. Using o(h) notation where necessary, compute the following for VERY small h:

i. P(Xh = 2|X0 = 1);

ii. P(Xh = 3|X0 = 1) (the answer to this is not zero);

iii. P(Xs+h = 1|Xs = 2);

iv. P(Xh = 3|X0 = 3).

c. Write down the transition matrix of the jump chain of the process {Xt;t ≥ 0}.

d. Does the jump chain of the process {Xt;t ≥ 0} have an equilibrium distribution? Justify your answer. If it does have an equilibrium distribution, find it. If it doesn’t, find all invariant distributions instead.

e. What is the probability that a bike currently being inspected will pass (and so be shipped to its new owner)?

f. How many times, on average, will the bike be inspected before eventually being shipped?

Let T be the time it takes from starting to assemble the bike through to shipping the bike. You can think of T having two independent parts: - the time, say T1, it takes from starting in ‘being assembled or reassembled’ to arriving in the state ‘being inspected’; - the time, say T2, it takes from arriving in the state ‘being inspected’ to arriving in the state ‘shipped’.

g. Calculate E[T] by carrying out the following steps:

i. Compute E[T1].

Hint - how is this connected to holding time in the state ‘being assembled or reassembled’?

ii. Compute E[T2]. Your answer should be a function of E[T].

Hint - you need to combine first step decomposition with the jump chain. You are starting in the state ‘being inspected’; where can you go from there and with what probability? When you apply first step decomposition here, it’s not always ‘+1’; you may need to add something else instead... something to do with holding time, perhaps? This is a test of how well you understand the underlying principles of first step decomposition!

iii. Find E[T] by using your answers to (i) and (ii) above.

h. On arrival at the factory, you find that the racing bike you ordered is currently being assembled. State the remaining expected time until it passes its inspection, justifying your answer carefully.