MATH 505b ASSIGNMENT 7Spring 2005, Prof. Alexander

Due Wednesday April 20.

Problems 4–7 are from Ross, Introduction to Probability Models.

(1) Let N(t) be a renewal process.(a) The distribution of N(t) given X1 = u is the same as the unconditional distribution

of N( ) + . (Fill in the blanks.) Assume u < t.(b) Express E(N(t)2 | x1 = u) as an unconditional expectation involving N(·). Use (a).(c) Show that v(t) = E(N(t)2) satisfies v = (v + 2m+ 1) ∗ F .(d) Show that v = m + 2m ∗ m. Use (c). HINT: Recall that m = F = m ∗ F . Take

Laplace transform, recalling that the Laplace transform of h ∗G is hG.(e) Use (d) to find v(t) when N(·) is a Poisson process.

(2) Let N(t) be a Poisson process with intensity λ, and let D(t) be the total life at time t,D(t) = E(t) + C(t).

(a) Find the distributios of E(t) and C(t). HINT: What is the probability there is norenewal in [t−x, t], or in [t, t+x]? But be careful—the distributions of E and C are different!

(b) Are E(t) and C(t) independent? Why or why not?(c) Show that D(t) has d.f. P (D(t) ≤ x) = 1− (1 + λ(t ∧ x))e−λx, x ≥ 0.(d) find E(D(t)) and E(X1). Are they the same? How much bigger is E(D(t)) (approx-

imately) if t is large?

(3) A Poisson process N(t) of radioactive particles arrives at a counter, but only even-numbered particles are detected. Let N(t) be the number of particles detected by timet. Find the corresponding renewal function m(t). HINT: For what event A is N(t) =12N(t)− IA? Also, to find P (N(t) is odd), look at the Taylor series of sinhx.

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(4) A machine in use is replaced by a new machine either when it fails or when it reaches theage of T years. Suppose the lifetimes of successive machines are independent with a commondistribution F having density f .

(a) Show that the long-run rate at which machines are replaced equals(∫ T0xf(x) dx+ T (1− F (T ))

)−1

.

(b) Show that the long-run rate at which machines in use fail equals

F (T )∫ T0xf(x) dx+ T (1− F (T ))

.

HINT: Use renewal-reward.

(5) Consider a single-server bank for which customres arrive in accordance with a Poissonprocess with rate λ. If a customre only will enter the bank if the server is free when hearrives, and if the service time of a customer has the distribution G, then what proportionof time is the server busy?

(6) The lifetime of a car has a distribution H and probability density h. Ms. Jones buys anew car as soon as her old car either breaks down or reaches the age of T years. A new carcosts C1 dollars and an additional cost of C2 dolars is incurred whenever a car breaks down.

(a) Assuming that a T -year-old car is working order has an expected resale value R(T ),what is Ms. Jones’ long-run average cost?

(b) If H is uniform in [2, 8] and C1 = 4, C2 = 1 and R(T ) = 4− T2, then what value of T

minimizes Ms. Jones’ long-run average cost in (a)?

(7) Each time a certain machine breaks down it is replaced by a new one of the same type.In the long run, what percentage of time is the machine in use less than one year old, if thelifetime distribution of a machine is:

(a) uniform in [0, 2];(b) exponential(1).

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