###### Introduction to Probability Models

238 SOLVED PROBLEMS

Question: 9.EX.6

## The minimal path sets are {1, 2, 4}, {1, 3, 5}, and {5, 6}. Give the minimal cut sets. ...

A minimal cut set has to contain at least one comp...
Question: 7.EX.42

## For an interarrival distribution F having mean μ, we defined the equilibrium distribution of F, denoted Fe, by Fe(x) =1/μ∫0^x[1-F(y)]dy (a) Show that if F is an exponential distribution, then F = Fe. (b) If for some constant c, F(x)={0, x<c 1, x ≥ c show that Fe is the uniform distribution ...

(a) F_e(x)=\frac{1}{\mu} \int_0^x e^{-y / \...
Question: 7.EX.35

## Satellites are launched according to a Poisson process with rate λ. Each satellite will, independently, orbit the earth for a random time having distribution F. Let X(t) denote the number of satellites orbiting at time t .(a) Determine P{X(t) = k}. Hint: Relate this to the M/G/∞ queue.(b) If at ...

(a) We can view this as an M / G / \infty[/...
Question: 7.EX.22

## The lifetime of a car has a distribution H and probability density h. Ms. Jones buys a new car as soon as her old car either breaks down or reaches the age of T years. A new car costs C1 dollars and an additional cost of C2 dollars is incurred whenever a car breaks down. Assuming that a T -year-old ...

Cost of a cycle =C_1+C_2 I-R(T)(1-I)[/latex...
Question: 7.EX.18

## Compute the renewal function when the interarrival distribution F is such that 1−F(t) = pe^−μ1t + (1 −p)e^−μ2t ...

We can imagine that a renewal corresponds to a mac...
Question: 7.EX.8

## A machine in use is replaced by a new machine either when it fails or when it reaches the age of T years. If the lifetimes of successive machines are independent with a common distribution F having density f , show that (a) the long-run rate at which machines are replaced equals ...

(a) The number of replaced machines by time t cons...
Question: 7.EX.3

## If the mean-value function of the renewal process {N(t), t ≥ 0} is given by m(t) = t/2, t ≥ 0, what is P{N(5) = 0}? ...

By the one-to-one correspondence of m(t) and F, it...
Question: 7.35

## Compute E[T], the expected time until the pattern h, h, h, t , h, h, h appears, when a coin that comes up heads with probability p and tails with probability q = 1− p is continually flipped. ...

Define a renewal process by letting the first rene...
Question: 7.27