Queues Applied to Telecoms

9Chapter 1 Exercises

EXERCISE 1.1.– Radioactive particle

We use Τ to denote the random variable equal to the lifetime of the radioactive particle.

Τ follows an exponential distribution with parameter λ, so its expected lifetime is equal to E(Τ) = 1/λ.

The probability that it will still be alive after its expected lifetime is equal to .

Its expected lifetime given that the particle is still alive after its expected lifetime is equal to E(Τ | Τ > 1/λ). Its value is still equal to 1/λ given the amnesia property of the exponential distribution.

We can also demonstrate this in another way. We first determine the probability distribution for the variable Τ | Τ > 1/λ.

Now, we can write that ℙ(Τ > t, Τ > 1/λ) = ℙ(Τ > t + 1/λ) since the exponential distribution is memoryless: ℙ(Τ > t, Τ > 1/λ) = ℙ(Τ > t + 1/λ) = .

Τ | Τ > 1/λ again follows an exponential distribution with parameter λ. Its expected value is thus 1/λ

EXERCISE 1.2.– Discretization of exponential distribution

X is a random exponential variable with parameter λ.

We use S to denote the ceiling function of X: S = ⌈X⌉ = n is equivalent ...