Question

36 Alice walks into a post office with 2 clerks. Both clerks are in the midst of serving customers, but Alice is next in line. The clerk on the left takes an Expo(A1) time to serve a customer, and the clerk on the right takes an Expo(A2) time to serve a customer. Let Ti be the time until the clerk on the left is done serving their current customer, and define T2 likewise for the clerk on the right Transformations 411 (a) If A 2, is T1/T2 independent of T +T2? T2))/(T2/(Ti T2)) Hint: Note that Tı/T2 = (Tı/(Ti (b) Find P(Ti < T2) (do not assume Xi = \2 here or in the next part, but do check that your answers make sense in that special case). (c) Find the expected total amount of time that Alice spends in the post office (assuming that she leaves immediately after she is done being served).

Answer 1

(a)

Let W = T1/(T1+T2).

Then T1/T2 = W/(1-W).

By the bank–post office story,Wis independent of T1+T2.

So T1/T2 is also independent of T1+T2.

(b)

We known that P(T1< T2) = λ1/(λ1+λ2).

In the special case λ1=λ2, this gives P(T1< T2) = 1/2, which must be true by symmetry.

Another way to derive this result is to apply the bank–post office story.

This story requires two Gamma r.v.s with the same rate parameter λ, so we will first represent

T1 = X1/λ1,

T2 = X2/λ2 with X1, X2 i.i.d. Expo(1), which is Gamma(1,1).

Then P(T1< T2) = P(X1/X2) < (λ1/λ2)

= P(X1/(X1+X2)) < (λ1/(λ1+λ2))

= λ1/(λ1+λ2),

since X1/(X1+X2)∼Beta(1,1), which is Unif (0,1).

(c)

The expected time spent waiting in line is 1/2λ.

Since the minimum of two independent exponential is exponential with rate parameter the sum of the two individual rate parameters. The expected time spent being served 1/λ. So the expected total time is 1/2λ + 1/λ = 3/λ