In the message transmission problem, Problem 5.3.2, the PMF for the number of transmissions

when message i is received successfully is

Let J₃ = K₃ - K₂, the number of transmissions of message 3; J₂ = K₂ - K₁, the number of transmissions of message 2; and J₁ = K₁, the number of transmissions of message 1. Derive a formula for PJ(j), the PMF of the number of transmissions of individual messages.

Problem:

A wireless data terminal has three messages waiting for transmission. After sending a message, it expects an acknowledgement from the receiver. When it receives the acknowledgement, it transmits the next message. If the acknowledgement does not arrive, it sends the message again. The probability of successful transmission of a message is p independent of other transmissions. Let K = [K₁ K₂ K₃] be the 3-dimensional random vector in which K_i is the total number of transmissions when message i is received successfully. (K₃ is the total number of transmissions used to send all three messages.) Show that

(a) Find the PMF of K.

(b) For each j ∈ {1, 2,..., n - 1}, find the marginal PMF PK₁,K₂,...,K j(k₁, k₂,..., k_j)

(c) For each i ∈ {1, 2,..., n}, find the marginal PMF PKi(ki).