p³(1 − p)−3 k₁< k2 < k3; [p³ - Pk(k) = k₁ € {1,2...}, otherwise.

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 acknowledgment, it transmits the next message. If the acknowledgment does not arrive, it sends the message again. The possibility of successful transmission of a message is 'p' independent of other transmissions. Let K = [K1 K2 K3]^T be the three dimensional random vector in which Ki the phone ring is the total number of transmissions when a message i is received successfully. The PMF for the number of transmissions when message i is received successfully is below. 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 one. Derive a formula for PJ(j), the PMF of the number of transmissions of individual messages.