Assignment:
A transmission device receives packets to be forwarded over a certain link. Time is divided in time slots of T seconds each, and the device can transmit a single packet per time slot. Packets can be are buffered at the device output port. If the buffer is empty at the beginning of a time slot, the slot remains idle, otherwise the node transmits the head-of-the-queue packet. This type of multiplexing is called statistical time-division multiplexing and the transmitting device is generally referred to as statistical multiplexer (SMUX). The process Xn that gives the number of buffered packets at the beginning of the nth slot evolves as described by the following equation:
Xn+1 = Xn + vn -x{Xn > 0}
where vn denotes the number of packets generated during the nth time-slot, whereas χ {Xn > 0} is an indicator rv that equals one if the nth slot carries a packet transmission (the buffer was not empty at the beginning of the slot) and zero otherwise. Assuming that X0 = 0 and that the rvs {vn}, n = 0, 1,... are iid with PMD pv(k), k = 0, 1, 2,..., also independent of the number of buffered packets, prove that Xn is a MC and find its transition matrix P.