Question: Consider a window between an external source and the node by which that source is connected to the subnet. The source generates packets according to a Poisson process of rate λ. Each generated packet is accepted by the node if a permit is available. If no permit is available, the packet is discarded, never to return. When a packet from the given source enters the DLC unit of an outgoing link at the node, a new permit is instantaneously sent back to the source. The source initially has two permits and the window size is 2. Assume that the other traffic at the node imposes a random delay from the time a packet is accepted at the node to the time it enters the DLC unit. Specifically, assume that in any arbitrarily small interval δ, there is a probability μδ that a waiting packet from the source (or the first of two waiting packets) will enter the DLC; this event is independent of everything else.

(a) Construct a Markov chain for the number of permits at the source.

(b) Find the probability that a packet generated by the source is discarded.

(c) Explain whether the probability in part (b) would increase or decrease if the propagation and transmission delay from source to node and the reverse were taken into account.

(d) Suppose that a buffer of size k is provided at the source to save packets for which no permit is available; when a permit is received, one of the buffered packets is instantly sent to the network node. Find the new Markov chain describing the system, and find the probability that a generated packet finds the buffer full.