Aloha In a MAC system, users transmit equal length packets through a shared channel.

The transmission time for each packet is T seconds.Acollision occurs when two or more packet transmissions overlap in time, and a transmission is considered successful if there is no collision during transmission. An example of the packet transmission is given in Figure 7.20, where the middle packet is successfully transmitted and the rest are considered to have collided as they overlap some other packets at least in part. The pure Aloha protocol is used for MAC. The packet is transmitted immediately after its arrival. If a collision occurs, the packet is delayed by a random amount of time and transmitted again. To simplify the analysis, assume the scheduling point process, which includes the generation of new packets and the retransmission of previously collided packets, is a Poisson process with rate g.

(a) Apacket is scheduled to be transmitted at time t. Find the period during which no other packet can be scheduled for transmission for this packet to be transmitted successfully. This period is called the vulnerable period.

(b) Determine the probability of successful transmission.

(c) What is the throughput of the system?

(d) Instead of a pure Aloha protocol, a slotted Aloha protocol is used, in which packets are sheduled for transmission only at multiples of T seconds. In other words, the slot size is T, and packets can only be transmitted at the boundaries of slots. What is the throughput of this system?

(e) Find the value of gT that maximizes throughput in regular and slotted Aloha and compare the two values.