depicted in Figure 3.2. In this way if a when the contention period ends and the line goes idle. Let Z be a collision takes place the two packets coincide completely, eliminat- random variable defined as in Equation 3.98. ing the possibility of packet loss by partly overlapping packets. The probability of success becomes c-Y and the throughput Y e-Y. The collisions (3.98) Time between first and last colliding packet maximum throughput Observe that Z must be less than T. Given that there is at least one The above results must be treated with caution. The conclusions are correct but only if they are qualified. The reason for this is that colliding packet the distribution of time between the last packet to the analysis does not take into account the nature of the underlying arrive and the end of the vulnerable period is the same as the time Markov chain which it purports to analyse. Indeed if an infinite until the first packet transmission. This observation shows that the station model is adopted the equilibrium can be shown not to exist distribution of Z is given by expression 3.99. (Kelly, 1985)! The slotted ALOHA channel will jam with prob- ability 1. (3.99) References 3/17 Vulnerable period Line still appears busy Line sensed idle 1 + T Packet transmission starts Signal from last colliding packet ends Vulnerable period Colliding packets 1 + T Line sensed idle Packet transmission starts Figure 3.13 Transmission under the non-persistent CSMA protocol: (a) successful; (b) unsuccessful (3.102) From Equation 3.99 the mean value of Z is obtained, as in Equation 3.100. (3.100) The maximum possible throughput can now be found from exam- ining C over all possible values of a Other forms of CSMA protocol Packet transmissions and contention go in cycles. There are only are discussed in (Kleinrock, 1975b). Note, once again that these two kinds: results must be treated cautiously too; stability questions arise with CSMA protocols as well. Successful transmission, equal to the idle period plus packet 3.6 References transmissi