he following provide two examples monotonic increasing. Furthermore any maximum of H must be illustrating a typical approach to the problem of determining the unique since H is strictly concave. In addition the maximum must throughput. (See also Chapter 13.) 3/16 Queuing theory Even in the case where the station population is finite the channel may perform inadequately. This is because the system may have bistable operation with one stable point at low traffic and most stations idle giving high throughput and a second one where most stations are busy and the throughput is low because of the large number of stations all attempting together. The system may remain in this second state for long periods. The underlying difficulty is that the protocol does not adapt the number of transmission attempts according to the number of busy Collision caused if packet stations. The retransmission probability should be high if the num- transmits here ber of busy stations is small but low when there are large numbers of busy stations. This problem is overcome by having the stations keep an estimate of the backlog which is updated following each slot according to whether it was idle contained a successful trans- Figure 3.12 Vulnerable period for unslotted ALOHA mission or contained a collision (Gallager and Bertsekas, 1987). 3.5.2 Non-Persistent Carrier Sense Multiple Access 3.5.1 ALOHA models Under Carrier Sense Multiple Access (CSMA) mobiles in a com- munication network having packets to transmit do SO if they "sense" Consider first asynchronous (unslotted) ALOHA, as shown in Fig- the radio channel is not being used. We do not go into a description ure 8.12. All packets are the same length and the transmission of a of how this is achieved but suppose that all mobiles become aware packet will be successful provided another packet does not transmit of a packet transmission within T time units after it was begun. The in it's vulnerable period. The probability of a successful trans- period T is called the vulnerable period. mission 