For energy consumption in ideal channel, we know that the packet failure is only due to collision and there are no transmission errors. For the case of ideal channel
is zero. Based on the above concept of active and non active nodes we have three available states, that is,
transmit,
receive and
listen (idle/overhearing). We further define operations within the three states: (a) successful transmission; (b) successful reception; (c) overhearing (reception of packets intended for other stations); (d) idle listening (when the channel is idle); (e) unsuccessful (colliding) transmissions; and (f) reception of collisions. The probabilities of different operations in an ideal channel are classified as follows:
is the probability of successful reception of packet destined for Node
, and is equal to
;
is the probability of successful reception of packet not destined for Node
, and is equal to
;
is the probability of successful reception of packet destined for relay
, and is equal to
;
is the probability of successful transmission of a packet by Node
l, and is equal to
;
is the probability of reception of a collided packet, and is equal to
.
;
is the probability of collision on transmission of a packet by Node
, and is equal to
;
is the probability of idle slots, and is equal to
.
is successful reception of packets destined for node
provided that there is a transmission free from collision and error. Similarly
is the successful reception by relay with the same conditions. It is true as the relay is not involved in the contention process.
is the successful reception by all non active overhearing nodes. The term
ensures that only non active nodes are considered.
is the successful transmission of a packet provided there is a transmission without any failure.
is the transmission where there is no error and failure is due to collision.
is the reception of collided packet.
is the probability that there is no transmission.
The numerator in (5) is defined in expression (14). As evident from the nature of relay based MAC protocols, we use control packets to coordinate relays which are followed by the data and Acknowledgement (ACK) packets. As from the operations defined earlier in this section we know that in a transmission slot we have active nodes and non active nodes. In order to model this behaviour of transmitting and receiving (active nodes) or receiving only (non active nodes) control and data packets, we formulate (7) to (11) to show the working of the protocol. To calculate the energy consumed by nodes (active and non active) (7) to (11) shown above are used in (14). For ideal scenario where we have no transmission errors, it is possible to simplify (14) by substituting
. These equations are independent of the protocol. Also,
,
and
are the power consumed (in Watts) to transmit, receive and listen (idle/overhearing), respectively.
,
and
are the short, distributed and extended inter-frame times.
is the propagation delay and
is the slot time. In (7) and (8), reception and transmission of multiple packets is shown. Equation (
7) gives a generalized equation for determining
and
, which are probabilities of successful reception of packets by relay and destination (which are active nodes). Equation (
7) consists of the sum of energy consumed in receiving, transmitting and listening. Energy consumed in each of these states is the product of slot duration and respective power. Here the slot duration in transmitting and receiving of the control and data packets is the sum of their times. Where
and
are total number of control and data packets received. Similarly
and
are total number of control and data packets transmitted. The sum of
,
,
and
is the total number of control and data packets in a protocol. The same expression is used to determine
, where no transmission of packets is involved. In (8) successful transmission of a packet by an active node (source) is given. In (9) and (10),
is the time for collision of control packet (initiated from source to relay or destination) and
are
the probabilities of transmission and reception of collided packets. Equation (
11) shows the listening (idle) state as a product of idle slot and idle power. Equations (
7)–(11) are for the ideal case where there are no errors and are the same as in [
7]. This set of equations represent a generic model and is used to show performance of relay based MAC protocols and can easily be adapted to cater for 802.11 a/b/g physical layers, with the parameters changed appropriately.