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Suppose that N sensors are randomly placed in a circle with xed radius P The task is to subdivide the circle into n rings, and determine their widths R1, R2, , Rn, so that the network lifetime is optimized, where R1 R2 Rn P (that is, the sum is xed) Optimization variables are therefore n, R1, R2, , Rn The sink is in the center of the rst circle, of radius R1 It is assumed that the energy required for communication between two sensors (or sensor and sink) is proportional to d a c, for some constants a (2 a 6) and c For instance, two particular models considered in ref [7] are a 2, c 104, and a 4, c 108 For simplicity, the energy is charged fully to the transmitting node.

If between R and Y there is more than one SM that modulates one input of Y, the resulting modulating value will be the product of the individual modulations in the path X is a selector node (S) if its output selects one of its direct descendants as the branch to follow, short-circuiting the others An S is just a particular case of an AM that always modulates the actuators of a single descendant with a value of 1 and the actuators of the other descendants with a value of 0 The descendant modulated by 1 can be different for each input values combination, but it is always unique X is a virtual sensor (VS) if it de nes a mapping function from a sensor space to another sensor, real or not, space, which is used by some action or decision module.

Suppose that the sensor distribution is uniform, and therefore the number of sensors Ni in the ith ring is proportional to the areas of the rings That is, Ni/Nj R2/R2, and the sum N1 N2 Nn N i j is xed, equal to the total number of sensors It is assumed that each sensor helps a proportional number of sensors from the rings farther from the sink in retransmitting To send a message from sensor in the ith ring to a sensor in the (i 2 1)th ring, we assume that the energy needed is proportional to Ra c, as an average i amount with respect to ring size, or worst-case amount for the rst ring (transmitting to sink, which is the zeroth ring) We also assume that the maximum transmission radius is limited, equal to T.

The sensors in the rst ring spend (in the worst case) energy proportional to Ra c to send their own message directly to the sink, and to retransmit each 1 message In addition, each of them retransmits a proportional number of messages from sensors in all other rings The number of sensors it helps is therefore (N2 Nn )=N1 (N N1 )=N1 , and the total number of messages it sends is 1 (N 2 N1)/N1 N/N1 P 2/R2 Thus the energy needed for these trans1 missions is (Ra c)P 2/R2 This is a function of one variable, which has the mini1 1 mum at r1 (2c/(a 2 2))1/a For a 2, the energy is minimized for the maximal transmission range The (unrealistic) case c 0 is easy to discuss We therefore continue the discussion only for the case c 0 and a 2.

10 ] .

The only requisite of this function is that it has to be de ned for every possible value of the original sensor space X is a virtual actuator (VA) if it de nes a mapping function from an actuator, real or not, space used by some action or decision module to another actuator space The only requisite on this function is that it has to be de ned for every possible value of the original actuator space..