Power Consumption Model

The power consumption of a wireless transmitter can be divided into two parts: circuit power and transmission power. The transmission power usually dominates since long-range communications (over 100m) are common in wireless networks. In order to maintain the same transmission rate, the required transmission power needs to increase with the distance between the transmitter-receiver pair to offset the propa- gation loss. In addition, the circuit power is expected to decrease as the IC technology advances. This part of power only occupies a small portion of the whole power con- sumption. So in our work, we assume the transmission power dominates the negligible circuit power.

In our power model, we assume the channel is slowly time-varying, which means the channel condition will not change during transmission. Proper channel coding can reduce the energy consumption effectively during transmission. We take the Additive White Gaussian Noise (AWGN) channel model as an example, which explains how energy, rate, and data size are related. With optimal channel coding, the maximum transmission rate is [32]:

S = B

2 log2(1 + P0 N0B

), (3.1)

where S is the transmission rate, P0 is the received signal power, N0 is the spectral density, and B is the channel bandwidth. From this equation, we can describe the re- lationship between the transmission rate S and the received power P0 by the following equation:

P0 = N0B · (2

2S

B − 1). (3.2)

As we aforementioned, the power will increase with distance between transmitter and receiver in order to maintain the same transmission rate. Considering this power attenuation, we have: P = P 0 A = N0B A · (2 2S B − 1), (3.3)

where P is the transmission power and A is the attenuation factor for the transmitter- receiver pair. The attenuation factor A is generally inversely proportionally to a function of the distance, denoted by l. For example, this function could be a square

function, A ∝ 1/l2, in [32]. In this work, we do not assume any specific form of

the relationship between attenuation factor and distance except that all transmitter- receiver pairs have the same fading functions which are only affected by distance. It is easy to see that the required transmission power P is strictly increasing and strictly convex in the transmission rate S. This power function P (S) is continuous

in S though we only consider the discrete cases for this function in this work.

Let Pi denote the power consumption function for task τi. Let Ci and Si represent the size and rate of data transmission for τi, respectively. The transmission time to transmit data Ci equals to C_Si

i. Therefore it consumes Pi(Si)

Ci

Si units of energy. The

energy consumed for τi for transmission in one period, denoted by Ei, with data size Ci at transmission rate Si becomes

Ei(Ci, Si) = Pi(Si) Ci Si = N0B Ai · (22SiB − 1) ·Ci Si , (3.4)

where the coefficient Ai for each transmitter/receiver pair differs depending on the distance between them. Similar power models were defined in [121] [125], as well. As the channel states and receiving nodes are assumed to be static during the transmis- sion period, the power attenuator factor Ai is also static. We fix the bandwidth B and assume it is the same for all the streams. To simplify the problem, we assume the overhead of switch among different transmission rates is tiny and can be ignored.

3.1.3

Problem Formulation

We consider the transmission in a hyper-period T which is defined as the Least Common Multiple (LCM) of task periods T1, T2, . . . , TN. The consideration of a hyper-period ensures all tasks can finish their periodic transmissions at least once.

Let Emax represent the units of energy budget allocated to the transmitter during

this hyper-period T . Our objective is to maximize the total reward while all tasks meet their deadlines and the total energy consumption does not exceed the budget Emax. In other words, the optimization problem in this work is to find a speed and a data size for each task to maximize the overall rewards while satisfying delay and

energy constraints.

The reward Ri can be a function of multiple variables such as data size, transmis- sion rate, etc. In this work, we define reward function Ri as a generic function of data size cj_i to be transmitted, represented by Ri(cji). It is conceptually the same as the reward functions in [11][33] and the utility function in [92]. In this work, however, we do not assume any specific form of the reward function. It can be reduced to different forms in different application contexts. In its simplest form, the reward function Ri can be interpreted as the amount of data transmitted, with respect to the data size; the reward maximization problem is then reduced to throughput maximization, as in [43] [121]. In an image sensing application, it could be interpreted as the amount of information transmitted using different image formats [73]. In a file transferring application, it can be reduced to the probability of successful file delivery [110].

In general, we formulate the reward maximization problem as

maximize N X i=1 T Ti Ri(Ci) (3.5) subject to N X i=1 T Ti · Ci Si ≤ T (3.6) N X i=1 T Ti Ei(Si, Ci) ≤ Emax (3.7) Si ∈ {s1, s2, . . . , sM}, 1 ≤ i ≤ N (3.8) Ci ∈ {c1i, c 2 i, . . . , c K i }, 1 ≤ i ≤ N. (3.9)

Constraint (3.6) guarantees that all the data streams can be completed under the EDF scheduling [74]. Whenever there is a schedule that maximizes the reward and can guarantee all streams transmitted under constraints, we can always convert that

schedule to EDF scheduling with the same reward, according to the proof of EDF optimality in [75]. When all the packets in a hyper period are ready for transmission at time 0, (3.6) reduces to a discrete case as in [121]. If we enable continuous data size and transmission rate in (3.8) and (3.9), respectively, this problem is reduced to a continuous case, similar to that in [121]. In constraint (3.9), we enable multiple choices of data sizes; the problem is reduced to the one in [97] when we fix available data size of each task i in constraint (3.9) to be {c1

i, c2i}, with c1i = 0 and c2i 6= 0, for all 1 ≤ i ≤ N . Theorem 1 shows the reward-maximization problem for periodical tasks defined by (3.5)-(3.9) is NP-hard.

Theorem 1. The reward-maximization problem for periodical tasks defined by (3.5)- (3.9) is NP-hard.

Proof. Let xij and xil denote two 0-1 decision variables, respectively. Let Ωi = {s1, s2, . . . , sM} and Λi = {c1i, c2i, . . . , cKi } for all 1 ≤ i ≤ N . If a task i is assigned to transmit data at rate j ∈ Ωi, then xij = 1; otherwise, xij = 0. If a task i is assigned to transmit data with data size l ∈ Λi, then xil = 1; otherwise, xil = 0. The optimal problem ((3.5)-(3.9)) can be rewritten as:

maximize N X i=1 X l∈Λi T Ti Ri(Cil)xil (3.10) subject to N X i=1 X j∈Ωi X l∈Λi T Ti · Cil Sij xijxil ≤ T (3.11) N X i=1 X j∈Ωi X l∈Λi T Ti Ei(Sij, Cil)xijxil ≤ Emax (3.12) X j∈Ωi xij = 1, 1 ≤ i ≤ N (3.13) X l∈Ωi xil = 1, 1 ≤ i ≤ N. (3.14) By viewing _TT

iRi(Cil) as the profit of item l out of class i, terms

T Ti ·

Cil

Sij and

T

TiEi(Sij, Cil) as the corresponding weights, the optimal problem formulation ((3.10)-

(3.14)) becomes to an instance of well-known NP-hard Multidimensional Multiple- Choice Knapsack Problem (MMKP) [65].

3.2

Branch-and-Prune for the Optimal Solutions

A general method of solving the optimal MMKP problem is to search the solution space until an optimal solution is found and confirmed [65]. We can use breadth- first search to generate partial solution along with the sequence of receivers. This algorithm enumerates all possible data sizes and transmission rate for each receiver. This process can be visualized as a state space branch where each non-leaf node in this tree has M × K children if there are M transmission rate levels and K data size levels for each receiver. Therefore a naive algorithm would generate (M × K)i

nodes at level i. The state space can grow exponentially with the task number. To reach the solution in practical run time, most researchers relied on heuristics to obtain approximated solutions [61][79][93] or adapted approaches to reduce the computational complexity [34][25][123]. These approaches are not readily applicable

to our problem as our problem involves more decision factors. In the following,

we develop a branch-and-prune algorithm for the reward maximization optimization problem with 2-dimension multiple choices of data size and transmission rate.