P- CSCF
A.3 Energy States
Parameter Description
z Number of paths a node has in its cache (# of routes to the sink)
∆α /∆T Rate of change of energy measured in some interval T G Set of 1-hop neighbors (|G| = number of 1-hop neighbors) dp Distance of node from sink on path p
d Average distance of node from sink across z paths
R∗ Best value of R among all paths known by the node (without adding the node own Ri) dmin the minimum distance, in number of hops, across all paths to the sink from node i h1, h2 Protocol constants greater than 1, with h1> h2
ε1 Threshold (on transmission energy) below which we want to prioritize transmission ε2 Threshold (on transmission energy) above which we want to prioritize encryption
Table A.2: Parameters used when deviating from fair allocation.
lifetime is defined as the contiguous period of time during which more than X% of the nodes in the network can route packets to the sink after completing all required rounds of encryption.
In the following, for ease of notation, we drop the subscript i although we are still referring to computations at a single node. The additional parameters used to decide when and how to deviate from fair allocation are shown in Table A.2.
A.3 Energy States
In the following we discuss the different energy states a node may be in and the reason behind the fudge factor K1.
When deviating from a fair allocation of energy, we take away from one type of energy and give to the other. Somehow, we need to bind the amount of energy swapped in some reasonable way. From the routing protocol point of view, we have a rule that says that BMEP is subject to a fudge factor K1, whereby a newly heard path NP will replace the existing BMEP only if MENP > K1∗ MEBMEP. This implicitly says that we don’t care about a new path if this has a minimum-energy node with transmission energy that is not at least K1 times more than the ME of the existing path. Thus, the proposal is to bound the amount of energy taken away, g(α), using
A B C
D B E
F G H
<< 1
<
>>
C’
dmin> 1
dmin= 1
dmin> 1
Figure A.1: Energy-state transition graph for non-stub nodes (i.e., |G| > 1).
the same factor of K1. Specifically, we restrict the amount of energy taken away to be min{g(α), (1 − K1) ∗ αFT X} when we take away from transmission energy. In doing so, taking away energy from transmission energy won’t trigger “artificially created”
updates in BMEP when in phases 1 and 2.
When taking away from encryption energy (we only do this in state G), we use min{g(α), (1 − K1) ∗ αFR} as well. In such case, however, we don’t care about artifi-cially gaining up to a K1fudge factor of energy from encryption, (which doesn’t have anything to do with the protocol as no fudge factors are used for BPR) but rather, we use the same metric just to be reciprocally fair with the previous case when giving to transmission energy and taking away from encryption energy.
Let us look at the two different scenarios of the node having more than one 1-hop neighbor and the node having only one 1-hop neighbor (i.e., the node is a stub node in the network).
A.3. Energy States 117
A.3.1 Non-stub node: |G| > 1
Fig. A.1 shows the energy-state transition graph for non-stub nodes. Let us describe each state in detail, including the node’s energy assignments.
State A: α >> ε2; ∆α/∆T > 0 Energy allocation:
g(α) = |∆α/∆T | ∗ ∆T ∗ z,
αT X = αFT X− min{g(α), (1 − K1) ∗ αFT X}, αR= αFR+ min{g(α), (1 − K1) ∗ αFT X} where:
z> 1: there are 1 or more entries in the node’s cache, α >> ε2: the node has high energy,
∆α /∆T > 0: the rate at which energy is growing (energy-harvesting nodes).
Rationale: deviate from fair allocation by taking from transmission energy and giving to encryption energy. The more paths there are in the cache, the more we take from transmission energy, but the larger the magnitude of growth in energy (in T), the more we take away from transmission energy.
State B: ε1< α < ε2; ∆α/∆T − > 0 or ∆α/∆T > 0
Energy allocation: Fair allocation as per Eq. A.1 and Eq. A.2
where:
z> 1: there are 1 or more entries in the node’s cache, ε1< α < ε2: moderate energy,
∆α /∆T − > 0 or ∆α /∆T > 0: the rate at which energy changes it is slightly negative (i.e., going down slowly) or slowly growing.
State C: α << ε1; ∆α/∆T > 0; dmin> 1 Energy allocation:
g(α) = |∆α/∆T | ∗ ∆T ∗ (zh1∗ dh2), αT X = α − min{g(α), (1 − K1) ∗ α}, αR= min{g(α), (1 − K1) ∗ α}
where:
z> 1: there are 1 or more entries in the node’s cache, α << ε1: the node has low energy,
|G| > 1: the node has more than 1 neighbor, d>> 1: the node is on average far from the sink,
∆α /∆T > 0: the rate at which energy changes it is either 0 or slightly growing.
Rationale: initially give all energy to transmission, then take away some and gi-ve it to encryption energy. The more paths there are in the cache or the farther we are from the sink, the more we take away from transmission energy, and give to encryption. The higher the growth in energy (in T), the more we take away from transmission.
State C’: α << ε1; ∆α/∆T > 0 or ∆α/∆T → 0−; dmin= 1 Energy allocation:
αT X = α αR= 0 where:
dmin= 1: the node is 1 hop from the sink.
All other parameters are the same as in state C.
Rationale: since the node is 1 hop away from the sink, in a low-energy scenario such as this one, the node chooses transmission over encryption and allocates all of its
A.3. Energy States 119
energy to transmission. In order words, such node is not willing to do any encryption operations.
State D: α >> ε2; ∆α/∆T → 0− Energy allocation:
g(α) = z/(1 + |∆α/∆T | ∗ ∆T ),
αT X = αFT X− min{g(α), (1 − K1) ∗ αFT X}, αR= αFR+ min{g(α), (1 − K1) ∗ αFT X} where:
z> 1: there are 1 or more entries in the node’s cache, α >> ε2: the node has high energy,
∆α /∆T → 0−: the rate at which energy changes is very slow or 0.
Rationale: deviate from fair allocation by taking from transmission and giving to encryption energy. The more paths there are in the cache, the more we take from transmission energy, but the larger the magnitude of decline in energy (in T), the less we take out from transmission energy.
State E: α << ε1; ∆α/∆T → 0−; dmin> 1 Energy allocation:
g(α) = (zh1∗ dh2)/(1 + |∆α/∆T | ∗ ∆T ), αT X = α − min{g(α), (1 − K1) ∗ α}, αR= min{g(α), (1 − K1) ∗ α}
where:
z> 1: there are 1 or more entries in the node’s cache, α << ε1: the node has low energy,
|G| > 1: the node has more than 1 neighbor, d>> 1: the node is on average far from the sink,
∆α /∆T → 0−: the rate at which energy changes it is 0 or slightly declining.
Rationale: initially give all energy to transmission, then take away some and give it to encryption energy. The more paths there are in the cache or the farther we are from the sink, the more we take away from transmission energy, and give to encryption. The higher the decline in energy (in T), the less we take away from transmission.
State F: α >> ε2; ∆α/∆T << 0
Energy allocation: Fair Allocation as per Eq. A.1 and Eq. A.2
where:
z> 1: there is more than 1 entry in the node’s cache, α >> ε2: node has very high energy,
∆α /∆T << 0: the rate at which energy changes it is very negative (i.e., energy is going down fast).
Rationale: since energy is declining rapidly we do not take from transmission and give to encryption even though there is a lot of energy.
State G: ε1< α < ε2; ∆α/∆T << 0 Energy allocation:
g(α) = |∆α/∆T | ∗ ∆T /(zh1∗ dh2),
αT X = αFT X+ min{g(α), (1 − K1) ∗ αFR}, αR= αFR− min{g(α), (1 − K1) ∗ αFR} where:
z> 1: there are 1 or more entries in the node’s cache, ε1< α < ε2: the node has moderate energy,
∆α /∆T << 0: the rate at which energy changes it is very negative (i.e., energy is
A.3. Energy States 121
going down fast).
Rationale: deviate from fair allocation by taking from encryption energy and gi-ving to transmission energy. The amount taken is directly proportional to the magni-tude of change of energy, and inversely proportional to the number of paths in the cache and the average distance of the node i from the sink. That is if energy is de-clining rapidly, more energy is taken away from encryption; but if there are a large number of paths in cache or the node is very far away from the sink on average, a les-ser amount of energy is taken away from encryption (recall that there is a moderate amount of energy available, it just happens to be getting used up fast). The relati-ve weighting factors are such that number of paths has more impact than arelati-verage distance when they are both bigger than 1.
State H: α << ε1; ∆α/∆T << 0
Energy allocation:
αT X = α, αR= 0
where:
z> 1: there are 1 or more entries in the node’s cache, α << ε1: node has very low energy available,
|G| > 1: node has more than 1 neighbor,
∆α /∆T << 0: the rate at which energy changes it is very negative (i.e., energy is going down fast).
Rationale: since the node has very little energy, give all energy to transmission and no energy to encryption.
P P Q
R S Q
T S Q
<< 1
<
>>
Figure A.2: Energy-state transition graph for stub nodes (i.e., |G| = 1).
A.3.2 Stub node: |G| = 1
Fig. A.2 shows the energy-state transition graph for stub nodes. Let us describe each state in detail, including the node’s energy assignments.
State P: α >> ε2; ε1< α < ε2; ∆α/∆T > 0 Energy allocation:
g(α) = |∆α/∆T | ∗ ∆T ,
αT X = αFT X− min{g(α), (1 − K1) ∗ αFT X}, αR= αFR+ min{g(α), (1 − K1) ∗ αFT X} where:
A.3. Energy States 123
α >> ε2or ε1< α < ε2: high or moderate energy,
∆α /∆T > 0: the rate at which energy changes is slightly positive (i.e., available energy grows slowly).
Rationale: since harvesting is increasing the available energy, the stub node takes away from transmission and gives energy to encryption.
State Q: α << ε1; ∆α/∆T > 0 or ∆α/∆T → 0− or ∆α/∆T << 0 Energy allocation: Fair Allocation as per Eq. A.1 and Eq. A.2 where:
α << ε1: node has very low energy available,
∆α /∆T > 0 or ∆α /∆T → 0− or ∆α/∆T << 0: energy either grows slowly or decreases very fast.
Rationale: since the node already has very little energy and its energy can go down very fast or slowly increase, follow fair allocation.
State R: α >> ε2; ∆α/∆T → 0− Energy allocation:
g(α) = (z + 1)h1/(1 + |∆α/∆T | ∗ ∆T ), αT X = αFT X− min{g(α), (1 − K1) ∗ αFT X}, αR= αFR+ min{g(α), (1 − K1) ∗ αFT X} where:
α >> ε2: the node has high energy,
∆α /∆T → 0−: the rate at which energy changes is very slow or zero.
Rationale: since this is a stub node, it should take away more energy from tran-smission than a non-stub in state D and give it to encryption.
State S: ε1< α < ε2; ∆α/∆T → 0−or ∆α/∆T << 0 Energy allocation:
g(α) = 1/(1 + |∆α/∆T | ∗ ∆T ),
αT X = αFT X− min{g(α), (1 − K1) ∗ αFT X}, αR= αFR+ min{g(α), (1 − K1) ∗ αFT X } where:
e1< α < e2: the node has moderate energy,
∆α /∆T → 0− or ∆α/∆T << 0: the rate at which energy changes is either zero or very negative (i.e., energy levels decrease either slowly or very fast but do not in-crease).
Rationale: since this is a stub node, it behaves as a state-D non-stub node except that we drop the variable z from the equations since z is either 1 or 2 for stub nodes.
State T: α >> ε2; ∆α/∆T << 0
Energy allocation: Fair Allocation as per Eq. A.1 and Eq. A.2
where:
α >> ε2: node has very high energy,
∆α /∆T << 0: the rate at which energy changes it is very negative (i.e., energy is going down fast).
Rationale: since energy is declining rapidly we do not take from transmission and give to encryption even though there is a lot of energy.