Energy required to establish connection depends on state of Network Interface Card which is the end-point for communication. if Node is in sleep mode then amount of energy required to establish communication different from the node if it is in active mode. like wise same thing for standby mode, overhear mode etc..

Lets assume our Energy consumption model is given by the following:

E = TX(d) + RX 

Where RX is the energy required to receive the packet (spent by the receiver) , and TX(d) is energy spent by the transmitter which is distance dependant. The distance is (d)

If you are interested in transmit power and you believe in a Poisson point process model for the nodes and Rayleigh fading, the calculation is relatively straightforward, see Section III in

M. Haenggi, “On Routing in Random Rayleigh Fading Networks,” IEEE Transactions on Wireless Communications, vol. 4, pp. 1553-1562, July 2005.

An attempt to include the characteristics of the power amplifier was made here:

M. Haenggi, “The Impact of Power Amplifier Characteristics on Routing in Random Wireless Networks,” in IEEE Global Communications Conference (GLOBECOM'03), (San Francisco, CA), Dec. 2003

Both analyses do not account for receive energy, which, in some cases may exceed the transmit energy due to the characteristics of the low-noise amplifier at the receiver.

This is a deeper question than you might think.  What Shannon actually pointed out in his capacity theorem is not that "energy is needed to communicate".  That's a physical question, and information theory is not a theory of physics. What he showed is the capacity of a channel in terms of its randomness (entropy) where in the standard form, energy is detectable with an accuracy limited by an uncertainty assumed to be due to a gaussian energy distribution whose entropy is proportional to the width of the band.  (noise entropy per hertz).

So the question here depends on the "noise" that cannot be avoided in the system used to signal from source to destination.   In a single hop channel, it is usually assumed that all of the noise is due to an uncorrelated source *heard by the receiver* and superposed on the signal.   The noise observable at all places between the sender and the receiver are *irrelevant*.

however, one can make an *assumption* that noise is the same everywhere (not true if there are jammers that are only at particular points in space), then repeating the signal over multiple hops conserves transmission energy compared to using all the energy over many hops.  But that is largely *only* an assumption, and far from true in the real world.

There are many other problems in thinking about actual wireless forwarding and sensor networks.  

In my personal opinion (and experience measuring the actual channel characteristics indoors and in outdoor, near-ground environments) the idea that there is a reasonable closed-form model of communications efficiency in a multihop channel is *absurd*.

That said, the math is fun for people who can disregard the actual physical environment.  But it's pretty non-productive, and a bit frightening that whole theory departments waste their time on pure-free-space, simplistic models of realistic networks.  (you can't even get a *first-order* accurate model that way).

Measurements are far better.  Bell Labs did a lot of measurements on microwave indoor propagation in real life, and also on microwave indoor noise, both done back in the "good old days".   Very few measurements are done and published, and the measurements are disregarded when they contradict that "theory department's" convenient math.  Sadly, most propagation measurements attempt to focus on the outdoor equivalent of "almost free-space", because that is what the cellular companies want to fund.

There is nothing absurd about modeling, and abstractions have proven very useful. And sure, measurements are important, but they can only be taken once a network is deployed, and as engineers we need to design networks, so we need to have an idea of the performance of a network before building it. Once we are in the position to take measurements, it is often too late to change the basic architecture of the system.

Dr. Hanggi - obviously I didn't mean to apply the term "absurd" to the idea of modeling.   However, if you are modeling physical phenomena, you really ought to study the actual physics, both via (in this case) electromagnetic theory (classical or QED) and experimental validation of your models.  While a *particular* system might need to be built before it can be measured, it's quite reasonable to measure physical phenomena to validate aspects of one's model - *especially* key assumptions.

My comments about absurdity refer to the idea of "optimizing" (in closed form equational form) an incredibly weak model that may not even be that applicable.

The first time some folks at Berkeley measured wifi propagation within a few feet of the ground in a grassy field, the divergence from expectations was not a few percent or a few dB.  They had assumed free space would apply.  It did not.  Period. Full stop.  Now I think one can design systems that are resilient to such deviations from models.  After all, WiFi kind-of works as expected.   But perhaps the reason it seems to work is that expectations are so confused in the first place?  (I say this as someone who has designed and tested front-ends for WiFi SDR implementations in the field, and as someone who has studied RADAR techniques with massive ground clutter).

So the question asked - looking for a "closed form" optimizable model - is a purely academic one, one that must be qualified by extremely impractical assumptions if one is designing a *real* energy limited sensor network for a real physical deployment.

I'm not devaluing theory - I use it and do it.  But I am questioning the concept that one can optimize a closed-form equation with the hope of understanding this particular real world issue.

I will quote from A. Einstien "Everything should be made as simple as possible, but no simpler". And my humble opinion coincides with this quote, that the secrets of nature always lies in the abstracted models that try to represent the nature.

We should all remember everytime we are talking on our mobiles, that it has all started by a paper and a pen of J.C. Maxwell.

Sometimes it seems to us that diving deeper in the abstracted theory might be a waste of time, well... it might... Because no one can tell whether it is surely going to benefit, but also, no one can tell that it is surely wast of time. Many lessons from science history can be learnt, and many great applications came from the most abstract theories. However, I understand the point that closed forms for highly specific scenarios are sometimes un-useful, and I do agree that many of the wasted scientific efforts goes into this direction. But "scenario-specific" minimum energy is not the aim of my question.

I was not intending to divert the topic of this question into a Philosophical argument, I am still eager to hear your answers about the theoretical minimum possible energy. 

Many thanks Dr. Heanggi for referring your paper. I already know your work, that is quite unique and remarkable in this area. Also your book on "Stochastic Geometry" is very helpful and interesting.

THat the paper cited (co-author Balakrishnan is a friend and colleague) does not include a noise model makes it incorrect for determining the "lowest energy" needed to transmit a particular rate between points.

If all you need is any equation that appears in the literature for a citation, that's perhaps enough.  However in any proposed system, the noise model is the most crucial aspect of determining achievable information rate (even if you are forced to use a particular modulation scheme).

In a wireless system in space, noise is not position independent.  It is not constant.

I would have no problem with deriving results from such simple models, except that the practice in much publication is to avoid publishing the actual domain of applicability of such results.

Since the model used by the cited paper is neither justified by experimental measurement in typical sensor deployments, nor by a rigorous analysis of expected spatially varying noise sources and propagation of same, one would expect a paper that cited that paper to state (in the title and abstract, as well as conclusions) that the derivation depends on an equation that is justified only by appearing in one paper in the literature, where other papers exist that show evidence for quite different situations.

There is a vast literature on energy propagation models going back decades.  This is a classic (the one I mentioned above) and its citations and papers that cite it provide a rich set of models of the energy propagation that might be expected in a complex physical environment typical of "sensor networks":

Saleh and Valenzuela - A statistical model for indoor multipath propagation.

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=1146527

But that paper does not have a similar "noise" model that can be used to discover the *minimum* energy needed to sustain a particular instantaneous data rate using a particular modulation scheme.

It would be wrong to assume "noise" is uniformly distributed in a sensor network environment without some kind of justification as a way of discussing minimum energy needed for either modulation, transmission, or demodulation.  Low-duty-cycle systems, like some sensor networks, spend a lot of energy looking for a detectable signal - which I'm sure you know is why synchronized clocks are used to minimize receiver energy expended in trying to detect non-existent signals against noise.

Revised answer.

If I were to tackle this problem, I might try a combination of graph theory, specifically minimal cost flow algorithm, and Shannon's equation combined with either free space propagation, or a variant such as the Hata model, if you want to do some sensitivity analysis of the results.

First, you need to know how much information a sensor must transmit, in b/s, and the channel bandwidth used by the sensors. Using Shannon's equation, you derive the threshold SNR required, a value that will apply to each arc in the graph. You also need to know the sensitivity of each receiver (assume the same sensitivity at each node).

The distance between each node pair gives the path loss in each arc of the graph. So now you know the threshold SNR required to transmit sensor data, the receiver sensitivities, and the path loss at each arc in the graph, so you can compute the transmit power required for each arc in the graph. This transmit power becomes your path cost.

Now you run the min cost flow algorithm, to determine the lowest cost path from A to B. This should give what you call "theoretical minimum possible energy."

By using different propagation loss models, you can see the effects on total transmit power required. It's possible that he optimal path will change, based on the propagation model. You would expect loss to increase more rapidly with distance, with anything other than free space propagation, which will tend to favor more short hops as the optimal solution.

(In principle, you should be able to derive one ugly equation that describes this process.)

Dr. Reed, in some cases, analytical expressions can be obtained that do not require the assumption of a specific path loss model. These general expressions can be simplified once a concrete model is available, perhaps through measurements. If such expressions don't adequately reflect reality, it is not the fault of the theory itself but of the inadequate models used. So the question is how to extract a good model from thousands of measurements that (in the wireless case) often show rather random fluctuations. In many cases, a distance-dependent deterministic component together with one or two random variables (accounting for small-scale fading and shadowing, for example) gives a very good fit to the measurements (and are widely used by the wireless industry also for simulations), and adding any more parameters would limit the applicability of the model to one specific scenario (one set of measurements), and that is not the purpose of analysis. 

Related, I think analytical expressions are very valuable even if they are not in strict closed-form but involve an integral or a sum.

Dr Haenggi - I agree with the pragmatics of using simple analytical models (with appropriate understanding of their limits).

My concern here is that the simple analytical models become the "conventional truth" and are used far outside their domain of applicability. This is unfortunately quite common, especially given the relationship between professors and students where the pressure is to find a "publishable result". Unfortunately, relationship to practical, real situations often is omitted by authors/researchers, not checked by "peer reviewers" (who are often not the skilled practitioners, but graduate students to whom a professor delegates his/her reviewing without checking the work of those students.  I know this firsthand from being on program committees for major conferences... and being laughed at for my lack of delegation of what is thought to be drudge work. :-( ).

As an example, over the last few years, it has been common for me to listen to Ph.D. "experts" tell the public that short-wavelength radio transmission won't travel as far as longer wavelengths.  As a context-free statement that is UTTERLY false, and the "expert" should be (in my opinion) stripped of their PhD for not qualifying their statement.  [on the face of it, since microwaves and light are preferred for communications in space missions because they propagate "farther", there are some engineers who must suffer from holding two contradictory ideas in their mind at the same time.]  Those of us who understand the physics and engineering issues know that this description is misleading at minimum, and wrong except in certain "special" situations (such as broadcasting near the ground or in the troposphere where ground waves and ducting facilitate lower frequency propagation along the earth's curvature, beyond the horizon).

So my point was being made in the specific context of this question: calculating the minimum energy to communicate between point A nd point B in a "sensor network".  As far as I know, a "sensor network" is not a mathematical structure in itself.  And there are lots of engineering choices related to modulation, antennas, that can be made (some still the subject of deep research that I follow myself and occasionally pursue experimentally when I have time free from my current startup).  The minimum over all potential approaches is truly unknown, especially in the context of "sensor networks".

So I am trying to suggest two things:

1) the choice of models in any theory that is thought to be applicable here make a huge difference. It's not just a matter of picking any random equation out of the literature, removing it from its domain of applicability;

2) Simple models are quite useful, as you suggest, because they are both tractable and because they can remove irrelevant issues allowing focus.  But the key issue is what is the relevant issue?   If you are trying to minimize energy consumption, focusing on radiated energy is far from the dominant issue in most systems. Real transceivers in real situations consume energy that far exceeds the radiated energy, and for sensor networks, the major energy used is actually in receiving, not sending.... [I'm tempted to point out that simple (impulse-radio-like) UWB detection using CMOS itself could be far cheaper in energy consumption, if the protocol were sorted out, than are conventional receivers, and UWB transmitters can be similarly low energy given the lack of really wideband correlated noise sources].  You won't find the optimum by modeling the transmissions as if they were continuous broadcasts way above the noise floor.  This is like the "drunk looking for keys under the lamppost" because that's where the only light is.

Assuming homogeneous wireless sensor network, two decision variables (data rate and transmission power), I set the problem of finding minimal energy route under the reliability constraint, see formula (12), Z Nikola, G Dimic, and D Bajic. "Channel loss based energy consumption model for low-power wireless communications." New Technologies, Mobility and Security (NTMS), 2011 4th IFIP International Conference on. IEEE, 2011.

The answer you need is in the solution of that formula. The formula is pretty general and allows you to respect all the facts regarding modelling given in previous answers.

The transmission of the power consumed in transmitting and receiving a k-bit message, a distance d (using this radio model), can be computed as:

E_Tx (k,d)=E_elec× k + ε_(amp )×k×(d)^2   and   E_Rx (k)=E_elec× k 

Here you can see that transmission uses the most amount of power. Also, how dependent it is on  thedistance and  to some extent the the size of the message.

Problem is, that equation does not optimize the A to B path, in a (potentially) multi-hop mesh, which was the premise in the original post ("routed via other nodes").

I'm not sure that one can categorically state that the transmitter will use more power than the receiver. You might conceive of receivers with multiple antennas and tuners, fancy equalizers, and error correction algorithms, that could draw more power than the transmitter.

Indeed, in some of the most common sensor network hardware, reception is more costly than transmission, even if the transmit power is set to the maximum.

Well, d is the distance the packet was sent to the next hop. It is not from the source to the destination. As far as the equations, certainly. However, given the research question some assumptions must be made and criteria set prior to the computations.

Maybe this will help.

Heinzelman, W. R., Chandrakasan, A., & Balakrishanan, H. (2000). Energy-efficient communication protocol for wireless microsensor networks. Proceedings of the 33rd Hawaii International Conference on System Sciences, 1-10.

http://pdos.csail.mit.edu/decouto/papers/heinzelman00.pdf

I think LEACH protocol was a good start in the direction of energy efficient routing. However, it uses a quite simplistic energy consumption model (Free Space Loss). Major leaps took place in the literature after LEACH. Maybe hundreds of "energy efficient" routing and clustering protocols are available today. However, I am not aware if any work till now has analytically identified the theoretical minimum energy, that accounts for both the transmit and receive energy required to deliver a packet.

I anticipate that Stochastic Geometry tool will lead to discover the answer of this question. It would be great if you can refer some work in this regards.

No,  this is what I found to be pervasive in the literature. Again, more recent.

Xu, J., Lei,Wang, L., Ma, C., & Shu, L. (2012). Impacts of duty-cycle on overlapping Multi-hop clustering in wireless sensor networks. IEEE ComSoft E-Letters, 1(1), 4.

http://committees.comsoc.org/commsoft/E-letter1.pdf

Well, there are number of angles one can investigate here.

The Xu et. al. paper goes beyond the original question, by introducing the idea of sleep cycles and clustering.  Certainly, if some of the nodes go into sleep mode, they will not be able to participate in the mesh to connect points A and B. But there will be a cluster head node whose job is to keep the mesh connected.

The researchers are investigating the effect on wake times for the nodes. One would think that increasing the wake times should require fewer clusters, for an optimal solution. They are interested in investigating this effect with different network densities.

Fine, but the propagation loss model is not mentioned, and it seems hard to deny that the path loss model would have an effect on results. So Akram's question remains unanswered. Furthermore, if you are interested in minimizing the number of sensors required in any given area, it's not clear to me that you can allow any of them to sleep. One can make the case that a sleeping sensor node is not being of much use, and could be eliminated altogether.

Let's take the case where the nodes are active always. Most of these papers consider the case of the sensor mesh having to communicate with a basestation. Using the min cost flow algorithm I mentioned previously, any clustering that actually improves efficiency will happen naturally. "Clustering" would be manifested by the fact that certain nodes in the mesh will become merge points for paths between other nearby nodes, on the way to the base station. Essentially, the base station becomes the root of a spanning tree consisting of capacity-constrained arcs.

These merge points will be transmitting a higher rate of packets, up to their capacity limit, which the min cost flow algorithm takes into account. This higher packet rate results in transmit power required over a longer duty cycle, hence more energy than what a node out there on its own would need. And not to forget that there's also energy required for reception. Receive power would probably be a lot less variable than transmit power, for the arcs of this graph, and adding it in may be an interesting exercise. It could make the optimization less convincing. 

But still, changing the propagation model will change the optimal answer. None of these papers would say otherwise, that I can tell.

in any network energy is transmitted from one node to another but according to the question,  it depends on the size of the packets and  there flag status of its header, means encoding style, which is to be transmitted as you know the geometric path you can calculate the total energy transmitted.