Maximizing the Functional Lifetime of Sensor Networks Arvind Giridhar by juanagui

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									       Maximizing the Functional Lifetime of Sensor Networks
                                   Arvind Giridhar                                                   P.R. Kumar
                            Coordinated Science Laboratory                                 Coordinated Science Laboratory
                            University of Illinois, Urbana.                                 University of Illinois, Urbana.
                              Email: giridhar@uiuc.edu                                       Email: prkumar@uiuc.com




   Abstract— The functional lifetime of a sensor network is defined as the      the total energy consumed may not be optimal for network lifetime,
maximum number of times a certain data collection function or task can         due to the distributed nature of the communication burden as well as
be carried out without any node running out of energy. The specific task
                                                                               energy supplies.
considered in this paper is that of communicating a specified quantity
of information from each sensor to a collector node. The problem of               Slight variants of this problem, with essentially the same modeling
finding the communication scheme which maximizes functional lifetime            assumptions and notion of lifetime (i.e., time until the first node
can be formulated as a linear program, under “fluid-like” assumptions           failure) have been considered in a number of papers in the literature,
on information bits. This paper focuses on analytically solving the linear     in the context of energy aware or maximum lifetime routing. Chang
program for some simple regular network topologies.
   The two topologies considered are a regular linear array, and a             and Tassiulas [2] consider the problem of maximizing lifetime
regular two-dimensional network. In the linear case, an upper bound            given a certain set of source-destination information rates that must
on functional lifetime is derived, as a function of the initial energies and   be supported. Similar approaches, leading to linear and/or integer
quantities of data held by the sensors. Under some assumptions on the          programming formulations which are basically the same as in this
relative amounts of the energies and data, this upper bound is shown to
be achievable, and the exact form of the optimal communication strategy
                                                                               paper, have been employed in [3–5]. Most of these papers have
is derived. For the regular planar network, upper and lower bounds on          focused on finding distributed algorithms to find the optimal routing
functional lifetime, differing only by a constant factor, are obtained.        strategy to maximize lifetime, without specifically analyzing what
   Finally, it is shown that the simple collection scheme of transmitting      this solution is.
only to nearest neighbors, yields a nearly optimal lifetime in a scaling
                                                                                  In this work, we consider a restricted version of the problem
sense.
                                                                               considered in the above papers. Two regular spatial topologies are
                           I. I NTRODUCTION                                    studied, a linear array and a planar circularly symmetric network.
                                                                               Each of these consist of many sensors and one collector. Under these
   Consider a network of sensors, each with a certain quantity of              restrictions, we are able to provide sharp analytical bounds, and in
data to be communicated to a designated collector node. The sensor             some cases exact solutions, to the functional lifetime problem. These
nodes are low power devices, while the collector node might be                 analytical results cannot directly be translated to practical distributed
an information processing center, or a more energy-rich device                 algorithms, since constructing the optimal schemes requires non-local
functioning as a gateway to a higher bandwidth wireless network                information. The advantage of an analytical approach however lies in
(such as in [1]). Information packets can be relayed - there is no             the structural insight that it provides; while the linear programs could
compression or processing allowed - through an arbitrary sequence of           equally well be solved numerically, the numbers may not provide
nodes before reaching the collector. There is an energy cost associated        such insight. Specifically, this paper attempts to shed light on natural
with each transmission, which consists of a cost to transmit, that is          questions of interest such as:
a function of the transmitter-receiver distance, as well as a cost to
                                                                                 •   How does lifetime depend on the relative quantities and distri-
receive.
                                                                                     bution among sensors of data that is to be collected?
   The above is a somewhat idealized model of a sensor network,
                                                                                 •   What is the structure of routing strategies that are optimal with
with a very simple definition of a data collection task. In this
                                                                                     respect to lifetime?
paper, we examine the notion of functional lifetime, which is defined
                                                                                 •   How does lifetime scale with the size of the network, and/or
as follows: Given a quantity of data and an energy budget (for
                                                                                     quantity of data to be collected?
transmitting and receiving) at each node, the functional lifetime is
                                                                                 •   How well do simple routing strategies perform in comparison
the maximum number of times the task of delivering all the data
                                                                                     to the optimal strategy?
to the collector node can be repeated before some node runs out
of energy. Alternatively, it is the maximum common scale factor by             We take the approach of addressing these questions for simple
which all the quantities of data at all the sensors can be scaled, while       network topologies, which nevertheless are fairly good representatives
ensuring that it can be delivered to the collector node without some           of multi-hop networks. Indeed, our results are indicative of what can
node running out of energy.                                                    be expected in the broader class of “more or less” regular networks
   In general, a sensor network may be simultaneously performing a             as well. Our main contributions can be summarized as follows:
variety of tasks, including sensing, processing and communication.               1) We provide an upper bound on the functional lifetime for
In such a context a natural question to ask is the “cost” to longevity              regular linear networks, which is valid for arbitrary energy
associated with carrying out a specific task of interest, and how                    profiles and data distribution.
such a cost is to be minimized. The per task functional lifetime                 2) Under some restrictions on the energy profile and data distribu-
defined above addresses such a question for communication tasks;                     tion, this upper bound is shown to be achievable, and the exact
its inverse represents the maximum fraction of any node’s energy,                   form of the optimal solution is given. This optimal solution
i.e., or “fraction of lifetime” of the network, that is consumed in                 causes all nodes to die simultaneously.
performing the task. The important point to note is that minimizing              3) Similar results are obtained for a class of regular planar
     networks.                                                                     Collector
  4) The optimal solutions are compared to a simple suboptimal                         1 1 1 1 1 1 1 1 1 1 1
                                                                                       0 0 0 0 0 0 0 0 0 0 0
                                                                                       1 1 1 1 1 1 1 1 1 1 1
                                                                                       0 0 0 0 0 0 0 0 0 0 0
     routing strategy, which consists of each node forwarding all                            1 2 3         n
     data to its nearest neighbor in the direction of the collector.                          Fig. 1.    A regular linear sensor network.
     The results indicate that the simple strategy is nearly optimal
     in a scaling sense.
All the above results depend on a specific property of the energy cost         Consider the following linear program:
function, which is derived in Lemmas 3 and 4 in the appendix.                Min z                                                                     (1)
   Analytical upper bounds on lifetime are also derived in [6]. These
                                                                               subject to:
bounds are not very sharp for the “many-to-one” information flows
that we consider here, since they deal with total energy consumption          1
                                                                                             λij f (d(i, j)) +           λji fR     ≤ z,     1 ≤ i ≤ n (2)
rather than per-node energy consumption.                                      Ei
                                                                                     0≤j≤n                       1≤j≤n
   One minor point needs to be noted. While most of the afore-
mentioned papers have considered lifetime as being in actual time                                               λij −           λji = bi ,   1 ≤ i ≤ n (3)
units, as a function of sets of desired information rates, we have                                      0≤j≤n           1≤j≤n

defined functional lifetime as being the number of times a task can                                           λij ≥ 0, 1 ≤ i, j ≤ n.
be repeated. At one level, this is merely an issue of semantics; one
                                                                            Here λij denotes the amount of information transmitted from node i
definition can be mapped to the other by mapping a time unit to
                                                                            to node j, and z is essentially the maximum of the normalized costs,
a “round,” which is the time taken for one repetition of the data
                                                                            or energy cost divided by initial energy, incurred by each node. The
collection task. However, in the former approach one also needs to
                                                                            set of λij ’s satisfying the constraints specifies a communication flow
address the feasibility of a particular communication flow, which is
                                                                            from the set of sensors to the collector node.
specified in bits per time unit, given the bandwidth and interference
constraints of a wireless network.                                              This formulation is in a different form than the linear program
                                                                            described in [2], but the two can be mapped to each other via simple
   In our formulation, however, there is no limitation on the time
                                                                            change of variables. The following lemma establishes the connection
taken to perform the required communication. For instance, we could
                                                                            between the functional lifetime problem and the linear program (1).
allow for the extremely inefficient strategy of only one transmitter
                                                                                Lemma 1: The solution to the optimization problem (1) yields
in the entire network being active at a time. Such a strategy might
                                                                            the strategy that maximizes the number of times the information
have an extremely large associated delay, but the energy consumption
                                                                            collection process described above can be repeated before some node
would be as specified by our modeling assumptions. Thus, our
                                                                            dies. The functional lifetime is given by z1 , where z ∗ is the optimal
                                                                                                                          ∗
formulation clearly separates the issue of lifetime optimization from
                                                                            value of the objective function.
scheduling, delay, and achievable throughputs. In practice, however,
                                                                            Proof. Assuming the fluidity of information, any feasible solution to
sensor network routing algorithms must jointly consider all these.
                                                                            (1) can be translated to a communication scheme with appropriate
Characterizing the tradeoffs between lifetime, throughput and delay
                                                                            scheduling, since the flow conservation constraints (3) are satisfied.
remains an open problem.
                                                                            If λ∗ is the solution to the optimization problem, with optimal value
   The outline of the rest of the paper is as follows. Section II                                                   bi                  λ∗
describes the model and linear programming formulation. Sections            z ∗ , one can replace all the bi ’s by z ∗ and the λ∗ ’s by zij . Then,
                                                                                                                                 ij       ∗

III and IV deal with the linear and planar networks, respectively.          from the constraints (2), we have that for each i,
Section V obtains scaling laws for the optimal lifetime, followed by
                                                                                      1
a discussion of conclusions and future work.                                                         λij f (d(i, j)) +            λji fR     ≤ Ei ,    (4)
                                                                                      z∗
                                                                                             0≤j≤n                        1≤j≤n


   II. M ODEL , A SSUMPTIONS AND F ORMULATION OF L INEAR                    meaning that no sensor runs out of energy. Thus, the optimal
                          P ROGRAM                                          communication scheme can repeat the operation z1 times without
                                                                                                                                 ∗
                                                                            any sensor running out of energy.
   The general problem setting we consider is very similar to those            On the other hand, any communication scheme to repeat the
considered in [2–5]. Suppose there are sensors 1, 2, . . . , n located on   operation K times, i.e., transmit Kbi bits from each node to the
a plane, along with a collector node labeled 0. Sensor i located at         collector, must involve sending bits between nodes in such a way so
(xi , yi ), has bi bits to send to the collector node, and has initial      as to satisfy the flow conservation constraints (3). Furthermore, the
energy level Ei . The energy consumed by node i in sending m                total energy consumed by each node i must be no more than Ei ,
units of information to node j, which is a distance d(i, j) away,           so a constraint of the form (4) must be satisfied as well. Thus, the
is mf (d(i, j)), while the energy consumed by j (for j = 0; we do           constraints (3) and (2) are necessary as well as sufficient, and z1 is
                                                                                                                                              ∗

not count the energy consumed by the collector node) in receiving           the maximum number of times the operation can be repeated.
m units of information, is fR m, for a given constant fR ≥ 0. We               The linear program given above can be numerically solved for any
seek the communication scheme that maximizes the number of times            number of nodes, any spatial placement, and energy function. The
the information collection process, i.e., communicating bi bits from        rest of the paper, however, will be devoted to explicitly solving this
each sensor to the collector node, can be repeated before one of the        problem in some restricted cases.
sensors dies, i.e., has no remaining energy.
                                                                             III. F UNCTIONAL L IFETIME OF R EGULAR L INEAR NETWORKS
   We make the simplifying assumption that information is infinitely
divisible and incompressible. As a consequence of this “fluid-like”            Consider the uniform linear configuration (Figure 1), where n
assumption, flow conservation is preserved.                                  sensors spaced evenly at a distance d from each other on a line, with
a single collector node located at the leftmost point. The collector        z ∗ is lower bounded as follows:
node is located at the origin, and sensor i at the point (id, 0).                                                              n                     f
                                                                                                                                            1− f1
                                                                                                              n
  The energy cost function f (·) is given by                                                                     b             j=i+1                     j
                                                                                                              i=1 i             n             f
                                                                                                                                           1+ fR
                                                                                                  z∗ ≥
                                                                                                                                j=i                  j
                                                                                                                                n                     f
                                                                                                                                                                 .              (7)
                            f (x) = Et xα eγx ,                       (5)                                     n   Ei            j=i+1
                                                                                                                                            1− f1
                                                                                                                                                          j
                                                                                                              i=1 fi             n            f
                                                                                                                                           1+ fR
                                                                                                                                   j=i               j
where α ≥ 2 is the path loss exponent, γ ≥ 0 the absorption
coefficient, and Et is some positive constant. We make the following         Proof. We find a feasible solution to the dual program of (6) with
assumption on the constant fR , which represents the energy to receive      objective function equal to the RHS of (7). The result then follows
a unit of information:                                                      by weak duality.
   Assumption: fR ≤ Et dα (1 − (1/2)α−1 ).                                    Associating the first n constraint equations in (6) with vari-
   This is a fairly reasonable condition; for α = 3, it states that         ables y1 , y2 , . . . , yn , and the last n constraint equations with
                                                  3
the power to receive must be no more than 4 times the power to              w1 , w2 , . . . , wn , the dual linear program has the following form:
transmit to the nearest neighbor. As stated, it is a sufficient condition
for Lemma 4 to hold; we expect that a somewhat weaker condition                           Max b1 y1 + b2 y2 + . . . + bn yn                                                     (8)
would also suffice. Nonetheless, the structure of the energy cost                           subject to:
function, together with this assumption, are critical to all the results
                                                                                                 E1 w1 + E2 w2 + . . . + En wn                                       ≤     1
in this paper. The only specific property we require of the energy
cost function is stated in Lemma 4, and this property is essential                                                         −f1 w1 + y1                               ≤     0
for any of our results to hold. Essentially, what it describes is the                               −f1 w1 − fR w2 + y1 − y2                                         ≤     0
relative costs of short and long hops; the disadvantage of a long hop                               −f2 w1 − fR w3 + y1 − y3                                         ≤     0
is that due to the sharp degradation in signal power with distance, it is
                                                                                                                        ···
more inefficient in terms of transmit power than multiple short hops
(see [7]). However, multiple short hops involve more receptions and                              −fn−1 w1 − fR wn + y1 − yn                                          ≤     0
thus increase the power spent in receiving, and in addition increase                                                       −f2 w2 + y2                               ≤     0
the energy consumption of the relaying nodes. The optimal strategy                                  −f1 w2 − fR w1 + y2 − y1                                         ≤     0
described in Subsection II B optimally balances these two effects.
                                                                                                    −f1 w2 − fR w3 + y2 − y3                                         ≤     0
However, if power to receive is too high (i.e., the assumption on fR
is violated), then relaying becomes too expensive, and such a strategy                                                  ···
is no longer optimal.                                                                            −fn−2 w2 − fR wn + y2 − yn                                          ≤     0
   The form of the cost function f (·) given by (5), and the assumption                                                                      ···
on fR will be implicitly assumed for the rest of the paper.                                                             −fn wn + yn                                  ≤     0
   Denote fi := f (id) for brevity. The linear program (1) has the                               −fn−1 wn − fR w1 + yn − y1                                          ≤     0
following form:
                                                                                                                        ···
    Min z                                                             (6)                   −f1 wn − fR wn−1 + yn − yn−1                                             ≤     0
      subject to:                                                                                            w1 , w2 , . . . , wn ≥ 0.

          E1 z −            λ1j f|j−1| +           λj1 fR   ≥    0            Consider the following choices of yi ’s and wi ’s: For each 1 ≤ i ≤
                    0≤j≤n                  1≤j≤n
                                                                            n, set
                                                                                                         n             f
                                                                                                                  1− f1
                                                                                                         j=i+1             j
                                                                                                          n        f
          E2 z −            λ2j f|j−2| +           λj2 fR   ≥    0                                       j=i
                                                                                                                1+ fR
                                                                                                                     j                                                   yi
                    0≤j≤n                  1≤j≤n                                          yi =                   n                 f
                                                                                                                                            and wi =                        .   (9)
                                                                                                                               1− f1                                     fi
                                           ···                                                      n   Ei       j=i+1
                                                                                                                  n        f
                                                                                                                                       j
                                                                                                    i=1 fi              1+ fR
                                                                                                                  j=i              j
         En z −             λnj f|j−n| +           λjn fR   ≥    0
                    0≤j≤n                  1≤j≤n                               Then,      b y = z ∗ in equation (7), so the result is proved if
                                                                                         i i i
    λ10 + λ12 + . . . + λ1n − λ21 − λ31 − . . . − λn1       =    b1         this set of choices actually constitutes a feasible solution to the dual.
    λ20 + λ21 + . . . + λ2n − λ12 − λ32 − . . . − λn2       =    b2         Clearly, the wi ’s are all non-negative. Now,
                                           ···                                                                           n
                                                                                                                                       1− f1
                                                                                                                                             f
                                                                                                                Ei       j=i+1                   j
     λn0 + λn1 + . . . + λnn−1 − λ1n − . . . − λn−1n        =    bn                                             fi        n       f
                                                                                                                               1+ fR
                                                                                                                           j=i      j
                       λij ≥ 0, for each 1 ≤ i, j ≤ n.                                    Ei wi =                            n                           f
                                                                                                                                                                     = 1.
                                                                                                                                             1− f1
                                                                                      i              i         n   Ei              j=i+1                     j
                                                                                                               i=1 fi               n          f
                                                                                                                                            1+ fR
                                                                                                                                   j=i                   j
A. Upper Bound on Lifetime
                                                                            Also, −fi wi + yi = 0 for each 1 ≤ i ≤ n, by definition of wi . Next,
   We now give the first main result, which provides an upper bound          it must be proved that for each 1 ≤ j ≤ n, 1 ≤ k ≤ n,
to functional lifetime via linear programming duality.
   Theorem 1: Let z ∗ be the optimal value of the linear program (6).                         −f|k−j| wj − fR wk + yj − yk ≤ 0.
For k > j, we have
                                                         k                                       1 1 1 1 1 1 1 1 1 1 1
                                                                                                 0 0 0 0 0 0 0 0 0 0 0
                                           fk−j          i=j+1
                                                               1 − f1
                                                                   fi                                        k       n
      −fk−j wj + yj        =       1−                     k−1
                                                                         yk                  Collector
                                            fj                1 + fR
                                                          i=j     fi
                                                                                               Fig. 2.      The optimal communication “flow” for node k.
                                           fR
                           <       1+             yk .
                                           fk
  Thus, the only remaining case is in which k < j, for which we                        Suppose that the equations are indeed solvable and that the
need to prove that                                                                   appropriate solutions are all nonnegative. We can then explicitly
                                                                                     calculate the objective function z ∗ . Solving (11) and (14), we get
           fj−k                         fR
      1−            yj     ≤       1+            fk                                                                            E1 ∗
                                                                                                                                  z         +    b1 fR
            fj                          fk                                                                                     f1                 f1
                                                                                                                 λ∗
                                                                                                                  10   =                     fR
                                                                                                                                                            ,                           (17)
                                                         j                                                                          1+
                                      fR                 i=k+1
                                                                1 − f1
                                                                    fi                                                                       f1
                           =       1+                                     yj .                                                 E1 ∗
                                      fk                   j−1
                                                               1 + fR                                                          f1
                                                                                                                                  z         − b1
                                                           i=k     fi                                            λ∗
                                                                                                                  21   =                    fR
                                                                                                                                                   .                                    (18)
In other words,                                                                                                                   1+        f1


                           fj−k
                                        j
                                              1−         f1                          Substituting these in (12) and (15) and solving for λ∗ , we have
                                                                                                                                          32
                                        i=k+1            fi
                    1−          ≤       j−1
                                                              .
                            fj                1+         fR                                                                E1
                                                                                                                              (1   −  f1
                                                                                                                                         )                  b1 (1 −    f1
                                                                                                                                                                          )
                                        i=k+1            fi                                    fR                E2        f1         f2                               f2
                                                                                     λ∗ (1 +
                                                                                      32          ) = z∗            +               fR
                                                                                                                                                    −                 fR
                                                                                                                                                                               − b2 .
This is proved in Lemmas 3 and 4 in the Appendix.                                              f2                f2          1+     f1
                                                                                                                                                                1+    f1
  The crucial step in this proof is the validity of Lemma 3, which
guarantees the feasibility of the conjectured dual solution. The                     Repeating this calculation, we obtain the following expression for
validity of this Lemma depends on the exact structure of the cost                    λ∗
                                                                                      kk−1 ,
function f (·), as will be evident in its proof.                                                      k−1         k−1              f1        k−1                k−1             f1
                                                                                                            Ei    j=i+1
                                                                                                                           1−      fj                           j=i+1
                                                                                                                                                                          1−    fj
B. Attaining the Dual Upper Bound                                                    λ∗
                                                                                      kk−1   =z   ∗
                                                                                                                    k−1
                                                                                                                                        −              bi           k−1
                                                                                                                                                                                     . (19)
                                                                                                            fi            1+   fR
                                                                                                                                                                          1+   fR
                                                                                                      i=1           j=i        fj            i=1                    j=i        fj
  It turns out that the RHS of (7) is the objective function value
corresponding to a particular assignment of variables for the primal                 By setting λ∗
                                                                                                 n+1n = 0 (since there is no node n + 1), we then obtain
problem, which is however not always feasible. But when it is, it is                 the following expression for z ∗ :
automatically optimal due to strong duality. We now describe how to                                                                     n               f
                                                                                                                                                  1− f1
obtain this assignment of variables.                                                                                      n
                                                                                                                             b          j=i+1      j
                                                                                                                          i=1 i          n     f
                                                                                                                                            1+ fR
  Consider the following 2n equations:
                                                                                                             z∗ =
                                                                                                                                        j=i      j
                                                                                                                                        n         f
                                                                                                                                                                ,                       (20)
                                                                                                                                              1− f1
                   Feasible flow constraints                                   (10)                                        n   Ei        j=i+1       j
                                                                                                                          i=1 fi         n      fR
                               λ∗ f1 + λ∗ fR
                                10      21               =    z ∗ E1          (11)                                                       j=i
                                                                                                                                             1+ f
                                                                                                                                                  j

                   λ∗ f2
                    20     +   λ∗ f1
                                21     +   λ∗ fR
                                            32           =    z ∗ E2          (12)   which is the same as the upper bound derived in Theorem 1. The
                                                ···                                  above calculations show that the equations (10) do indeed admit
            λ∗ fk + λ∗          ∗
                                                              z ∗ Ek                 solutions. However, these solutions correspond to a valid flow only
             k0      kk−1 f1 + λk+1k fR                  =                    (13)
                                                                                     if the λ∗ ’s and λ∗
                                                                                              k0           kk−1 ’s are non-negative. Duality thus yields the
                                                ···
                                                                                     following result.
                           λ∗ fn + λ∗
                            n0      nn−1 f1              =    z ∗ En ,                  Theorem 2: If the set of equations (10) have non-negative solu-
                                       λ∗
                                        10   − b1        =    λ∗
                                                               21             (14)   tions, the corresponding communication flow, in which each node i
                               λ∗ +    λ∗    − b2        =    λ∗              (15)   communicates only to the two nodes, node i − 1 and the collector
                                21      20                     32
                                                                                     node 0, is optimal with respect to the primal linear program (6). The
                                                ···
                                                                                     normalized cost incurred by each node under this flow is the same,
                  λ∗         ∗
                   k−1k−2 + λk−10 − bk−1                 =    λ∗
                                                               kk−1           (16)   and consequently all nodes in the network die at the same time. The
                                                ···                                  optimal functional lifetime is given by the inverse of the expression
                  λ∗         ∗
                                                              λ∗                     in (20).
                   n−1n−2 + λn−10 − bn−1                 =     nn−1
                                                                                        Figure 2 shows the form of the optimal strategy. The proof of
                           λ∗
                            nn−1   +   λ∗
                                        n0   − bn        =    0,
                                                                                     Theorem 2, depending as it does on the construction of the dual
with unknowns      λ∗
                    ii−1 and    λ∗
                                 i0for each 1 ≤ i ≤ n, and z ∗ . If                  solution and the solubility of the set of equations (10), seemingly
these equations are solvable, and if the λ∗ ’s and λ∗ ’s are all
                                                ii−1         i0                      gives no intuitive reason why such a particular combination of long
nonnegative, then the solution to these equations clearly yields a                   and short hops is optimal. In fact, it is possible to construct a
feasible solution to the original linear program (6) (after setting all              direct proof that such a strategy is optimal, by employing variational
the other flows to zero). The communication flow defined by this                        arguments to show that any other strategy is suboptimal. Such a proof
solution consists of each node sending a part of its information to                  is much more lengthy than the one provided here. It turns out that
the collector node, and the rest to its nearest neighbor in the direction            the inequality proved in Lemma 4 is one of the key steps in this
of the collector. The exact quantities are chosen in such a way as to                alternate proof.
equalize the normalized costs incurred by each node (if possible,                       Theorem 2 provides sufficient conditions which are purely in terms
otherwise the solution is not valid). This corresponds to all nodes                  of the bi ’s and Ei ’s. However, the form of the conditions is somewhat
depleting their energy supply at the same time.                                      involved. The following lemma gives simpler sufficient conditions.
                                                                                                                                     1 1 1
                                                                                                                                     0 0 0
  Lemma 2: If bEfi and Ei are non-decreasing in i, then the set of
                 i
                                                                                                                                     1 1 1
                                                                                                                                     0 0 0
                                                                                                                                   0 0 0 0 0
                                                                                                                                   1 1 1 1 1
                                                                                                                                   1 1 1 1 1
                                                                                                                                   0 0 0 0 0
                   i
equations (10) have non-negative solutions.                                                                                            0
                                                                                                                                       1
Proof. We first prove that λ∗                                                                                                      00 0 0 0 0 0
                                                                                                                                  11 1 1 1 1 1
                            k+1k ≥ 0 for all k. For convenience,                                                                  11 1 1 1 1 1
                                                                                                                                  00 0 0 0 0 0
                           j
                                   1− f1
                                        f                                                                                              0
                                                                                                                                       1
                                                                                                                                  000000000
                                                                                                                                  111111111
denote g(i, j) :=          l=i+1            l
                                                . Substituting (19), this is the same                                             000000000
                                                                                                                                  111111111
                            j        fR
                                                                                                                                       1
                                                                                                                                       0
                                                                                                                                  11 1 1 11 1
                                                                                                                                  00 0 0 00 0
                                   1+ f
                                                                                                                                  00 0 0 00 0
                                                                                                                                  11 1 1 11 1
                               l=i     l
as proving that
                                                                                                                                   0 0 0 0 0
                                                                                                                                   1 1 1 1 1
                                                                                                                                   0 0 0 0 0
                                                                                                                                   1 1 1 1 1
                                                 k
                                                    b g(i, k)
                               z∗ ≥              i=1 i
                                                               .                                                                     0 0 0
                                                                                                                                     1 1 1
                                                k   Ei                                                                               1 1 1
                                                                                                                                     0 0 0
                                                                                                                                       0
                                                                                                                                       1
                                                i=1 fi
                                                       g(i, k)
                                                                                                                            Fig. 3.    The regular planar network
Cross multiplying and canceling the common terms, we need to prove
that
           n                           k
                                                 Ei                                                              +fR bk
                 bi g(i, n)                         g(i, k)                                                             f1
                                                 fi                                                               1 − fk−1
         i=k+1                         i=1
                                                                                                            =              fR
                                                                                                                                  (Ek−1 z ∗ − (fR + f1 )λ∗
                                                                                                                                                         k−1k−2 + fR bk−1 )
                                                 n
                                                       Ei
                                                                             k                                        1+   fk−1
                                  ≥                       g(i, n)                  bi g(i, k)      .                                              fR f1
                                                       fi                                                                  fk−1 + fR − fR +       fk−1
                                           i=k+1                             i=1
                                                                                                                 +bk−1                      fR
                                                                                                                                                          + bk fR
   Now, for any k + 1 ≤ l ≤ n, and 1 ≤ m ≤ k, by assumption we                                                                        1+   fk−1
have bl Em ≥ bm El . This implies that the lmth cross term in the
        f
          m
                   fl
                                                                                                            ≥    fR bk + f1 bk−1                                           (23)
LHS is greater than or equal to the lmth cross term in the RHS, for                                         ≥    0,
each l and m. This proves that λ∗ k+1k ≥ 0 for all k ≥ 0.
   What is left to prove is that λ∗ ≥ 0 for all k. From (17), it is
                                  k0                                                                    where (22) follows from the assumption that Ek ≥ Ek−1 , and (23)
clear that λ∗ > 0. Solving for λ∗ in (13) and (16), we obtain that
            10                    k0                                                                    from the induction hypothesis.
for k ≥ 2,
                  fR     1
        λ∗ (1 +
         k0          )=    (Ek z ∗ − (fR + f1 )λ∗
                                                kk−1 + fR bk ).                                          IV. F UNCTIONAL L IFETIME OF A R EGULAR P LANAR N ETWORK
                  fk    fk
Thus, we need to prove that for 2 ≤ k ≤ n,                                                                 Consider the planar network shown in Figure 3. The center node
                       ∗
                  Ek z − (fR +          f1 )λ∗              + fR bk ≥ 0.                                is the collector, and there are N concentric circles, each containing
                                             kk−1
                                                                                                        nodes along its circumference. The ith ring, of radius iR, contains
We prove this using induction on k. For the base case k = 2, we                                         M i nodes, equally spaced along the circumference. There are thus
have                                                                                                    a total of N(N+1) M nodes. This particular configuration for the
                                                                                                                        2
                                                                                                        network is chosen simply as a convenient example of a regular planar
            E2 z ∗ − (fR + f1 )λ∗ + fR b2
                                21
                                                                                                        network, due to its circular symmetry. Any other regular arrangement
                                                            E1 ∗
                                                            f1
                                                               z     − b1                               of nodes would admit similar results, though the analysis might be
                     ≥ E1 z ∗ − (fR + f1 )                            fR
                                                                                 + fR b2 (21)           more cumbersome.
                                                                1+    f1
                                                                                                           The direct approach of the last section is considerably harder in
                     = b1 f1 + b2 fR ≥ 0,
                                                                                                        the planar case. However, we can use the following simple idea to
where (21) follows by substituting (18) and the assumption that E2 ≥                                    obtain an upper bound. Suppose that all intra-ring communication,
E1 . Suppose now that the induction hypothesis is true for k − 1.                                       i.e., communications between nodes belonging to the same ring, have
Substituting for λ∗
                  k−1k−2 in (19), we get that                                                           zero cost. Further, let the cost of a unit of transmission from any
                                 f1                                 Ek−1 ∗                              node in ring i to any node in ring j = i be f ((i − j)R), i.e. as if the
                       1−       fk−1                                fk−1
                                                                         z   − bk−1                     distance between the two nodes were equal to the distance between
        λ∗
         kk−1    =                fR
                                                λ∗
                                                 k−1k−2         +             fR
                                                                                        .
                       1+                                               1+                              the two rings (when in fact the former is larger than or equal to the
                                fk−1                                         fk−1
                                                                                                        latter). Since all link or hop communication costs are thereby either
Then,                                                                                                   reduced or remain the same, it is clear that such a modified cost
                                                                                                        network would have functional lifetime greater than or equal to the
 Ek z ∗ − (fR + f1 )λ∗
                     kk−1 + fR bk
                                                                                                        original planar network. Therefore, an upper bound on the lifetime
                                                         f1                                             of this modified cost network is also an upper bound on the lifetime
                                                 1−
    ≥     Ek−1 z ∗ − (fR + f1 )
                                                        fk−1
                                                                    λ∗                                  of the original network.
                                                           fR        k−1k−2
                                                 1+     fk−1                                               The treatment of this modified cost network is simplified by the
                                 Ek−1 ∗                                                                 fact that it is equivalent to the following linear network: Replace
                                      z          − bk−1
          −(fR + f1 )
                                 fk−1
                                                                + fR bk                          (22)   the ith ring with a super-node si , located at a distance iR from the
                                                 fR
                                    1+          fk−1
                                                                                                        collector node. The initial energy of this super-node is     j
                                                                                                                                                                       Eij , and
                         fR                           fR                 f1                             the number of bits initially held by it is equal to ij bij , where Eij
            Ek−1 + Ek−1 fk−1 −                           E
                                                     fk−1 k−1
                                                                    −       E
                                                                        fk−1 k−1
    =                                                                                       z∗          and bij are the amount of energy and number of bits respectively
                                                      fR
                                        1+           fk−1                                               of the j th node in ring i of the original planar network. The set of
                                1−      f1                                                              super-nodes thus forms a linear array with inter-node distance R. The
                                       fk−1                             (fR + f1 )bk−1
          +(fR + f1 )                   fR
                                                       λ∗
                                                        k−1k−2 +               fR
                                                                                                        upper bound of Theorem 1 can therefore be directly applied (with
                                1+     fk−1
                                                                          1 + fk−1                      fi now denoting f (iR)) to obtain the following upper bound on the
                                          a                                                            dividing j by i, i.e., j = pi + q. This node communicates a fraction
                                                                                                          Bi
                                                                                                       Ai +Bi
                                                                                                               of its total load to the collector node, and the remaining to ring
                                                                                                       (i−1). Of these remaining bits, fractions i−1 and i−1−q respectively
                                                                                                                                                      q
                                                                                                                                                               i−1
                                                                                                                                    (p(i−1)+q−1)2π
                                                                                                       are transmitted to nodes         M (i−1)
                                                                                                                                                     and (p(i−1)+q)2π of the i −
                                                                                                                                                            M (i−1)
                                                 θ                                                     1th ring.
                                                                                                          This ensures that for each i ≥ 2, each node in the (i − 1)th ring
                        e                 b                   d                                                                i
                                                                                                       receives a total of i−1 Bi bits from ring i. To see this, suppose that
                                                                                                       each node in the ith ring sends the same number of bits to the (i−1)th
                                                                                                                                                ′        ′
                                                                                                       ring. In the (i − 1)th ring, node (p (i−1)+q )2π receives a fraction
                                                                                                                                                  M (i−1)
                                          f                                                            q ′ +1
                                                                                                                from node    (p′ i+q ′ +1)2π
                                                                                                                                               in ring i, and a fraction   i−1−q ′
                                                                                                                                                                                     from
                                                                                                       i−1                         Mi                                        i−1
                                                                                                                 ′     ′
                                                                                                                (p i+q )2π                               i
                                                                                                       node             in ring i, for a total of     times the number of bits
                                          c                                                                         Mi                                  i−1
                                                                                                       sent by each node in ring i to ring i − 1. Backwards induction from
                      Fig. 4.      Bounding the constant K.                                            the outermost node, together with the flow conservation constraints,
                                                                                                       complete the proof.
                                                                                                          What is the normalized cost incurred by each node? The distance
                         1
functional lifetime      ∗
                        zp
                             of the regular planar network:                                            between each node in the ith ring and the collector node is iR.
                                     Mi                  n                      f
                                                                                                       However, the distance between each such node and the node(s) it
                                                                    1− f1
                        N            j=1
                                           Eij
                                                         j=i+1                      j                  transmits to in the (i − 1)th ring is greater than R. Let the largest
                                                          n           f
            1
                        i=1           fi
                                                             j=i
                                                                   1+ fR
                                                                             j
                                                                                                       such distance for any ring be KR. In that case, each node in the ith
             ∗
               ≤                                         n                  f
                                                                                            .   (24)   ring incurs a normalized cost that is less than K times the normalized
           zp            N           Mi
                                                                   1− f1
                         i=1
                                        b
                                     j=1 ij
                                                         j=i+1
                                                          n       f
                                                                                j
                                                                                                       cost incurred by the ith super-node under the optimal scheme derived
                                                               1+ fR
                                                         j=i                j                          above (because total load and energy are scaled by the same factor
This upper bound is valid for any set of Eij ’s and bij ’s.                                            M i).
  To provide lower bounds on functional lifetime, however, we will                                        The factor K can be bounded as follows. Consider Figure 4. Note
                                                                                                                            2π
have to impose further restrictions on the energies and information                                    that the angle θ = M i , |af | = iR, and |f c| = R. The two nodes in
                                                                                                            th
quantities of the sensor nodes. Suppose that it is further true that for                               the i ring nearest to node j must both be within the arc ef d. Thus,
each ring i,                                                                                           the maximum distance between j and any of these nodes is no more
                                                                                                       than |cd|. We then have,
            Eij = Ei and bij = ¯i , for all 1 ≤ j ≤ M i,
                  ¯            b                                                                (25)
and that
                ¯i fi
                                                                                                                     |cd|2   =    |bc|2 + |bd|2
                b            ¯
                        and iEi are non-decreasing in i.                                        (26)
                  ¯
                 Ei                                                                                                          =    (iR(1 − cos θ) + R)2 + (iR sin θ)2 ,
Then, Lemma 2 guarantees the achievability of the following func-                                                            =    R2 (1 + 2i(i + 1)(1 − cos θ)),
tional lifetime for the linear network of super-nodes:                                                                       =    R2 (1 + 2i(i + 1)2 sin2 (θ/2),
                                                     n         f                                                                                       π 2
                             N      ¯
                                    Ei M i           j=i+1
                                                           1− f1
                                                                 j                                                           ≤    R2 (1 + 2i(i + 1)2(      ) ,
                             i=1      fi              n      fR                                                                                       Mi
               1                                          1+ f                                                                               π
                  ≥
                                                      j=i     j
                                                                                        .       (27)                         ≤    R2 (1 + 8( )2 ).
              zp∗                                    n
                                                              1− f1
                                                                    f                                                                       M
                             N      ¯i M i
                                    b                j=i+1
                                                      n         f
                                                                        j
                                                                                                         Thus,
                             i=1                             1+ fR                                                                                       π
                                                     j=i           j                                                                    f (R      1 + 8( M )2 )
   Now we construct a routing scheme for the actual planar network                                                               K≤                             .                    (28)
                                                                                                                                                  f (R)
which achieves this solution to within a constant factor. This scheme
                                                                                                          The results of this section are summarized in the following
attempts to mimic the optimal scheme for the linear super-node
                                                                                                       theorem.
network, with each ring equally dividing the flow that it transmits
                                                                                                          Theorem 3: Consider the regular planar network of Figure 3. Its
to the next inner ring among all the nodes in that inner ring. The
                                                                                                       optimal functional lifetime is upper-bounded by the right hand side
constant factor gap arises because the distance between two nodes in
                                                                                                       of (24). If the initial energies and information quantities in the planar
successive rings is larger than the inter-ring distance.
                                                                                                       network further satisfy conditions (25) and (26), then there exists a
   Let λ∗             ∗
          ii−1 and λi0 be the respective amounts of information                                                                                              1
                                                                                                       routing strategy achieving a functional lifetime of K times the right
communicated from super-node si to super-node si−1 and to the
                                                                                                       hand side of (27), where K is given by (28).
collector node respectively, under the optimal flow. Define Ai :=
λ∗ii−1             λ∗                                                                                  It should be noted that as m → ∞, the factor K approaches 1,
  Mi
       , and Bi := M i .
                     i0
                                                                                                       yielding asymptotic optimality. This indicates that the routing strategy
   The nodes in the first ring communicate all their bits to the                                        described above is nearly optimal in large dense networks.
collector. This scheme will ensure that each node in the ith ring
has a total load of Ai + Bi bits, i.e., each node receives an equal                                                              V. S CALING I MPLICATIONS
amount of incoming flow, and further that each such node will incur                                        We now evaluate the simple routing strategy consisting of commu-
the same cost.                                                                                         nicating only with the nearest neighbor in the direction of the collec-
   Consider the ith ring, for i ≥ 2. Index each node by the angle of                                   tor. Because it involves only short hop communication, this scheme is
the radial line connecting the node to the origin. These angles are                                    an appealing one for operational reasons, such as minimizing medium
multiples of M i . Consider the node located at the angle 2πj . Let
               2π
                                                              iM
                                                                                                       access contention. However, from the point of view of lifetime, it is
p and q respectively be the quotient and remainder obtained upon                                       not clear whether it is a good routing strategy, because it places
a higher load on nodes close to the sink. Thus, it is worthwhile to        more powerful communication infrastructure, are more expensive to
compare it with the optimal strategy derived in this paper. Henceforth,    maintain, whereas sensor nodes are cheap. On the other hand, the
it will be referred to as simply the nearest neighbor strategy.            number of sensor nodes per collector node is limited by such lifetime
   Consider first the linear case. Let α = 2, γ = 0, and fR = 0.            and throughput considerations. One of the motivations for performing
These are the smallest values within the range we consider, so the         in-network processing is precisely to limit the communication burden
lifetime obtained will be the largest possible. Let Ei = E and bi = b      on nodes which are close to the collector.
for all i, and let the inter-node distance be d = 1. Under the nearest
neighbor strategy, the node next to the collector is the the bottleneck,                 VI. C ONCLUSIONS AND F UTURE W ORK
and it will transmit nb bits. Hence, the functional lifetime under this
              E                                                               The main contribution of this paper is to provide sharp bounds,
scheme is bf1 n .
   Now, consider the lifetime corresponding to the optimal strategy.       and in some cases exact solutions, to the functional lifetime problem
We have                                                                    for spatially regular networks. Even here, some gaps remain. The
        n                  n                                               characterization of the optimal solution holds only under certain
                  f1              (j − 1)(j + 1)   i(n + 1)                conditions on the energies and traffic requirements of the network.
             1−      =                           =          .
                  fj                    j2         (i + 1)n                Nevertheless, we believe that these conclusions hold broadly for large
     j=i+1               j=i+1
                                                                           sensor networks.
Substituting in (24), we get
                                                                              Another interesting extension involves considering more general
                                           n       1
                   1               E       i=1 i(i+1)
                                                                           cost functions. In practice, power levels belong to a discrete set,
                          =                 n      i                       and all pairs of nodes may not be connected. Such effects could be
                   z∗             bf1       i=1 i+1
                                           n     1     1
                                                                           modeled by suitably changing the cost function. The key property
                                   E       i=1 i
                                                    − i+1                  of the cost function that underlies all our results is the inequality in
                          =                n           1
                                  bf1      i=1
                                                 1 − i+1                   Lemma 4. It would be worth investigating if a similar result holds
                                   E    1      1                           for more general functions.
                          ≈                           .                       One major limitation of our model is that we treat information as
                                  bf1 n + 1 1 − log n
                                                  n
                                                                           incompressible, whereas in-network processing [8] and compression
   Thus, not only is the optimal lifetime of the same order as the         of correlated sources might substantially reduce the relaying burden.
lifetime under the nearest neighbor strategy, but the ratio of the two     Indeed, our results suggest the need for such techniques to increase
converges to 1 as the size of the network goes to ∞. Thus, in the          lifetime.
linear case at least, the nearest neighbor strategy is nearly optimal.
   A similar calculation can be performed for the regular planar
                                                                                                    ACKNOWLEDGMENTS
network of Figure 3. The nearest neighbor strategy in this case
involves each node in the ith ring transmitting to its two nearest            This material is based upon work partially supported by
neighbors in the (i − 1)th ring just as in the scheme described in the     AFOSR under Contract No. F49620-02-1-0217, NSF under Con-
last section. The nodes in the first ring would then incur the greatest     tract Nos. NSF ANI 02-21357 and CCR-0325716, USARO un-
                                                    N
                                           Kf (R)         ib
normalized cost, which would be             i=1
                                                   = N(N+1)bf (R) .        der Contract Nos. DAAD19-00-1-0466 and DAAD19-01010-465,
                                          E                2E
The functional lifetime associated with this scheme is thus upper          DARPA/AFOSR under Contract No. F49620-02-1-0325, and DARPA
                  2E
bounded by KN(N+1)bf (R) . On the other hand, the optimal functional       under Contact Nos. N00014-0-1-1-0576.
lifetime can be bounded by substituting γ = 0 and α = 2 in (24), to
get                                                                                                      R EFERENCES

                                        N
                                                                           [1] A. Mainwaring, D. Culler, J. Polastre, R. Szewczyk, and J. Anderson,
                                              1
             1             E            i=1 i+1                                “Wireless sensor networks for habitat monitoring,” in WSNA ’02: Pro-
                  =                     N                                      ceedings of the 1st ACM international workshop on Wireless sensor
             z∗          bf (R)              i2
                                        i=1 i+1                                networks and applications. ACM Press, 2002, pp. 88–97.
                                            N     1                        [2] J.-H. Chang and L. Tassiulas, “Maximum lifetime routing in wireless
                           E                i=1 i+1                            sensor networks,” IEEE Trans. Netw., vol. 12, no. 4, pp. 609–619, 2004.
                  =                     N
                         bf (R)                      1
                                            i − 1 + i+1                    [3] A. Sankar and Z. Liu, “Maximum lifetime routing in wireless ad-hoc
                                        i=1
                                                                               networks,” in Proceedings of IEEE Infocom ’04, Hong Kong, 2004, pp.
                           E        2                 logN
                  ≈                             N−1       log N
                                                                               1089–1097.
                         bf (R) N (N + 1)       N+1
                                                      − N(N+1)             [4] K. Kalpakis, K. Dasgupta, and P. Namjoshi, “Efficient algorithms for
                               2E                                              maximum lifetime data gathering and aggregation in wireless sensor
                  ≈                      logN.                                 networks,” Comput. Networks, vol. 42, no. 6, pp. 697–716, 2003.
                         bf (R)N (N + 1)                                   [5] R. Madan and S. Lall, “Distributed algorithms for maximum lifetime
There is thus a log N factor improvement in this case. Note that for           routing in wireless sensor networks,” in Proceedings of IEEE Globecom
                                                                               ’04, Dallas, 2004, pp. 748–753.
larger values of α, the ratio between the gap in performance between
                                                                           [6] M. Bhardwaj, A. Chandrakasan, and T. Garnett, “Upper bounds on the
the optimal scheme and the nearest neighbor scheme will further                lifetime of sensor networks,” in Proceedings of IEEE ICC 2001, Helsinki,
reduce.                                                                        Finland, 2001, pp. 785–790.
   Some conclusions follow from these scaling results. First, the          [7] S. Narayanaswamy, V. Kawadia, R. Sreenivas, and P. Kumar, “Power
lifetime scales down sharply with the size of the network. This is             control in ad-hoc networks: Theory, architecture, algorithm and im-
                                                                               plementation of the COMPOW protocol,” in Proceedings of European
not very surprising in itself, but underscores one of the problems             Wireless Conference 2002, Florence, Italy, 2002, pp. 156–162.
with the “many sensors one collector” network configuration. This           [8] A. Giridhar and P. Kumar, “Computing and communicating functions over
seems to be one of the major tradeoffs to be made in sensor networks,          sensor networks,” IEEE Journal on Selected Areas of Communication, to
since collector nodes, which are essentially gateways to a larger and          appear.
                                A PPENDIX                                                We can cancel out the common factor eγa . Letting L(a) :=
  Lemma 3: For any 1 ≤ k < j,                                                            e−γa dRHS(a) , R(a) := e−γa dLHS(a) , we have
                                                                                                 a                      a


                           fj−k
                                         j
                                               1−   f1
                                                    fi
                                                                                               L(a)       =    (α + γ(a + b))(a + b)α−1 b(b + c)α−1 eγb
                                         i=k+1
                   1−           ≤        j−1
                                                         ,                                                     +γbc(a + b)α−1 (b + c)α−1 eγb
                            fj                 1+   fR
                                         i=k+1      fi
                                                                                                               +(γa + α)c(a + b)α−1 (b + c)α−1 eγb
where fi := (id)α eγ(id) , with α ≥ 2 and γ ≥ 0.
                                                                                                               −(α + γ(a + b))(a + b)α−1 cα ).                   (30)
Proof. We prove this for fixed k, by induction on j. For j = k + 1,
the LHS and the RHS are the same, so the inequality is true with                         Let the four summands in (30) be L1 , L2 , L3 , L4 . Now,
equality. Suppose it is true for some j > k. From the induction
hypothesis,                                                                                     R(a)      =     (γ(a + b) + α)(a + b + c)α−1 eγa bα eγb
          j+1          f1                                      f1                                               +γc(a + b + c)α−1 eγa bα eγb
                1−                      fj−k             1−   fj+1
          i=k+1
          j
                       fi
                              ≥      1−                       fR
                                                                         .                                      +K(α + γ(a + b + c))(a + b + c)α−1 .             (31)
                1+     fR                fj              1+   fj
          i=k+1        fi
                                                                                         Let the three summands in (31) be R1 , R2 , R3 . Since (a+b)(b+c) >
It is thus enough to prove that                                                          (a + b + c)b, we have L1 > R1 and L2 > R2 . It remains to prove
                              1−    f1                                                   that L3 + L4 ≥ R3 . Now,
                fj−k               fj+1                fj+1−k
           1−                               ≥   1−                   .
                 fj           1+    fR                  fj+1                                    L3    ≥       (α + γa)(1 + γb)c(a + b)(b + c)α−1
                                    fj
                                                                                                      =       (α + γa + γαb + γ 2 ab)c(a + b)(b + c)α−1
Cross multiplying and rearranging terms, this is equivalent to proving
that                                                                                                  ≥       (α + γ(a + b + c))c(a + b)(b + c)α−1 ,             (32)

 fj−k f1 −f1 fj +fj fj+1−k ≥ fj+1 fj−k +fR (fj+1 −fj−k+1 ). (29)                         since α ≥ 2 and b ≥ c. Thus,

Let a := k, b := j − k, c := 1. Adding and subtracting fj+1−k fj−k ,                     L3 + L4 > (α + γ(a + b + c))c(a + b)α−1 ((b + c)α−1 − cα−1 ).
and canceling out the common factor d2α (recall that fi = f (id)),
                                                                                         Since c ≥ 1, it remains to prove that
this is equivalent to proving that
                                                                                             (a + b)α−1 ((b + c)α−1 − cα−1 ) ≥ K(a + b + c)α−1 .                 (33)
     ((a + b)α eγ(a+b) − bα eγb )((b + c)α eγ(b+c) − cα eγc ) ≥
                           α γ(a+b+c)            α γ(b+c)          α γb           ′                                             (a+b)α−1 ((b+c)α−1 −cα−1 )
          ((a + b + c) e                 − (b + c) e          )(b e          +   fR ),   Consider the expression P (a) :=             (a+b+c)α−1
                                                                                                                                                           .   Differ-
         ′       fR                                                                      entiating with respect to a, we get
where fR :=     E t dα
                       .   This in turn is proved in Lemma 4, which
follows.                                                                                       dP (a)                      c(a + b)α−2 ((b + c)α−1 − cα−1 )
                                                                                                          =      (α − 1)
                                                                                                da                                  (a + b + c)α
  Lemma 4: Given any real numbers a ≥ 0, b ≥ c ≥ 1, α ≥ 2,                                                ≥      0.                                              (34)
γ ≥ 0, K ≤ 1 − (1/2)α−1 ,
                                                                                         Therefore, the ratio is minimum at a = 0. Thus,
     ((a + b)α eγ(a+b) − bα eγb )((b + c)α eγ(b+c) − (c)α eγ(c) ) ≥
                                                                                                P (a)     ≥      P (0)
          ((a + b + c)α eγ(a+b+c) − (b + c)α eγ(b+c) )(bα eγb + K).                                                             cα−1
                                                                                                          =      bα−1 (1 −              )
                                                                                                                             (b + c)α−1
Proof. The common factors eγ(b+c) can be cancelled out from both                                                   1
                                                                                                          ≥   1 − ( )α−1 ≥ K.
sides. Denote the left hand side and right hand side of the resulting                                              2
inequality, as functions of the variable a, respectively by LHS(a),                      Note: The conditions on a, b, c and K for the above inequality to hold
and RHS(a).                                                                              may seem somewhat arbitrary. This is mainly due to the presence of
   F (a) := LHS(a) − RHS(a) is a differentiable function of a,                           the constant K. If K = 0, the inequality has a more symmetric
and F (0) = 0. It is enough to prove that dF (a) > 0 for all a > 0.
                                              da                                         structure, and is true for any a, b, c ≥ 0, and α > 1. If, further,
We have                                                                                  the constant γ = 0, the inequality follows in straightforward fashion
dLHS(a)                                                                                  from Jensen’s inequality, due to the convexity of the function xα
            = (γ(a + b) + α)(a + b)α−1 eγa ((b + c)α eγb − cα ).
    da                                                                                   (but not only from its convexity; the inequality does not hold for all
Also,                                                                                    convex functions). We are not aware if any form of this inequality is
dRHS(a)                                                                                  already known.
        = (γ(a + b + c) + α)(a + b + c)α−1 eγa (bα eγb + K).
   a

								
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