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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Connectivity, Power, and Energy in a Multihop
Cellular-Packet System
Sayandev Mukherjee, Senior Member, IEEE, Dan Avidor, Member, IEEE, and Katherine Hartman
Abstract—In this paper, we study a large network of subscriber stations (SSs) with certain common wireless capabilities
and base stations (BSs) having direct connections to the wired
infrastructure in addition to common wireless capabilities. SSs
can communicate with the “outside world” only through the BSs.
Connections to SSs without a direct (i.e., a single hop) wireless
connection to any BS are established, if possible, through other
SSs serving as wireless repeaters. The locations of the SSs and the
BSs follow independent homogeneous planar Poisson processes.
The propagation channels exhibit signal attenuation with distance
and log–normal shadowing. We evaluate exactly the probability
of an SS to have a direct wireless connection to any of the BSs
and a lower bound on the t-hop (t arbitrary) outage probability
of an SS. We then define the minimal hop-count routing algorithm
and calculate the mean number of hops for routes connecting SSs
to BSs, when a maximum hop-count constraint is imposed. We
compute next the probability distribution of the transmit power
under the assumption of perfect power control. We conclude by
calculating a bound for the total mean transmit energy required
to transfer a data packet from an SS to a BS over a minimal
hop-count route and show that this energy is significantly lower
than the corresponding value in a single-hop network operating at
the same outage probability.
Index Terms—Multihop network, outage probability, routing,
transmit energy.
I. I NTRODUCTION
E STUDY a large network of wireless transceivers that
we call nodes. Two types of nodes are involved: subscriber stations (SSs) with certain common wireless capabilities
and base stations (BSs) having direct wideband connection to
the wired infrastructure in addition to the common wireless
capabilities. SSs can communicate with the “outside world”
only through the BSs. To augment connectivity, connections to
SSs without a direct (i.e., a single hop) wireless connection to
any BS are established, if possible, through other SSs serving
as wireless repeaters, as long as the number of hops does not
exceed a prescribed limit. Regular SSs are mobile or installed
at customer premises, and their locations cannot be predicted
ahead of time; therefore, we assume that their locations are
random. We further assume that owing to practical constraints,
W
Manuscript received January 19, 2005; revised December 1, 2005 and
March 11, 2006. Parts of this work were presented at the IEEE VTC 2005Fall, Dallas, TX, and IEEE SECON 2005, Santa Clara, CA. The review of this
paper was coordinated by Dr. Q. Zhang.
S. Mukherjee and D. Avidor are with Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974 USA (e-mail: sayan@lucent.com; avidor@
lucent.com).
K. Hartman is with the Massachusetts Institute of Technology, Cambridge,
MA 02139 USA (e-mail: khartman@mit.edu).
Digital Object Identifier 10.1109/TVT.2007.891428
availability of high-speed wired connections, and economic
considerations, BSs are sparse and often cannot be positioned
based on coverage considerations only. To account for this
reality, we assume that the BSs, like the regular SSs, are also
placed randomly over the service area.
This paper focuses on two related issues. The first is the
probability that an SS in a fixed arbitrary location (or while
passing such a location) has a “working” wireless connection
to any of the BSs, evaluated as a function of the densities of
the BSs and SSs, the statistical properties of the propagation
channels, and the limit set on the maximum number of hops
if any. Such a limit is typically set due to delay and possibly
capacity considerations. We derive analytical results and lower
bounds when exact results are not obtainable. To assess the
tightness of the bounds, we compare them with simulation
results. We focus next on the well-known minimal hop-count
routing algorithm and calculate the mean number of hops for
routes connecting SSs to BSs. We then turn to the second
issue, which is the probability distribution of the transmit power
under the assumption of perfect power control, i.e., transmitters
transmit just enough power to be “properly” received. We conclude by calculating an expression for the total mean transmit
energy required to transfer a data packet from an SS to a
BS. We show that this energy is significantly lower than the
corresponding value required in a single-hop network operating
at the same outage probability. Multihopping can, therefore,
save SS battery power.
This paper is organized as follows: In Section II, we briefly
discuss prior research work on connectivity in data networks. In
Section III, we define the nomenclature to be used in this paper
and the Poisson process that controls the spatial distribution
of SSs and BSs in our system. In Section IV, we define the
propagation model assumed in this paper. In Section V, we
derive the distribution of the number of nodes (BSs or SSs,
as the case may be) with a direct connection to an SS and an
exact expression for the single-hop probability of an SS outage.
We then proceed to derive a lower bound for the probability of
t-hop outage. In Section VI, we describe an application where
the derivations presented in the previous sections are utilized.
The well-known minimal hop-count routing algorithm is presented in Section VII, where we calculate the mean number of
hops conditioned on a preset limit on the maximum number of
hops. Section VIII considers the transmit power on multihop
links in a system with perfect power control. In Section VIII-A
and -B, we derive the cumulative distribution function and the
mean transmit power on a single hop from an SS to a BS
or another SS, respectively. In Section VIII-C, we calculate
the mean total transmit energy per packet on a minimum-hop
0018-9545/$25.00 © 2007 IEEE
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
route when at most tmax hops are allowed. In Section VIII-D,
we compare, by way of example, the mean total energy per
packet with that of a single-hop system operating at the same
outage probability. Section IX presents the results and explains
each figure and the way the data was derived. Section X
presents our conclusions.
II. R EVIEW OF R ELATED W ORK
The issue of connectivity of wireless ad hoc networks has
attracted many researchers. In this kind of network, the interesting question is the probability that the network of terminals
forms a connected graph of nodes so that a route can be
found from any node to any other node. A related issue is
that of coverage: Can a wireless mobile moving randomly
throughout the service area maintain continuous connectivity
with a network of fixed wireless terminals? Several researchers
focused on packet radio networks, in which transmitters with
fixed range R are distributed over a d-dimensional space
A = [0, l]d according to a homogeneous Poisson point process
with intensity D. In [1], the authors prove that in a 2-D
space and for
a given D, limA→∞ P{area is covered} = 0
when R = (1 − ǫ) ln A/(πD) for any ǫ > 0 and that
limA→∞ P{area is covered} = 1 (i.e., all points of A are
within distance R from a Poisson point) when R =
(1 + ǫ) ln A/(πD) for any ǫ > 0. They were also able
to
prove that P{network is fully connected} = 0, if R =
(1 − ǫ) ln A/(πD) for any ǫ > 0.
Santi and Blough [2] study the topic of connectedness with
high probability (WHP) for n independently and uniformly
placed transceivers. In their formulation, n is fixed and not a
random variable. They take the size A = [0, l]d of the deployment region as a parameter of the model and study the critical
transmitting range required to achieve connectivity WHP when
l goes to infinity. For a 1-D space of length l, they prove that,
if Rn ≥ 2l ln l, the network is connected WHP, where R is the
common radio range of the transceivers. The network is not
connected WHP if Rn ≤ (1 − ǫ)l ln(l), for 0 < ǫ < 1. They
generalize the sufficient condition for connectedness WHP for
the 2-D and 3-D cases and give a necessary condition for
connectedness WHP that is weaker than in the 1-D case. They
show that, for 2-D and 3-D networks, reducing the communications range yields progressively “less connected” networks,
making itself evident by the appearance of isolated nodes.
Their results can be applied to both dense and sparse networks.
Besides analytical results of static scenarios, they also present
simulation results of mobility scenarios. They show that for
a large range of parameters, mobile networks are effectively
stationary as far as connectivity is concerned. For 2-D and 3-D
spaces, they further investigate through simulations the minimal
values of R and n that ensure either a connected graph or the
formation of a single connected component that includes a large
fraction (e.g., 90%) of the nodes.
The above studies focused on pure ad hoc networks. Realizing that the capacity of ad hoc networks does not scale
well with the number of nodes, some researchers have recently
examined “hybrid networks” [3], [4], where a sparse network of
BSs is placed on a regular grid within an ad hoc network. The
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BSs are all connected via a high-data-rate wired infrastructure
and serve as relays for data packets generated by the nodes of
the ad hoc network. Dousse et al. [3] study hybrid networks,
where BSs positioned on a fixed square grid are added to a
low-density ad hoc network with nodes placed according to
an isotropic Poisson point process. They take the transmission
range R to be fixed and allow the intensity of the Poisson
process to change. Otherwise, their system model is basically
the same as in [1], but they also consider 1-D space and the
case of a strip of infinite length and fixed finite width. Like
[1], [2], and [5], they allow any number of hops. As aforementioned, the objective is to achieve connectivity between
any two arbitrarily chosen nodes, allowing the use of a pair
of BSs as repeaters. For the hybrid 1-D model, they provide
a bound on the maximum spacing between consecutive BS to
keep the probability of finding a node with no connection below
a prescribed value. For the 2-D and the “strip” models, they
provide simulation results of the probability of having a disconnected node.
Assuming nodes distributed in the plane according to a 2-D
spatial Gaussian distribution around the origin, Miller [6]
provides an analytical approximation, as well as a curvefitting function derived by regression to simulated data, for the
probability of a double-hop connection between two randomly
selected nodes. He also provides an upper bound to the probability of a t-hop connection. However, Miller’s model deals
with a single cluster of nodes. Chandler [7] considers a network
with node locations given by points of a homogeneous Poisson
process and derives an expression for the probability that two
nodes at a distance D apart cannot communicate in t hops or
less. Although not stated explicitly in his paper, his expressions
for t ≥ 3 are actually lower bounds on the exact probabilities
(see [8] for details).
The above studies do not consider fading. For the case of
nodes distributed according to a planar homogeneous Poisson
process with log–normal shadowing, Bettstetter and Hartmann
[9] derive an expression for the node isolation probability
(single-hop outage) but do not obtain a closed form for this
expression. Orriss and Barton [10] derive an exact closed-form
expression for the node-isolation probability with log–normal
shadowing for arbitrary densities. They also allow two-slope
propagation models and varying spatial densities (in this paper,
we provide a different and independent derivation of an exact
closed-form expression (18) for the single-hop outage probability, which matches the one in [10]).
All of the above studies ignored the effects of cochannel
interference. Zorzi and Pupolin [11] consider randomly located
cochannel interferers and account for log–normal shadowing
and Rayleigh fading. However, in this paper, the authors address
a different issue, namely, the probability that a data packet is
received correctly by the intended receiver under interference
caused by concurrent transmissions of a random set of other
nodes with no central control, no power control, and under the
assumption that all the nodes have the same probability of being
on (i.e., transmitting) or off (i.e., not transmitting). They apply
the results to the computation of the throughput of a packet
radio network. It is, therefore, conceptually different from the
multihop connectivity that we study in our paper.
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
III. L OCATION M ODEL FOR BS S AND SS S
We begin with the following model assumptions.
1) BSs are points of a homogeneous Poisson process on the
plane with intensity λBS , i.e.,
i) The number of BSs in any finite region A, NBS (A), is
a Poisson random variable with mean given by λBS ×
area (A), i.e.,
n
[λBS × area(A)]
,
n!
SS
KSS
SS
PT,max
δ
rij
SS
10zij /10 > Pmin
⇔ rij < rSS 10zij /(10δ) , with rSS ≡
P{NBS (A) = n} = e−λBS ×area(A)
×
6) SS i has a direct connection to (i.e., is one hop away from)
another SS j if, and only if, the received power exceeds
SS
, i.e., if, and only if
some given threshold Pmin
SS
SS
PT,max
KSS
SS
Pmin
1/δ
(4)
n = 0, 1, . . . . (1)
ii) Conditioned on a given number of BSs in a
chosen region, say NBS (A) = n, the locations
(X1 , Y1 ), . . . , (Xn , Yn ) of these n BSs are independent and identically distributed (i.i.d.) uniformly over
that region: (X1 , Y1 ), . . . , (Xn , Yn ) i.i.d. with common probability density function (pdf):
1/area(A), (x1 , y1 ) ∈ A
fX1 ,Y1 (x1 , y1 ) =
(2)
0,
otherwise.
iii) The numbers of BSs in any two disjoint finite regions
A and B are independent, i.e.,
A ∩ B = ∅ ⇒ (∀m)(∀n)P{NBS (A) = m, NBS (B) = n}
= P{NBS (A) = m} × P{NBS (B) = n}.
where δ is the distance-loss exponent, 10zij /10 is the
shadow fade between SSs i and j, rij is the distance
SS
between the SSs, and KSS
is a constant taking into
account the parameters of an SS-to-SS link like antenna
gain, antenna height (again assumed equal for all SSs),
etc. Observe that (4) reduces to rij < rSS if there is no
shadow fading, i.e., if zij ≡ 0.
7) An SS i has a direct connection to a BS b if, and only if,
the received power at the BS on the uplink exceeds some
BS
given threshold Pmin
and the received power at the SS on
SS
the downlink exceeds Pmin
, i.e., if, and only if
BS
KSS
SS
PT,max
δ
rib
(3)
2) SSs are points of a homogeneous Poisson process on the
plane with intensity λSS , i.e., (1)–(3) hold with NSS (·)
and λSS in place of NBS (·) and λBS , respectively.
3) The BS process is independent of the SS process, i.e.,
for any two regions A and B (which may not be disjoint
in general), NBS (A) and NSS (B) are independent, and
given NBS (A) = n, the distribution (2) of the location of
the BSs in A is the same, regardless of the number and
location of the SSs in B.
IV. R ADIO P ROPAGATION M ODEL
We make the following assumptions.
1) There is attenuation with distance and log–normal
shadow fading.
2) The shadow-fade attenuations between any BS and SS,
and between any two SSs, are i.i.d. log–normal.1
3) The shadow-fade attenuation between any two nodes is
the same, regardless of which node is the transmitter and
which is the receiver.
4) All SSs are identical, and the maximum transmit power
SS
of any SS is PT,max
.
5) All BSs are identical, and the maximum transmit power
BS
of any BS is PT,max
.
1 The i.i.d assumption is widely used in the literature, even though field
measurements seem to indicate that the shadow fades between two links with
a common node are correlated [12]–[15]. However, the authors are not aware
of any correlation model, which has been widely endorsed by the scientific
community.
BS
KSS
BS
10zib/10 > Pmin
and
BS
PT,max
δ
rib
SS
10zib/10 > Pmin
⇔ rib < rBS 10zib /(10δ)
(5)
BS
rBS ≡ KSS
min
SS
BS
PT,max
PT,max
,
BS
SS
Pmin
Pmin
1/δ
(6)
where 10zib /10 is the shadow fade between the BS and
the SS, rib is the distance between the BS and the SS, and
BS
SS
KSS
is defined similarly to KSS
but takes into account
the antenna gains and heights of both the BS and the
SS. Observe that (5) reduces to rib < rBS if there is no
shadow fading, i.e., if zib ≡ 0.
V. O UTAGE P ROBABILITY
In this section, we derive expressions for the probability that
an arbitrarily located SS cannot communicate with any BS or
with any SS. We then show how to obtain lower bounds on the
probability that an arbitrary SS cannot communicate to any BS
in t hops or less.
It will be seen that the analysis of connectivity that follows
applies equally to the connectivity with BSs and with other
SSs. For brevity, in the derivation of these results, we use the
term “node” for either “BS” or “SS.” Accordingly, wherever
a subsequent result has a quantity with the subscript “n” (for
“node”), the corresponding result is true if “n” is replaced with
either “SS” or “BS” throughout the expression.
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
Now, observe that as r0 → ∞, we have
A. Distribution of the Number of Nodes With a
Direct Connection to an SS
lim r02 pn (r0 ) = lim EZ min r02 , rn2 exp(2hZ/δ)
Define the disk with radius r centered at (x, y) by
′
′
′
′
2
2
B(x, y; r) = (x , y ) : (x − x) + (y − y) ≤ r
r0 →∞
2
and the punctured disk obtained by deleting the center
= rn2 exp(2h2 σ 2 /δ 2 )
B (x, y; r) = B(x, y; r)\ {(x, y)} .
For brevity, we shall write B(r) and B ′ (r) in place of B(0, 0; r)
and B ′ (0, 0; r), respectively.
Recall our use of the subscript “n” in place of either “SS” or
“BS” throughout an expression. Consider an SS at (0, 0), and
let Nn′ (r0 ) ≡ Nn (B ′ (r0 )) be the number of nodes (BSs or SSs,
as the case may be, with range rn and corresponding intensity
λn ) whose distance from this SS is at most r0 (where r0 is
assumed to be very large). From the theory of homogeneous
Poisson point processes, we know that
Nn′ (r0 ) ∼ Poiss λn πr02 .
Let R denote the distance of an arbitrary node in B ′ (r0 ) from
the SS at (0, 0). Since the node is in B ′ (r0 ), it follows from
the theory of homogeneous Poisson processes that its location
must be uniformly distributed in B ′ (r0 ) and independent of the
location of any other node(s) in B ′ (r0 ). Thus, the pdf of R is
2r
, 0 < r ≤ r0
0,
r > r0 .
r02
(7)
Let 10Z/10 be the shadow-fade attenuation on the link between an arbitrary node in B ′ (r0 ) and the SS at (0, 0). Since the
shadow fades are assumed i.i.d. across links and the locations
of the nodes in B ′ (r0 ) are also i.i.d., it follows that the event
that there is a connection between the arbitrary node and the SS
at (0, 0) is also i.i.d. across nodes, with the common probability
being given by
pn (r0 ) = P{R < rn 10Z/(10δ) }
= P{R < rn exp(hZ/δ)}
= EZ
min{r0 ,rnexp(hZ/δ)}
0
where h =
ln 10
10
Nn′′ ∼ Poiss(µn exp(2α2 ))
(12)
µn = λn πrn2
(13)
where
α=
hσ
.
δ
(14)
Note that µn is the mean number of connected “neighbors” in
the absence of fading (σ = 0).
From (12), it follows that the probability that an SS at
(0, 0) has no direct connection to any of the “nodes” in the
network (where the “nodes” can be either the BSs or the SSs) is
given by
qn ≡ P {Nn′′ = 0} = exp −µn exp(2α2 ) .
(15)
Thus, taking the “nodes” to be the BSs, the probability that an
SS has no direct connection to any BS is given by
where
′′
qBS = P {NBS
= 0}
= exp −µBS exp(2α2 )
(16)
2
µBS = λBS πrBS
(17)
and similarly, taking the “nodes” to be the SSs, the probability
that an SS has no direct connection to any other SS is given by
(8)
2r
dr
r02
where
(9)
It follows from the theory of Poisson processes [16, Sec. 5.1,
para. 3, p. 53] that Nn′′ (r0 ), which is the number of nodes in
B ′ (r0 ) with a connection to the SS at (0, 0), has the following
Poisson distribution:
Nn′′ (r0 ) ∼ Poiss λn πr02 pn (r0 ) .
(11)
where we use the fact that Z ∼ N (0, σ 2 ) and the known
moment generating function of the Gaussian distribution.
From (11) and (10), we see that Nn′′ , which is the number of
nodes in the entire plane with a connection to the SS at (0, 0),
has the following distribution:
1
= 2 EZ min r02 , rn2 exp(2hZ/δ) .
r0
r0 →∞
= EZ rn2 exp(2hZ/δ)
′
fR (r) =
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(10)
′′
qSS = P {NSS
= 0}
= exp −µSS exp(2α2 )
(18)
2
µSS = λSS πrSS
.
(19)
B. Application: Triangulation-Based Positioning
The above analysis may be applied to the case where the
location of a mobile SS in a region served by multiple BSs
is to be determined through triangulation. For this, we require
the SS to have a connection to at least three BSs. Taking the
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
SS’s location to be the origin, the probability that the SS’s
location cannot be determined is just the probability that
′′
< 3} = qBS
P {NBS
2
µBS exp(2α2 )
.
1+ µBS exp(2α )+
2
2
therefore, the complementary cdf F̄T (t) ≡ 1 − FT (t). Clearly,
F̄T (1) = qBS . For arbitrary t ≥ 2, we can write
F̄T (t) = P{T > t}
′′ > t − 1}
= P{T > 1, T1 > t − 1, . . . , TNSS
′′ > t − 1|T > 1}
= qBS × P{T1 > t − 1, . . . , TNSS
(21)
C. Probability Distribution of the Distance Between an SS and
a Node With a Direct Connection to the SS
For future reference, we derive here the pdf of the distance
between an SS and a node with a direct connection to the
SS. Recall the definition of the indicator function 1A (·) of an
arbitrary set A:
1, if x ∈ A
.
1A (x) =
0, if x ∈ A
For the same situation as analyzed in Section V-A, the pdf
of the distance R between the node and the SS, conditioned
on there being a direct connection between them, i.e., R <
rn exp(hZ/δ), is given by
fR|R<rn exp(hZ/δ) (r)
P {R ≤ min[r0 , r], R < rn exp(hZ/δ)}
d
lim
dr r0 →∞
pn (r0 )
ln(R/rn )
1
E
(R)Q
R
(0,min[r
,r])
0
α
d
=
lim
dr r0 →∞
pn (r0 )
=
1
d
lim 2
=
dr r0 →∞ r0 pn (r0 )
2rQ
=
ln(r/rn )
α
rn2 exp(2α2 )
r
0
2r′ Q
ln(r′ /rn )
α
dr′
=
∞
n=0
[from (11)]
≥
∞
x
=
exp(−u2 /2)
√
du.
2π
D. t-Hop Outage Probability of an SS
In hybrid multihop ad hoc networks, with a limit imposed on
the maximum allowed number of hops, the probability that an
SS is in outage is the probability that it cannot reach a BS in
a given maximum number of hops. We call this the multihop
outage probability, as opposed to the single-hop outage probabilities computed in the previous sections.
Let T be the minimum number of hops required for an
arbitrary SS (assumed located at the origin) to connect to a
BS. Then, T is a random variable taking values in the set
{1, 2, . . .}. Let the cumulative distribution function (cdf) of
T be FT (t). The t-hop outage probability of the SS is,
′′
P {NSS
= n}
′′
× P {T1 > t − 1, . . . , Tn > t − 1|T > 1, NSS
= n}
(20)
where
Q(x) =
′′
where NSS
∼ Poiss(µSS exp(2α2 )) is the number of SSs with
′′
which the SS at (0, 0) has a direct connection, and T1 , . . . , TNSS
are, respectively, the minimum number of hops required for
′′ (which have a direct connection to the SS at
SS1 , . . . , SSNSS
the origin) to connect to a BS.
′′
For any NSS
= n, it is clear that T1 , . . . , Tn are identically
distributed. However, we know from (20) that these SSs are
located in the geographical proximity of the SS at the origin
and, thus, are in the vicinity of one another as well. Consequently, T1 , . . . , Tn are not independent, and the probability
that all of them are larger than t − 1 is higher than it would have
been if they were randomly located somewhere on the (infinite)
plane. Nevertheless, in order to proceed further, we make the
following crucial assumption.
Assumption (Conditional Independence): Conditioned on
the SS at the origin having no direct connection to any BS
(and not counting paths crossing the SS at the origin), the event
that any SS with a direct connection to the SS at the origin
cannot connect to any BS in ≤ t hops is independent across
all such SSs (and has the same probability for all such SSs, by
symmetry) for all t.
Note that this assumption leads, as discussed above, to a
lower bound on the probability of t-hop outage. We have
′′ > t − 1|T > 1
P T1 > t − 1, . . . , TNSS
∞
′′
P {NSS
= n} (P{T1 > t − 1|T > 1})n
n=0
1−P{T >t−1|T >1}
qSS 1
(22)
′′
where we use the fact that NSS
∼ Poiss(−ln qSS ).
In the Appendix, we prove that
P{T1 > 1|T > 1} = βqBS = βP{T1 > 1}
(23)
where, from (77)
β=
∞
0
2r
1
2 exp(2α2 ) Q α ln
rSS
× exp
λBS
2π
∞
−v 2/2
e
∞
r
rSS
2
e−w /2
−∞
−∞
αw
× g(r; rBS e
αv
, rBS e )dwdv dr
(24)
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
able to connect to a BS is lower bounded by limt→∞ qt . This
may be obtained numerically by solving for q from the following transcendental equation obtained by setting qt−1 = qt = q
in (29):
and g(r; a, b) is given by (72). Then, we may write
P{T1 > t − 1|T > 1}
= P{T1 > 1|T > 1}P{T1 > t − 1|T1 > 1, T > 1}
= β P{T1 > 1} P{T1 > t − 1|T1 > 1, T > 1}.
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1−βq
q = qBS qSS
.
(25)
Furthermore, we note that the SS at (0, 0) is a neighbor of
SS 1. Suppose SS 1 has no direct connection to any BS.
Then, knowing in addition that the SS at (0, 0) also has no
direct connection to any BS, increases the probability that
SS 1 has no connection to any BS in t − 1 hops or less over
its corresponding value if we do not know that the SS at (0, 0)
has no direct connection to any BS. In other words, we have
P{T1 > t − 1|T1 > 1, T > 1} ≥ P{T1 > t − 1|T1 > 1}.
This gives us
P{T1 > 1} P{T1 > t − 1|T1 > 1, T > 1}
≥ P{T1 > 1} P{T1 > t − 1|T1 > 1}
(31)
Note that q = 0 is not a solution of the above equation. In other
words, a randomly located SS has a nonzero probability of not
being able to connect to any BS, even if it is allowed an arbitrary
number of hops to do so.
Remark: Observe that the sequence {qt }∞
t=1 in (29) is
monotonically decreasing, as can be shown by mathematical
induction.
1−βq1
1−βq1
<
1) From (78), we see that qSS
< 1, so q2 = q1 qSS
q1 .
2) Suppose qt < qt−1 < · · · < q1 ≡ qBS for some t.
1−βq
1−βqt
1−βqt
3) Then, qSS
< qSS t−1 , so qt+1 = qBS qSS
<
1−βqt−1
= qt , i.e., qt+1 < qt < qt−1 < · · · < q1 ≡
qBS qSS
qBS .
= P{T1 > t − 1}
= F̄T (t − 1)
(26)
where, in the final step, we make use of the fact that the
unconditional t-hop outage probability of SS 1 is the same as
that of the SS at (0, 0).
Substituting (26) into (25), we thus obtain
P{T1 > t − 1|T > 1} ≥ β F̄T (t − 1).
(27)
Substituting (27) into (22), we may rewrite (21) as follows:
1−β F̄T (t−1)
F̄T (t) ≥ qBS qSS
.
(28)
Note that (28) is a sharper bound than the one previously
derived in [17] and can itself be generalized to SSs located
according to a “clumped Poisson” process, as shown in [18].
We have previously proven that the single-hop outage probability is given exactly by qBS . Define
q1 = qBS ,
1−βqt−1
qt = qBS qSS
,
t = 2, 3, . . . .
(29)
Then, it is easy to show by induction that we have the probability of t-hop outage in (28) to be lower bounded by qt , as defined
by (29):
F̄T (t) ≡ 1 − FT (t) ≥ qt ,
t = 2, 3, . . . .
1−β F̄T (t)
Consider the same model as discussed in Sections III and IV,
but the SSs, aside from their wireless capabilities, are sensors
designed to monitor certain localized events and report to any of
the BSs whenever an event is detected. Multihopping is acceptable, but there is a limit tmax on the number of hops allowed.
tmax reflects the “real time” requirement of the application
and the need to flush out old reports, which might otherwise
clog the system. Nothing is known about possible locations of
events, and we therefore assume that the location of each event
is uniformly distributed over the area and is independent of the
SS process. Assume further that each sensor can detect (with
probability 1) an event occurring within a fixed (nonrandom)
distance rsensor from it but cannot detect further events. The
task is to estimate the probability that an event will not be
reported to BS.
Consider an event occurring at an arbitrary location and
denote it as the origin. The probability that this event cannot
be detected by any sensor is just the probability that there is no
sensor within rsensor of the point (0, 0). As we have seen above,
this probability is
2
.
exp −λSS πrsensor
(30)
Proof: Since 1 − FT (1) = q1 , it follows from (28) that
1−βq1
1 − FT (2) ≥ qBS qSS
= q2 , so (30) holds for t = 2. Now,
suppose it holds for some arbitrary t, so that F̄T (t) ≥ qt . Then,
from (28)
1 − FT (t + 1) ≥ qBS qSS
VI. A PPLICATION : S ENSOR N ETWORKS
FOR E VENT D ETECTION
1−βqt
≥ qBS qSS
= qt+1
which completes the proof of (30) by induction.
As t → ∞, i.e., as we allow an SS to connect to a BS through
more and more hops, the probability that the SS will not be
(32)
Next, suppose there are NSS (rsensor ) = n sensors in
B(rsensor ). In this case, the event is detected by all of these
n sensors. However, no report will reach a BS if none of these
sensors can reach a BS in tmax hops. Allowing n to take all
possible values, the probability of this event is
∞
n=1
P {NSS (rsensor ) = n} × P{None of these n SSs
can connect to a BS in ≤ tmax hops}.
(33)
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2
Since NSS (rsensor ) ∼ Poiss(λSS πrsensor
), we can follow the
same sequence of steps as in the derivation of (28) to show that
(33) can be lower bounded for every n by
∞
n=1
P {NSS (rsensor ) = n} (P{T > tmax })n
2
= exp −λSS πrsensor
(1 − P{T > tmax }) .
(34)
Let pnr be the probability that an event is not reported to any
BS. Then, pnr is the probability that either the event is not
detected at all by any sensor or that it is detected by one or more
sensors, none of which can report it to any BS within tmax hops.
Using (32), (34), and (30), we obtain the following lower bound
on pnr :
2
2
+ 1 − exp −λSS πrsensor
pnr ≥ exp −λSS πrsensor
2
(1 − qtmax )
(35)
× exp −λSS πrsensor
connection possible to any BS, the SS chooses a minimum
power connection to one of the SSs to which it knows it can
connect (from the route-exploration phase).
Note that in the routing-path-exploration and setup phase,
if each SS at each hop-distance from some BS broadcasts the
minimum transmit energy per packet to reach a BS starting
from itself, then each SS could have chosen the minimum total
transmit-energy routing path instead of the minimum hop-count
routing path to a BS. The minimum hop-count and minimum
transmit-energy routing paths are two extreme choices in the
spectrum of possibilities arising from trading off transmit energy (affecting the battery life of SSs and self-interference,
which affects the throughput) versus latency (as measured by
the hop count). It is possible to conceive of a whole family of
routing algorithms that choose paths that optimize a weighted
combination of transmit energy and hop count, within a certain
level of complexity. However, in this paper, we focus only on
the minimum hop-count routing algorithm.
where qtmax is given by (29) with t = tmax .
B. Properties of the Minimum Hop-Count Routing Path
VII. N UMBER OF H OPS ON A P ATH F ROM AN SS TO A BS
Recall that T is the hop count of the minimal hop-count route
between an SS and a BS, and the maximum transmit power
SS
capability of the SS is PT,max
.
A. Routing Scheme
In this section, we define the minimal hop-count routing
algorithm, some properties of which we then analyze in subsequent sections. This same routing scheme was studied extensively before, for instance in [19]. For the reader’s convenience,
we present here one simple implementation. We begin with
each BS broadcasting (with maximum power) a beacon or pilot
signal that, when received by any SS, indicates to that SS that it
is just a single hop away from the particular BS. These SSs now
broadcast signals (also with maximum power) giving their own
identification and saying that they are one hop away from a BS.
The SSs that receive these signals, and have not received any
BS directly, now know that they are two hops away from a BS,
and so on. This models the signaling in the IEEE 802.16-2004
mesh-networking mode, for example, in [20].
A given SS may receive such signals from multiple SSs that
advertise themselves as being at some number of hops away
from a BS. The given SS focuses on those SSs that advertise
the minimum number of hops and establishes a routing path to
the particular SS among those that it can reach with minimum
transmit power. This process guarantees that if a path exists
at all between the given SS and BS, one that contains the
minimum number of hops will be found. If the given SS has
no path connecting it to any BS, then it is in outage and will be
assumed not to transmit at all.
While the route exploration phase requires beacon broadcasts
with maximum power, the actual route followed by a packet
from any given SS is as follows: The SS first checks to see
if it can connect directly to a BS (even if this requires using
the maximum power available). If it can, it connects to the
BS that requires least transmit power to do so. If there is no
Recall that a given SS is in t-hop outage if there is no path
at all connecting this SS to any BS in t or fewer hops and
that the probability of this event is lower bounded by qt , as
computed earlier. In other words, for a given t ∈ {1, 2, . . .}, the
probability of t-hop outage is the probability that the number
of hops required for the SS at (0, 0) to connect to a BS is at
least t + 1, or P{T > t}. Thus, the values of the t-hop outage
probability for t = 1, 2, . . . give us the complementary cdf of
the random variable T . However, we do not know the exact
distribution of T , since (29) is only a lower bound on the t-hop
outage probability.
Note that we may write the expected value of T as follows:
E[T ] =
∞
i P{T = i}
i=1
=
∞
i
P{T = i}
i=1 t=1
=
∞
∞
P{T = i}
t=1 i=t
=
∞
t=1
=1 +
P{T ≥ t}
∞
t=2
=1 +
∞
P{T > t − 1}
P{T > t}
(36)
t=1
but this sum does not converge, since P{T > t} ≥ qt and,
as shown above, limt→∞ qt > 0. In other words, since an SS
has a nonzero probability of never being able to connect to a
BS, regardless of the number of hops allowed, the expected
minimum number of hops required to connect to a BS is infinite.
This situation happens if, and only if, the SS belongs to a
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
connected cluster of SSs that are isolated from all the BSs and
is modeled by setting T = ∞.
Suppose now that we permit a maximum of tmax hops for an
arbitrary SS to connect to some BS. The limit tmax is expected
to be set by delay considerations. An SS that cannot reach a BS
in at most tmax hops is not served. The quantity of interest is,
therefore, E[T |T ≤ tmax ]. Clearly, if tmax = 1, then E[T |T ≤
tmax ] = 1. For tmax > 1, E[T |T ≤ tmax ] may be computed
as follows:
i=1
tmax
i P{T = i|T ≤ tmax }
i
=
i=1 t=1
tmax tmax
t=1 i=t
tmax
=
t=1
=1 +
=1 +
tmax −1
t=1
=1 +
tmax −1
t=1
=
rSS (p) ≡
P{T ≥ t|T ≤ tmax }
t=1
P{t < T ≤ tmax }
1 − P{T > tmax }
rBS (p) ≡
P{T > t} − P{T > tmax }
1 − P{T > tmax }
(37)
1
1 − tmax P{T > tmax }
1 − P{T > tmax }
tmax −1
tmax −1
P{T > t}
(42)
t=1
qt .
(43)
t=1
SS
pKSS
SS
Pmin
1/δ
= rSS
p
SS
PT,max
1/δ
.
Furthermore, assume that if an SS can reach a BS on the
uplink, then the BS can always reach the SS on the downlink.
Then, from (6), the range of the SS to the BS is given by
P{T > t|T ≤ tmax }
+
1
1 − qtmax
≥
P{T = i|T ≤ tmax }
tmax −1
tmax −1
1
1 − P{T > tmax }
E[T |T ≤ tmax ] ≥
From (4), we see that if an SS transmits with maximum
power p, then in the absence of fading, its range to another SS
is given by
P{T = i|T ≤ tmax }
=
then (40) reduces to
VIII. M EAN T OTAL T RANSMIT E NERGY PER P ACKET ON
A P OWER -C ONTROLLED M INIMUM -H OP P ATH
F ROM AN SS TO A BS
tmax
E[T |T ≤ tmax ] =
825
P{T > t} .
BS
pKSS
BS
Pmin
1/δ
= rBS
p
SS
PT,max
1/δ
.
(44)
A remark on notation: Observe that rBS is identical to
SS
rBS (PT,max
) as defined in (44) and, similarly, for rSS . Simi′′
larly, we shall shortly define NBS
(p), µBS (p), T (p), qBS (p),
qSS (p), and µSS (p) in such a way that the quantities rBS , rSS ,
′′
NBS
, µBS , T , qBS , qSS , and µSS defined in the previous sections
may now be interpreted as the corresponding functions of p
SS
.
evaluated at the argument p = PT,max
t=1
(38)
If we estimate P{T > t}, t = 1, . . . , tmax for some tmax from
simulations, then (38) can be used to estimate E[T |T ≤ tmax ].
Alternatively, we could look at the effect of tmax on E[T |T ≤
tmax ] as follows: If tmax is such that the probability of tmax -hop
outage is small, i.e., if
P{T > tmax } < ǫ
(39)
for some small ǫ, then from (38), we have
E[T |T ≤ tmax ] ≥
1
1 − P{T > tmax }
t
× 1 − tmax ǫ +
Consider an SS that can receive from one or more BSs on the
downlink. What is the minimum transmit power on the uplink
from this SS to any of these BSs? If the SS transmits with power
p, the number of BSs to which it can connect is given by
′′
2
NBS
(p) ∼ Poiss µBS (p) exp(2α2 ) , µBS (p) ≡ λBS πrBS
(p).
Let P0 be the minimum required transmit power to connect
the SS directly to at least one BS. Then, the probability that
P0 > p is
F̄P0 (p) = P{P0 > p}
max −1
t=1
P{T > t} . (40)
In particular, if ǫ ≤ 1/tmax , i.e., if
P{T > tmax } <
A. Minimum Transmit Power From an SS to a BS
1
tmax
(41)
′′
= P {NBS
(p) = 0}
= qBS (p) ≡ exp −µBS (p) exp(2α2 )
(45)
2/δ
p
2
= exp −µBS exp(2α )
SS
PT,max
2/δ
p̃
= qBS
(46)
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where
where
p̃ ≡
p
SS
PT,max
.
γ(a, z) =
z
e−v v a−1 dv
(48)
0
Suppose that the SS always transmits just enough power to
establish a connection to a BS, if at all possible, and does not
SS
transmit at all if P0 > PT,max
, i.e., if T > 1. Thus, the actual
SS
and zero
transmit power is P0 if, and only if, P0 ≤ PT,max
otherwise (i.e., if T > 1, which occurs with probability qBS ).
Thus, the cdf of the power transmitted by the SS is
FP0 |P0 ≤PTSS,max (p) = FP0 |T =1 (p)
1 − F̄P0 (p)
1 − qBS
p̃2/δ
qBS
1
SS
−
1−qBS , 0 ≤ p ≤ PT,max
= 1−qBS
SS
1,
p > PT,max
=
fP0 |P0 ≤PTSS,max (p)
=
ζa (x) =
γ (a + 1, ln(1/x))
1
,
(1 − x)
[ln(1/x)]a
we may rewrite (47) as
SS
E P0 |P0 ≤ PT,max
SS
PT,max
d
SS
F
(p)
dp P0 |P0 ≤PT ,max
1 2/δ p̃2/δ
2
1
1
SS
ln qBS
qBS , 0 ≤ p ≤ PT,max
1−qBS δ
p p̃
lim ζa (x) = 0.
lim ζa (x) = lim
x→1
y→0
y→0 ay a−1
= lim
y→0
=
=
PTSS
,max
pfP0 |P0 ≤PTSS,max (p)dp
1
2
1
p̃2/δ
dp̃
p̃2/δ qBS
ln
δ
qBS
1
=
1 − qBS
1
× ln
=
=
0
1
=
1 − qBS
uδ/2 exp [−u ln(1/qBS )]
0
qBS
1
du
[u = p̃2/δ ]
ln(1/q
BS )
(1 − qBS ) [ln(1/qBS )]δ/2
0
1
γ 2δ + 1, ln qBS
1
=
δ/2
(1 − qBS )
1
ln qBS
y a exp(−y)
[1 − exp(−y)] + y a exp(−y)
1
1+
a exp(y)−1
y
1
1 + a limy→0
exp(y)−1
y
1
1 + a limy→0
1
1
<
a+1
2
exp(y)
1
for all
[l’Hôpital’s rule again]
a>1
which means that for any δ > 2, even if qBS is very close
to 1, the average transmit power when the SS does transmit
and reach a BS is less than half the maximum transmit power
available to it.
Remark: It follows from the discussion in Appendix I that
when the BS and SS location processes are independent, if the
BSs are not located according to a homogeneous process, then
the probability of SS isolation increases. Since ζa (x) increases
monotonically with x (as proven in Appendix II), this also
implies that if the BSs location process is not homogeneous,
i.e., if the BSs are not spread uniformly, but are, for instance,
clustered, the mean minimal transmit power of the SS is larger
than if the BS location process is homogeneous.
0
1
[y = ln(1/x)]
[l’Hôpital’s rule]
SS
PT,max
SS
PT,max
γ(a + 1, y)
y a [1 − exp(−y)]
= lim
SS
E P0 |P0 ≤ PT,max
=
(50)
x→0
Thus, the mean power transmitted by the SS (normalized by the
SS
) is given by
maximum transmit power PT,max
1
= ζδ/2 (qBS ).
(49)
From (49), we see that
SS
p > PT,max
.
0,
0<x<1
From (50), it follows that for any x and a, ζa (x), being the
ratio of the average of P0 to its maximum, must be ≤ 1. In fact,
we have
with the corresponding pdf given by
=
is the incomplete Gamma function.2
Defining
v δ/2 e−v dv
(47)
2 Some
authors use this name for the function Γ(a, z) =
[21, eq. (8.350), p. 890].
!∞
z
e−v v a−1 dv
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
B. Minimum Transmit Power From an SS to Another on the
Minimum-Hop Route
Consider an SS, say SS∗ , that is not within t − 1 hops from
any BS. Given this fact, its location is not independent from
those of the BSs; SS∗ tends to be somewhat removed from BSs,
especially if t is large. We are interested in the mean minimum
transmit power required for SS∗ to connect, if at all possible,
with an SS that, in turn, is exactly t − 1 hops away from a
BS. Note that in computing the mean of the minimum transmit
power, we also need to average over all arbitrary locations
(X, Y ) of SS∗ .
Let Pt∗ denote the minimum power for SS∗ to transmit to one
of the SSs that are exactly t − 1 hops from a BS (assuming
no constraint on the maximum transmit power). Then, the
probability that Pt∗ > p is the probability that transmitting at p,
SS∗ cannot reach any SS that is exactly t − 1 hops away from
a BS. Although these SS locations do not form a homogeneous
process, we know from the remark at the end of Appendix I
that the outage probability of SS∗ from these SSs is lower
bounded by that obtained by assuming that these SSs are points
of a homogeneous Poisson process (with intensity λSS P{T =
t − 1}), which is qSS (p)P{T =t−1} . Therefore, we have
P{T =t−1}
P {Pt∗ > p} ≥ qSS (p)P{T =t−1} = qSS
p̃2/δ
.
Recall again that SS∗ actually transmits if, and only if, there
is at least one such SS to which it can connect, the probability
P{T =t−1}
of which is 1 − qSS
. Following the same steps as in the
calculation of (50) shows that the mean SS transmit power
SS
(normalized by its maximum value PT,max
) has the following
lower bound:
E [PT∗ |can reach SS for which T = t − 1]
SS
PT,max
P{T =t−1}
. (51)
≥ ζδ/2 qSS
C. Computation of Mean Total Transmit Energy
per Packet on a Minimum-Hop Path With at
Most tmax Hops From an SS to a BS
Consider a given SS and another SS, which is t ≤ tmax hops
away from the “closest” BS in the routing scheme described
above. Denote it by SSt . Let SSi , i = 1, . . . , t − 1 be i hops
away from the BS on the path connecting SSt to the BS. For
tmax = 2, and when the distance between the given SS and the
BS is known, the authors have computed an exact expression for
cdf of the total transmit energy per packet [22]. Here, we deal
with the general case where tmax is arbitrary and the distance
between the SS and the BS is not known.
From (50), SS1 transmits to some BS with average-to-peak
transmit power ζδ/2 (qBS ). In order to calculate the mean transmit power from SSi+1 to SSi , i = 1, . . . , t − 1, recall from the
definition of the minimum-hop routing algorithm that SSi+1
cannot connect to a BS directly and also cannot directly connect
827
to any SS that is ≤ i − 1 hops away from a BS, so it then looks
among the SSs that are i hops away from a BS and chooses the
one (SSi ) that it can reach with minimum transmit power. From
(51), we see that SSi+1 transmits with average-to-peak power
P{T =i}
).
lower bounded by ζδ/2 (qSS
Assume that the transmission of a single packet on each hop
occupies a single time slot of duration Ts . Then, conditioned on
T = t ≤ tmax , the expected value of the total transmit energy
Etot for the packet in the path has the following lower bound:
E[Etot |T
SS
= t] ≥ Ts PT,max
t−1
ζδ/2
ζδ/2 (qBS ) +
i=1
P{T =i}
qSS
so the expected total transmit energy per packet conditioned on
the path being at most tmax hops is lower bounded as follows:
E[Etot |T ≤ tmax ]
= ET {E[Etot |T ]|T ≤ tmax }
=
1
1 − P{T > tmax }
tmax
t=1
SS
ζδ/2 (qBS ) +
≥ Ts PT,max
tmax t−1
×
t=1 i=1
tmax −1
i=1
tmax −1
i=1
1
1 − P{T > tmax }
P{T =i}
ζδ/2 qSS
SS
= Ts PT,max
ζδ/2 (qBS ) +
×
1
1 − P{T > tmax }
P{T =i}
P{T = t}
ζδ/2 qSS
SS
= Ts PT,max
ζδ/2 (qBS ) +
×
E[Etot |T = t]P{T = t}
tmax
P{T = t}
t=i+1
1
1 − P{T > tmax }
(P{T > i} − P{T > tmax })
P{T =i}
× ζδ/2 qSS
.
(52)
We can estimate P{T > i}, i = 1, . . . , tmax from simulation
and use these values to evaluate the right-hand side of (52).
D. Comparison of Mean Total Transmit Energy per Packet for
Single-Hop and for Minimum-Hop Count-Routed MultiHop
We will show now, by way of example, that allowing multihop makes it possible to obtain the same connectivity as in a
single-hop system but with a significant reduction in transmit
energy per data packet.
Suppose we have a system with a given ratio of intensities λSS /λBS . We assume that the SSs have maximum transSS
mit power capability PT,max
= PT . We now compare the
following.
1) We allow a multihop with a maximum of tmax hops.
Then, the tmax -hop outage probability is P{T > tmax },
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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
where we emphasize that qBS is as defined in (54) for
transmit power PT .
The mean energy per packet over this single hop is then
and the mean energy per packet under the minimum hopcount routing scheme is lower bounded by (52)
SS
Ēmultihop ≥ Ts PT,max
ζδ/2 (qBS ) +
×
1
1 − P{T > tmax }
tmax −1
i=1
P{T =i}
(P{T > i} − P{T > tmax }) ζδ/2 qSS
(53)
Ēsingle-hop = Ts PT′ ζδ/2 (P{T > tmax })
δ/2
ln P{T > tmax }
= Ts PT ζδ/2 (P{T >tmax })
.
ln qBS
where
(58)
qBS = exp −λBS π exp(2α2 )rBS (PT )2
qSS = exp −λSS π exp(2α2 )rSS (PT )2
(λ
= qBSSS
(54)
/λBS )[rSS (PT )/rBS (PT )]2
(55)
and
rBS (p) =
Thus, the ratio of the mean energy per packet for the single-hop
versus the multihop path with the same outage probability is
upper bounded by
Ψ
BS 1/δ
KSS
BS
Pmin
p1/δ ,
rSS (p) =
SS 1/δ
KSS
SS
Pmin
p1/δ .
(56)
SS
PT,max
Note that because
= PT , rBS (PT ) = rBS and,
similarly, for rSS .
In practice, SS-to-SS links often do not have the same
range capability as SS-to-BS links. The reasons are likely
to be better placements, higher towers, better antennas
(for instance, diversity reception at the BSs), and other
enhancements that are common at BSs but might be
impractical at the SS side of the link due to cost or space.
As a result, rSS might be significantly smaller than rBS
[see (4) and (6)]. Note that the ratio
ρ≡
rSS (p)
=
rBS (p)
SS
KSS
SS
Pmin
BS
Pmin
BS
KSS
1/δ
is not a function of SS transmit power. We may then
rewrite (55) as
(λ
qSS = qBSSS
/λBS )ρ2
.
This value of qSS can then be substituted into (53) to
obtain a lower bound on Ēmultihop .
2) Now, suppose we aim to achieve the same outage probability P{T > tmax } on a single hop to a BS by increasing the maximum SS transmit power capability from
PT to PT′ .
From (45), we see that in order to have the single-hop
outage probability match the tmax -hop outage probability
P{T > tmax }, the maximum SS transmit power capability must be increased to PT′ such that
2
exp −λBS π exp(2α2 )rBS (PT′ ) = P{T > tmax }. (57)
Comparing (54) and (57) and using (56), we see that
PT′ = PT
ln P{T > tmax }
ln qBS
δ/2
λSS
, qBS , tmax , δ
λBS
≡
Ēsingle-hop
Ēmultihop
ζ δ (P{T > tmax })
2
≤
tmax −1
ζ δ (qBS ) +
2
"
i=1
ln P{T >tmax }
ln qBS
(P{T >i}−P{T >tmax })ζ δ
2
δ2
ρ2 λSS P{T =i}
λBS
qBS
.
1−P{T >tmax }
(59)
In the absence of the exact expression for the distribution of T ,
(59) cannot be computed. However, we can estimate P{T > i}
from simulation and substitute in (59) in order to evaluate Ψ.
Alternatively, Ψ could be estimated directly from simulations
but with an obvious cost in terms of running time.
IX. A NALYTIC AND S IMULATION
R ESULTS —D ESCRIPTION AND D ISCUSSION
In this section, we plot analytic results and bounds, describe
the simulation setup, and present some simulation results.
BS
BS
SS
SS
) = (PTSS KSS
/Pmin
)=
/Pmin
We set δ = 4 and (PTSS KSS
BS BS
SS
12
(PT KSS /Pmin ) = 10 , such that from (5) and (4), respectively, we have rBS = rSS = 1000 m. In other words, when
σ = 0 dB, the limiting reception range is 1000 m on both
SS-to-SS and SS-to-BS links (this choice is kept true for all
subsequent figures, except Fig. 4, where we specify otherwise).
We present in Fig. 1 the probability of an SS not having
a direct connection to any BS versus λBS , where the linkfading model is defined by σ = 0, 4, and 8 dB. These results
were derived from (16). Note that σ = 0 dB corresponds to the
geometric model, where the connectivity is deterministically
ensured for all nodes within the range rBS . We notice that
as σ increases, the probability of an SS not having a direct
connection to any BS, i.e., the single-hop outage probability,
decreases for a given choice of λBS . The shadow-fading blocks
connections to some close-by BSs, while permitting direct
connections to some BSs that are farther away than rBS , with
positive-net effect on connectivity.
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
829
Fig. 1. Plot of the single-hop outage probability given by (16) versus λBS for
σ = 0, 4, and 8 dB.
Fig. 3. Plots of the lower bound on t-hop outage probability given by qt
from (29) for two different choices of the parameters (qBS , qSS ) and β = 1.
Fig. 2. Simulation results for the t-hop outage probability for t = 2, 3, and 4
compared with the corresponding analytic lower bounds given by (30).
Fig. 4. Plots of the analytical lower bound (30) on double-hop outage probability for a fixed SS-to-SS link budget and three cases of SS-to-BS link budget,
equal to that of the SS-to-SS link, 10-dB better, and 20-dB better.
To find the t-hop outage probability for t = 2, 3, and 4 hops
versus λSS and compare the results with the corresponding
analytic lower bounds, we ran simulations as described below.
For each choice of λSS , the simulation makes repeated placements of 100 λSS a2 SSs over a large square of size 10a for
some chosen a. This corresponds to a “partial” [23] realization
of the Poisson SS process, where we choose a certain number
of SSs to be placed in the region of interest, and these SSs are
now placed independently and uniformly over the given region.
Similarly, for each choice of λBS , we place 100 λBS a2 BSs over
the large square. There are i.i.d. shadow fades between all pairs
of nodes. For each placement of SSs and BSs, we count the
fraction of SSs that are more than t hops away from all BSs and
average this fraction over all placements to yield our estimate
of the t-hop outage probability. To eliminate the dependence on
edge effects, we collected statistics only on the nodes in a small
square (with side of length a) at the center of the large square.
Fig. 2, derived from simulations, plots the t-hop outage
probability for t = 2, 3, and 4 hops versus λSS and compares
them to the bound computed in (29) for σ = 8 dB. Note that
β does not depend on λSS and only needs to be computed
numerically once from (77). We see that the benefit of allowing
additional hops diminishes quickly. This is supported by Fig. 3,
derived analytically from the lower bound on t-hop outage
probability given by (29). For both examples, three hops are
sufficient to achieve most of the possible benefit. Note that once
qBS and qSS are given, link parameters like σ, λBS , and λSS are
immaterial.
Fig. 4 compares the lower bound on double-hop outage
probability given by (30) for a fixed SS-to-SS link budget given by rSS = 1000 m and three different values of
rBS for the SS-to-BS links. The upper curve corresponds
BS
BS
BS
SS
to the case (PTSS KSS
/Pmin
) = (PTBS KSS
/Pmin
) = 1012 or
830
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Fig. 5. Plot of E[T |T ≤ tmax ] for tmax = 5 given by (38) versus the ratio λSS /λBS , as estimated in two ways. (a) By obtaining P{T > t} via simulation
and (b) by approximating P{T > t} by the lower bound qt , as given by (29). We have also shown the ratio of the latter (approximate) value to the former (exact)
value. The BS intensity was fixed at λBS = 10−7 m−2 , and the other parameters of the simulation were σ = 8 dB, δ = 4, and rBS = rSS = 1 km.
rBS = 1000 m, the middle curve corresponds to rBS =
1013/4 m and the lower curve to 1014/4 m. In other words, the
upper curve corresponds to the case where the SS-to-BS link
budget is equal to that of the SS-to-SS link, the middle curve
is 10 dB better, and the lower curve is 20 dB better. This figure
allows us to compare the power of different techniques with
respect to their ability to enhance connectivity. For instance,
it allows us to compare the multihop scheme discussed in this
paper to using higher towers or better antennas at the BSs in a
single-hop system.
In Fig. 5, we fix λBS = 10−7 m−2 and plot E[T |T ≤ tmax ]
versus λSS /λBS for tmax = 5 hops, as calculated from (38)
in two ways: 1) using values for P{T > t} obtained from
simulation and 2) approximating P{T > t} by qt as given by
(29). The other parameters are σ = 8 dB, δ = 4, and rBS =
rSS = 1 km. Note that the use of qt in place of P{T > t},
t = 2, . . . , tmax , provide estimates that are mostly within 10%
of the correct values.
In Fig. 6, we plot ζδ/2 (q) given by (49) as a function of q,
for several choices of the distance-loss exponent δ. The ratio of
average to peak SS transmit power required to connect to a BS
is given by ζδ/2 (qBS ) if the peak SS transmit power is chosen
such that the probability of the SS having no direct connection
to any BS is qBS . Note that (50) provides the exact value of the
desired ratio and is not a bound.
In Fig. 7, we compare the plots of E[Etot |T ≤ tmax ] and
the right-hand side of (52) versus λSS for σ = 8 dB, tmax =
5 hops, λBS = 10−7 m−2 , δ = 4, and rBS = rSS = 1 km, so
that qBS = qSS = 0.6187. The right-hand side of (52) was calculated employing the values of P{T > t} obtained from simulation, while E[Etot |T ≤ tmax ] was obtained directly through
simulation. The reason for the behavior of E[Etot |T ≤ tmax ]
Fig. 6. Plot of the function ζδ/2 (q) given by (49) as a function of q for
several choices of the distance-loss exponent δ. The ratio of average-to-peak
SS transmit power required to connect to a BS is given by ζδ/2 (qBS ) if the
peak SS transmit power is chosen such that the probability of the SS having no
direct connection to any BS is qBS .
versus λSS is as follows: We are only considering SSs that
have a connection to a BS within at most tmax hops. When
λSS is very low, SSs are few and far between, so connections
to BSs, when present, are principally just one hop in length.
As λSS grows, more SSs gain a connection to a BS, but they
require a larger number of hops in order to do so, which leads
to a higher total transmit-energy requirement per data packet.
As λSS keeps growing further, better routes to BSs can be
found, thereby reducing the total transmit-energy requirement
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
831
Fig. 7. Comparison of E[Etot |T ≤ tmax ]/(Ts PT , maxSS ) (derived directly from simulations) and the right-hand side of (52) (calculated using the values of
P{T > t} obtained from simulation) when both are plotted against λSS for σ = 8 dB, tmax = 5 hops, λBS = 10−7 m−2 , δ = 4, and rBS = rSS = 1 km,
so that qBS = qSS = 0.6187.
Fig. 8. Plot, derived from the simulation, of the tmax -hop outage probability
P{T > tmax } for tmax = 5 hops and the ratio Ψ of total transmit energy per
packet for single-hop versus minimum hop-count-routed paths with the same
outage probability for different choices of the ratio of intensities λSS /λBS and
δ = 4. In each hop, the probability of not being able to connect to any BS is
qBS = 0.6187.
from before, while the number of SSs with a connection to a
BS within tmax hops saturates (as P{T > tmax } becomes very
small). We note that the bound in (52), while not tight, tracks
the exact average transmit energy per packet reasonably well
over the range of SS intensities simulated.
In Fig. 8, we plot P{T > tmax }, which is obtained from
the simulation versus λSS /λBS for tmax = 5, ρ = 1, and
qBS = 0.6187, the latter being obtained from (16) with λBS =
10−7 m−2 , rBS = rSS = 1 km, σ = 8 dB, and δ = 4. We also
plot the ratio Ψ of the mean single-hop to multihop transmit
energies as derived from the simulations. It is seen that at least
for this choice of parameters, the minimum hop-count multihop
path requires much less total transmit energy per packet than the
single-hop path for the same outage probability.
Fig. 8 demonstrates that by replacing a single-hop system
with a multihop system, where both operate at the same outage
probability, it is possible to reduce the total mean transmit
energy required per packet by a large factor. This happens
even though some packets undergo repeated transmissions and
receptions. Saving on transmit energy is very important in
systems where the SSs are battery operated. Good examples are
the mobiles of the cellular system and certain sensor systems.
Another aspect of the same feature is the reduction in the
mean interference emission generated per packet. This could
be exploited in many ways, but we do not discuss this subject
further here.
X. C ONCLUSION
In this paper, we investigate the possible benefits of multihopping in cellular packet-data networks, where the SSs’
and BSs’ locations follow independent homogeneous planar
Poisson processes and the propagation channels exhibit signal attenuation with distance and log–normal shadowing. We
calculate the probability that an arbitrary SS will not have
access to any BS and show how it depends on the density
of BSs and SSs and how it diminishes as more hops are
allowed. We develop exact analytical expressions and concise
832
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
lower bounds when this is not possible. In the later case, we
produce simulation results for comparison with the analytic
bounds. Our bounds are concise and lend themselves to easy
evaluation. We demonstrate that the use of SSs as repeaters
is particularly beneficial in fading environments. We calculate
the mean number of hops achieved by a minimum hop-count
router when only routes containing at most tmax hops are
allowed. Then, we calculate the mean transmit energy per
packet and show, by way of example, that with multihop, it
is possible to achieve the same outage probability as a singlehop system with reduced total transmit energy per data packet,
which is an important consideration for battery operated SSs.
Our derivations allow experimentation with different channel
models, for instance, macrocell models for the SS-to-BS links
and picocell models for SS-to-SS links. We also provide several
example applications, showing how the developed theory can
be put to use.
The common probability that each of these nodes in B(x, y; r0 )
has a direct connection to (x, y) is given by
A PPENDIX I
P ROBABILITY OF I SOLATION OF AN A RBITRARY SS F ROM
A LL N ODES W HEN N ODE L OCATION P ROCESS I S
I NHOMOGENEOUS P OISSON
µ (B(x, y; r0 )) pn (x, y; r0 )
Suppose the node location process is Poisson but not homogeneous, i.e., the number of nodes Nn (A) in a given region A
is Poisson with mean
λn (x′ , y ′ )dx′ dy ′
(60)
µ(A) ≡
(x′ ,y ′ )∈A
for some intensity function λn (·, ·) instead of λn × area(A) for
a constant intensity λn , as in the homogeneous Poisson case.
For any region A, given Nn (A) = k, these k nodes are i.i.d.
over the region A with the pdf of the location (X ′ , Y ′ ) of any
one of them being
′
′
fX ′ ,Y ′ (x , y ) =
λn (x′ ,y ′ )
µ(A)
0
(x′ , y ′ ) ∈ A .
otherwise
(61)
Compare (60) and (61) with (1) and (2), respectively, for
the homogeneous Poisson process. As before, we assume that
the numbers of nodes in two disjoint regions are independent
Poisson random variables and that the node location process is
independent of the SS location process.
Consider an SS at location (x, y). The number of nodes in
the disk B(x, y; r0 ) is Poisson with mean
µ (B(x, y; r0 )) ≡
2πr0
λn (x + r cos θ, y + r sin θ)rd rdθ.
0 0
pn (x, y; r0 ) =
(62)
EZ
pn (x, y; r0 ) = P {R < rn exp(hZ/δ)}
where R is the distance of the node from (x, y), and Z is the
shadow fade on the link between this node and the SS at (x, y).
Note that if (r, θ) are the polar coordinates representing the
location of this node relative to (x, y), then from (61), we have
rλ (x+r cos θ,y+r sin θ)
n
, r ∈ [0, r0 ], θ ∈ [0, 2π)
µ(B(x,y;r0 ))
fR,Θ (r, θ) =
0,
otherwise.
(64)
Substituting (64) into (63), we obtain the expression shown at
the bottom of the page. Applying the same argument that led to
(10) now yields Nn′′ (B(x, y; r0 )), which is the number of nodes
in B(x, y; r0 ) with a direct connection to the SS at (x, y), to be
Poisson with mean
2π
= EZ
dθ
0
0
min{r0 ,r
n exp(hZ/δ)}
dr rλn(x+r cos θ, y+r sin θ) .
0
Letting r0 → ∞, we finally obtain Nn′′ (x, y), which is the
number of nodes in the plane with a direct connection
to the SS at (x, y) to be Poisson with mean given by
EZ [µ(B(x, y; rn exp(hZ/δ)))].
Recognizing rn exp(hZ/δ) as the range of the SS at (x, y),
we now define the locally averaged intensity of the node
location process around the SS location (x, y) as follows:
λ̄n (x, y)
! !
2π r exp(hZ/δ)
EZ 0 0 n
rλn (x+ r cos θ, y+ r sin θ)dr dθ
! !
≡
2π r exp(hZ/δ)
EZ 0 0 n
r drdθ
! !
2π r exp(hZ/δ)
rλn (x + r cos θ, y + r sin θ)drdθ
EZ 0 0 n
.
=
πrn2 exp(2α2 )
(65)
Thus, the probability of isolation of an SS at (x, y) from the
nodes of the inhomogeneous Poisson process is given by
P {Nn′′ (x, y) = 0} = exp −πrn2 exp(2α2 )λ̄n (x, y) . (66)
From (66), we see that the isolation probability of an SS in
the presence of nodes located according to such an inhomogeneous Poisson process is not independent of the location of
this SS. Note that the SS could be located at any arbitrary point
(X, Y ) in the plane. Equivalently, the locally averaged intensity
! !
2π min{r0 ,rn exp(hZ/δ)}
0
(63)
rλn (x + r cos θ, y + r sin θ)dr dθ
µ (B(x, y; r0 ))
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
of the node location process around the random SS location
may be seen as the random variable V , which is the following
function of the random SS location coordinates (X, Y ):
V ≡ λ̄n (X, Y ).
(67)
From (66), the probability that an SS at a random location
(X, Y ) is isolated may be written as exp[−πrn2 exp(2α2 )V ],
where V is as defined in (67).
Thus, qn the probability that an arbitrarily located
SS is isolated from all such nodes is the average of
exp[−πrn2 exp(2α2 )V ] over all possible values of V ; the locally averaged intensity of the node-location process around the
random SS locations:
qn ≡ EV exp −πrn2 exp(2α2 )V .
(68)
Noting that the negative exponential is strictly convex in its
argument, we see from Jensen’s inequality applied to (68) that
qn ≥ exp −πrn2 exp(2α2 )E[V ] .
(69)
Now, suppose that this SS is a point of a homogeneous Poisson
point process and that we know that it is located in a given
(possibly large) region, say the disk B(0, 0; r1 ). Then, the pdf
of its location (X, Y ), fX,Y (·, ·) is given and constant over
B(r1 ). From (69), we know that of all possible intensities
λn (·, ·) of the node-location process, the SS outage probability
from the nodes of this process qn , is minimized if 1) λn (·, ·)
maximizes E[V ] for the given fX,Y , and 2) we have an equality
in (69). To find λ̄n (·, ·) achieving 1), note that we can write
E[V ] = EX,Y λ̄n (X, Y )
=
λ̄n (x, y)fX,Y (x, y) dx dy
B(r1 )
≤
B(r1 )
×
B(r1 )
(69). From 1) and 2), it follows that qn is minimized if λ̄n (·, ·)
is a constant.
Remark: From (65), λ̄n (·, ·) is constant if, and only if,
λn (·, ·) is constant, i.e., if, and only if, the node-location
process is homogeneous Poisson. Equivalently, if the nodelocation process is inhomogeneous, then qn is larger than the
value obtained by assuming that the node-location process is
homogeneous Poisson.
A PPENDIX II
P ROOF T HAT ζa (x) I S M ONOTONICALLY I NCREASING
IN x FOR A NY a > 0
We use the well-known result that the expected value of a
random variable is the area under the curve of the complementary cdf of that random variable in order to obtain
SS
E P0 |P0 ≤ PT,max
ζa (qBS ) =
SS
PT,max
PTSS
,max
=
=
1/2
2
fX,Y
(x, y) dx dy
1
SS
PT,max
1
SS
PT,max
0
1
0
1 − FP0 |P0 ≤PTSS,max (p) dp
1/a
p̃
qBS
− qBS
dp̃.
1 − qBS
Therefore, to prove that ζa (x) is monotonically increasing in x
for a given a, it is sufficient to prove that for any w ∈ [0, 1], the
function
hw (x) ≡
xw − x
1−x
is monotonically increasing for all x ∈ (0, 1). This result follows from the observation that
1/2
[λ̄n ]2 (x, y) dx dy
833
h′w (x) =
fw (x)
wxw−1 − 1 − wxw + xw
=
(1 − x)2
(1 − x)2
where
(70)
where the final inequality is obtained by applying the Schwarz
Inequality to the functions λ̄n (·, ·) and fX,Y (·, ·), which are
viewed as elements of the Hilbert space L2 (R2 ) of functions on
R2 with the Euclidean norm. The Schwarz Inequality also says
that the inequality in (70) is replaced by an equality, i.e., E[V ] is
largest, if, and only if, λ̄n (·, ·) and fX,Y (·, ·) are identical except
for a multiplicative constant, i.e., if, and only if, the locally
averaged intensity of the node-location process about each SS
location exactly matches (to within a multiplicative constant)
the pdf of the SS location itself. Since the latter is constant, we
require λ̄n (·, ·) to be constant.
Then, 2) follows immediately because if λ̄n (·, ·) is a constant,
then so is V , which, from Jensen’s Inequality, yields equality in
fw (x) = (1 − w)xw +
w
x1−w
−1
is monotonically strictly decreasing for all x ∈ (0, 1) because
w(1 − w) w(1 − w)
1
w(1 − w)
′
fw (x) =
1−
−
=
<0
x1−w
x2−w
x1−w
x
for all x ∈ (0, 1). Thus, for all x ∈ (0, 1), we have
fw (x) > fw (1) = 1 − w + w − 1 = 0
which in turn implies that
h′w (x) > 0
for all
x ∈ (0, 1)
which proves that hw (x) is monotonically increasing for all
x ∈ (0, 1).
834
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
A PPENDIX III
A REA OF I NTERSECTION OF T WO D ISKS W ITH
A RBITRARY R ADII
Define, for any x, y, r, a, b > 0 θ ∈ [0, 2π)
A(x, y; r, θ; a, b) ≡ B(x + r cos θ, y + r sin θ; a) ∩ B ′ (x, y; b).
(71)
Then from [24, eq. (11)], we see that the area of this region
does not depend on x, y, and θ and is given by (72), shown at
the bottom of the page.
A PPENDIX IV
D ERIVATION OF P{T1 > 1|T > 1}
Suppose we have an SS at (0, 0) and another SS, labeled
SS 1, at a distance of R̃ = r̃ from (0, 0). Without loss of
generality, we may take the location of SS 1 to be (r̃, 0).
SS 1 is given to have a connection to the SS at (0, 0), i.e., the
cdf of R̃ is given by (20)
2r
Q
fR̃|R̃<rSS exp(hZ̃/δ) (r̃) = 2
rSS exp(2α2 )
1
r̃
ln
α rSS
where Z̃ is the shadow fade (in decibels) on the link between
the SS at the origin and SS 1.
Consider the BSs in B(r0 ), where r0 is very large. The
number of BSs in B(r0 ) is NBS (r0 ) ∼ Poiss(λBS πr02 ).
Consider an arbitrary BS in B(r0 ). Let the distance of this
BS from (0, 0) be R and its distance from SS 1 be R′ . Denote
the shadow fade (in decibels) between this BS and the SS at
(0, 0) by Z and the shadow fade (in decibels) between this
BS and SS 1 by Z ′ . The BS has a connection to SS 1 if, and
only if, R′ < rBS exp(hZ ′ /δ), i.e., if, and only if, it lies in
B(r̃, 0; rBS exp(hZ ′ /δ)), and it has a connection to the SS at
(0, 0) if, and only if, it lies in B(0, 0; rBS exp(hZ/δ)).
Since the shadow fades on all links are i.i.d., the event that
an arbitrary BS in B(r0 ) has a connection to SS 1 but not
to the SS at (0, 0) is independent across all BSs in B(r0 ).
Furthermore, p′BS (r0 ) which is the probability that an arbitrary
BS in B(r0 ) has a connection to SS 1 (at distance r̃ from the
origin) given that this BS does not have a connection to the
SS at (0, 0), is, therefore, the probability that this BS lies in
B(r̃, 0; rBS exp(hZ ′ /δ)) \ B(0, 0; rBS exp(hZ/δ)). Since the
location of the BS is uniformly distributed over the disk B(r0 ),
this probability is just the fractional area of the above region
′
area B r̃, 0; rBS ehZ /δ \B rBS ehZ/δ
′
pBS (r0 ) = EZ,Z ′
area [B(r0 )]
′
EZ,Z ′ area B r̃, 0; rBS ehZ /δ \B rBS ehZ/δ
=
πr02
(73)
where we assume that r0 is large enough to avoid edge effects.
′
(r0 ), which
It follows from [16, Sec. 5.1, para. 3, p. 53] that NBS
is the number of BSs in B(r0 ) that have a connection to SS 1 but
not to the SS at (0, 0), is Poiss(λBS πr02 p′BS (r0 )). Furthermore,
since
lim πr02 p′BS (r0 )
′
= EZ,Z ′ area B r̃, 0; rBS ehZ /δ \ B rBS ehZ/δ
2
= EZ,Z ′ πrBS
exp(2hZ ′ /δ)
r0 →∞
#
′
− area B r̃, 0; rBS ehZ /δ ∩ B rBS ehZ/δ
2
exp(2hZ ′ /δ)
= EZ ′ πrBS
′
− EZ,Z ′ g r̃; rBS ehZ /δ , rBS ehZ/δ
′
2
= πrBS
exp(2α2 ) − EZ,Z ′ g r̃; rBS ehZ /δ , rBS ehZ/δ
(74)
where g(r̃; a, b) is as defined in (72), we see that (conditioned
′
on SS 1 being at r̃ from the origin) NBS
, which is the number of
BSs in the plane that have a connection to SS 1 but which does
not have a connection to the SS at the origin, has the following
distribution:
′
∼ Poiss µBS exp(2α2 )
NBS
#
hZ ′ /δ
hZ/δ
, rBS e
− λBS EZ,Z ′ g r̃; rBS e
.
Thus, P{T1 > 1|R̃ = r̃, T > 1}, which is the probability that
there is no BS in the plane with a connection to SS 1 but
g(r; a, b) ≡ area [A(x, y; r, θ; a, b)]
= area [A(0, 0; r, 0; a, b)]
π (min{a, b})2 ,
if r < |a − b|
2
−1 r 2 +a2 −b2
2
−1 r 2 +b2 −a2
+ b cos
2ra
2rb
= a cos
1
−
(−r
+
a
+
b)(r
+
a
−
b)(r
−
a
+ b)(r + a + b), if |a − b| ≤ r < a + b
2
0,
if r ≥ a + b
(72)
MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM
which does not have a connection to the SS at the origin, is
′
the probability that NBS
=0
P{T1 > 1|R̃ = r̃, T > 1}
′
= P {NBS
= 0}
′
.
= qBS exp λBS EZ,Z ′ g r̃; rBS ehZ /δ , rBS ehZ/δ
(75)
Averaging over the values of R̃, we have
P{T1 > 1|T > 1} = βqBS
where
β = ER̃|R̃<rSS exp(hZ̃/δ)
′
× exp λBS EZ,Z ′ g R̃; rBS ehZ /δ , rBS ehZ/δ
(76)
=
∞
0
dr̃
1
2r̃
2 exp(2α2 ) Q α ln
rSS
× exp
λBS
2π
∞
−v 2 /2
dv e
∞
r̃
rSS
dw e−w
2
/2
−∞
−∞
αw
× g (r̃; rBS e
#
, rBS e ) .
αv
(77)
Note that from (77)
1 < β = ER̃|R̃<rSS exp(hZ̃/δ)
′
× exp λBS EZ,Z ′ g R̃; rBS ehZ /δ , rBS ehZ/δ
1
2
e2hZ/δ =
≤ exp λBS EZ πrBS
qBS
i.e.,
qBS < qBS β ≤ 1.
(78)
835
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ACKNOWLEDGMENT
The authors would like to thank P. Polakos and C. Papadias
for their continued support of this research.
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MD, Mar. 2001.
Sayandev Mukherjee (S’92–M’97–SM’05) was
born in Bangalore, India, in 1970. He received the
Bachelor of Technology degree from the Indian Institute of Technology, Kanpur, India, in 1991, and
the M.S. and Ph.D. degrees from Cornell University,
Ithaca, NY, in 1994 and 1997, respectively, all in
electrical engineering.
Since 1996, he has been a member of the Technical
Staff with the Wireless Research Laboratory, Bell
Laboratories, Lucent Technologies, Murray Hill,
NJ. His research interests include stochastic models, wireless-system simulations, and connectivity issues in ad hoc wireless
networks.
836
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Dan Avidor (M’06) was born in Tel Aviv, Israel,
on October 18, 1936. He received the B.Sc. degree
in electrical engineering from the Technion–Israel
Institute of Technology, Haifa, Israel, in 1958, and
the M.Sc. and Ph.D. degrees in engineering from the
University of California, Los Angeles, in 1970 and
1981, respectively.
After completing his studies for the Ph.D. degree,
he returned to Israel, where he served as a Research
and Development Department Head in the Israeli defense forces. He is currently a distinguished member
of the Technical Staff with the Wireless Research Laboratory, Bell Laboratories,
Lucent Technologies, Murray Hill, NJ. His current research interests include intelligent antennas, scheduling in high-speed downlink packet access (HSDPA)
systems, ad hoc and sensor networks, and digital processing.
Katherine Hartman was born in Albany, NY, in
1986. She is currently a sophomore working toward
the B.Sc. degree in materials science and engineering
at the Massachusetts Institute of Technology (MIT),
Cambridge.
She was an Intel Science Talent Search Finalist in
2004, with a research project in astronomy. She is
currently an undergraduate member of an MIT team
researching the electrodeposition of iron–tungsten.
Her current interests include solar-cell research and
carbon nanotubes.