175
Wireless Networks 1 (1995) 175^185
Mobile users: To update or not to update?
a;1
Amotz Bar-Noy
a
b
, Ilan Kessler
b;2
and Moshe Sidi
c;3
IBM T.J. Watson Research Center, P.O.Box 218, Yorktown Heights, NY 10598, USA
Qualcomm Israel, MATAM - Advanced Technology Center, P.O.Box 719, Haifa 31000, Israel
c
Electrical Engineering Department, Technion, Haifa 32000, Israel
Received November 1994
Abstract. Tracking strategies for mobile users in wireless networks are studied. In order to save the cost of using the wireless
links mobile users should not update their location whenever they cross boundaries of adjacent cells. This paper focuses on three
natural strategies in which the mobile users make the decisions when and where to update: the time-based strategy, the number
of movements-based strategy, and the distance-based strategy. We consider both memoryless movement patterns and movements
with Markovian memory along a topology of cells arranged as a ring. We analyze the performance of each one of the three strategies under such movements, and show the performance differences between the strategies.
users to the backbone network whenever the users move
1. Introduction
from cell to cell. The focus in these studies is on the
Future wireless networks will provide ubiquitous
trade-off between updating and retrieving information
communication services to a large number of mobile
from directories in the backbone network. Thus, the
users. The design of such networks is based on a cellular
issue considered in these works is the cost of utilizing
architecture [6^9] that allows efficient use of the limited
the wired links of the backbone network for manage-
available spectrum. The cellular architecture consists
ment of directories.
of a backbone network with fixed base stations intercon-
Another important issue is the cost of utilizing the
links for the actual tracking of mobile users. To
nected through a fixed network (usually wired), and of
wireless
mobile units that communicate with the base stations
illustrate this issue consider the simple tracking strate-
via wireless links. The geographic area within which
gies known as the Always-Update, in which each mobile
mobile units can communicate with a particular base
user transmits an update message whenever it moves
station is referred to as a cell. Neighboring cells overlap
into a new cell, and the Never-Update, in which the
with each other, thus ensuring continuity of communi-
users never send update messages regarding their loca-
cations when the users move from one cell to another.
tion. Clearly, under the former strategy the overhead
The mobile units communicate with each other, as well
due to transmissions of update messages is very high,
as with other networks, through the base stations and
especially in networks with small cells and a large num-
the backbone network.
ber of highly mobile users, but the overhead for finding
One of the important issues in cellular networks is
users is zero since the current location of each user is
the design and analysis of strategies for tracking the
always known. With the latter strategy there is no over-
mobile users. In these networks, whenever there is a
head for updating but whenever there is a need to find a
need to establish communication with any particular
particular user, a network-wide search is required, and
user, the network has first to find out which base station
this overhead is very high.
can communicate with that user. This is due to the fact
These two simple strategies demonstrate the basic
that the users are mobile and could be anywhere within
trade-off that is inherent to the problem. Thus, there are
the area covered by the network. This issue was consid-
two basic operations that are elemental to tracking
ered in [1,6,2].
mobile users ^ update and find (or search/locate).
In [1] and [6] efficient data structures for manipulat-
Associated with each one of these operations there is a
ing the information regarding the locations of the
certain cost. For instance, frequent updates cause high
mobile users were investigated. In these works, it is
power consumption at the mobile units and load the
assumed that this information is sent by the mobile
wireless network. Network-wide searches load both the
backbone and the wireless networks. As demonstrated
1
This author is currently with the Electrical Engineering Depart-
ment,
Tel-Aviv
University,
Tel
Aviv
69978,
Israel
(E-mail:
[email protected]).
2
The work of this author was done when he was with IBM, T. J.
Watson Research Center, Yorktown Heights, NY, USA.
3
Part of the work of this author was done when he was visiting
IBM, T. J. Watson Research Center, Yorktown Heights, NY, USA.
Ä J.C. Baltzer AG, Science Publishers
by the two examples above, increasing one cost leads to
a decrease in the other one. In fact, the above simple
strategies are the two extreme strategies, in which one
cost is minimized and the other cost is maximized. Existing systems use either one of these two strategies or the
following combination of them (used in current cellular
176
A. Bar-Noy et al. / Mobile users
telephone networks). Each user is affiliated with some
ficult one since the mobile users need information
geographic region, referred to as its home-system.
about the topology of the cellular network.
Within the home-system, the Never-Update strategy is
In this paper we consider both memoryless move-
used. When the user moves into another region, then it
ment patterns and movements with Markovian memory
must manually register (update), and then the Never-
along a topology of cells arranged as a ring. We analyze
Update strategy is used within that region. The partition
the performance of each one of the three strategies
into regions is static, and is based on the market areas
under such movements, and show the performance dif-
where licenses for cellular systems can be granted to
ferences between the strategies. In the memoryless case,
operator companies by the FCC.
we prove that the distance-based strategy is the best
Recently, the problem of utilizing efficiently the wire-
among the three strategies, and that the time-based
less links was considered in [2], where a new approach
strategy is the worst one among them. In the Markovian
according to which a subset of all cells is selected and
case, we show that this is not always true, as for certain
designated as
reporting cells is presented. The idea is
types of movement patterns the time-based strategy is
that mobile users will transmit update messages only
better than the movement-based strategy. In both the
upon entering a reporting cell, and a search for any user
memoryless and the Markovian movement patterns, the
will always be restricted to the vicinity of a reporting
numerical results obtained show that the difference
cell ^ the one to which the user lastly reported. This
between the movement-based and the time-based strate-
strategy is static in the sense that the location of the
gies is small, and that this difference vanishes as the
reporting centers is fixed. Also, the strategy is global in
update rate decreases. The results also show that the dif-
the sense that all mobile users transmit their update
ference between the distance-based strategy and the
messages in the same set of cells. Note that it might hap-
other two strategies is more significant.
pen that a mobile user will transmit frequent update
Recently it has come to our attention that a dis-
messages, even if it does not move far since it can move
tance-based strategy has been independently considered
in and out a reporting center frequently. It can also
in [5]. In [5] it is assumed that the users are paged with
happen that a mobile user will not transmit update mes-
a high rate. Since whenever the user is paged and found
sages for long periods of time, even if it moves a lot,
the location of this user is updated, they consider the
since it does not enter a reporting center. The question
evolution of the system between two successive pagings.
of where to place the reporting centers so that some cost
Using dynamic programming approach, an iterative
function is optimized is answered in [2] for various cell
algorithm for computing the optimal value for the
topologies.
parameter D is given.
In this paper we focus on dynamic strategies in which
The paper is organized as follows. In section 2 we
the mobile users transmit update messages according
define the model. In section 3 we present the analysis
to their movements and not in predetermined cells.
and results for the memoryless movement pattern. In
These strategies are local in the sense that the location
section 4 we present the analysis and results for the Mar-
update may differ from user to user and the users them-
kovian movement pattern. Section 5 is devoted for dis-
selves decide when and where to transmit their update
cussion and open problems.
messages.
The simplest dynamic strategy that is considered is
the time-based update in which each mobile user updates
2. The model
its location periodically every T units of time, where T
is a parameter. Another dynamic strategy considered is
Consider a cellular wireless radio system with N cells.
the movement-based update in which each mobile user
Two cells are called neighboring cells if a mobile user
counts the number of boundary crossing between cells
can move from one of them to another, without crossing
incurred by its movements and when this number
any other cell. In this paper we study a ring cellular
exceeds a parameter M it transmits an update message.
topology in which cells i and i
The last dynamic strategy considered is the distance-
(in the following all arithmetics with cell indices are
‡ 1 are neighboring cells
based update in which each mobile user tracks the dis-
modulo N ). Thus, a mobile user that is in cell i, can only
tance it moved (in terms of cells) since last transmitting
move to cells i
an update message, and whenever the distance exceeds
a parameter D it transmits an update message.
‡ 1 or i ÿ 1 or remain in cell i.
To model the movement of the mobile users in the
system we assume that time is slotted, and a user can
It is not difficult to realize that from an implementa-
make at most one move during a slot. The movements
tion point of view the time-based strategy is the simplest
will be assumed to be stochastic and independent from
since the mobile users need to follow only their local
one user to another. For each user, we analyze two types
clocks. The movement-based strategy is more difficult
of movements ^ the i.i.d. (independent and identically
to implement since the mobile users need to be aware
distributed) movement and the Markovian movement.
when they cross boundaries between cells. The imple-
In the i.i.d. model, if a user is in cell i at the beginning of
mentation of the distance-based strategy is the most dif-
a slot, then during the slot it moves to cell i
‡ 1 with
177
A. Bar-Noy et al. / Mobile users
ÿ 1 with probability p, or
ÿ 2p …0 p 1 2†,
cells in any slot are assumed to start after all updates
independently of its movements in other slots. In the
movements are completed by the end of the slot. A
Markovian model, during each slot a user can be in one
search is assumed to take place only after the system has
probability p, moves to cell i
remains in cell i with probability 1
<
<
=
of the following three states: (i) The stationary state
…S† (ii) The right-move state …R† (iii) The left-move state
…L†. Assume that a user is in cell i at the beginning of a
and searches associated with that slot are done, and all
reached a steady- state, and the time of search, ts , is chosen independently of the evolution of the system. This
implies that the probability that the system is in a parti-
slot. The movement of the user during that slot depends
cular state at ts is the stationary probability of that
on the state as follows. If the user is in state S then it
state.
remains in cell i, if the user is in state R then it moves to
‡ 1, and if the user is in state L then it moves to
cell i ÿ 1. Let X …t† be the state during slot t. We assume
that fX …t† t ˆ 0 1 2
g is a Markov chain with tranˆ Prob‰X …t ‡ 1† ˆ l X …t† ˆ kŠ
sition probabilities p
ˆ p ˆ q, p ˆ p ˆ , p ˆ p
as follows: p
ˆ p, p ˆ p ˆ 1 ÿ q ÿ and p ˆ 1 ÿ 2p (see
cell i
;
;
;
;...
=
k;l
R;R
L;S
R;S
L;L
L;R
v
R;L
v
S ;R
S ;L
S ;S
Fig. 1).
The performance measures that we consider in this
paper are the expected number of update messages per
slot transmitted by a user, denoted by
U
x,
and the
expected number of searches necessary to locate a user,
denoted by
S
x,
2 fD M T g denotes the strat-
where x
;
;
egy under which the measure is taken, i.e., distancebased, movement-based and time- based, respectively.
Obviously, for update and search strategies to be good,
We consider three dynamic update strategies: (i)
Time-based update: Each mobile user transmits an
both measures should be kept small, as explained in the
introduction.
update message every T slots; (ii) Movement-based
Since the movements of different users are indepen-
update: Each mobile user transmits an update message
dent, it suffices to calculate the above performance mea-
whenever it completes M movements between cells; (iii)
sures
Distance-based update: Each mobile user transmits
presented in the sequel we refer to a particular user
an update message whenever the distance (in terms of
(arbitrarily chosen).
for
one user.
Thus,
throughout
the
analysis
cells) between its current cell and the cell in which it last
reported is D (the act of sending an update message by
a user is referred to as reporting). We assume that N is at
3. The I.I.D. model
least twice the parameter of the update strategy.
The search strategy for finding a user is simple:
3.1. Distance-based update
Whenever a user is looked for by the system, it is first
searched at the cell it last reported to, say i. If it is not
found there, then it is searched in cells i
j
ˆ1
;
2; . . . ; N =2, starting with j
6 j for every
ˆ 1 and continuing
6 j (for a spe-
… † be the distance between the cell in which
Let Y t
the user is located at time t and the cell in which it last
transmitted an update message. Notice that here there is
upward until it is found. A search in cells i
no need to distinguish whether the user is on the left or
cific j) is counted as a single operation. We assume that
on the right of the cell in which it last transmitted an
the search operation is fast, and terminates within one
update message. Clearly,
slot.
kov chain. Let Qd
We assume that updates and searches are done at
the beginning of slots, such that if a user is both updat-
...;
D
…†
fY …t† t ˆ 0 1 2 g is a Marˆ lim !1 Prob‰Y …t† ˆ d Š d ˆ 0 1
;
;
;
;...
;
t
;
;
ÿ 1, be the stationary probability distribution of
Y t . The balance equations for these probabilities are
ing and being searched in the same slot then the update
ˆ pQ ‡ pQ
2pQ0
operation precedes the search. Movements between
1
2pQ1
2pQd
D
ˆ 2pQ ‡ pQ
2 ;
0
ˆ pQ ÿ ‡ pQ ‡
d
ÿ1 ;
1
d
2pQD
ÿ1
2
1 ;
ˆ pQ
D
d Dÿ2
;
ÿ2 :
Solving these equations and normalizing the stationary
probabilities to sum to one, we have
Q0
ˆ D1
Qi
;
ˆ 2…DDÿ i†
2
from which it follows that
U ˆ pQ
D
Fig. 1. State diagram of the Markov walk.
D
ÿ1
ˆ D2p
2
;
S ˆ1‡
D
;
X
D
d
1
i Dÿ1
ÿ1
ˆ1
dQd
;
1
ˆ 1 ‡ D3 ÿ 3D
:
178
A. Bar-Noy et al. / Mobile users
The expression for
U
is due to the fact that an update
D
S ˆ 1 ‡ M1
M
message is transmitted only when the distance of the user
from the cell in which it last reported is D
ÿ 1, and the
user moves in the direction that increases this distance
(which happens with probability p). The expression for
S
D
is true since if at the time of search the distance of the
ˆ1‡M
1
user from the cell in which it last reported is d , then it
takes d
‡ 1 searches until it is found.
ÿ1
X
where M
m
ˆ1
1
m
2
ÿ
X 12…jÿ1† 2j M 1
2
j
ˆ1
j
2
j
m
2
‡
ÿ1 ÿ1
… 3†
6m7
M 1
M
2
M
M ;
… 4†
2
ˆ 1 if M is even and ˆ 0 otherwise.
M
3.3. Time-based update
3.2. Movement-based update
Since in each slot a user makes a move with probabil-
With time-based update we immediately have that
ity 2p and remains in the same cell with probability
1
mÿ1 M
U ˆ T1
ÿ 2p, it takes on average M =2p slots to complete M
movements. Therefore,
T
since the user transmits an update message every T slots.
U ˆM :
2p
To compute the expected number of searches required
M
to locate the user we use the fact that the probability that
… 0 t T ÿ 1†
The computation of the expected number of searches
the user makes m movements in t slots
required to locate a user is more complicated in this case.
is given by
We start with the fact that given that the user has made
ˆ mjslots ˆ tŠ
ˆ m …2p† …1 ÿ 2p† ÿ ; 0 m t :
Using (1) and the fact that Pr‰…t modT † ˆ tŠ ˆ 1=T for
all t ˆ 0; 1;
; T ÿ 1, we obtain
m
…0 m M ÿ 1† movements since last transmitting
an update message at some cell, the probability that its
distance from that cell is d is given by
‰
ˆ d jmovements ˆ mŠ
m
1
;
1 d m , m+d even ,
‡
… 1†
2
Prob distance
8 <2
ˆ:
m
m
‰
Prob movements
t
T
otherwise .
2
‡ d after m
ˆ0 mˆ0
t
m
X
d
movements, the number of movements that the user
‡ d must exceed by exactly d the
T ÿ1 X
t
t
X
S ˆ 1 ‡ T1
The explanation for (1) is as follows. Assume that the
makes towards cell i
ˆ1
d
m
m
… †
is 1=2, which accounts for the term 1=2
m
S ˆ 1 ‡ T1
T
. The factor of
2
2 is due to the fact that in order to be at distance d the
‡ d or in i ÿ d .
L…t† ˆ maxf tj the user reported in slot g.
user can be either in i
Let
… † be the number of movements that the user has
made during the slots L…t†; L…t† ‡ 1;
; t ÿ 1 (if
t ÿ 1 < L…t† then I …t† ˆ 0). Recall that t is the slot in
Let I t
...
s
which a search occurs. Since a search occurs at random,
1
2
t
m
… 5†
;
‡ d is even. Changing the order of summation in
(5) and using (3) we obtain
that it moves in either one of the two possible directions
m
where the latter sum is taken only for values of d such
that m
‡ d must be …m ‡ d †=2. This accounts for the binomial
…2p† …1 ÿ 2p† ÿ
mÿ1
2
It follows that the number of movements towards cell
coefficient. Given that the user moves, the probability
m
‡d
number of movements that it makes away from that cell.
i
m
s
d
user last reported in cell i. To be at cell i
t
...
2
0;
m
T ÿ1 X
T ÿ1 X
t
…2p† …1 ÿ 2p† ÿ
ˆ0 tˆm
mÿ1 1
mÿ1
m
m
6 m7
2
2
ÿ1
m
t
m
m
… 6†
:
This will be used in the next subsection. In Appendix B
we show that (5) simplifies to
S ˆ 1 ‡ T2
T ÿ1
X
2j
T
j
ˆ1
ÿ 2 T …ÿ1† ‡ p : …7†
jÿ1
j‡1
j
1
j
the probability that it will occur after m movements
ˆ 0; 1; 2; ; M ÿ 1) is uniformly distributed. There‰ … † ˆ mŠ ˆ 1=M for all m ˆ 0; 1; 2; ;
M ÿ 1 and we have,
(m
...
fore, Prob I ts
S ˆ 1 ‡ M1
...
ÿ1 X
m
X
M
M
m
ˆ0 d ˆ1
d
m
m
‡d
2
mÿ1
1
2
;
… 2†
3.4. Comparison
In this section we compare the performance of the
three strategies analyzed in the previous subsections.
We first show that the movement-based strategy is better than the time-based strategy in the sense that for the
where the latter sum is taken only for values of d such
same expected update rate, the expected number of
that m
searches required to locate the user is greater in the
‡ d is even. In Appendix A we show that the two
sums above simplify to the following:
time-based strategy. We then show that the distance-
179
A. Bar-Noy et al. / Mobile users
based
strategy
is
better
than
strategy in a similar sense.
U ˆ
Since
2p
M
=M
U ˆ
and
T
the
with the movement-based strategy. Since
1
=T ,
it follows that in
order to have the same expected update rate, the param-
ˆ
M
2p
ˆ M=
T
eters of the two strategies should satisfy
Proposition "1". For T
We now turn to compare the distance-based strategy
movement-based
S S
, we have
M
mÿ1 1
m
6 m7
2
2
T ÿ1 X
t
… †ˆ
a m
ˆm
m
t
ÿ
ÿ
D
…†
;
8
… p† ‡ … ÿ p† ÿ
m
2
1
1
t
2
m
…†
:
9
S ˆ ‡M
1
1
M
S ˆ ‡M
We first show that
… †ˆ
T
a m
1
1
T
… †<
a m
ˆ
ˆ
S ‡D
ˆ
< … 2 p †m‡
1
1 X
i
ˆ … p†
2
m
second
‡
1
1
ˆ
m
[3].
and
1
… p†
2
m
‡
1
2
ˆ
i
1
ˆ
1
i
2
S ˆ ‡M
1
1
M
‡M
1
1
ˆ ‡M
1
1
‡M
1
1
ˆS
T
ÿ
X
1
ˆ
M
ÿ
X
m
0
1
ˆ
M
ÿ
X
m
0
1
ˆ
M
ÿ
X
m
0
1
m
:
ˆ
0
,
: :
1 56
1
2
j
ˆ
D
2
j
ÿ
X
j
2
1
1
p
p eÿ j 1 ‡ 1D:89
jˆ
2
4
1
1
6
2
2
1
:
1
D2
‡ p …D ÿ †
1
2
1
‡
Z
3
2
2
D
2
ÿ
2
1
p
!
xdx
1
… †
;
10
p
… † ˆ ‡… : =D †… ‡… = †…D ÿ † = †
g… †
h… D † ˆ S ˆ ‡ D = ÿ = D
> ˆ h… † g… † ˆ : > ˆ h… †
0
0
g …D † h …D †
D g D
:
1
D
2 89
1
2
0 63
1
3
1 ;
2
1
1
1 3
.
3 94
for all
Part
(b)
pj is
2
Note
3
2 .
2
that
1
3 2
1
Further-
2, which completes the
Pn
j
ˆ
1
2
3
n
…j p
j … j = e† †
obtained
approximation [3]
=
3 2
!
from
(4)
by
j
2
.
using
Stirling's
p
and the fact that
Part (b) of Proposition 2 indicates that when the
equality
is
by
[3]
(page
199,
eq. (5.56)). Thus we have
M
2
proof of part (a).
;
1
ˆD
.
2
more,
‡ i … ÿ p†
m
2
0 63
Define
m
0
1
=M
where in the second inequality we used Stirling's bound
… p† ‡ … ÿ p†
2
D
1
ˆ ‡
0
1. This is true since
m
0
… †;
0
1
2p
ÿ
X 12…jÿ1† 2j 1
1
M
f m
… † … †:
2
8
D
‡D
a m f m
ÿX
mÿ
m‡i
i
the
m
T ÿ1
X
m
M
1
ÿ1
X
M
Proof. From (4) we have that
From (3) and (6) it follows that
M
ˆD
M
M
D
1
U ˆ
, it follows that in order to have the
S S ; p
S =S ˆ = !1
(a)
1
2
same expected update rate, the parameters of the two
(b) lim
… †ˆm
where
D
=D
Proposition "2". For M
Proof. Let
f m
2p
strategies should satisfy
2p.
:
T
U ˆ
and
update rate decreases, the expected number of searches
necessary to locate the user in the movement-based
… † … †‡M
1
a m f m
ÿ
X
M
m
ˆ
1
0
… † … † ‡ f …M ÿ † M
1
1
… † … † ‡ f …M ÿ † M
a m f m
1
T ÿ1
X
m
ˆM
ÿ1
X
M
lently that if the two strategies have the same expected
… ÿ a…m††
ˆ
Tÿ
X
m
1
m
number of searches necessary to locate the user, then
when this number increases, the expected update rate in
0
1
1
1
… † … †‡M
strategy by a factor of about 1.56. One can show equiva-
1
a m f m
a m f m
strategy is larger than this number in the distance-based
… ÿ a…m††f …m†
ˆM
… †
a m
… †… †
a m f m
The first and the second inequalities follow from the
fact that
… †
f m is a non-decreasing function of m. This can
be seen by using (8) to obtain
… ‡ †ˆ ‡
f …m †
f m
1
1
1
=m
m even ,
m odd .
1
The
second
equality
PT ÿ1 a…m†
1.
mˆ0 MP
M ÿ1
1
ˆ ÿ
1
ˆ
M
m
ˆ
0
This
fact
… †ˆ
a m
follows
1
M
from
implies
PM ÿ 1
m
ˆ
0
that
the
1
… ÿ a…m†† :
1
M
fact
PT ÿ 1
m
that
ˆM a… m †
p
Fig. 2. Cost of search
S
x
vs. update rate
p
ˆ
:
0 05.
U
x in the i.i.d. model for
180
A. Bar-Noy et al. / Mobile users
2pQ0;S
ˆ …1 ÿ q ÿ †…Q ÿ ‡ Qÿ… ÿ †
‡ Qÿ ‡ Q †
ˆ …1 ÿ q ÿ †…Q ÿ ‡ Q ‡ †
1 d Dÿ2
‡ qQ ÿ ‡ Q ‡
1 d Dÿ2
v
1;R
2pQd ;S
D
1;R
1;L
;
v
d
D
1;R
d
1 ;L
1;L
;
;
Qd ;R
ˆ pQ
Qd ;L
ˆ pQ ‡
d ;S
d
d ;S
vQd
2pQD
QD
Fig. 3. Cost of search
S
x
vs. update rate
p
ˆ 0 25.
U
x
ˆ …1 ÿ q ÿ †Q
v
;
ÿ1 L
ˆ pQ
;
1
;
ˆ pQ
QD
the movement-based strategy is larger than this number
;
ÿ1 S
;
1;L ;
d
ÿ1 R ‡ qQd ‡1 L ;
ÿ1 R
in the i.i.d. model for
:
v
1;R
D
D
D
d Dÿ2
;
;
ÿ2 R ;
;
ÿ1 S ‡ qQDÿ2 R ;
;
;
ÿ1 S ‡ vQDÿ2 R ;
;
;
and from symmetry considerations we have
in the distance-based strategy by a factor of about
Qd ;R
2.43.
ˆ Qÿ
Qd ; S
Numerical results
ÿ…D ÿ 1† d …D ÿ 1†
d ;L ;
ˆ Qÿ
1
d ;S ;
d … D ÿ 1†
;
:
Solving these equations and normalizing the stationary
The trade-off between the two costs considered
probabilities so that their sum is one we have
above, i.e., the expected number of update messages per
slot transmitted by a user,
U
x,
and the expected number
of searches necessary to locate a user,
S
x,
Qd ;R
is depicted in
Figs. 2 and 3. It can be seen that the distance-based
example, to achieve a cost of search equals to 3, the
Qd ;L
than 2.5 times the update rate under the distance-based
strategy (the same is true for the time-based strategy). It
ment-based and the time-based strategies is not signifiQd ;S
rate decreases.
v
v
v
v
d Dÿ1
ˆ D…1 ‡ 2p ÿ pq…Dÿ ÿ†‰dD†……11 ÿÿ qq ‡‡ †† ‡ 2…q ÿ †Š
v
v
v
v
d Dÿ1
ˆ D……11ÿ‡q2pÿÿ†‰…q Dÿ ÿ†‰dD†……11 ÿÿ qq ‡‡ †† ‡‡ 2…q…qÿÿ †Š†Š
;
;
v
v
v
1
;
;
1
is also evident that the difference between the movecant, and that this difference is vanishing as the update
v
0
strategy is better than the two other strategies. For
update rate under the movement-based strategy is more
ˆ D…1 ‡p‰…2pDÿÿqdÿ†…1†‰ÿDq…1‡ÿ †q‡‡2…†q‡ÿ2…†Šq ÿ †Š
d Dÿ1
v
v
Q0;S
v
4.1. Distance-based update
v
;
;
ˆ D…1 1‡ÿ2pqÿÿq ÿ †
4. The Markovian model
v
:
The user reports at slot t if and only if either Y …t ÿ 1†
… † be the distance between the cell in which ˆ D ÿ 1 and X …t ÿ 1† ˆ R or Y …t ÿ 1† ˆ ÿ…D ÿ 1† and
X …t ÿ 1† ˆ L. Therefore U ˆ 2Q ÿ
, i.e.,
the user is located at time t and the cell in which it last
transmitted an update message. In the Markovian
U ˆ D…1 ‡ 2p ÿ q ÿ2p†‰…1D‡…1qÿÿq ‡† † ‡ 2…q ÿ †Š
model it is important to distinguish whether the user is
on the right (positive Y …t†) or on the left (negative Y …t†)
Let Y t
D
D
1;R
v
D
of
the
cell
in
which
it
last
reported.
Clearly,
f…Y …t† X …t††, t ˆ 0 1 2, g is a Markov chain. Let
ˆ lim !1 Prob‰Y …t† ˆ d , X …t† ˆ xŠ, d ˆ 0 1, ,
Q
D ÿ 1, x 2 fS R Lg be the stationary probability distri;
d ;x
;
;
...
;
t
;
...
;
bution of this Markov chain. The balance equations
for these probabilities are
Q0;R
ˆ pQ ‡ qQ ÿ ‡
‡ qQÿ ‡ Q
0;S
D
1;R
v
1;R
1;L ;
ÿ…Dÿ1† L
vQ
;
v
v
v
:
Since the number of searches required to locate a user is
just
X j j…
X… ‡
D
S ˆ1‡
D
i
ˆÿ…Dÿ1†
Dÿ1
ˆ1‡2
i
we have that
ÿ1
ˆ1
i
i Qi;S
Qi ;S
‡Q ‡Q †
Qi;R
i ;R
i ;L
‡Q †
i ;L
;
181
A. Bar-Noy et al. / Mobile users
SD ˆ
1‡
…D ÿ
1†‰D ‡ 1 ÿ …q ÿ
3‰D ÿ …q ÿ
v†…D ÿ 2†Š
v†…D ÿ 2†Š
;
:
0
0
Let Pm …d xjx † ˆ Prob‰Y…tm † ˆ d , X…tm † ˆ xjX…t0 † ˆ x Š
(note that from the definition of tm it follows that Y …t0 †
ˆ
;
;
; Rjx † ‡P^
;
^ …d xjx † ˆ P …d xjx † ‡ P …ÿd xjx †
0). Define P
m
m
m
for d
> 0, and let
^
P
0
0
0
m …d j x
^ …d
† ˆ P
m
0
m …d
^ …d jL†,
ˆ P
m
Let Y …t† be the distance between the cell in which
the user is located in slot t and the cell in which it last
transmitted an update message. As before, positive
(negative) Y …t† indicates the user is on the right (left) of
>0
for all d
and m 0. The probability
that a user will be at an absolute distance d (from the
cell in which it last transmitted an update message) at
time tm , namely after m movements, is therefore,
^ …d † ˆ P
^ …d jR†Prob‰X …t † ˆ RŠ
P
m
m
0
the cell in which it last transmitted an update message.
g. Let I …t† be the number of
movements that the user has made during the slots L…t†;
L…t† ‡ 1; . . . ; t ÿ 1
(if
t ÿ 1 < L…t†
then
I …t† ˆ 0).
the user reported in slot
Clearly, 0 I …t† M ÿ 1.
To compute the expected number of update messages
; ; ;
;
ˆ
limt!1 Prob‰I …t† ˆ m
; ;
X
M ÿ1
Prob‰Y …ts † ˆ d Š ˆ
Prob‰Y …ts † ˆ d jI …ts † ˆ mŠ
mˆ0
1
;
X
ˆ
lowing balance equations will be used shortly.
…1 ÿ
qÿ
v†…Q
;
i ÿ1 R ‡
;
; ;
;
;
i ˆ 0 1 ... M ÿ1
Qiÿ1;L † ˆ 2pQi;S
;
Prob‰I …ts † ˆ mŠ
M ÿ1
X …t† ˆ xŠ, m ˆ 0 1, . . ., M ÿ 1, x 2 fS R Lg. The fol-
Qiÿ1;R ‡ Qiÿ1;L ˆ Qi;R ‡ Qi;L
; ;
mˆ0
;
;
an update message at slot t if and only if I …t ÿ 1†
M ÿ 1 and X …t ÿ 1† is either R or L. Therefore,
QM ÿ1;R ‡ QM ÿ1;L ˆ Q0;R ‡ Q0;L ˆ
ˆ
ˆ
1
M
1
M
ÿ
ÿ
1ÿqÿ
vÿ
2p
1ÿqÿ
2p
v
1
M
QM ÿ1;R ‡ QM ÿ1;L
UM
SM ˆ
ÿ
1‡
1
…13†
M
XX
M ÿ1 M ÿ1
;
1
;
;
;
ˆ …1 ‡ q ÿ v†=2 and ˆ …1 ÿ q ‡ v†=2
the quantities Pm …d RjR† and Pm …d LjR† for 0 d m
M ÿ 1.
Defining
we have that,
;
;
;
;
P0 …0 RjR† ˆ
balance equations. The third equality follows from the
=
fact that Qm;R ‡ Qm;S ‡ Qm;L ˆ 1 M for all m, which
;
P0 …d xjR† ˆ 0
;
Pm …d RjR† ˆ
follows from symmetry considerations. Solving for U M
we have
v†
^ …d †
dP
m
mˆ0 d ˆ1
P0 …0 LjR† ˆ
M …1 ‡ 2p ÿ q ÿ
:
To complete the computation we only need to have
Q0;S
:
2p
M
where the latter sum is taken only for d ‡ m even.
The second and fourth equalities follow from the above
UM ˆ
1
Therefore, the expected number of searches required to
where i ÿ 1 is computed modulo M. The user transmits
UM ˆ
^ …d † 1
P
m
…12†
locate the user is
i ˆ 0 1 ... M ÿ 1
ˆ
…11†
have
Markov chain f…I …t† X …t††, t ˆ 0 1 2 . . .g with the staQm;x
:
Returning now to the original Markov chain, we
per slot transmitted by the user, U M , we focus on the
;
^ …d j R †
LŠ ˆ P
m
^ …d jL†Prob‰X …t † ˆ
‡ P
m
0
Clearly, ÿ…M ÿ 1† Y …t† M ÿ 1. Let L…t† ˆ maxf t j
probabilities
; Ljx †.
^ …d jR†
By symmetry considerations we have that P
m
4.2. Movement-based update
tionary
0
0
mÿ1 … d ÿ
m 1
;
Pm …d LjR† ˆ
:
P
P
;
;
;
;
1 RjR† ‡
ÿ…M ÿ
mÿ1 … d ÿ
m 1
;
;
d 6ˆ 0 x 2 fR Lg
;
mÿ1 … d ‡
;
1 LjR†
1† d M ÿ 1
1 RjR† ‡
ÿ…M ÿ
P
;
P
mÿ1 … d ‡
;
;
1 LjR†
1† d M ÿ 1
;
;
:
To compute the expected number of searches neces-
It is obvious that these probabilities can be computed
sary to locate a user, S M , we will consider an embedded
recursively from the above relations. Using transform
Markov chain that ignores the states in which the user
techniques, one can obtain (rather complicated) closed-
;
0
does not move. To that end, let J …t t † be the number of
form expressions for these probabilities.
movements that the user has made during the slots
;
;
;
0
L…t† L…t† ‡ 1 . . . t ÿ 1. Recall that ts is the slot in
;
which a search occurs. Let tm ˆ maxft L…ts †jJ …ts t†
ˆ
;
4.3. Time-based update
mg. The embedded Markov chain is f…Y …tm † X …tm ††,
; ;
m ˆ 0 1 2, . . .g.
As in the i.i.d. case, with time-based update we have
182
A. Bar-Noy et al. / Mobile users
U ˆT
1
T
since the user transmits an update message every
We
now
compute
the
expected
number
of
T slots.
searches
required to locate the user. As in the previous subsection,
…†ˆ
L t
let
…†
max
f t j
the user reported in slot
g
. Let
Y t be the distance between the cell in which the user is
t and the cell in which it last reported.
located in slot
ts is the slot in which the user is being
Recall that
XXj j
searched. Then we have
S ˆ ‡T
1
1
T
ÿ
T
1
t
ˆ d ˆÿt
t
‰ … †
d Prob Y ts
0
T † ˆ tŠ :
ˆ d j…t
T † ˆ tŠ
Prob‰Y …t † ˆ d j…t
Q …d †ˆ Prob‰Y …t †
T † ˆ tŠ
t ˆ ; ;
ˆ d jX …L…t †† ˆ x; …t
;T ÿ
d ˆ ÿt; ÿ…t ÿ †;
; ;
;t
T † ˆ tŠ ˆ …S †Q …d †
Prob‰Y …t † ˆ d j…t
‡ …R†Q …d † ‡ …L†Q …d † ;
…x†
s mod
The
probabilities
determined
as
Let
s mod
s
...
s mod
x
t
s
follows.
.
1 and
1
...
Then
for
0 ...
, we have,
L
t
is the stationary probability of being in state
x, i.e.,
1
1
1
The probabilities
T
ÿ
1 and
x
t
0 ...
; ;
0 1
...
;
, via the follow-
… † ˆ … ÿ p† Q ÿ … d † ‡ p Q ÿ … d † ‡ p Q ÿ … d † ;
S
1
S
t 1
2
L
t 1
R
R
t 1
… †
…d † ˆ … ÿ q ÿ v † Q ÿ …d ÿ † ‡ q Q ÿ …d ÿ †
… †
‡ v Q ÿ …d ÿ † ;
…d † ˆ … ÿ q ÿ v † Q ÿ …d ‡ † ‡ q Q ÿ …d ‡ †
… †
‡ v Q ÿ …d ‡ †
S
t 1
1
L
t 1
L
Qt
R
t 1
1
1
1
15
S
t 1
1
R
t 1
L
t 1
1
1
1
ˆ
:
0 1,
x in the Markovian model
:
p
0 1.
4.4. Numerical results
The
trade-off
between
the
two
costs
considered
above, i.e., the expected number of update messages per
slot transmitted by a user,
U
x , and the expected number
of searches necessary to locate a user,
S
x , is depicted in
ment-based strategy (see Fig. 5). It seems that this phenomenon
occurs
when
the
movement
``restless and erratic'', i.e.,when
pattern
is
p and q are small, and
v
is close to 1. Note also that as in the i.i.d. case, the differ-
14
Qt
v
in which the time-based strategy is better than the move-
ing recursion.
Qt d
:
0 8,
the i.i.d. model, in the Markovian model there are cases
2
...
ˆ
tegies. However, it is interesting to note that unlike in
are computed for
1
q
distance-based strategy is better than the other two stra-
;
2
U
ˆ
x vs. update rate
Figs. 4 and 5. As in the i.i.d. case, it can be seen that the
…S† ˆ ‡ ÿp qÿÿq vÿ v
p
…R† ˆ …L† ˆ
‡ pÿqÿv :
Q …d †
t ˆ
d ˆ ÿt; ÿ…t ÿ †;
; ;
;t
S
for
0 1
S
t
R
t
Fig. 4. Cost of search
s
s mod
s
where
are
ence between the movement-based and the time-based
strategies seems to vanish as the update rate decreases.
5. Discussion
In this paper we focus on three natural dynamic
tracking
strategies,
namely,
the
distance-based,
the
16
initiated with
… † ˆ Q …d † ˆ Q …d † ˆ …d † ;
…d † ˆ
d ˆ
…d † ˆ
R
Q0 d
where
1 if
L
S
0
0
0 and
0 otherwise.
The explanation for (14) is as follows. Given that in
slot
… †
L ts
(the slot in which the user last reported) the
user is in state
… †‡
S , the user will be in slot L ts
S with probability
… ÿ p†
1
2
, in state
1 in state
L with probability
p, and in state R with probability p. Given that in slot
L…t † ‡
T† ˆ t
…t
1 the user is in state
s
s mod
will
be at
reported is
are similar.
distance
ÿ …d †
x
Qt
x (x
2 fS ; R ; L g
, the probability that in slot
1
) and that
ts the user
d from the cell in which it last
. The explanations for (15) and (16)
Fig. 5. Cost of search
S
for
x vs. update rate
q
ˆ
:
0 1,
v
ˆ
:
0 8,
U
ˆ
p
x in the Markovian model
:
0 1.
183
A. Bar-Noy et al. / Mobile users
movement-based and the time-based strategies. We con-
ÿ1
X
M
ˆ 1 ‡ M1
sidered a network in which the cells are arranged in a
2
ˆ0
M
ÿ1 1m
X
ˆ 1 ‡ M1
Markovian movement patterns.
In the memoryless case, we proved that the distancebased strategy is the best among the three strategies,
and that the time-based strategy is the worst one among
them. In the Markovian case, we showed that this is
not always true, as for certain types of movement patterns the time-based strategy is better than the move-
difference
vanishes
as
the
update
rate
decreases. The difference between the distance-based
strategy and the other two strategies is more significant.
There
are
a
number
of
problems
that
are
not
further research. In this paper we considered only a ring
topology. However, the ideas and techniques used here
seem to be extendable to other topologies, such as a grid
results would change (if at all) if this cost is defined differently.
l
Next,
ˆ 2l for an integer l we have
ÿ ÿ 2l ÿ 1
1 X
1
2l
ˆ
:
l
l ÿ1
M
2
l
ˆ
ˆ1
1
2l
2
M 1
2
l
summing
the
odd
2l
1
1
values
of
m,
i.e.,
ˆ 2l ÿ 1 for an integer l we have
ÿ 1 X
1
2l ÿ 2
1 X
1
ˆ
…
2l ÿ 1†
l
M
2
l ÿ1
M
2
ˆ
ˆ
taking
m
M
2
2l
M
2
2
1
l
2l
ÿ1 2l l
1
:
The odd case is identical to the even case when M is
odd. When M is even, only the odd case contributes to
the summation. Thus we obtain that (2) simplifies to
(4),
S ˆ1‡M
1
X12…jÿ1† 2j ÿ
M 1
2
M
In particular, one could adopt a combination of all or
reasonable possibility. It is interesting to see how the
X
M
other walking patterns of the users, and other strategies.
the cost defined here for the search operation is only one
2
2l ÿ1 ÿ
M 1
2
1
graph. Another possible extension could be to consider
some of the strategies considered in this paper. Finally,
m
2
expression, i.e., taking m
l
addressed in this work, which could be the subject of
1
(3). Summing first only even values of m in the above
obtained show that the difference between the movethis
ˆ1
m
2
k
the second summation. This shows that (2) simplifies to
Markovian movement patterns, the numerical results
that
m
m
ÿ 1 ÿ m ÿ 1
kÿ1
k
ˆd e
ÿ mÿ1 6 7
…17†
ÿ1 :
The last equality follows from the telescopic nature of
ment-based strategy. In both the memoryless and the
ment-based and the time-based strategies is small, and
mÿ1 X
m
m
1
m
ring topology, and analyzed the performance of each
one of the three strategies under a memoryless and a
m
where M
j
2
ˆ1
j
‡
j
M 1
M
2
M
M ;
2
ˆ 1 if M is even and ˆ 0 otherwise.
M
Appendix B
The goal is to simplify (5),
Appendix A
XX
S ˆ 1 ‡ T1
The goal is to simplify (2),
S ˆ1‡M
1
M
M
m
ÿ1
m
ˆ0 d ˆ0
T ÿ1 X
t
t
X
T
d
m
m
mÿ1
1
‡d
2
2
2
;
ˆ0 mˆ0
t
m
X
d
ˆ0
m
d
m
m
…2p† …1 ÿ 2p† ÿ
m
mÿ1
1
‡d
m
;
2
2
t
where the sum is taken only for values of d such that
where the sum is taken only for values of d such that
m
m
‡ d is even.
Let m ‡ d ˆ 2k with k an integer. Then,
S ˆ1‡M
1
M
ˆ1‡M
1
2
M
ÿ11mÿ1 X
m
X
ˆ0 2
k ˆd
M
ÿ1 1 mÿ1
X
m
m
2 42m
ˆ1‡M
1
e
ˆ0
m
m
X
ÿ1
X
ˆ0
m
k
m
2
mÿ1 X
m
m
1
2
k
ˆd
m
2
e
2
k
3
5
ÿ 1 ÿ m
kÿ1
k
T ÿ1 X
m X
T ÿ1
t X
T
ˆ 1‡T
m
2
M
S ˆ1 ‡ T
2
2
ÿ1 ÿm
kÿ1
ˆd e
ˆd e
m
k
2
m
X
m
k
m
2
Rearranging (5) we obtain
m
…2k ÿ m†
‡ d is even.
ˆ 1‡T
2
ˆ 1 ‡ T2
m
ˆ1 d ˆ1 tˆm
d
m
m
ˆm
mˆ1 d ˆ
d
m
ˆ1 d ˆ1 tˆm
ˆ1 mˆ1 d ˆ1 tˆj
‡d
m
m
‡d
2
j
T ÿ1 X
m X
T ÿ1
X
X
j
p
m
p
m
…1 ÿ 2p† ÿ
t
tÿm
X
t
2
m
d
‡d
m
m
m
T ÿ1 X
m X
T ÿ1
t X
d
m
2
T ÿ1 X
m X
T ÿ1
t X
1 t
m
m
m
‡d
2
j
ˆ0
ÿ m…ÿ2p†
X
t
t
j
ÿ m…ÿ2† ÿ
jÿm
ˆm
t
t
m
j
j
j
ÿ m…ÿ2† ÿ
jÿm
j
m
m
p
j
j
:
p
184
A. Bar-Noy et al. / Mobile users
Since we need the above expression to simplify to (7) we
Since …l ‡ i†
have to show that,
hand side is
XXX
j
T ÿ1
m
d
m‡d
mˆ1 d ˆ1 tˆj
2j ÿ 2
jÿ1
…ÿ1†
jÿm
m
2
ˆ
t
tÿm
m
1ÿm
2
j ÿm
…18†
Pi
2i ÿ1
To simplify the left hand side of (14) we prove the
tˆj
jÿm
m
ˆ
j
T
m
j‡1
2i
Proof. Identity (19): By [3] (page 174, eq. (6)) we get
ÿ t 1 tÿm
ÿj1 t
…j ÿm † ˆ m …j †, and therefore,
that
m
T ÿ1 X
t
tÿm
T ÿ1 X t
j
ˆ
jÿm
m
tˆj
m
tˆj
that
P T ÿ1
tˆj
t
j ‡1
j
X
p
T
…j † ˆ …j ‡1 †.
y
2
2i ÿ1
ÿy1
and
i
X
x
ˆ
2i ÿ1
1
ÿi
2
, it follows that
l ˆ1
2i
2
2i
2i
ÿ
i
ÿ
2
2
2i
ÿ
1
i
2
2i
2
mˆ1
j
m
…ÿ1†
1ÿm
2
j ÿm
m
X
d
m
2i ÿ 1
X 1 2i j
2i
j
2
2i
j
j
ÿ2
i
i
2
i ˆ1
i
2i
m‡d
2j ÿ 2
X 1 2i 2i
j
2
2
j
i
2
j
2i ÿ 1
i
j
ÿ
ˆ
2i
2j ÿ 2
jÿ1
j ÿ1
X
2
2
j
We now partition the left hand side of (20) into two
integers
and
in
the
odd
d ˆ 2l ÿ 1
case
2i 2
i ˆ1
2
2i
2l
l ˆ1
X 1 2iÿ1
j ‡1
2
‡2
j
i ˆ1
j
2i ÿ 1
2
2i
2i ÿ 1
!
ˆ
j ‡1
2i
…24†
:
jÿ1
2
j
ÿ
2j ÿ 2
i
X
X 1 2iÿ1 2i
l
2i
…2l ÿ 1†
2i ÿ 1
i‡l ÿ1
l ˆ1
ˆ
i‡l
l ˆ1
i
X
…2l ÿ 1†
l ˆ1
i
2i
2
2i ÿ 1
i‡l ÿ1
2
:
ˆ
j
i‡l
i
2
i
2i
i
ÿ1
2
2
ˆj
jÿ
2
ÿi
i
X
!
2
…26†
:
j
2
2i
:
i
…22†
ÿ2i1ÿ
i
1
j
2iÿ1
ˆ 2j
ÿ2iÿ21ÿ j ÿ1 1
i ÿ1
2i ÿ2
it follows
y
. Therefore, the left hand
side of (25) is equal to
X 1 2iÿ2 2i ÿ 2
j
2
‡ i ÿ i and
x
Proof. Identity (25): By the definition of
that i
2
l ˆ1
…25†
3
i ˆ1
i
2
iÿ1
jÿ1
:
2…i ÿ 1†
We now use the following identity [3] (eq. (5.38)),
;
get that the left hand side is
…l ‡ i †
j
!
3
jÿ
ˆj
2i ÿ 1
i
X 1 2iÿ1 2i
j
i ˆ1
…21†
;
Proof. Identity (21): First we replace l by l
2i
j
i
i
2
i ˆ1
j
j
the second identity follows similar arguments.
l ˆ1
i
similar we are going to prove only the first one.
i‡l
identities. We prove only the first identity, the proof for
i
X
j
need the following two identities. Again since they are
To simplify this expression we need the following two
i
X
;
We only prove (23), the proof for (24) is similar. We
i
X 1 2i j X
j
ÿ2
2i
j‡1
4
m ˆ 2i ÿ 1 where l and i are integers. Thus, the left hand
j
i
2
j‡1
‡
and
side of (20) becomes
1
i ˆ1
cases: In the even case d ˆ 2l and m ˆ 2i where l and i
are
and for an odd j
…20†
:
jÿ1
2
d ˆ1
:
…23†
ˆ
.
p
:
i
Consequently, it remains to show that
i
It remains to show that for an even j
i ˆ1
j
X
ÿ2i11
i
ˆ
1
ˆ
i ‡l
2
:
y
y ÿx
Pi
2i
i‡l
l ˆ1
and that
2
2i i ˆ1
The claim follows since by [3] (p. 174, eq. (9)) we get
ÿi
Using identities (21) and (22), the left hand of (20)
2
:
2i ÿ 1
becomes
2
j
2
…19†
:
it follows that the left
i ‡l ÿ1
Therefore, the left hand side is
following identity:
2
ˆ
i ‡l ÿ1
2i ÿ1
i‡l ÿ1
Py ÿ y 1
ˆ
xˆ1 x
Since
l ˆ1
T ÿ1 t ÿ m
X
t
i
X
l ˆ1
:
j‡1
ˆ 2i
i ‡l
2i
T
2i
2i
i‡l
X
n
2
:
kˆ0
n
2k
2k
k
2
ÿ2k
ˆ
nÿ
45
n
2
!
1
2
:
185
A. Bar-Noy et al. / Mobile users
Amotz Bar-Noy
p
We obtain identity (25) by replacing k with i ÿ 1 and n
by j ÿ 1.
mathematics
received the B.A. degree in
and
computer
science
in
1981
and the Ph.D. degree in computer science in
1987, both from the Hebrew University of Jerusalem, Israel. He was a post-doctoral fellow
Plugging identities (25) and (26) in (23), it remains
to prove that
"
3
j ÿ
jÿ1
j2
j
2
!
ÿ
ÿ 1
ˆ
2j ÿ 2
ˆ
2
1989 he has been a member of the Communication Network Group at IBM T. J. Watson
:
j ÿ 1
…27†
x
jÿ
we get
y
j
2
3
j
jÿ1
2
2
jÿ
ber 1987 to September 1989. Since October
!#
j
Again by the definition of
1
3
j ÿ
2
3
2
ÿ
ÿ1
jÿ
3
j2
jÿ1
j ÿ
3
!
2
j
j ÿ 1
ˆ
2j ÿ 2
Center,
Yorktown
Heights,
New
and distributed algorithms. His other research interests include parallel algorithms and combinatorial algorithms.
2
E-mail: [email protected]
Ilan Kessler
:
j ÿ 1
2
Research
York. His main research interests include communication networks
2
j
, and thus it remains to prove that
2
at Stanford University, California, from Octo-
received the B.Sc., M.Sc., and
D.Sc. degrees in electrical engineering from
…28†
the Technion, Israel Institute of Technology,
Haifa, Israel, in 1979, 1986, and 1990, respectively. From 1979 to 1983 he was a technical
officer in the Israeli Defense Force, where he
Proof. Identity (28):
2j ÿ 2
j ÿ 1
ˆ
was
…2j ÿ 2†…2j ÿ 3† 1 1 1 …j ‡ 1†…j†
1
…j ÿ 1†…j ÿ 2† 1 1 1 …2†…1†
j ÿ 1
responsible
for
the
development
and
deployment of mobile data networks. From
1990 to 1994 he was a Research Staff Member
j ÿ 1
at the IBM Thomas J. Watson Research Cen-
ˆ
j
j ÿ 1
1
1
2j ÿ 2
j
ˆ
111
j ÿ 2
j ‡ 2
j
2
ter, Yorktown Heights, NY, where he worked on design and analysis
of algorithms for mobile wireless networks and high- speed communi-
‡ 1
cations networks. In February 1994 he joined Qualcomm Inc., San
…2j ÿ 3†…2j ÿ 5† 1 1 1 …j ÿ 1†
j
j
2
2
… †…
ˆ
1
j ÿ 1
2j ÿ 4
j22
ÿ1
j ÿ 1
j2
2…j ÿ
j ÿ 1
5
j
2
2
2
j
j
2
2
… †…
j ÿ
1
Israel, where he has been leading a group working on CDMA based
3
†2…j ÿ
1
jÿ1
Diego, CA, and since August 1994 he is with Qualcomm Israel, Haifa,
ÿ 1† 1 1 1 …1†
3
!
2
j
† 1 1 1 2…
satellite networks.
ÿ
1
2
†
E-mail: [email protected]
ÿ 1† 1 1 1 …1†
Moshe Sidi
:
p
2
received the B.Sc., M.Sc. and the
D.Sc. degrees from the Technion, Israel Institute
of
Technology,
Haifa,
Israel,
in
1975,
1979 and 1982, respectively, all in electrical
engineering. In 1982 he joined the faculty of
Electrical
Engineering
Technion.
During
the
Department
academic
at
year
the
1983-
1984 he was a Post-Doctoral Associate at the
References
Electrical Engineering and Computer Science
[1]
31st Symposium on Foundation of Computer Science, October
A. Bar-Noy and I. Kessler, Tracking mobile users in wireless
[3]
R.L.
D.E.
Knuth
and
O.
Patashnik,
Concrete
[4]
D.J. Goodman, Cellular packet communications, IEEE Trans.
[5]
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wireless
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Honig
for
and
K.
personal
Steiglitz,
Optimization
communications
of
mobility
Research
Center, Yorktown
He served as the Editor for Communication Networks in the
from 1989 until 1993, and as
the Associate Editor for Communication Networks and Computer
until 1994. Currently he serves as an Editor of the IEEE/ACM Transactions on Networking and as an Editor of the Wireless Journal. His
research interests are in wireless networks and multiple access proto-
tracking, preprint.
S.J. Mullender and p.B.M. Vitanyi, Distributed match-making,
Algorithmica 3 (1988) 367^391.
[7]
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Networks in the IEEE Transactions on Information Theory from 1991
Commun. 38 (1990) 1272^1280.
M.L.
IBM Thomas
IEEE Transactions on Communications
Mathematics (Addison-Wesley, 1990).
Madhow,
at
Heights, NY. He coauthors the book ``Multiple Access Protocols: Performance and Analysis,'' Springer Verlag 1990.
networks, IEEE Trans. Inf. Theory 39 (1993) 1877^1886.
Graham,
Technology, Cambridge, MA. During 1986-1987 he was a visiting
scientist
1990, pp. 503^513.
[2]
[6]
Department at the Massachusetts Institute of
B. Awerbuch and D. Peleg, Sparse partitions, in: Proc. of the
I. Cidon and M. Sidi, A multi-station packet-radio network,
Perform. Eval. 8 (1988) 65^72 (see also INFOCOM'84, pp. 336^
343).
[8]
R. Steele, The cellular environment of lightweight hand held
[9]
R. Steele, Deploying personal communication networks, IEEE
portables, IEEE Commun. Mag. X (1989) 20^29.
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cols, traffic characterization and guaranteed grade of service in highspeed networks, queueing modeling and performance evaluation of
computer communication networks.
E-mail: [email protected]