SIMILAR REGULAR PLANS FOR MOBILE CLIENTS
John Tsiligaridis
Heritage University, Math and Computer Science, 3240 Fort Road, Toppenish, WA, 98948, U.S.A.
Keywords: Broadcasting, Broadcast plan, Mobile computing.
Abstract: The broadcast problem including the plan design is considered. The data are inserted and numbered into
customized size relations at a predefined order. The server ability to create a full, regular Broadcast Plan
(RBP) with single and multiple channels, after some data transformations, is examined. The Basic Regular
Algorithm (BRA) prepares an RBP and enables users to catch their items avoiding wasting energy of their
devices. In the case of multiple channels, a dynamic grouping solution is proposed, called Full Partition
Value Algorithm (FPVA) under a multiplicity constraint. The Similar Regular Plan Algorithm (SRPA)
provides faster service of the supreme sets with the use of fewer channels. The combination of FPVA and
SRPA provides flexibility for finding desired solutions. This last property, can be offered by servers today
providing channel availability and lower energy consumption. Simulation results are provided.
1 INTRODUCTION
An efficient broadcast schedule program minimizes
the client expected delay, which is the average time
spent by a client before receiving the requested
items. The expected delay is increased by the size of
the set of data to be transmitted by the server. A lot
of work has been done for the data dissemination
with flat and skewed design (Acharya et al., 1995,
Yee et al., 2002, Ardizzoni et al., 2005, Bertossi et
al., 2004). For the flat design when the cycle
becomes large the users have to wait for long until
they catch the data in case they had lost them
previously. For the skewed design ,the most
frequently requested data items should be put in fast
channels whereas the cold data can be pushed to
slow channels. Various methods have been
developed to partition the data according to their
popularity using dynamic programming (Yee et al.,
2002) , and the heuristic algorithm VFk (Peng et al.,
2000). The minimum time broadcast problem has
been addressed by computing the minimum degree
spanning tree of directed acyclic graphs in (Yao et
al., 2008). The Min-Power broadcast problem in
wireless ad hoc networks has been answered by
assigning transmission range to each node (Hashemi
et al., 2007).
When the broadcast cycle has long size, the flat
scheduling needs many channels to avoid the user
delay. The regular design with the equal spacing
property (Acharya et al., 1995) can provide
broadcasting for single and multiple channels with
average waiting time less than the one of the flat
design. It also offers channel availability, and less
energy consumption while there is no need for use of
channels with different speeds.
For the regular design, the system works with a
number of channels that could be of the same speed.
The users of all sets, except for the last one, can get
their data from the same channel. Only the users of
the last set (the most unpopular set) have to switch
to another channel. The data are considered
homogenous or heterogeneous with multiples of a
basic size. Data can be sent by a single channel or a
set of channels.
In this paper, we study the problem of finding
the number of channels that can send a group of
data, while ensuring equal spacing of repeated
instances of items. The FPVA with SRPA provides
a framework for a dynamic solution under
constraints, in the case of multiple channel
allocations. The FPVA can be extended to the case
of broadcasting updated data.
The paper is organized as follows. In Section 2,
the model description is given. The BRA is
developed in Section 3. In Sections 4,5 and 6 the
FPVA, the SRPA and their combination are
developed respectively. Finally, simulation results
are provided in Section 7.
69
Tsiligaridis J..
SIMILAR REGULAR PLANS FOR MOBILE CLIENTS.
DOI: 10.5220/0003602000690074
In Proceedings of the International Conference on Data Communication Networking and Optical Communication System (DCNET-2011), pages 69-74
ISBN: 978-989-8425-69-0
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
2 MODEL DESCRIPTION
2.1 The Relations in the Broadcasting
Plan
In our approach we consider three sets S
i
(i=1,2,3)
with their sizes S
is
so that S
3s
S
2s
S
1s
.The
possibility of providing full BP (it does not include
any empty slot) is examined iteratively using
relations starting from the last level of hierarchy S
3
.
The number of S
i
items (or items of multiplicity
(it_mu
i
)) will be sent at least one from S
3
,while for
the other two sets at least two. Given the size S
3s
,
S
2s
, S
1s
from the integer divisions of S
3s
, using
array (arr), we can create a set of relations S
div
(j< S
3s
), with different number of relations (n_rel) and
subrelations in each set (i-subrelation, i=1,2,3). We
create a set of relations including their subrelations
by considering items of different size from each set.
Each relation has three subrelations.
The following definitions are essential:
Definition 1: The size (or horizontal dimension) of a
relation (s_rel) is the number of items that belong
to the relation and it is equal to the sum of the size of
the three subrelations (s_rel=
=
3
1
_
i
i
subs
). The number
(or vertical dimension) of relations (n_rel) with
s_rel define the area of the relations (area_rel).
Example 1: The relation A=(a, b, c, d, f) has the
following three subrelations starting from the end
one; the 3-subrelation (f) with s_sub
3
= 1, the 2-
subrelation (b,c,d) with s_sub
2
= 3, and the 1-
subrelation (a) with s_sub
1
=1. The s_rel=5
Definition 2: The area of the i-subrelation
(area_i_sub) is defined from its size (s_sub
i
) and
the number of the relations (n_rel) that are selected.
It is given by (s_sub
i
) x (n_rel).
Example 2: From a relation with s_rel=5 and if
n_rel=5 then the area of this relation is 5x 5 .
Hence there are 25 locations that have to be
completed.
Example 3: If two relations are: (1,2,3,5,6,7),
(1,3,4,8,9,10) with s_sub
3
=3, s_sub
2
=2, then : 2-
subrelation
1
=(2,3) and 2-subrelation
2
=(3,4). The
last two subrelations ((2,3),(3,4)) comes from S
2
={2,3,4} having 3 as repeated item.
Definition 3: A BP is full if it provides at least 2
repetitions of items and it does not include empty
slots in the area_rel. A BP is regular if it is full and
provides equal spacing property (Acharya et al.,
1995).
Definition 4: The number of items that can be
repeated in a subrelation is called item multiplicity
(it_mu) or number of repetitions (n-rep).
Definition 5: A subrelation i (i-subrelation) that
belongs to set S
i
is strong if, in its area, it can
provide the same number of repetitions of all the
items of a set (without empty slots) for all the
relations. The strong i-subrelations create strong
relations.
Definition 6: Integrated relations (or integrated
grouping) is when after the grouping, each group
contains relations with all the data of S
2
and S
1
. This
happens when: ( (2_subrelation) = S
2
)
(
(1_subrelation) = S
1
). See example 7 for details.
Grouping length(g): The g is a divisor of S
ks
(1,..,k).
It is the n_rel that can provide homogenous
grouping. The g
k
stands for the various values of g.
Supreme sets (SS): are all the sets except the last
one. (SS =S
i
(i=1..n-1)\ S
n
). The last set (S
n
) can be
named as secondary set (SS). The divisors of SS are
named last set divisors (LD= {ld
i
}, for i=1..k}).
Some (or all) of the ld
i
values can provide an RBP
(homogenous grouping) and then ld
i
= g
k
. It is
possible all ldi values to be grouping values as well
(ld
i
= g
i
). Similar Regular Plans (SRP): are the plans
that have the same subrelations for the supreme sets
and different for the secondary set.
Example: Considering : (1) RBP
1
with s_sum
1
= 5,
s_sum
2
=5, s_sum
3
=8 and s_sum
4
=12,(2) RBP
2
with
s_sum
1
= 5, s_sum
2
=5, s_sum
3
=8 and s_sum
4
=6. The
two RBPs consist an SRP.
Partition value (pv): It is the common divisor of S
is
(i=1,.., k) and gl for a given size of s_sum
i
. Hence:
pv
i
| S
is
and pv
i
| gl. Each set must have its own pv.
Example 4: If S
3s
=40, g=20, considering that
s_sum
3
=8 then pv
3
=5 (=40/8) . Hence pv
3
| S
3s
and
pv
3
|g
The criterion of homogenous grouping(chg): when
pv
i
| g.
The criterion of multiplicity constraint(cmc): This
happens when: it_mu
i+1
< it_mu
i
(i= 1,..,n-1).
The PV criterion: when PV
i
> PV
i+1
The number of channels (nc): S
k
/ gl (where S
k
is the
last set)
It is considered that a|b (a divides b) only when b
mod a =0 (f.e. 14 mod 2=0). The relation with the
maximum value of n_rel provides the opportunity
of maximum multiplicity for all items of S
2
and S
1
and finally creates the minor cycle of a full BP. The
major cycle is obtained by placing the minor cycles
on line. Similar description of the relations model is
in (Tsiligaridis ,2009, Tsiligaridis et al., 2007).
DCNET 2011 - International Conference on Data Communication Networking
70
2.2 Some Analytical Results
Two basic Lemmas provide the possibility of the
FBP and RBP construction. The first deals with a
particular case of the S
2s
and S
3s
while the second is
a general case for every value of S
2s
, S
3s
. Proofs and
details for the case of empty slots BP are not
included in this work due to limited space.
After making sure that there is a RBP the data from
the array (the minor cycles for each array line) are
transferred to queues for broadcasting. For multiple
channels, the data from integrated relations are
grouping with GHA and then are broadcasting.
Example 5: The relation A= (a, b, c, d, f) has the
following three subrelations (s_sub
i
) starting from
the end one; the 3-subrelation (f) with s_sub
3
= 1,
the 2-subrelation (b,c,d) with s_sub
2
= 3, and the 1-
subrelation (a) with s_sub
1
=1. The size of relation
(s_rel) =5.
Lemma 1 (particular case): The basic conditions in
order from a set of data to have a regular broadcast
plan are: k= S
2s
/ S
3s
(1) and m= it_mu
2
= S
2s
/ k (2)
(item multiplicity).
Proof: For (1) if k= S
2s
| S
3s
then the k offered
positions can be covered by items of S
2s
and we can
take a full BP. From (2) m represent the number of
times (it_mu) that an item of S
2
will be in the
relation.
Example 6: (full BP) Consider the case of: S
1
= {1},
S
2
={2,3}, S
3
= { 4,5,6,7,8,9, 10, 11}. Moreover k=
S
2s
| S
3s
= 4(8/2) , and m=2(4/2) the it_mu
2
=2=4/2 .
The relations for the full BP are: (1,2,4,5), (1,3,6,7),
(1,2,8,9)(1,3,8,9). Since (s_sub
3
/ s_sub
2
) >1 we have
r_p =4 (2*2).
Example 7: Let’s consider S
1
= {1}, S
2
={2,3,4,5},
S
3
= {6,7,8,9, 10, 11,12,13}. Again, k=2(8/4), m=
it_mu
2
=2(4/2). Hence the FBP is (1,2,3,6,7),
(1,4,5,8,9),(1,2,3,10,11) ,(1,4,5,12,13). The
subrelations (2,3) (4,5).
Lemma 2 (general case): Given that S
2s
and S
3s
(and S
2s
? S
3s
) with k
1
, k
2
their common divisors as:
k
1
= n/S
2s
(3) and k
2
= n/S
3s
(4) (where n= common
divisors of S
2s
and S
3s
): (a) if k
2
< S
2s
and k
2
/S
2s
(5) then there is an RBP with it_mu
2
= k
2
/S
2s
(b) if
k
2
> S
2s
and S
2s
/k
2
(6) then there is an RBP with
it_mu
2
= S
2s
/k
2
The RBP will have for both cases k
2
relations.
Proof: From (3) we get that the number of S
2
items
in a line s_sub
2
= k
1
/ S
2s
. From (4) we have s_sub
3
= k
2
/ S
3s
. If (5) is valid then it means that the k
2
positions (offered by S
3
) can be covered by k
2
/S
2s
items (it_mu
2
). If (6) is valid then it means that the
k
2
positions (offered by S3) can be covered by S
2s
/
k
2
Example 8: S
1
= {1}, S
2
={2,..,13}, S
3
= { 15,..,32}
, S
2s
= 12, S
3s
= 18. If n =3, k
1
= 3/12 =4, k
2
=
3/18=6, and k
2
/S
2s
= 6/12 = 2. Hence we have 6
relations and the 2-subrelations are:
(….,2,3,4,5,…),(…,6,7,8,9…),(…,10,11,12,13,…),
(….,2,3,4,5,…),(…,6,7,8,9…),(…,10,11,12,13,…).
If n=2, k
1
= 2/12 =6, k
2
= 2/18=9, and from k
2
/S
2s
=we have 9 ? 12.
Theorem 1: Let us consider the case of multiple
channel allocation with different multiplicity of
sets (such as: S1, S2, S3). Then, the validity of chg
can be achieved when pv
i
= S
is
/ s_sub
i
. If pv
i
| n
k
then g = n
p
and the criterion of homogenous
grouping holds.
Example 9: Let’s consider again the same four sets
S1,S2,S3,S4 with S
1s
=10, S
2s
=20,S
3s
= 40, S
4s
=120
If gl =20 (20 is a divisor of 120) then S
1s
/ gl, S
2s
/
gl, gl / S
3s.
The chg exists. The number of channels
is: nc=120/20= 6. Considering s_sum1 = 5,
s_sum
2
=5,s_sum
3
=8 then pv1 = 10/5=2, pv
2
=
20/5=4, pv
3
=40/8=5 and since pv
1
|20 ,pv
2
|20,pv
3
|20
then there is an homogenous grouping. With g =20
Theorem 2: For the lower values of LD that can
offer an RBP, we have higher values of AWT
i
for
supreme set and more channels.
Proof: For lower LD values the PV
i
has also lower
values which means greater size of all the sub
relations (s_sub
i
) and finally greater values for
AWTi
2.3 The SRP
Theorem 3: For SRP increasing the LD number of
groups (gi) we have lower AWTi for supreme sets
and small number of channels.
Proof: Increasing ld
i
results to having less data for
the Sk (secondary set ) in the relations which
minimize the AWT
i
(i=1..k-1). This theorem can be
applied to FPVA.
The number of channels is determined by the S
ks
and
the g
k
(#channels= S
ks
/ g
k
).
Example 10: Lets consider S
1s
=10, S
2s
=20,S
3s
= 40,
S
4s
=120 and PVA
1
= 2, PVA
2
=4, PVA
3
=5, PVA
4
=5. For g
k
= 10 we have: s_sub
1
=5, s_sub
2
=5,
s_sub
3
=8, s_sub
4
=12 and AWT
1
=15 (30/2), n_ch =
12(120/10). For g
k
=20, s_sub
1
=5, s_sub
2
=5,
s_sub
3
=8, s_sub
4
=6 and AWT
1
=14 (24/2). , n_ch =
6(120/20).
From Theorem 3 it is obvious the SRP’s ability to
provide lower values of AWT for the SRP with
fewer channels ( 12 to 6). This is the advantage the
SRP offers for the RBP design.
For any BP the upper and lower bound of AWT
(UA, LA) is depending on the size of s_sub
i
(i=1..n).
For a SRP, the UA and LA depend on the value of
s_sub
4
. They can be defined considering the
possible upper and lower values of s_sub
4
.
SIMILAR REGULAR PLANS FOR MOBILE CLIENTS
71
Theorem 4: For any RBP there is always a LA with
the unit correspondence.
Unit correspondence is the case when in each
relation only one item of S4 is considered. In that
case we can have the same LA for different g values
but with different n_ch.
Example 11: Let’s consider LA and for: s_sum
1
=5,
s_sum
2
=5, s_sum
3
=8 and s_sum
4
=1 with broadcast
cycle = 19 (5+5+8+1). The AWT
3
= 52.5 ((5*19
+10)/2).
The AWT
3
remains the same independently of
the value g. So if g=60 , AWT
3
=52.5 while n_ch =
2 (120/60). On the other hand if g=20 , AWT
3
=52.5
,and n_ch = 120/20 =6. The UA for the SRP can be
found for the non unit correspondence. The
max(s_sum
4
) can provide an UA when the
conditions to create an RBP are valid.
The conditions are: pv
i
| g
3 THE BASIC REGULAR
ALGORITHM (BRA)
The BRA is based on the conditions to find a RBP
and provide opportunities for multiplicity on the
items of Si (i<n) and it is for a single channel
allocation.
From all the above the model steps are: (a)
partition of data according to their popularity using
probably dynamic programming (Yee et al., 2002),
(not shown in this work), (b) construction of FBP
and RBP, (c) grouping of data lines and (d) sending
them to a minimum number of channels.
4 THE FULL PARTITION VALUE
ALGORITHM (FPVA)
The FPVA focuses on solving a problem using as
many available channels as possible and
minimizing the AWT of the supreme set
(S
1
,S
2
,S
3
,S
4
) as in Theorem 2.For all the predefined
number of integrated relations (g) we try to
discover the values of pv
i
(in) so that the criterion
of homogenous grouping is valid and the
multiplicity constraint is satisfied.
It works with no grouping or BRA. When no
available channels exist (or when the system prefers
not to use all the available channels ) we move to the
next satisfactory solution. The LD has set the ld
i
at
an increasing order according to Theorem 2. It starts
from a maximum number of available channels
solution, tests for chg and cmc and continues until
it finds the solution with the most appropriate
number of channels. The AWT
i
is examined so that
when it goes above a threshold an LD increase is
made (new pace).
Example 12: Consider the sets: S
1s
=10,S
2s
=20,
S
3s
=40, S
4s
=120. The divisor of S
4s
are:
LD={20,30,40}. For ld
i
=20 the number of channels
(if an RBP can be created) , n_ch =120/20 =6.
The divisors of S
i
(in), D
3
= {8,5},
D
2
=(5,4),D
1
={5,2}. Taking : d
3
= 8,d
2
=5,d
1
=5 .
Considering as ld
i
=20 , s_sub
3
=8(=d3), s_sub
2
=5
(=d2), s_sub
1
=5(=d1) then we have:
DCNET 2011 - International Conference on Data Communication Networking
72
pv
3
=40/8=5, and it_mu
3
= 20/5=4
pv
2
=20/5=4, and it_mu
2
= 20/4=5
pv1=10/5=2, and it_mu1= 20/2=10
So the chg and the cmc are valid (it_mu
3
<it_mu
2
<it_mu
1
) and an RBP can be created with g
4
= ld
i
If the divisors of S4 are at a decreasing order
(e.i.,60,40,20) the n_ch will take the lower value.
This is used when the design of RBP is only for a
minimum number of channels.
The RBP for all the available channels can be
achieved when the divisor of S4 are at an
increasing order. This comes from the n_cl formula
(D
4
/ d
4
). In addition a new parameter (AWT) has be
considered and if it is above a threshold then two
choices come up: (a) apply FPVA-SRPA or (b) only
increase d
4
.
5 THE SIMILAR REGULAR
PLAN ALGORITHM (SRPA)
The SRPA focuses on providing a solution using
the available number of channels minimizing the
AWT of the supreme set (S1,S2,S3) as Theorem 3.
The new RBP can provide better AWT with less
number of channels. This is an economical solution
because with fewer channels we have faster server
service. SRPA can include all the AWT i (ik-1,
k=4) and not just the AWT
1
. A number of saving
channels (s_ch) is produced from the SRPA.
Example 13: Let us consider that: d
4
= 10, s_sub1=5,
s_sub2=5, s_sub3=8, s_sub4=12 and AWT
1
=15
(30/2), n_ch = 12(120/10). D
4
= {6,2} . For d
4
=20,
s_sub
1
=5, s_sub
2
=5, s_sub
3
=8, s_sub
4
=6 and
AWT
1
=14 (24/2). , n_ch = 6(120/20). We have
savings of 6 channels (12-6).
6 THE FULL PARTITION VALUE
ALGORITHM (FPVA-SRPA)
The combination of FPVA and SRPA provides the
opportunity to find a new RBP with possibilities to
change the parameters in order. First, the FPVA
finds an RBP and then the SRPA finds the most
desirable solution (lower number of channels) by
saving more available channels that could be used
for another broadcasting. It works as FPVA but it
uses an additional step, the SRPA.
7 SIMULATION
For our simulation, Poisson arrivals are considered
for the mobile users’ requests. The items are
separated into three categories according to their
popularity using Zipf distribution. Three scenarios
have been developed:
Scenario 1: In Fig. 1, data in various sizes with
equal spacing (RBP) from S
1
and S
2
sets, and flat
(for all the sets) (Acharya et al., 1995) with long
broadcast cycle size are depicted. For the data with
equal spacing the AWT is less than the one of the
flat data. It is considered a single channel service.
We will also take the same results of the RBP for the
users interested in data of S
1
, S
2
if more channels
were used.
Figure 1: The AWT for regular and flat data.
Scenario 2: Considering S
4s
= 120,
S
3s
=60,S
2s
=40,S
1s
= 20. The AWT for S
1
,S
2
,S
3
remain the same (Fig. 2) because PAV could find
the same values of pv
i
for all the number of
channels(6,3,2).The AWT
4
has increasing trend,
and it depends on the # of channels the PAV
discovers. The lower the # of channels is the greater
the AWT
4
. For S
4
it is considered that for each
relation there is one element (no repetitions). In
SIMILAR REGULAR PLANS FOR MOBILE CLIENTS
73
other words the s_sub
i
(i=1,..,4) remains the same
for all the cases of the number of channels.
Figure 2: The AWT for the same s_sub
4.
Scenario 3. This is the SPRA approach. Let us
consider: S
4s
= 120, S
3s
=60,S
2s
=40,S
1s
= 10 and same
values of s_sub
i
from all the sets except the s_sub
4
.
In that case s_sub
4
diminished from 4 to 2 with
increasing g (from 30 to 60). The n_ch is reduced by
a factor of two (from 4 to 2). As shown in Fig.3
there is almost a double increase of AWT
4
(from
339tu to 609tu). The AWT for the other sets
(S
1
,S
2
,S
3
) has only a small difference. This is due to
the combination of increasing g and diminishing
s_sub
4
.
Figure 3: New AWT for SPRA.
8 CONCLUSIONS
A new framework for a broadcast data model plan
with a set of algorithms is presented. Our proposed
model with parametric changes can guarantee faster
service with fewer channels for the supreme set.
Applying these algorithms the next generation
servers and their components with the scale up
possibilities, tools etc can enhance their self-
sufficiency, self-monitoring so that they may also
address quality of service, and other issues with
minimal human intervention.
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