Dynamic Data Broadcasting with Traffic Awareness
Chih-Lin Hu and Ming-Syan Chen
Department of Electrical Engineering
National Taiwan University
Taipei, Taiwan, R.O.C.
E-mail: [email protected]; [email protected]
Abstract
Data dissemination has significantly served as a scal-able data delivery mechanism in wireless networks. How-ever, even though the broadcast traffic has the nature of dy-namic changes, most previous research efforts were elabo-rated upon the premise of static workloads and access pat-terns without having proper traffic awareness. In this pa-per, we address the existence of client impatience and ac-cordingly devise an on-line traffic awareness mechanism based on a novel selective deferment and reflection technique (SDR) to estimate the dynamic workloads and access pat-terns in a granularity of a broadcast cycle. In comparison with prior probing and feedback approaches, our design is of practical usefulness in that it has low complexity and is light-weight without performance degradation. With various dynamic traffic scenarios, the experimental results show that with an increasing/decreasing workload, the real access fre-quency distribution is bounded by two specific estimated dis-tributions. This fact in turn suggests us to employ a trigono-metric tuning method to further enhance the estimation. In addition, we examine that the mean difference between the estimated access frequency distribution and the real one is very small, consequently indicating the feasibility and relia-bility of our proposed data broadcast mechanism with traffic awareness.
1
Introduction
With the limited bandwidth capacity, data broadcasting is a promising mechanism for the scalable information dis-semination in wireless network environments [2][5][12]. As its paradigm, a server applies a broadcast program and de-livers all data items in the database to clients through a shared medium periodically. The benefits are twofold: (1) in the client side, data broadcasting achieves a total sav-ing of all clients’ battery energy by avoidsav-ing per access re-quest transmission, and (2) an information server can
mod-erately mitigate the inherent performance and scalability problems. Thus, the access time and the energy consump-tion are primary performance measures in a data broadcast system. Accordingly, there are three major categories in the data broadcast research community. First, broadcast scheduling aims to arrange the contents (a series of items) of the broadcast data so as to minimize the mean access time [2][9][13][17][19]. Second, indexing broadcast data can fa-cilitate the clients’ tuning on the broadcast medium, and can thus save the energy consumption as well as reduce access duration [7][11][14]. Third, a hybrid data delivery exploits the commonality of access interests to minimize the average access time by delivering hot (i.e., popular) data on the push (i.e., broadcast) bandwidth and cold (i.e., unpopular) data on the pull (i.e., on-demand) bandwidth [3][10][18]. In addi-tion, the caching and pre-fetching mechanisms complement these three categories [1][22].
Note that in prior studies, there is a crucial weakness that a server does not possess traffic awareness against the nature of dynamic traffic. Explicitly, a server is unable to perceive the dynamic changes of workload, client population and ac-cess commonality because a client, in a sense, is passive and does not respond to his access, which is a challenging is-sue for a server to adjust the broadcast contents. Regardless, the previous research efforts were mainly based on that the workloads and access patterns are static and the prior knowl-edge of traffic changes is available. As a result, in order to attain dynamic adaptiveness and performance improve-ment, the probing and the feedback/piggyback techniques have been called for to cope with this issue [6][10][13][18]. A probing technique intentionally halts the broadcast service for a certain time period, which will in turn cause the clients to request data by the pull mode; meanwhile, the server can understand the push access interests from their correspond-ing pull requests. However, a broadcast miss will lead to many bursty requests and may congest the uplink channel. Incidentally, the partial probing and the sampling techniques are presented to alleviate the possible performance degrada-tion [8]. As for the feedback technique, there exists an
op-timistic assumption that all clients will willingly keep their broadcast access statistics and report them actively, requiring further justifications in practice. Moreover, an extra structure is needed to represent the feedback information [16], which will complicate the design of a data broadcast system. Since the mobile devices are usually resource-limited, the cost of extra computation is very undesirable. Therefore, it is dis-putable to employ these two techniques. More supplements and related works will be reviewed in Section 2.
In pursuit of the traffic awareness, we devise a selective deferment and reflection technique (SDR) by exploiting the behavior of client impatience to collect and detect the dy-namic workloads and access patterns on the broadcast chan-nel. Basically, it is derived from the observation that a client usually has a limited patience for his push access [15]. That is, a client will alternatively send an impatient (pull) request for a push item when the waiting time exceeds his patience. By counting the impatient requests on the uplink channel, the server can calculate the access frequency proportion of push items on the broadcast channel. In addition, the server will deliberately generate a “single item” broadcast miss, which causes the clients to declare their demands for that item. The basic idea of the SDR technique is as follows. First, the SDR technique heuristically selects a push item and defers its broadcast for a certain time period. At the same time, the clients do not know this intended deferment and hence continue their waiting for this item. After the corresponding waiting time expires, the clients will take this instance as a single item broadcast miss and subsequently change their ac-cess to this item from the push mode to the pull mode. Then, the server can count the exact access frequency for this item and further use it as the reflective base to estimate the ac-cess frequency in accordance with the relative proportion of the number of impatient requests for each push item. On the other hand, the exact access frequency of each cold item is directly available by counting the regular pull requests. Con-sequently, with the combination of the access frequencies of hot and cold items, the SDR technique can attain the esti-mated access frequency distribution, thereby providing the traffic awareness required to the server.
As mentioned, since the increased traffic due to a broad-cast miss is confined to the range of a single deferred item, we comment that the SDR technique can avoid the scalabil-ity problem in the probing technique, and can provide an op-portunity to serve as an on-line traffic awareness procedure. Within a deferment period, the server is able to calculate the access frequencies of all items simultaneously and adjust the broadcast contents dynamically for the next broadcast pro-gram so as to achieve better performance. This feature dis-tinguishes this paper from others. Note that adaptiveness is either absent or sensitive to a dedicated traffic factor in the previous works. With quantitative analyses, the SDR tech-nique is shown to be amenable to dynamic traffic awareness.
multiplexer
pull queue pull channel
push channel data data broadcast program mobile client uplink channel Database
Figure 1. A hybrid data delivery model.
To provide more insights into the SDR technique, we im-plement a system simulator with a variety of dynamic traf-fic factors and conduct several experiments for performance studies. It is shown by the experimental results that given an increasing or decreasing workload with a static access pat-tern, the real access frequency distribution is bounded by two estimated distributions which are calculated with two specific reflective bases. In view of this, we further devise a trigonometric tuning method to optimize the estimated ac-cess frequency distribution. In addition, we find that the SDR technique with a suggested reflective base is able to generate an access frequency distribution very close to the real one, showing the feasibility and reliability of our proposed data broadcast mechanism with traffic awareness.
The rest of the paper is organized as follows. Section 2 first models the data broadcast environment and gives some supplements and related works. Section 3 studies the SDR technique with its analysis and discussions. Section 4 de-scribes the simulation model and demonstrates the experi-mental results. This paper concludes with Section 5.
2
Preliminary
Section 2.1 presents a hybrid data delivery model where the traffic awareness is considered. Section 2.2 reviewed some prior studies with their problems to this model.
2.1
A Hybrid Data Delivery Environment
As depicted in Figure 1, the broadcast bandwidth is par-titioned into an upward channel and a downward channel. The former delivers clients’ pull requests, and the latter cor-responds to a series of interleaved data slots of equal size. Data slots are further classified as the push or pull mode. Logically, we view the push slots as a push channel and the pull slots as a pull channel, and these slots are multiplexed into a downward channel. The wireless information server contains a database including all broadcast items which are classified as hot or cold ones according to their previous ac-cess frequencies. The server broadcasts hot items over push slots periodically and delivers cold items by pull slots in
re-sponse to clients’ explicit access requests from the uplink channel. The assumptions adopted in this study are as fol-lows: (1) the bandwidth of an uplink channel is constant, (2) each item is self-identified and read-only, (3) a data slot is either in the pull or push mode, and can alternatively be switched, (4) the broadcast cycle is tunable, (5) the data clas-sification policy is based on their relative access frequencies, and (6) a client will submit an impatient pull request for a hot item when his waiting time has expired. This model is sim-ilar to those in [3][6][18], while having a graceful extension for the design of a general traffic awareness mechanism.
2.2
Related Work and Supplement
In a push delivery, the access time, i.e., a half of the broad-cast cycle on average, is dominated by the number of items in the database, whereas a pull response time in a pull delivery can be realized by an M/M/1 queueing model. Rather, a hy-brid delivery is called for to strike a compromise among the trade-offs; however, the adaptiveness of data classification and bandwidth allocation against dynamic traffic changes is a critical challenge. Without the traffic awareness mecha-nism, a hybrid delivery is not reliable. In the previous re-searches, few works were adaptive to dedicated traffic fac-tors. In [3], the slot allocation and the contents of push and pull sets are static. The work in [18] firstly applied the broad-cast miss, the original probing technique, to calculate the ac-cess statistics, but might suffer from the scalability problem. In contrast, the work in [6][10] used the feedback/piggyback technique to append the access statistics information into the clients’ regular pull requests, which complicates the design of a data broadcast system. As for broadcast scheduling, the work in [21] devised an on-line approach to pick the next broadcast item and also applied an off-line algorithm to gen-erate a fixed-length broadcast program. In [19], a priority index policy is proposed to select the next broadcast item. Other on-line scheduling works in [4][20] considered the number of pull requests and the duration time from the last broadcast, to determine the most “profitable” broadcast item. In addition, the work in [23] presented a statistic maximum
likelihood estimation to estimate the effectiveness of static
scheduling. Finally, the notion of “impatient user” was orig-inally addressed in [15] for broadcast scheduling, but only to improve the service ratio. Comparatively, our work exploits the “client’s impatience” to estimate the access frequency distribution. It is noteworthy that we aim at the design of a traffic awareness technique with all-inclusive traffic factors to offer a foundation for broadcast scheduling, indexing and the hybrid data delivery.
3
SDR Traffic Awareness
The abstraction of the SDR traffic awareness mechanism consists of the adaptive module, the pull procedure and the
push procedures. The adaptive module can dynamically ad-just data classification and bandwidth allocation. The pull and push procedures are mutual and responsible for the col-lections of the traffic information. The pull procedure re-sponds to the pull requests for cold data; on the other hand, the push procedure maintains a broadcast program including all hot data. For a push access, a client can submit alterna-tively an impatient pull request when the waiting time ex-ceeds his patience. Hence, the server has the exact number of regular pull requests for each cold item and the number of impatient requests for each hot item. Furthermore, the server defers the broadcast of a selected item purposely for a broadcast cycle; meanwhile, the pull procedure can count the exact number of requests for this deferred item. Accordingly, the reflection estimation can calculate the access frequencies of hot items. Consequently, with access frequencies of cold and hot items, the server can be aware of traffic changes and perform adaptive data broadcast.
3.1
Notation and Premise
Time is slotted equally and each time slott
uis equal to an
item slot interchangeably. The server has a databaseDthat
contains a number ofm=jDjdata items of equal sizes. Let
the request arrival rate
iof each item d
iform a Poisson
pro-cess with an aggregate arrival rate= P m i=1 iwhere 1 2 :::: m in a t
u. After data classification, the pull
item set U c
includes cold items d 1
;d 2
;:::;d
k and the push
item setU h
includes hot itemsd k +1
;d k +2
;:::;d
m. The
ag-gregate arrival rates are c = P k i=1 iand h = P m i=k +1 i
respectively. The length of a broadcast cycleLis the
num-ber of data slots used to deliver all items inU h
at lease once according to a specific scheduling policy. To maintain the generality, the SDR applies a flat broadcast programP for
a fundamental basis in comparison with others. Thus, Lis
equal to the number of hot items, m k, inU h
in a hy-brid data delivery. In addition, we assume that the index of a broadcast programPcan be transmitted to the clients ahead
of the push data either through the same downward channel or a distinct channel, so that a client can decide to access an item by the pull/push mode. When a client wants to access an item on the push channel, this client has a patience!in
waiting for this item. The number of impatient requests for itemd
i is denoted as
iin a time unit, and its aggregate in
a broadcast cycle is L
i . In practice, the total number of the
accumulative impatient requests is independent of whether an index is provided or not.
3.2
Pull Access Calculation
During a broadcast cycle, in a tu, the server can count
the exact access frequency
i of each d i in U c from an up-link channel. Thus, the sum of access requests ford
iin Lis L i = P 1jL i
... t waiting time d1 di ... dj dL d1 ... di di+1 ... dL a broadcast cycle: L di-1 ... t0 tL t2L
Figure 2. The client impatience with his impatient request.
broadcast cycle, and the mean access frequency ford iis i = X 1jL i (j)=L= L i =L; 1ik; (1)
wherej indicates thej-th slot inL. Moreover, the ratio of
mean access frequency among pull items is
1 ::::: i ::::: k = L 1 =L::::: L i =L::::: L k =L; whered i 2 U c
and1 i k. Moreover, the total pull
access frequency in aLis P 1ik P 1jL i (j).
3.3
Push Access Estimation
This subsection presents an explicit procedure to estimate the access frequencies of push items by three phases: (1) modeling the behavior of client impatience, (2) the selective deferment, and (3) the reflective estimation.
3.3.1 Client Impatience Modeling
Since each client has a patience in waiting for the arrival of a push item in the broadcast programP, the client will
submit an impatient request for that item over the uplink channel if the waiting time exceeds his patience. Let a ran-dom variablexpresent a client’s patience and be independent
of other clients’ patience. Asxis an exponential distribution
with a mean patience!, the probability density function of x is given by f (x) = 1 ! e 1 ! x where0 x < 1, and
the distribution function ofx isF (x) = 1 e 1 ! x where 0x<1, or otherwiseF (x)=0.
Figure 2 illustrates the behavior of a client’s push access. There are two possible situations: (1) the item will be cast later in this broadcast cycle, (2) the item has been broad-cast in this broadbroad-cast cycle so that the client has to wait until its broadcast in the next broadcast cycle. Given that a broad-cast cycle includesL slots and the arrival of a client with
interest in itemd
Lis located in the
j-th slot, this client has
to wait(L 1) j+ 1 2 slots where 1 2 indicates an average access delay in thej-th slot. However, with a push access for d
i, because item d
ihas already been broadcast in this cycle,
the client has to waitL+i j+ 1 2
slots until thed
i’s broadcast
in the next cycle. Accordingly, the probability that a client will generate an impatient request ford
ican be derived as F (i;j)= 1 e 1 ! (i j 1 2 ) ; i>j; 1 e 1 ! (L+i j 1 2 ) ; ij; (2) ... t d1 di ... dL d1' ... ... dL' t0 tL t2L di' a broadcast cycle: L patient clients the aggregate number
of push accesses for di in a broadcast
waiting time
The server calculates the mean access frequency at the end of a broadcast cycle L which di is deferred for.
t2
t1
impatient clients
Figure 3. The selective and deferment reflection (SDR).
where1 i;j L, the item is scheduled in thei-th slot,
and the client starts to access atj-th slot. Let i
(j)be the
access rate ford i in the
j-th slot. Then, the arrival rate of
impatient requests ford
iis given as i (j)= i (j)F (i;j); (3)
and the aggregate of
iin a broadcast cycle is L i = X 1jL i (j)F (i;j): (4)
We observe that if every client has an equal expected pa-tience!, the summation of the impatient probabilities in a
broadcast cycle is the same for each push item. Hence, L i is in proportion to P 1jL i (j). In addition, despite of
not knowing various i
(j), instead of resolving the i, the
server can calculate the mean of push access rate at the end of a broadcast cycle since
L
i is available from the uplink
channel. We can have
L i = X 1jL i F (i;j)= i (L e L 2 2! ); (5) where(L e L 2 2!
)is a constant in a broadcast cycle.
Fur-thermore, it is important to note that the ratio of the relative access frequencies of push items is equal to the ratio of the numbers of their impatient requests in a broadcast cycle.
L k +1 ::::: L i ::::: L m = k +1 ::::: i ::::: m ; (6) wherek+1imandd i 2U h . 3.3.2 Selective Deferment
The ratio in Equation (6), is not informative enough for the server to perform adaptation to comply with the traffic changes. In that case, the server employs a selective defer-ment to produce useful information for further reflective es-timation, rather than to evaluate! and
i
(j)directly. The
server intentionally makes a temporary “single item” broad-cast miss, compelling the clients to disclose their interests in push data. Explicitly, the server selects a single item and de-fers it for a broadcast cycle. Whether the interested clients
are patient or not, they will submit pull requests. Then, the server can calculate the access frequency of this item. Figure 3 illustrates the selective deferment procedure. According to the “equal spacing” property that for an item d
i, the
in-tervals
iin two consecutive broadcasts is fixed and equal to s j where d i 6= d j and d i and d
j are in the same
P. The
interval[t 0
;t L
]is equal to the broadcast cycleL. Att 1the
server selects an itemd
iand deliberately defers its broadcast
in the next broadcast cycle[t L
;t 2L
]. Since the clients are not
aware of the deferment, for those clients who are interested ind
iand arrive before t
1, they are satisfied in this broadcast
cycle. Aftert
1, all clients interested in d
iwill be satisfied at t
2, except some clients whose patience is less than the
wait-ing timet 2 t. In[t 1 ;t 2
], by Equation (3) the server will
receive a number of i
(j)F
r
(i;j)impatient requests for d iand thus L i is available at t
2. Moreover, when those
pa-tient clients find thatd
iis absent at t
2, and they will submit
regular pull requests ford
i. Therefore, the server can obtain
the exact access frequency L i ofd i by adding L i and the number of the regular pull requests ford
i. 3.3.3 Reflection Estimation In[t 1 ;t 2 ],d
i is selectively deferred as a reflective base,
and simultaneously other impatient frequencies of push data
< k +1 ; k +2 ;:::; m
> can be available similarly. From
Equation (6) with L
i, the SDR is able to estimate the access
frequency of each push item reflectively as follows.
x = x i L i L = x i i ; k+1xm: (7)
By Equations (1) and (7), the server can further obtain the dynamic access frequency distribution(x)of all data in a
broadcast cycle as follows.
< 1 ;:::; k ; k +1 i i ;:::; i 1 i i ; i ; i+1 i i ;:::; m i i >; where i
=(i)is the mean access frequency of the
reflec-tive base. In addition, the estimated meanW
loadof traffic workload is P 8di2U c i + P 8di2U h x i i :
3.4
SDR Analysis and Characteristic
We present several characteristics of the SDR technique with their theoretical and quantitative analyses under various traffic conditions. Without the loss of generality, the SDR technique assumes neither that the workload is in propor-tion to client populapropor-tion nor that the access pattern is static, in comparison with other previous works. First, Theorem 1 shows the accuracy of the SDR technique under a static traffic, and Lemma 1 presents a supplement similar to other works. Then, the other theorems and lemmas are derived to investigate the SDR technique with dynamic traffic changes. Proofs of theorems are omitted in this paper to save the page space.
Theorem 1 Given a static traffic, the estimated access
fre-quency distribution(x)by the SDR is equivalent to the real
distribution (x).
Lemma 1 Given aP with a mean patience!,
L
i is
deter-mined by the client population if the access probabilityp ifor
an itemd
iis static and a client can access a push item only
if this client has no pending push access.
Theorem 2 LetPbe a flat program with a length of L. If
i
inU h
is increasing in ascending order. Asd
1, the first item
inP, is selected as the reflective base, then 1
(i) ' (i)
with2iL.
Theorem 3 LetP be a flat program with a length ofL. If
i in
U h
is decreasing andd
i is the reflective base,
i (x)
/ (x)with1xL.
Theorem 4 Givend
iis the reflective base in
P,
i
(x)has
at least a cross with (x)where i
(k)= (k)andk=i.
With a prior knowledge of the access frequency distri-bution and the workload, the server can determine the item whose
L i
= L
i is the smallest as the reflective base. Lemma
2 provides a guideline for the SDR technique to select the reflective base. In the next section, we will compare the esti-mated results with Lemma 2 and Property 1.
Lemma 2 Let P be flat with a length of L. If the server
selectsd i, whose L i = L
i is the smallest, to be the reflective
base, then i
(x)has a cross with (x)at thex =i.
Fur-thermore, we can have L i = i (i)/ (i)if i is
contin-uously increasing during this broadcast cycle. In contrast,
L i = i (i)' (i)if
iis continuously decreasing during
this broadcast cycle.
Property 1 Given that P is flat, the item of the smallest
L i
= L
i is either the middle item d
d L 2 e
or an item very close tod
d L 2 e
ifPis scheduled in an order by the access frequency,
or otherwise the server can sort these items by their respec-tive numbers of impatient requests, and taked
d L
2 e
as the re-flective base ifP is scheduled randomly.
3.5
Trigonometric Tuning SDR Estimation
According to Properties 2 and 3, it is interesting that “weighting
0 iand
00
i” can possibly improve the estimation.
By Theorems 2, 3 and 4, in an ascending/descending work-load with a constant access pattern,
1
(i) can be slightly
higher/lower than (i) correspondingly, whereas d
L 2 e
(i)
can be slightly lower/higher than (i). Hence, we further
devise a tuning method based on the semi-monotonic feature of a trigonometric function to enhance the SDR estimation.
Property 2 We can narrow the inaccuracy of the estimation
of
ito a small range of a distance between 0 iand 00 i
esti-mated reflectively by deferringd
1and d d L 2 e respectively.
Property 3 Given an increasingly or decreasingly dynamic
workload with a static access pattern, the average of 1 (x) and d L 2 e
(x), denoted by(x), has two cross points with (x)ifPis scheduled in order of the access frequency.
With aP, we can transform a series of item slots in aL
into the range of [0;] in radians subsequently. Each slot
is then mapped to an angle by a one-to-one mapping. For instance, the angle ofd
1is 0, the angle ofd d L 2 e is 2 and for otherd iin P, its angleÆ xis x L where1xL. Given that FI
(x)is the estimation distribution by usingd 1as the
reflective base and MI
(x)is reflected by the middle item d
d L
2 e
, we can have a tuned distribution (x)as (x)=cos 2 Æ x FI (x)+sin 2 Æ x MI (x); (8) wherecos 2 Æ x +sin 2 Æ x
=1. We usesinandcosfunctions
because the value ofsinfunction is gradually up to1with the
argument from0to 2
, whereas acosfunction has an inverse
trend from1to0. Because MI
(x)has a cross with (x)
atd d L 2 e , we applysinto MI
(x). Likewise, we applycos
function to FI
(x) whose cross point with (x) is atd 1.
In light of this, we can tune the estimated value within the range between
FI
(x)and
MI
(x) with two cross points
at the pointÆ x = 2 wheresinÆ x = 1andcosÆ x =0with ( L 2 ) = MI ( L 2 ) = ( L 2
)and at the pointÆ x = 0with (1) = FI
(1). Furthermore, we will investigate
Equa-tion (8) and present the simulaEqua-tion results in SecEqua-tion 4.
4
Simulation and Results
Section 4.1 describes the simulation environment, includ-ing traffic generation, candidates of the reflective bases, ac-curacy measure of estimation, etc. Subsequently, the SDR technique is inspected by various traffic scenarios and the simulation results are summarized in respective subsections.
4.1
Simulation Model
Let the discrete spaceR
zof the one-to-one mapping
com-prise xs of all items in U h . Then, we generate R z by
the Zipf distribution with a skew coefficient where = P d x 2R z x and x = p x = ( 1 x ) = P 1xL ( 1 x ) . In this simulation, we design two dynamic traffic gen-erators, partially dynamic traffic and fully dynamic
traf-fic. The former is the general case where the workload increases/decreases gradually or the access frequencies of some items vary in ascending/descending order. The latter can be viewed as the bursty traffic where the access pattern
Table 1. Simulation parameters description
notation meaning value
m the number of items inP 501000
access frequency in a time unit 530
Zipf’s skew coefficient +2:0 5:0
! client patience time 503000
L the length of a broadcast cycle 501000
0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100
the position of an item in the broadcast program
th e ag g re g at e o f acce ss es f o r an i
tem real in a descending program
real in a randomized program estimated curve by SDR estimated curve by SDR Ψ(Χ)in a descending program Ψ(Χ)in a randomized program Φ(Χ)in a descending program in a randomized program Φ(Χ)
Figure 4. The estimation by the SDR under a static traffic.
in each slot is assigned arbitrarily by a Zipf distribution with a range of skew coefficients. Note that by the “equal space property,” we generate a broadcast program P and
sched-ule the items either in descending order by the access fre-quency or in an arbitrary order. With the ordered items in
P, three candidates for the reflective base are investigated in
our study: (1) the first item (d
FI), scheduled in the
broad-cast program, (2) the middle item (d
MI), scheduled in the
broadcast program, and (3) the last item (d
LI) in a
broad-cast program. To determine the reflective base, we utilize the difference mean and the variance to measure the estimation accuracy. Let
x be the difference between the estimated
and accurate access frequencies of an itemd
x, and the mean xis =E( x )= P R z L x L x
=Land the variance is 2 =E ( ) 2 = P R z ( x ) 2
=L. Table 1 lists the
simulation parameters.
4.2
Static Traffic
Figure 4 depicts the estimated curves by the SDR under a static traffic with! =200,L =100,=10and =0:5.
Given a broadcast programP of descending order, the SDR
can obtain an estimated access frequency distribution(x)
which exactly matches the real (x)for whichever item is
selected as the reflective base. This is because L
i is
deter-mined by the summation ofF (i;j)inLas iis static. As ! is constant, L i is in proportion to L i . Comparatively, given a
Pof random order,(x)is also exactly the same with (x).
The results show that under a static traffic, the estimation by the SDR is regardless of the order by whichPis scheduled.
0 10 20 30 40 50 60 70 0 10 20 30 40 50 the position of an item in the broadcast program
th e ag g reg at e o f p u sh acces s fo r an i te m real FI MI
(a) under an increasing workload
0 10 20 30 40 50 60 70 0 10 20 30 40 50 the position of an item in the broadcast program
th e ag g reg at e o f p u sh acce ss fo r an i te m real FI MI
(b) under a decreasing workload
Ψ(Χ) FI Φ (Χ) MI Φ (Χ) Ψ(Χ) FI Φ (Χ) MI Φ (Χ)
Figure 5. The real (x)versus the estimated FI (x)and MI (x), reflected byd FI and d MI.
4.3
Dynamic Workload & Trigonometric Tuning
Referring to the analyses in Section 3.5, we simulate the SDR with a downward/upward workload and various client patience, and evaluate the trigonometric tuning method.Figure 5(a) depicts the result with an increasing workload from=10to=20,=0:5,L=50, and! =100in
a broadcast cycle. Whend
FI is the reflective base, the
esti-mated FI
(x)is slightly higher than the real (x)and has
the maximal distance between-in at the middle item. Oppo-sitely,
MI
(x)is slightly lower than (x)and has a cross at d
MI when the server takes d
MI as the reflective base. The
reason is that for an item d
i which has been broadcast in
this cycle, an earlier interested client has a higher probability to submit an impatient request ford
ithan another interested
client who arrives later. Even thoughincreases gradually,
the probability of submitting an impatient request ford i is
lower in the end of the broadcast cycle because of a shorter waiting time. In addition, it is noted that the itemd
LI in
the end of a broadcast program faces the same circumstance withd
FI in that a broadcast program is cyclic. Inductively,
this circumstance is symmetric withd
MI as the symmetric
point. In view of this, we can understand that if an item is scheduled earlier thand
MI, the rate of its push accesses
to impatient requests can be more than those of other items scheduled lately. Comparatively, Figure 5(b) illustrates the result with a decreasing workload from= 20to =10.
Inversely, FI
(x)is lower than (x), whereas MI
(x) is
up (x)with a cross point atd
MI. This is because that for
an item that has been broadcast, the earlier a client arrives, the higher probability this client will submit an impatient re-quest. Sincedecreases gradually, the rate of
L FI to L FI is the smallest in a broadcast cycle and the estimated number of accesses for another item is lower than its real number. In contrast,
MI
(x)is mostly larger than (x).
Noticeably, Figure 5 demonstrates that (x)is bounded
by FI
(x)and
MI
(x)with an increasing/decreasing
work-load if the access pattern does not change. According
0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 the position of an item in the broadcast program
th e ag g reg at e o f p u sh acce ss fo r an i te m real FI MI trigonometric
(a) the estimated curves vs. the real curve
0 10 20 30 40 50 60 0 10 20 30 40 50 60 70 80 90 100 the position of an item in the broadcast program
th e ag g reg at e o f p u sh acces s fo r an i te m real trigonometric - w50 trigonometric - w100 trigonometric - w500
(b) the tuned curves with w=50, 100 and 500
Ψ(Χ) Φ∆(Χ) - w=50 - w=50 - w=50 FI Φ (Χ) MI Φ (Χ) Ψ(Χ) Φ∆(Χ) Φ∆(Χ) Φ∆(Χ) - w=50 - w=100 - w=500
Figure 6. The real (x)versus the estimated
(x)by
the trigonometric tuning.
to Equation (8), we can further obtain a tuned estimation
(x). Figure 6 displays the synthetic comparison among FI (x), MI (x),
(x)and (x)withfrom20to30,
L = 100and =0:2. As displayed in Figure 6(a), with a
mean difference0:603, MI
(x)is relatively closer to (x)
than FI
(x) which is above (x)with a mean difference
1:217. However, after the trigonometric tuning, we have
(x) very close to (x) with a mean difference 0:353.
Figure 6(b) depicts the results by the trigonometric tuning versus various! under a dynamic workload. By Equation
(7), although(x) by the SDR with a less information of
client impatience can slightly deviate from (x). However,
in this case even with a large!, the estimated result
(x)
is still close to the real distribution (x), showing very good
stability of the SDR technique.
4.4
Dynamic Access Pattern
Table 2 lists the experimental results with the suggested reflective bases. We examined
MI under a heavy workload
with 2000 accesses in a broadcast cycle. In most cases, d
MI is suggested as the reflective base and the
correspond-ing mean difference is smaller than1, except few extreme
cases with excessive changes of access patterns. For those cases in the sides of the diagonal, their mean differences are about0:2, meaning that the difference between the real and
the estimated aggregate of push accesses in a broadcast cy-cle is only about40. Although there are few cases whose
reflective bases are suggested with thed
LI, we find that their
mean differences are very close to those byd
MI. Therefore,
we observe that the mean difference will be larger if the item of a larger change of access frequency is selected as the re-flective base.
5
Conclusions
In this paper, we have devised a novel selective defer-ment and reflection (SDR) technique which can be aware of the dynamic traffic changes. Compared to prior probing
Table 2. The suggested reflective bases, whereL=200, =10,!=200andfrom 5to1. -5 -4 -3 -2 -1 0.2 0.4 0.6 0.8 1.0 -5 - LI LI LI MI MI MI MI MI MI -4 LI - LI MI MI MI MI MI MI MI -3 LI LI - MI MI MI MI MI MI MI -2 LI MI MI - MI MI MI MI MI MI -1 MI MI MI MI - MI MI MI MI MI 0.2 MI MI MI MI MI - MI MI MI MI 0.4 MI MI MI MI MI MI - MI MI MI 0.6 MI MI MI MI MI MI MI - MI MI 0.8 MI MI MI MI MI MI MI MI - MI 1.0 MI MI MI MI MI MI MI MI MI
-and feedback approaches, our design is of low complexity and light-weight without degrading the corresponding per-formance. We have conducted performance analysis with various dynamic traffic scenarios. It has been shown by the experimental results that the estimated access frequency dis-tribution is very close to the real one. Furthermore, we have found that with an increasing/decreasing workload, if the ac-cess pattern is static, the real acac-cess frequency distribution is bounded by two specific estimated distributions withd
FI
andd
MI as the reflective bases. In view of this, we have
de-vised a trigonometric tuning method to further optimize the estimation. Consequently, our proposed SDR traffic aware-ness mechanism is very feasible for dynamic data broadcast-ing to cope with the nature of a changbroadcast-ing traffic.
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