2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Dynamic Leveling: Adaptive Data Broadcasting in a Mobile
Computing Environment
WEN-CHIH PENG, JIUN-LONG HUANG and MING-SYAN CHEN∗ Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, ROC
Abstract. The research issue of broadcasting has attracted a considerable amount of attention in a mobile computing system. By utilizing broadcast channels, a server is able to continuously and repeatedly broadcast data to mobile users. From these broadcast channels, mobile users obtain the data of interest efficiently and only need to wait for the required data to be present on the broadcast channel. Given the access frequencies of data items, one can design proper data allocation in the broadcast channels to reduce the average expected delay of data items. In practice, the data access frequencies may vary with time. We explore in this paper the problem of adjusting broadcast programs to effectively respond to the changes of data access frequencies, and develop an efficient algorithm DL to address this problem. Performance of algorithm DL is analyzed and a system simulator is developed to validate our results. Sensitivity analysis on several parameters, including the number of data items, the number of broadcast disks, and the variation of access frequencies, is conducted. It is shown by our results that the broadcast programs adjusted by algorithm DL are of very high quality and are in fact very close to the optimal ones.
Keywords: broadcast disks, mobile computing, broadcast programs, multiple broadcast channels
1. Introduction
In a mobile computing environment, a mobile user with a power-limited mobile computer can access various informa-tion via wireless communicainforma-tion. Applicainforma-tions such as stock activities, traffic reports and weather forecast have become in-creasingly popular in recent years [21,22]. It is noted that mo-bile computers use small batteries for their operations without directly connecting to any power source, and the bandwidth of wireless communication is in general limited. As a result, an important design issue in a mobile system is to conserve the energy and communication bandwidth of a mobile unit while allowing mobile users of the ability to access informa-tion from anywhere at anytime [2,4,10].
The research issue of broadcasting has attracted a consid-erable amount of attention in a mobile computing system. By utilizing broadcast channels, a server is able to continu-ously and repeatedly broadcast data to mobile users. These broadcast channels are also known as “broadcast disks” from which mobile users can obtain the data of interest efficiently and only need to wait for the required data to present on the broadcast channel [1,9,20]. The corresponding waiting time is called the expected delay of that data item. One objective of designing proper data allocation in the broadcast disks is to reduce the average expected delay of data items. Broad-casting schemes in this context have been extensively studied [3,8,12,15,18].
Note that a lot of research effort has been elaborated on exploring multiple broadcast channels for data dissemination [13,14,16,17]. The advantages of utilizing multiple broadcast channels can be found in [16,17]. Organizing data in multi-ple broadcast channels raises a number of new research prob-∗Corresponding author.
lems. A system of multiple broadcast channels can be viewed as a broadcast disk array. The broadcast disks in a broadcast disk array can be categorized according to the speed of broad-cast disks, where the speed of a broadbroad-cast disk corresponds to the expected delay for the data items in that broadcast disk. The data items in each broadcast disk are sent out in a round robin manner. Clearly, as the number of data items in a broad-cast disk increases, the expected delay of those data items in-creases. As a result, the data items that are more frequently re-quested by mobile users should be put in fast broadcast disks, whereas cold data items can be pushed to slow broadcast disks to minimize the average expected delay of data items in the broadcast disk array. Thus, it has been recognized as an im-portant issue to develop algorithms to allocate data items to the broadcast disk array according to their access frequencies so as to minimize the average expected delay of data items [14,16].
The study in [14] explored the problem of generating hi-erarchical broadcast programs with the data access frequen-cies and the number of broadcast disks in a broadcast disk array given. Specifically, the problem of generating hierar-chical broadcast programs is first transformed into the one of constructing a channel allocation tree with variant-fanout. By exploiting the feature of tree generation with variant-fanout, a heuristic algorithm to minimize the expected delay of data items in the broadcast program is developed. The tree ob-tained in [14] is called channel allocation tree (or abbreviat-edly as allocation tree) where the depth of the allocation trees corresponds to the number of broadcast disks, and those leaf nodes in the same level of the allocation tree correspond to those data items to be put in the same broadcast disk. Fig-ure 1 shows a hierarchical broadcast program with its channel allocation tree where the upper channel is allocated with two data items and each of the three lower channels is allocated
Figure 1. The broadcast program and its allocation tree.
with three data items. As such, the data items in the fast disks (i.e., the upper broadcast channel) spin faster than those data items in the slow disks (i.e., the lower broadcast channels). Note, however, that the algorithm in [14] is designed for the situation where the data access frequencies and the number of broadcast channels are given. In practice, the data access frequencies may vary as time advances. For example, the ac-cess frequencies of the traffic data increase drastically during rush hours and decrease beyond the rush hours. Clearly, with-out adapting to the change of access frequencies, the broad-cast program determined off-line will unavoidably lead to de-graded performance. Thus, with the broadcast programs gen-erated by [14], it is important for the broadcast programs to dynamically adapt to the change of the data access frequen-cies so as to retain the performance of data broadcasting. This is the very problem that we shall address in this paper.
The problem we study can be best understood by the il-lustrative example in table 1. Assume that the data items Ri, 1 i 11 are of the same size and the number of broad-cast channels is 4. Denote that the access frequency of data item Ri as Pr(Ri). Four sets of access frequencies of data items are given in table 1 and drawn in figure 2 for clarity.1 The average expected delay in table 1 is obtained by mul-tiplying the access frequency of each data item by the ex-pected delay of that data item and summing up the results, i.e.,11i=1dRi · Pr(Ri). Same as in [1,19], the expected
de-lay for each data item in the broadcast disk i is formulated asNi
x=1(Ni− x)/Ni, where Ni is the number of data items allocated in the broadcast disk i. It can be verified that the expected delays of data items R1, R3, R5 and R9 in fig-ure 1 are dR1 = (1 + 0)/2 = 0.5, dR3 = (2 + 1 + 0)/3,
dR6 = (2 + 1 + 0)/3 and dR9 = (2 + 1 + 0)/3, respectively.
At time t0, with the access frequencies of data items and the number of broadcast channels given, the initial alloca-tion trees obtained by the work in [14] is shown in fig-ure 1. The average expected delay of data items at time t0 is11i=1dRi · Pr(Ri) = 0.8712. Assume that the allocation
tree will remain the same as the access frequencies of data items vary with time. With the allocation tree determined at time t0, the average expected delay of data items at time t4 is11i=1dRi· Pr(Ri)= 0.7371. Notice that with the access
frequencies changed, this average expected delay at time t4is 1The access frequencies of data items are generated by Zipf distribution
which will be described later in this paper.
much larger than its optimal value2(which is 0.5557 in this case). Thus, it is an important issue to dynamically adjust the broadcast program to reflect the change of access frequencies. Consequently, date items should be moved among levels of a given allocation tree to adapt to the change of access frequen-cies of data items.
In this paper, by shuffling data items among different levels in the allocation tree, we devise an algorithm to dynamically adjust the broadcast programs in response to the change of data access frequencies. This algorithm is referred to as al-gorithm DL (standing for dynamic leveling). Clearly, a naive approach to reach a new configuration would be re-executing the algorithm in [14] again, which is however costly. Algo-rithm DL is so designed that the new configuration for effi-cient broadcast programs can be reached with the purpose of minimizing the number of data movements. Notice that once the change of access frequencies is larger than the predeter-mined value (Such a value is called fluctuation factor), algo-rithm DL should be executed to dynamically adjust broadcast programs. Explicitly, the process of algorithm DL can be de-composed into two phases, namely (1) the casual adjustment phase and (2) the fine adjustment phase. In the casual adjust-ment phase, algorithm DL reaches an initial adjustadjust-ment for data items among broadcast channels. Then, for fine tuning, algorithm DL is designed to adjust the data items between neighboring levels in the fine adjustment phase with the ob-jective of minimizing the total cost of these two neighboring levels. Performance of algorithm DL is analyzed and a sys-tem simulator is developed to validate our results. Sensitivity analysis on several parameters, including the number of data items, the number of broadcast disks, and the variation of ac-cess frequencies, is conducted. It is shown by our simulation results that the broadcast programs achieved by algorithm DL are of very high quality and are in fact very close to the op-timal ones. This feature and the efficiency of algorithm DL justify the practical importance of algorithm DL.
The rest of this paper is organized as follows. Problem description is given in section 2. In section 3, we develop algorithm DL to adjust the allocation tree to reflect the change of access frequencies. Performance studies are conducted in section 4. This paper concludes with section 5.
2. Problem description
Table 2 shows the descriptions of symbols used in this paper. Denote the total number of data items as n, and a data item as Ri, 1 i n. The number of broadcast disks in a broadcast disk array is K. Recall that Pr(Ri)is the access frequency of
Ri and n
i=1Pr(Ri)= 1. Theoretically, generating a broad-cast program can be viewed as a partition problem for data items. Given the number of broadcast disks in a disk array and the access frequencies of all data items, we shall deter-mine the proper set of data items that should be allocated to each broadcast disk in a broadcast disk array with the purpose 2Such optimal values can be obtained by exhaustive searches for broadcast
Table 1
Access frequencies of data items.
Time Access frequency Average expected delay
Pr(R1) Pr(R2) Pr(R3) Pr(R4) Pr(R5) Pr(R6) Pr(R7) Pr(R8) Pr(R9) Pr(R10) Pr(R11) in figure 1 optimal
t1 0.126 0.123 0.116 0.11 0.10 0.095 0.0869 0.0777 0.0673 0.055 0.0388 0.8712 0.8712
t2 0.16 0.152 0.137 0.122 0.10 0.09 0.0763 0.061 0.0458 0.0305 0.0152 0.8338 0.7805
t3 0.2226 0.201 0.163 0.129 0.09 0.07 0.0504 0.0323 0.0181 0.008 0.002 0.7746 0.7148
t4 0.276 0.239 0.194 0.102 0.08 0.05 0.0298 0.0153 0.0064 0.0019 0.0002 0.7371 0.5557
Figure 2. Data access frequencies vary as time advances.
Table 2 Description of symbols.
Description Symbol
Number of broadcast disks in a broadcast disk array K
Number of data items within broadcast disk i Ni
The expected delay of data items within broadcast disk i di
The j th data item Rj
The access frequency of data item Rj Pr(Rj)
of minimizing the average expected delay of all data items (i.e.,ni=1dRi · Pr(Ri)). The problem of generating
broad-cast programs for K broadbroad-cast channels can be viewed as a discrete minimization problem: Given a list of n data items with their access probabilities, partition them into K parts so that the average expected delay of all data items is minimized. The minimization problem is known to be NP-hard [11]. As pointed out in [14], a broadcast program for a broadcast array of K broadcast disks can be represented as a channel alloca-tion tree with a height of K. Note that the leaf nodes in the same level of the allocation tree correspond to a set of data items to be put in the same broadcast disk.
To facilitate the presentation of the costs for an allocation tree, we have the following definition.
Definition 1. Suppose that level v in the allocation tree has j− i + 1 data items, Ri, Ri+1, . . . , Rj. The cost of level v in an allocation tree is defined as Ci,j =
j−i+1
k=1 ((j− i + 1) −
k)/(j− i + 1)jq=iPr(Rq). In essence, the value of Cij is related to the average expected delay of leaf nodes in level v.
This paper investigates the problem of adjusting the broad-cast program to match the access frequencies of data items. In order not to distract readers from the main theme of this paper for dynamically adjusting broadcast programs, readers interested in the details of collecting access frequencies are referred to [6,7,18,23]. Once the change of access frequen-cies is larger than the predetermined value, algorithm DL will be executed to reach the new configuration close to the op-timal one. Figure 3 shows the opop-timal allocation trees with the access frequencies given in table 1. In accordance with the access frequencies of data items at time t1and the num-ber of broadcast channels given, the allocation tree was deter-mined by the algorithm in [14]. It can be seen in figure 3, at time t2, t3and t4, the optimal allocation trees differ from the one at time t1due to the change of access frequencies. Con-sequently, date items should be moved among levels within the given allocation tree in response to the change of access frequencies of data items. Clearly, such movements have an impact on the average expected delay of all data items. The problem we shall study in this paper can be stated as follows.
Problem of adjusting allocation trees. Given an allocation
tree, we shall adjust data items among the broadcast disks when the access frequencies vary with the purpose of mini-mizing the expected delay of data items.
With the problem described above, we should devise an al-gorithm to determine the level of the allocation tree to start the adjustment and identify the movements of data items among levels in the allocation tree.
3. Algorithm DL: adjusting allocation tree by dynamic leveling
In section 3.1, we devise algorithm DL to adjust allocation trees which explores the features of the casual adjustment and the fine adjustment. Then, the execution scenario of algorithm DL is illustrated in section 3.2.
3.1. Design of algorithm DL
We devise in this paper an algorithm, referred to as algorithm DL, to dynamically adjust the broadcast programs by shuf-fling data items among different levels in the allocation tree. The process of algorithm DL can be decomposed into two phases, namely (1) the casual adjustment phase and (2) the fine adjustment phase. In the casual adjustment phase, algo-rithm DL moves data items among levels so as to enable the
Figure 3. The optimal allocation trees under different times. costs of most levels in the allocation tree to be smaller than
or equal to average cost. Then, for fine tuning, algorithm DL adjusts the data items between neighboring levels with the ob-jective of minimizing the total cost of these two neighboring levels. Note that algorithm DL is greedy in nature and is of time complexity O(K + n). The algorithmic form of algo-rithm DL is described below.
Algorithm DL.
Input: The status table (ST) with K rows, where K is the
number of broadcast disks in a broadcast disk array.
Output: The resulting allocation tree. begin
1. for each row i in table ST
2. ST(i)· D = ST(i) · C − ST(i) · P ;
3. ST(i)· G = false; /* ST(i) · G is the flag for casual checking*/
/*Array δ has K− 1 elements which record the cost difference between two neighboring level*/
4. for each element i in array δ
5. δ[i] = |ST(i) · C − ST(i + 1) · C| 6. casual_checking= 0;
/*The casual adjustment phase*/
7. Choose the row i from table ST such that ST(i)· D is maximal 8. repeat 9. begin 10. if (i== 1) 11. casual_tunning(i, i+ 1); 12. else if (i== k) 13. casual_tunning(i, i− 1); 14. else
15. {choose the row j where j ∈ (i − 1, i + 1) such that ST(j )· G is false and δ[j] is maximal; 16. casual_tunning(i, j );}
17. casual_checking++;
18. update table ST and array δ accordingly;
19. choose the row i from ST where ST(i)· C is maximal and ST(i)· G is false;
20. end
21. untilcasual_checking== K − 1; /*the fine adjustment phase*/
22. Construct a priority queue PQ;
/*A priority queue is a data structure which returns the element with the minimal value when one is to remove an element from the priority queue*/
23. for each element i in array δ 24. Insert δ[i] into the PQ; 25. while (PQ is not empty) 26. begin
27. remove the element i from PQ; 28. if (ST(i)· C < ST(i + 1) · C)
/*if there is no movement between level i and level i+ 1, moving equals to −1*/
29. moving=push_up(i, i+ 1); 30. else
31. moving=push_down(i, i+ 1);
32. if (moving= −1) /*some data movements occur*/ 33. Update the elements in PQ and table ST accordingly; 34. end
end
Procedurecasual_tuning(level i, level j ) {
sort those data items in level i according their access probabilities;
if (i < j ) begin
while (ST(i)· C >Ki=1ST(i)· C/K)
move the data item in the rightest side of level i to level j and update ST(i)· C accordingly;
end else
begin
while (ST(i)· C >Ki=1ST(i)· C/K)
move the data item in the leftest side of level i to level j and update ST(i)· C accordingly;
end
}
Table ST (standing for status table) is created to record the cost of each level in the allocation tree, and the number of rows in table ST is equal to the number of broadcast disks in a broadcast disk array (from line 1 to line 3). Note that in
table ST, the value of ST(i)· P is the cost of nodes in level i previously, whereas the value of ST(i)· C is the cost of nodes in level i when the latest access frequencies were collected. ST(i)· D stores the cost difference associated with level i, i.e., ST(i)· C − ST(i) · P . Also, ST(i) · G is used to indi-cate whether the casual tuning is performed or not. Array δ has K− 1 elements that record the cost difference between two neighboring levels (from line 4 to line 5). As can be seen in causal adjustment phase (i.e., from line 7 to line 21), al-gorithm DL makes sure that most levels of the allocation tree satisfy the requirement of the casual adjustment. Since the ca-sual adjustment intends to let the total cost of allocation tree be evenly allocated to all levels, it is possible that some data nodes would move back and forth between neighboring lev-els. For execution efficiency, the number of runs for the casual adjustment is limited to be K− 1. Procedurecasual_tuning is developed to move data items in level i so as to satisfy the purpose of the casual adjustment.
By exploiting the casual adjustment, data items are roughly allocated to each level of an allocation tree with the costs of most levels are smaller than or equal to average cost. Then, algorithm DL employs the fine adjustment to adjust data items between neighboring levels. As can be seen from line 22 to line 34 of algorithm DL, neighboring levels are examined on finding potential movements with the purpose of minimizing the total cost of neighboring levels. Specifically, in line 27 of algorithm DL, the sequence of performing the fine tuning is determined by identifying the largest cost difference among those between neighboring levels (i.e., the largest value in δ). After identifying the neighboring levels (e.g., level i and level i+ 1) to perform the fine tuning, one should determine the data movements between these levels. Note that there are two kinds of movements, i.e., pushing up and polling down. Judi-ciously applying these movements is able to reduce the total cost of these two neighboring levels. Clearly, if the cost of level i is smaller than level i+ 1, we should move data items from level i+ 1 to level i and vice versa. After deciding the direction of data movements, we should determine the num-ber of data items to move among levels in an allocation tree. Explicitly, we develope procedurepush_upandpull_downto determine such a number. To facilitate the presentation of algorithm DL, the procedures ofpush_upandpull_downare described in detail later. From line 26 to line 34, algorithm DL adjusts data items in neighboring levels iteratively with the objective of minimizing the total cost of neighboring lev-els until there is no further adjustment required (i.e., queue PQ is empty). As such, the allocation tree is adjusted so as to minimize the total cost of the allocation tree.
Once we identify the direction of data movements to per-form, we should determine the number of data items to move among levels. Suppose that data items in each level of an allo-cation tree are sorted according to the descending order of ac-cess frequencies. In order to evaluate the cost reduction by the movement of pushing up, we have the following definition.
Definition 2. Suppose that level i has k − i + 1 nodes, Ri, Ri+1, . . . , Rk and level i + 1 has j − k data nodes,
Rk+1, Rk+2, . . . , Rj. The reduction gain achieved by pushing
p nodes (i.e., Rk+1, Rk+2, . . . , Rk+p) up to level i, denoted by u(p), can be formulated as u(p) = (Ci,k + Ck+1,j)−
(Ci,k+p+ Ck+p+1,j).
In light of definition 2, we devise a procedurepush_upto identify the group of nodes in level i + 1 to be moved up-ward to level i so as to maximize the reduction gain between these two neighboring levels. Figure 4 shows the scenario of pushing up.
Procedurepush_up(level i, level i+ 1) {
Determine p∗such that u(p∗)= max1pj−k{u(p)}; /*determine the maximal value of u(p) when p varies from 1 to j− k*/
if u(p∗) >0
push nodes Rk+1, Rk+2, . . . , Rk+p∗to level i in the tree;
else
p∗= −1; /*no movement is performed since there is no cost-effective movement*/
}
Similar to the operation of pushing up, we have definition 3 and procedurepull_downbelow to evaluate the group of nodes in level i to be moved downward to level i+ 1 with the pur-pose of reducing the total cost of these two neighboring levels. Figure 5 illustrates the scenario of pulling down.
Definition 3. Suppose that level i has k− i + 1 data nodes, Ri, Ri+1, . . . , Rk and level i + 1 has j − k data nodes,
Rk+1, Rk+2, . . . , Rj. The reduction gain achieved by pulling
pnodes (i.e., Rk−p+1, Rk−p+2, . . . , Rk) down to level i+ 1, denoted by d(p), can be formulated as d(p) = (Ci,k +
Ck+1,j)− (Ci,k−p+ Ck−p+1,j).
Figure 4. A scenario of pushing up.
Table 3
The profile of an illustrative example.
Time R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11
t1 0.126 0.123 0.116 0.11 0.1 0.095 0.0869 0.0777 0.0673 0.055 0.0388
t2 0.276 0.239 0.194 0.102 0.08 0.05 0.0298 0.0153 0.0064 0.0019 0.0002
Procedurepull_down(level i, level i+ 1) {
Determine p∗such that d(p∗)= max1pk−i+1{d(p)}; /*determine the maximal value of d(p) when p varies from 1 to k− i + 1*/
if d(p∗) >0
pull nodes Rk−p∗+1, Rk−p∗+2, . . . , Rkdown level i+ 1 in the tree;
else
p∗= −1; /*no movement is performed since there is no cost-effective movement*/
}
3.2. An example execution scenario of algorithm DL
Consider the profile in table 3 where the number of data items nis 11 and the number of broadcast disks is 4. The initial al-location tree is shown in figure 6(a), where the alal-location tree is generated according to the access frequencies at t1. The values in table ST and their changes made in accordance with the execution of algorithm DL are shown in table 4. First, al-gorithm DL chooses the maximal ST(i)· D to start the casual adjustment. In this example, level 1 is chosen since ST(1)· D is the largest among all levels. As can be seen in table 4(a), since the cost of level 1 is larger than the average cost of all levels (i.e., ST(1)· C = 0.2575 > 4i=1ST(i)· C/4 = (0.2575+ 0.376 + 0.0951 + 0.0085)/4 = 0.184275), data items in level 1 should be pulled down to level 2 to meet the criterion of the casual adjustment. Thus, data item R2 is moved to level 2 and table ST is updated accordingly. Fol-lowing the same procedure, algorithm DL performs proce-dure casual_tuningiteratively until casual_checkingequals K − 1 (i.e., 4 − 1 = 3). Table 4(b) shows the values of table ST and the corresponding values in array δ after the ca-sual adjustment. The configuration of the allocation tree in figure 6(a) becomes the one shown in figure 6(b). Then, in the phase of the fine adjustment, each element of array δ with its value is inserted into queue PQ and algorithm DL performs the fine tuning between neighboring levels. From table 4(b), since δ[3] is the largest, the fine adjustment will be executed between level 3 and level 4. As ST(3)· C is smaller than ST(4)· C, those data items in level 4 should be pushed up to level 3. It can be verified that data item R5should be pushed up to level 3 so as to reduce the total cost of level 3 and level 4. Table 4(c) shows the values of table ST and the corresponding values in array δ after the fine tuning is performed between level 3 and level 4. Figure 6(c) shows the configuration of the allocation tree after the first fine tuning. From table 4(c), the next fine tuning is performed between level 2 and level 3. Following the fine adjustment, algorithm DL stops when there is no further data movement required. Table 4(d) shows the
Table 4
An execution scenario under algorithm DL. (a) Run 1 of algorithm DL, (b) table ST after the phase of the casual adjustment, (c) table ST after the
first fine tuning is performed, (d) the final result of table ST. (a)
Level i ST(i)· P ST(i)· C ST (i)· D ST(i)· G
1 0.1245 0.2575 0.133∗ false
2 0.326 0.376 0.05 false
3 0.25 0.0951 −0.1549 false
4 0.1611 0.0085 −0.1526 false
(b)
Level i ST(i)· P ST(i)· C ST(i)· D ST (i)· G δ
1 0.1245 0 −0.1245 true 0
2 0.326 0 −0.326 true 0.148
3 0.25 0.148 −0.102 true 0.4028∗
4 0.1611 0.5508 0.3897 false
(c)
Level i ST (i)· P ST(i)· C ST (i)· D ST(i)· G δ
1 0.1245 0 −0.1245 true 0
2 0.326 0 −0.326 true 0.376∗
3 0.25 0.376 0.126 true 0.117
4 0.1611 0.259 0.0979 false
(d)
Level i ST(i)· P ST(i)· C ST (i)· D ST(i)· G
1 0.1245 0 −0.1245 true
2 0.326 0.2165 −0.1095 true
3 0.25 0.252 0.002 true
4 0.1611 0.1072 −0.0539 false
values of table ST after performing algorithm DL. The final allocation tree is shown in figure 6(d). Note that in this case the final allocation tree happens to be the optimal one shown in figure 3(c).
4. Performance evaluation
In order to evaluate the performance of algorithm DL, we have implemented a simulation model of the broadcast en-vironment. Specifically, the simulation model is described in section 4.1. Then, we examine the impact of adjusting broad-cast programs in section 4.2. Performance of algorithm DL is analyzed in section 4.3. In section 4.4, algorithm DL and the work in [14] is comparatively analyzed.
4.1. Simulation model
Table 5 summarizes the definitions for some primary simula-tion parameters. The number of data items to be broadcasted
Figure 6. An execution scenario of algorithm DL: (a)–(c) the adjustment of the allocation tree, and (d) the resulting allocation tree. Table 5
The parameters used in the simulation.
Notation Definition
n total number of data items to be broadcast
K number of broadcast disks in a broadcast disk array
θ Zipf parameter
f fluctuation factor
no_adjust scheme which does not adjust the broadcast program OPT scheme to generate the optimal broadcast program VFK scheme to generate the broadcast programs statically
in a broadcast disk array is denoted by n and the number of broadcast disks in a broadcast disk array is K. The access fre-quencies of broadcast data items are modelled by the Zipf dis-tribution. Let Pr(Ri)= ((N − i)/N)θ/
n
j=1((N− j)/N)θ, where θ is the parameter of Zipf distribution [5]. It can be ver-ified that the access frequencies become increasingly skewed as the value of θ increases. Specifically, the initial Zipf pa-rameter, denoted by θ0, is set to 0.5. θcurrent is the Zipf pa-rameter for the current access frequencies, whereas θprevious is the Zipf parameter collected last. The number of f called the fluctuation factor is used to determine whether algorithm DL will be executed or not. If the difference between θcurrent and θprevious is larger than the value of f , DL will be exe-cuted to adjust the broadcast program in order to retain the performance. For comparison purposes, a scheme,no_adjust, which does not adjust the broadcast program in response to the change of access frequencies, is implemented. To ob-tain the optimal solutions for comparison, we implemented scheme OPT by using the technique of branch and bound [11]. For interest of brevity, the implementation details of
OPT are omitted in this paper. Notice that though scheme OPTis able to find the optimal broadcast program, the exe-cution time of OPT is prohibitively large due to its exponen-tial time complexity. For comparison purposes, we also im-plemented schemeVFK, which is able to generate broadcast program with the number of broadcast disks and the number of data items given.
4.2. The impact of adjusting broadcast programs
To show the advantage of adjusting broadcast programs when the access frequencies vary, we set the value of n to 50, the value of K to 4 and the value of f to 1. The expected delays of data items underno_adjust, DL andOPTare examined with the value of θ varied. Without loss of generality, assume that all the data items are of the same size which is used as one unit of waiting time. The initial broadcast program is generated by schemeOPT. The resulting expected delays of data items by runningno_adjust, DL andOPTare shown in figure 7. It can be seen from figure 7 that the access frequencies become in-creasingly skewed as the value of θ increases and the average expected delay of DL decreases since the broadcast program is properly adjusted by DL in accordance with the change of access frequencies of data items. Note that the difference be-tween expected delay of DL and that ofno_adjustbecomes larger as the θ increases, indicating the necessity of adjust-ing broadcast programs while the access frequencies of data items vary. It is worth mentioning that though algorithm DL is applied in 5 times, the expected delays of DL andOPTare still very close in figure 7, showing the good quality of con-figurations adjusted by algorithm DL.
Figure 7. The average expected delays ofno_adjust, DL andOPTwith the value of Zipf parameter varied.
4.3. The performance of algorithm DL
We now investigate the quality of solutions obtained by DL andOPT. Note that algorithm DL will dynamically adjust the broadcast program while the Zipf parameter varies. In order to evaluate the impact of increasing the value of f , we set the value of n to 50 and the value of K to 4. Figure 8 shows the performance results ofOPTand DL. As can be seen in figure 8, the difference between expected delay of DL and that ofOPTis almost negligible, showing the very high quality of the solutions obtained by algorithm DL. Note that as the value of f increases, the solutions obtained by algorithm DL are all very close to the optimal ones, indicating the robustness in algorithm DL.
Next, the experiments of varying the value of K forOPT and DL are conducted where we set the value of n to be 50 and the value of f to be 2.5. Figure 9 shows the average expected delays of OPT and DL with the value of K var-ied. As the value of K increases, the expected delays ofOPT and DL decrease. This agrees with our intuition since as the number of broadcast channels increases, the number of data items in each broadcast channel decreases, thereby reducing the expected delay of data items. Notice that the difference between the expected delay of DL and that ofOPTis very small, again showing the good quality of solutions obtained by DL. The performance of DL with the value of n varied is examined where we set the value of K to 5 and the value of f to 2.5. The average expected delays ofOPTand DL with the value of n varied are shown in figure 10. As the number of data items to be broadcast increases, the expected delays of data items resulted byOPTand DL increase linearly as we anticipate. Also, the difference between the expected delays resulted byOPTand DL is negligible.
4.4. Comparative analysis forVFKand DL
In [14],VFK is designed for the situation where the data ac-cess frequencies and the number of broadcast channels are given. The experimental results show that the broadcast pro-gram generated by VFK is of very high quality. However,
Figure 8. The average expected delays ofOPTand DL with the value of f varied.
Figure 9. The average expected delays ofOPTand DL with the value of K varied.
Figure 10. The average expected delays ofOPTand DL with the value of n varied.
in practice, the data access frequencies may vary as time ad-vances. It is important for broadcast programs to adapt to the change of the data access frequencies so as to retain the performance of data broadcasting. In this section, our exper-imental results show that algorithm DL is more efficient to achieve new configuration without re-executingVFK.
To evaluate the impact of increasing the value of n, we set the value of K to 15 and the value of f to one. Figure 11 shows the execution times incurred by VF and DL. In
fig-Figure 11. The execution times incurred byVFKand DL with the value of n varied.
Figure 12. The execution times incurred byVFKand DL with the value of
Kvaried.
ure 11, the execution times incurred byVFK and by DL in-crease as the number of data items inin-creases. Note that the execution time incurred byVFK is larger than that incurred by DL, showing that DL is able to achieve the new configu-ration more efficiently when the data access frequencies vary. It is also observed that the curve ofVFK in figure 11 is not as smooth as that of DL due to the lack of dynamic allocation adjustment ofVFK.
Next, we examine the impact of increasing the value of K. Without loss of generality, we set the value of n to 5 and the value of f to one. The execution times incurred byVFK and DL with the value of K varied are shown in figure 12. No-tice that when the value of K is smaller than 5, the execution time incurred byVFK is smaller than that incurred by DL. However, as the value of K increases, the execution time in-curred by VFK is significantly larger than that incurred by DL. This indicates that when the value of K increases, the advantage of algorithm DL over the approach of re-executing algorithm VFK increases. In all, algorithm DL is able not only to adjust the broadcast program efficiently in response to the change of access frequencies of data items but also to produce the solutions of very high quality. Note that the capa-bility of adjusting broadcast programs dynamically should be viewed as an enhanced feature rather than a limitation. In fact, when the value of f is set to be infinite, the initial broadcast
program generated byVFK will not be adjusted according to the change of access frequencies. Thus, the performance de-grades as the access frequencies vary, justifying the necessity of algorithm DL.
5. Conclusions
We explored in this paper the problem of adjusting broadcast programs to cope with the data access frequencies varied. By exploiting the features of the casual adjustment and the fine adjustment, we developed a heuristic algorithm DL to adjust broadcast programs when the access frequencies of data items change. Performance of algorithm DL was analyzed and a system simulator was developed to validate our results. Sen-sitivity analysis on several parameters, including the number of data items, the number of broadcast disks, and the varia-tion of access frequencies, was conducted. It was shown by our simulation results that the broadcast programs achieved by algorithm DL are of very high quality and are in fact very close to the optimal ones. This feature and the efficiency of al-gorithm DL justify the practical importance of alal-gorithm DL.
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Wen-Chih Peng biography and photo not available at time of publication. E-mail: [email protected]
Jiun-Long Huang biography and photo not available at time of publication. E-mail: [email protected]
Ming-Syan Chen biography and photo not available at time of publication. E-mail: [email protected]
