Minimizing mean client serving time and broadcast ...

Minimizing mean client serving time and broadcast schedule cost in wireless push systems c. K. Liaskos, Member, IEEE and G. I. Papadimitriou, Senior Member, IEEE Aristotle University, Department of Informatics, Thessaloniki, Greece 54124, Email: {cliaskos.gp}@csd.auth.gr

Abstract-The classic broadcast scheduling problem defines the optimal common schedule as the repeated sequence of data items that minimizes the clients' mean waiting time as well as the broadcast schedule's mean cost, over an infinite time horizon. In this context, a data item's cost is an independent variable representing generic losses incurred to the system with each transmission of the data item. While the cost parameter is essential for taking into account realistic factors such as mandatory copyright charges and client disapproval risks, related research has ignored it due to the proven NP-hardness of the classic scheduling problem. In this paper an algorithmic solution to the classic problem is presented, which yields almost perfect convergence with the optimal solution extracted via brute force.

Index Terms-wireless push systems, classic broadcast prob­

lem, cost, algorithmic solution.

I.

INTRODUCTION

Advances in wired and wireless communications have boosted the growth of the Internet, forming an interconnected and interactive mesh of persons and devices. Through this growth of the client base it has been made clear that large-scale common needs and preferences exist, and their exploitation is an efficient way for preserving network resources. This fact has inspired application authoring entities to undergo a paradigm shift, from personalized per-client serving to per-preference serving. Common examples include media on demand and social networking services. This fact has spurred the research interest in the area of broadcasting and push-based systems. A typical wireless push-based system architecture is shown in Fig. 1. A central server maintains a database of pages, i.e. data items. Each data item is described by its size, its request probability by the clients, and its cost. The latter serves as a metric of the generic losses incurred to the system with each transmission of the data item. Having a good estima­ tion of the pages' request probabilities available, the server creates a finite periodic broadcast schedule that minimizes the clients' mean waiting time [1, 2] and the mean schedule cost (classic broadcast scheduling problem). The clients read needed data from the broadcast stream, while supplying some sort of feedback in order for the server to detect changes in their preferences. This feedback is passed to an adaptation process [3] which extracts the renewed probability distribution estimate. Broadcast rescheduling is finally performed in order to match the clients' preferences once more.

The classic broadcast problem is proven to be NP-hard in [4]. In this study, the authors formulate the classic schedul­ ing problem as the minimization of the sum of the mean waiting time and the mean scheduling cost. Analysis-derived lower bounds of this quantity are given. Scheduling algorithms are also proposed, highlighting the fact that these bounds cannot be strictly achieved. In [5] the authors discuss the generalized maintenance problem (a known NP-hard problem), and prove the broadcasting of equally-sized items to be a subcase of it. A lower bound for the clients' mean waiting time is also provided. [6] proves the existence of a possible solution for teletext systems, and also defines a lower bound for the clients' mean waiting time in this case. To the best of the authors' knowledge, [4] terminated the research attempts towards solving the classic problem. All subsequent broadcast scheduling algorithms that have been proposed since then fo­ cused solely on the minimization of the clients' mean waiting time [1], or directly related metrics, such as the variance of the clients' waiting times [7], the mean clients' impatience [8] and the mean data utility [9]. In [2], it was shown that there exist (and can be easily calculated) schedules of minimal size that achieve the theoretical lower bounds of the mean waiting time. In the present work, the observation of [2] is extended to correspond to minimal overall schedule cost instead of minimal size. A scheduling algorithm able to achieve any of the possible mean waiting time/schedule cost combination is presented. Comparison with brute force solutions of the broadcast scheduling problem shows that the algorithm can achieve the optimal results in every test case, with insignificant degree of divergence. The remainder of this paper is organized as follows: Sec­ tion II states the assumptions made for the purposes of this study. The proposed cost-aware scheduling algorithm is presented in Section III and its evaluation through simulation takes place in Section I V. Conclusion is given in Section V. II.

SETUP O F THE EXAMINED SYSTEM

The conventional topology and assumptions are employed for the purposes of the present work: In an area covered by a wireless broadcast network, a number of clients are interested in a common set of N data items. A data item is a piece of information that can be request through one single query.

TABLE I NOTATION SUMMARY DATA BROADCAST SCHEDULING NOTATION Symbolfferm Description N The total number of available data items. i = 1 . .. N The index of the ith data item (arbitrarily enumerated). The request probability (popularity) of item i: E [0,1]

Pi

2:�lPi

Ii

L

2:�1 Vi'

Wi

Topology layout and operation of a typical wireless push-based

2:�1 Vi . Ci. i ]

Wi

i ax

Mobile Cliants

Fig. I. system.

i.

The total broadcast of the schedule: C = A random variable representing the waiting time for item i. is uniformly distributed in [0,w ax The maximum waiting time for item i. It is equal to the interval between two consecutive occurrences of item i in the periodic broadcast schedule (constant). STANDARD MATHEMATICAL NOTATION The mean value of random variable (.) with regard to probability distribution ( * ) .

C

,./

1

k The broadcast cost of data item

Ci

w

=

The number of occurrences of item i in a periodic broadcast schedule. The size of data item i. The size of the whole periodic broadcast schedule: L =

{li,Pi,

A scheduler uses the tuples cd; i=1 ... N in order to produce an optimal, periodic broadcast schedule [2] that is then cyclically transmitted to the clients. Each item i is repeated E N* times inside the schedule. The interval be­ tween consecutive occurrences of an item is held constant [2]. typically depends on ( [1]). The total size of the periodic broadcast schedule is given by [2]:

Vi

A data item can be of any nature: e.g. a video/audio file or a package containing a complete web page. The broadcast scheduling procedure typically utilizes per item metrics only, such as the request probability of an item and its size [1, 2, 7]. Therefore parameters such as the exact number of clients and the physical coverage of the network are not examined. The only assumption is that every client that can express interest in a data item must be able to read the data stream itself with at least a predefined signal quality level. (i.e. One must be able to listen to the stream in order to express preference for a contained item). Every item i = 1 ... N is characterized by its size (in bytes), its broadcast cost (arbitrary units) and its popularity metric While is a given, is approximated through the use of a lightweight feedback scheme [10]. It is commonly assumed [1-13] that i = 1. .. N represent item request =1. In the context of probabilities, and as such this work, it is assumed that the feedback mechanism has acted, and the probabilities i = 1 ... N are known to the scheduler. Related studies [1-13] make the same assumption. It must be clarified that in the examined family of wireless broadcast systems [1-13], client queries are never posted to the scheduler. A client retrieves a needed item simply by waiting until the needed item is broadcasted. It is also worth of note that in broadcast systems the clients need not know their specific needs in advance. A client typically expresses interest in a category of information via a subscription scheme. Preference for specific item (s) of this category is usually expressed a posteriori, i.e. after the item has been processed and found useful. Television schedules are typical examples of this routine.

Pi.

Ci

li

Pi,

li

Pi

L�l Pi Pi,

Vi

Pi, li, Ci

N

L i=l v·l ' ,

L='"'

(1)

The total cost is similarly calculated:

G=

N

Vi·Ci L i=l

(2)

However, since a schedule is designed for a good number of periodic repetitions, it holds that G= CiwM =G, M being the number of repetitions. Thus G and G coincide. Therefore:

G=G=

N

Vi·Ci L i=l

(3)

Notice however that in this paper the mean cost is extracted over all schedule repetitions, while e.g. in [4] it is extracted over all broadcasted items. Assuming that > 0, Vi, equation (3) is minimized when =1,Vi:

Ci

Vi

Gmin=

N

Ci L i=l

A client query for any item i may be answered in time

(4)

Wi

E

[0,wiax ] (uniformly distributed [1]). As discussed in [7], for the periodic schedules in discussion, it holds that:

,

max w·

L =-

Vi

(5)

The mean value of Wi,i 1 . . . N with regard to probability distribution Pi, is expressed as ( [1]): =

E [WiJ P

N

=

wmax ( L...- �=1 �2 ) Pi·

�.

-

L

=

N

�. 2

L...- �=1

i P Vi

(6)

The Vi parameters are chosen to satisfy some predefined optimization criteria. A popular goal is the minimization of Ep[WiJ. It has been shown [1] that this criterion is satisfied only through periodic schedules where the Vi parameters satisfy the condition:

� /f:lii

Algorithm 1

The proposed scheduling scheme.

The pages Pi : {7ri' li,Ci},i 1 . . . N and a stopping predicate criteria Cr(Ep[wiJ, C). Output: The Ui that correspond to Cr(Ep[wiJ, C). 1: Set vi,i 1 . . . N by eq. (7). 2: Set Vi 1,i 1 . . . N.

Input:

=

=

=

-

° v· "'-'

minimal mean waiting time of equation (8). Due to equation (10), this algorithm can thus be adapted to produce schedules of minimal cost and waiting time.The proposed process is summarized as Algorithm 1.

(7)

The " ( . ) 0" notation denotes optimal values and "",-, " propor­ tionality. Since vi must be integers, relation (7) usually holds by approximation. By setting Vi vi in equation (6) the lower bound of Ep[WiJ is derived ( [1]): =

=

3: repeat

4: 5:

6: 7: 8: 9:

Find page j : I Vj -

vj l ?: IVi- vi i , Vi 1 calculate Ep[WiJ by eq. (6). calculate C by eq. (3). until (Cr(Ep[wiJ,C» Pass the produced Vi,i 1 . . . N to the scheduler of [2]

Vj

=

Vj

+

=

and obtain the returned final schedule.

Ep[WiJmin

=

0.5·

(2::1�)

2

(8)

which is achievable for any � ratio through the scheduling algorithm of [2]. The employed notation is summarized in Table I. III.

CONNECTING THE MEAN WAITING TIME THE MEAN COST

Ep[WiJ AND

C

The present work is based on the noticeable similarity between equations (1) and (3). The total size L of the schedule and the mean cost C are expressed as linear equations of the items' occurrences in the schedule, Vi. Furthermore, the linear parameters are positive real numbers in both cases (li and Ci respectively). Therefore, equations (1) and (3) are homomorphic. We proceed to make an assumption of linearity between L and C, as the basis of the present work. This assumption is also supported by the fact that the optimal number of page occurrences vi is proportional to a fixed quantity, expressed by equation (7). The requirement for integer vi values may cause the assumption to be less accurate for small broadcast schedules (i.e. where N, ViVi, or both are numerically small values). For larger schedules however (Le. large L values) it is guaranteed to hold [1]. The linearity assumption means that should the relation f(.) between Ep[wd and L be (9) the corresponding relation between Ep[wd and C would be (10) where a is a constant value. In other words, the minimiza­ tion of the schedule's size leads to the minimization of its mean cost and vice versa. In [2] an algorithm was presented that can minimize the size of a periodic broadcast schedule, while still achieving the

The algorithm requires a targeted relation Cr between Ep[WiJ and C) as an input. This relation expresses the require­ ments of the scheduling authority. An exemplary expression could be:

I Ep[WiJ- Ep[WiJminl ( Ep[WiJmin