- Email: [email protected]
Scheduling objects in broadcast systems with energy-limited clients
Computer Communications 27 (2004) 1036–1042 www.elsevier.com/locate/comcom
Scheduling objects in broadcast systems with energy-limited clients D.N. Serpanosa,*, A.P. Traganitisb a
Department of Electrical and Computer Engineering, University of Patras, Patras GR-26500, Greece b Department of Computer Science, University of Crete, GR-71110 Heraklion, Crete, Greece
Abstract Broadcast systems are the popular infrastructure in push-based information distribution environments, where information objects are transmitted to subscribing clients, who are randomly switched on. The main problem in these systems is the construction of a periodic schedule (in cycles), where in every cycle each information object is transmitted several times; the number of appearances of each object is a function of its size and popularity. Existing algorithms try to transmit each object in a cycle with perfect periodicity, i.e. all object instances (appearances) in the cycle are equally spaced in time. These algorithms construct optimal schedules for environments with memory-less clients, optimizing the aggregate access delay for objects, and thus, minimizing client energy consumption accordingly. In this article, we provide an analysis of broadcast systems for memory-equipped (caching) clients. We change the scheduling optimization criterion to include actual object reception time in addition to access time, and thus, we provide a more realistic model for estimation of actual client power consumption. We prove that memory-equipped clients change the system model significantly, and allow for reduced object reception time, leading to improved energy consumption by clients. We give a simple proof of the fact that perfect periodicity in object transmission within scheduling cycles is necessary for optimal schedule, and calculate the conditions that optimal schedulers must satisfy. Since perfect periodicity is practically impossible to achieve (the problem is NP-hard), we analyze heuristic modifications of the broadcast schedule in order to achieve perfect periodicity for the more popular objects; heuristics include object transmission interleaving, preemptive transmission and exchange of object transmission order. We prove that interleaving should always be avoided and we calculate the conditions under which preemptive transmission and exchange of object transmission order result in reduced aggregate object reception delay, in systems with memory-less and memory-equipped clients. q 2004 Elsevier B.V. All rights reserved. Keywords: Broadcast systems; Broadcast scheduling; Energy-limited clients; Memoryless clients; Memory equipped clients
1. Introduction Broadcast systems are becoming increasingly important and popular for the delivery of a wide range of services to an increasing population of mobile and wireless users; importantly, many of the offered services are demanding in terms of bandwidth and user processing power, such as multimedia applications. Broadcast systems are attractive to deliver services to large client populations, because they are scalable in terms of user population and end-user bandwidth, while accommodating heterogeneous user technologies, e.g. mobile and wireless [1]. Their importance is increasing, due to the continuous development and deployment of thin, personalized user terminals that are capable to store, process, transmit and receive information; typical examples of such terminals include mobile phones * Corresponding author. E-mail addresses: [email protected] (D.N. Serpanos); tragani@ csd.uoc.gr (A.P. Traganitis). 0140-3664/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2004.01.022
and personal digital assistants (PDA). The powerful processing and communication characteristics of these client devices enable a wide range of services and applications, which deliver information to users efficiently. However, these terminals have limited resources, and one of the most important ones is energy: considering their reliance on batteries, it is necessary to optimize their operation taking into account their limited amount of power. There exist several architectures for broadcast systems. We focus on a powerful and popular system architecture, which is an asymmetric, push-based system with receiveonly clients (i.e. without uplink bandwidth), who are mobile, or in general, connected occasionally. This architecture has been used for several deployed systems, such as Pointcast [2], Traffic Information Systems [3], Stock, Weather and News dissemination systems, DirecPc by Hughes [4] and Internet-over-Satellite (IoS) by Intracom [5]. In our work, we consider this architecture and in analogy to a web-type environment, we assume that application information is composed of logical objects
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
(e.g. pages, pictures, text segments), where each object is characterized by a different popularity, depending on the number of clients that need to receive it. The system transmits objects, so that their mean aggregate reception delay is minimized for all users. This criterion, the minimized mean aggregate reception delay, is important not only for performance, but for energy consumption as well: minimized delay implies that client devices are switched on for a minimized time, and thus, energy consumption of client devices is minimized. In push-based broadcast systems servers transmit objects in periodic cycles [1,6,7]. The main problem in these systems is the construction of the schedule in every cycle, i.e. the order in which the objects are transmitted. Several scheduling algorithms exist to construct the exact transmission schedule in a cycle, given as optimization criterion the minimization of the mean aggregate access time, where access time is defined as the delay to start receiving an object. The schedule of a cycle (period) may include multiple transmissions of each object, depending on its popularity [6 –8]. Importantly, it has been proven that, when an object is transmitted several times within a cycle, the optimal mean access time is achieved when all transmissions (appearances) of an object are equally distanced within the cycle [6]. Most of the broadcast systems analyzed up to date, consider clients who are memory-less, i.e. they are not equipped with any storage. However, the recent advances in consumer electronics and embedded systems have resulted to the availability of clients with memory today. Importantly, their memory sizes are continuously increasing; in our work, we call this memory a cache, because it is not used for long-term storage. The existence of caching clients changes the model of a broadcast system, because such a client can start collecting an object even if it is turned on during the transmission of the desired object; this can be achieved provided that each object transmission is done using packets that have headers with the appropriate packet information. This provision is the state of the art though, because in modern wireless communications systems, such as GPRS [9] and DVB [10], transmission is performed with fixed size cells (also called radio units), where each cell has an appropriate header that includes sequence information. In this article, we present three main results. First, we provide a simple proof that the instances (appearances) of an object within a cycle must be equally distanced (periodic within the cycle). We provide this proof, because the existing one [6] uses arguments that are quite complex. Furthermore, in the course of our proof we show that, for the same broadcast schedule, caching clients have shorter waiting times for the complete reception of an object. Our second result is a solution to the scheduling problem for caching clients. As we show, the existence of cache not only reduces the reception time of an object, but leads to an optimal broadcast schedule that is different from the optimal
1037
schedule for traditional, cache-less clients. Specifically, we calculate the reduced reception delay and we derive the property that the optimal schedule for caching clients is based on, which is different from the one for cache-less clients. In our work, we use as optimization parameter the mean aggregate reception delay, i.e. the sum of the access delay and the actual object reception time, which is also known as tuning time [7]. Prior analyses [7,8,11] used models similar to teletext, and did not consider caching clients, because it was assumed that access time is significantly larger than actual object reception delay. However, this assumption is not valid in modern systems, because it is often necessary to transmit large objects often, i.e. objects with high popularity. It is important to note that, our optimization parameter reflects power consumption more realistically than existing models and calculations. This occurs because objects have variable sizes and thus, reception delay is not equal for all objects; however, prior work does not include this extra delay (and corresponding power consumption), since they consider cache-less clients for whom the access time is the only variable time. Finally, our third result is the analysis of a number of techniques to improve the scheduling for broadcast systems with or without caching clients. Considering that achievement of perfect periodicity of the multiple transmissions for all objects within a cycle is an NP-hard problem [12], we prove that the mean aggregate tuning time of an object may be further reduced, if we allow pre-emption (interruption) of an object’s transmission in order to transmit on schedule another more popular one; this is opposite to the usual practice to transmit objects without interruption. We describe the conditions under which pre-emption is advantageous and we show that switching the transmission order of two neighboring objects is also beneficial in some cases. Finally, we prove that interleaving the transmission of two neighboring objects is not advantageous under any conditions; such interleaving seems like a reasonable method, considering object packetization and the popularity of interleaving in ATM networks. The article is organized as follows. Section 2 presents our system model and notation, while Section 3 presents the simple proof of the perfect periodicity requirement. Section 4 describes the analysis for clients with caches, including the calculation of the optimum separation between instances of an object for the minimization of the aggregate reception delay. Finally, Section 5 describes the conditions under which pre-emptive scheduling and switching object transmission order provide improved results; furthermore, it presents a proof that interleaving is not beneficial under any conditions.
2. Model We consider an environment where a server S broadcasts information objects to a set of clients (users); clients are
1038
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
considered to have subscribed for reception of specific objects off-line. The server transmits objects from a set O ¼ {O1 ; O2 ; …; OM ; } where each object Oi is characterized by a popularity pi ; popularity is a probability that quantifies the number of clients expecting a specific object. Objects have arbitrary lengths, and transmission is done using fixed size cells, also called radio units. The length li of an object Oi is measured in radio units. In broadcast systems, transmission is performed in periodic cycles with period T: Each object is transmitted at least once in every cycle, and possibly multiple times, depending on its popularity and length. Appearance of an object Oi in a cycle is denoted as an instance of Oi : The spacing sij between two consecutive instances of Oi is the time between the beginning of its jth and its ðj þ 1Þth instance. Clients are devices that are switched on arbitrarily in time. When a client is switched on, it remains on until it receives the desired object(s), and then it is turned off. The activity to switch on a client can be modeled as posting a request for the object, which the client is waiting for. Hence, in the following, we will refer to the client switchon as a request. We assume that client requests for an object Oi appear uniformly distributed during a broadcast cycle T: We consider two different types of clients in our system: caching and cache-less. Cache-less clients have no storage capability and need to receive an object from beginning to end, in order to consume it. In contrast, caching clients have storage and can store partial object information; so, they can store the last part of an object first and then wait for the reception of its beginning later. We define as tuning time for an object the access time of the object (the time until the beginning of its reception) plus its actual reception time [8]. Thus, we define waiting time differently from others, who consider the waiting time equal to the access time. Our definition does not affect the construction of the optimal schedule for cacheless clients, who are considered in prior work. We use for optimization the mean aggregate tuning time, which is defined as the average of the mean tuning times of all P users, i.e. D ¼ M p D i¼1 i i ; where Di is the mean tuning time for Oi :
3. Periodic object transmission in a cycle In regard to broadcast systems with cache-less clients, there exists a proof that, in an optimal schedule all instances of an object must be equally spaced within a cycle T; an optimal schedule is one that minimizes the mean aggregate access delay [6]. In Lemma 1 below, we provide a simple proof of this requirement, in contrast to the existing proof [1] which is quite obscure. Furthermore, we present a proof that the same requirement exists for optimal scheduling with caching clients.
Fig. 1. Calculating mean tuning delay of Oi :
Lemma 1. The spacing si between any two consecutive instances of the same object Oi should be equal in a transmission cycle T. Proof. : We consider cache-less and caching clients separately. Cache-less clients: assume that object Oi must be broadcast fi times in a broadcast cycle T: Then, the instances of Oi within P1 a T are at spacings si1 ; si2 ; …; sifi ; where fj¼1 sij ¼ T; Fig. 1 shows the jth spacing, sij ; between the jth and ðj þ 1Þth transmission of Oi (with length li ). If there is one request for Oi during T; then the probability that it will appear during the spacing si1 is sTi1 ; and the mean delay to receive all of Oi is ðli þ ðsi1 =2ÞÞ: The same calculation holds true for every spacing sij : Thus, the average delay Di to receive Oi during a broadcast cycle T is: sif s Di ¼ ð1=TÞ si1 li þ i1 þ · · · þ sifi li þ i 2 2 ¼ ð1=TÞ{ðs2i1 =2Þ þ ðs2i2 =2Þ þ · · · þ ðs2ifi =2Þ þ li ðsi1 þ si2 þ · · · þ sifi } ¼ ð1=TÞ{ðs2i1 =2Þ þ · · · þ ðs2ifi =2Þ þ li T} Given that the sum of squares of numbers with a constant sum is minimized when the numbers are equal, it follows that Di is minimum for si1 ¼ · · · ¼ sif1 ¼ si ¼ T=fi : So: Di min
f ¼ i 2T
T fi
2
þli ¼
si þ li 2
Caching clients: under the same assumptions as above, the average delay to receive Oi in a cycle T is: 1 {l s þ ðsi1 2 li Þ½ðsi1 2 li Þ=2 þ li þ li si2 T i i1 þ ðsi2 2 li Þ½ðsi2 2 li Þ=2 þ li þ · · · þ li sifi 8 f1 1< X s2 li sij þ i1 þ ðsifi 2 li Þ½ðsifi 2 li Þ=2 þ li } ¼ T : j¼1 2
Dcachei ¼
9 f1 f1 = X X s2ifi s2i2 l2i 2 þ ··· þ 2 li sij þ fi þ li sij 2 li fi þ ; 2 2 2 j¼1 j¼1 8 9 f1 X s2if 1 < s2i1 l2 = ¼ þ · · · þ i þ li sij 2 i fi T: 2 2 2 ; j¼1 ( ) s2if 1 s2i1 l2 ¼ þ · · · þ i þ li T 2 i fi T 2 2 2
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
This expression is minimized when si1 ¼ · · · ¼ sifi ¼ si ¼ T=fi ; and so: Dcachei
min
¼ fi ðT=fi Þ2 =ð2TÞ þ li 2 ðl2i =2Þðfi =TÞ
¼ pi li þ pi s2i =li ¼ constant A
The last expression shows that, for the same broadcast schedule (same si ), local caching at a client leads to shorter delay to receive object Oi : The scheduling algorithm presented in Refs. [7,8], which requires that s2i pi =li ¼ constant does not take cache into account. If one wants to take advantage of a client’s cache, one must modify accordingly the condition on which the scheduling algorithm is based. As we prove in Section 4, the optimal broadcast schedule for caching clients requires that s2i pi =li þ li pi ¼ constant. The difference is significant, if there exist large (lengthy) objects with high popularities in the system.
4. Broadcast system with caching clients Theorem 1. The optimum broadcast schedule in a system with cache-enabled clients requires that: s2i pi þ li pi ¼ constant li Proof. : our proof is analogous to the one for cache-less clients [8]. Let us assume that, the requirement for equalspacing between successive transmissions of the same object within a cycle T is satisfied. Thus, the spacing si between all pairs of consecutive transmissions of the same object Oi within a cycle T is equal (as we will see this is not always possible, but this is approximately true for large broadcast cycles). Then, the average delay to receive object Oi is: Dcachei ¼ ðsi =2Þ þ li 2 ðl2i fi =ð2TÞÞ. Hence, the mean aggregate delay for all objects is: Dcachei pi ¼
i¼1
¼
M X
i¼1
li pi þ ð1=2Þ
i¼1
¼
M X
M X
M X
ðsi pi =2Þ þ
M X
li pi 2
i¼1
M X
ðl2i fi pi =ð2TÞÞ
i¼1
pi ½si 2 ðl2i fi =TÞ
i¼1
li pi þ ð1=2Þ
i¼1
M X
pi li ½ðsi =li Þ 2 ðli fi =TÞ
i¼1
We denote qi ¼ li fi =T: Clearly: P PMwith qi the 21quantity PM M i¼1 qi ¼ i¼1 ðli fi =TÞ ¼ T i¼1 li fi ¼ 1: Then, si =li ¼ si fi =li fi ¼ T=li fi ¼ q21 i : Thus: Dcache
M X i¼1
pi li þ ð1=2Þ
M X
22 p1 l1 ð1 þ q22 1 Þ ¼ · · · ¼ pM lM ð1 þ qM Þ ¼ constant
2 pi li ð1 þ q22 i Þ ¼ pi li þ pi li ðsi =li Þ
¼ Di min 2 ðl2i fi =ð2TÞÞ A
M X
DM : So, we find that the following relation must hold:
This leads to:
¼ ðsi =2Þ þ li 2 ðl2i fi =ð2TÞÞ
Dcache ¼
1039
pi li ½q21 i 2 qi
i¼1
We set ð›Dcache =›qi Þ ¼ 0; for i ¼ 1; 2; …; N; in order to identify the qi that minimize the mean aggregate delay
Considering the condition of the theorem, one can easily construct a scheduling algorithm that calculates the optimal schedule for caching clients following the steps in Ref. [11].
5. Pre-emptive object scheduling The optimum schedule requires that, for every object Oi ; its consecutive instances must be equally spaced within a broadcast cycle. This property is highly desirable, especially for energy-limited users, because it reduces busy waiting1 for a desired object. The design of an optimum schedule for several objects is an NP-hard problem [12], and this leads to adoption of heuristics which achieve sub-optimal (‘nonperfect’) schedules. Thus, it is very probable, as shown in Fig. 2, that an instance of an object (O2 with probability p2 and length l2 ) will be broadcast after the appropriate time (for perfect periodicity, in order to achieve the optimal schedule), because the transmission of another lengthy object (O1 in the figure) is taking place and must be completed first; even worse, this length object may have low popularity. One can attempt to improve the schedule with the following interventions: 1. Interleave transmission of O1 and O2 ; i.e. interrupt transmission of O1 ; transmit part of O2 ; complete the transmission of O1 and then of O2 (Fig. 3). As we prove below, this approach is ineffective, because it always increases total delay. Clearly, the same conclusion holds true, if we attempt a finer interleaving. 2. Interrupt transmission of O1 ; transmit O2 completely, and then resume and complete transmission of O1 (Fig. 4); this is an example of pre-emptive transmission. Below, we prove that this approach may decrease the total delay under the right circumstances. 3. Transmit O2 first (ahead of schedule) and then O1 (behind schedule), as shown in Fig. 5). As we prove below, this approach decreases the total delay under certain conditions. The results for each one of the above interventions hold for both client models, cache-less or caching. In Section 5.1, we present the complete analysis for cache-less clients, because it is much easier to present. In Section 5.2 we provide, for comparison, the analysis of intervention #2 1
As opposed to stand-by or doze waiting.
1040
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
Fig. 2. Impossibility of perfect periodicity.
(preemptive transmission) for the case of clients with caches and derive the condition under which the total delay is reduced. 5.1. Cache-less clients We analyze the effect of all three interventions for cacheless clients. In all cases analyzed below, only the delays of clients waiting for O1 and O2 are influenced. Remaining clients do not experience any difference in performance. Case 1. : Interleaving First, we calculate the increase of the average delay for O1 ; as shown in Fig. 3: requests for object O1 appearing in the interval ½t0 ; t2 will experience a delay increase to receive O1 ; this increase is equal to l21 : Requests in all other time intervals are unaffected. Therefore, the increase of the mean aggregate delay is equal to: p1 s1 l21 =T: Second, we calculate the increase of the average delay for O2 : requests for O2 that appear in the interval ½t3 ; t4 will experience a delay increase to receive O2 equal to: t6 2 t5 ¼ s22 ( ¼ spacing between instances #2 and #3). All other requests are unaffected. Given that the interval ½t3 ; t4 has length l12 ; the increase of the mean aggregate delay is equal to: p2 s22 l12 =T: Combining the two results above, we see that it is not beneficial to interrupt the broadcast of O1 in order to transmit part of O2 ; because this choice always increases the mean aggregate delay by the amount: Total delay increase ¼ ½p1 s1 l21 þ p2 s22 l12 =T . 0 Case 2. : Pre-emption In this case, transmission pre-emption means that the transmission of O1 is interrupted, O2 is transmitted completely and then, the transmission of O1 is concluded,
Fig. 3. Interleaving objects O1 and O2 :
Fig. 4. Pre-emptive transmission of O2 :
as depicted in Fig. 4. In order to calculate the effect of preemption, we need to calculate the increase of the average delay for O1 and for O2 : We calculate the average delay increase for O1 as follows. Requests for O1 appearing in the interval ½t0 ; t2 will experience a delay increase equal to l2 ; in order to receive O1 : All other requests remain unaffected. Therefore, the increase of the mean aggregate delay is equal to: p1 s1 l2 =T: In regard to O2 ; requests for the object that appear in the interval ½t1 ; t3 (with length s21 2 l12 ) will experience a delay decrease to receive O2 equal to l12 : Requests made during ½t3 ; t4 will experience a delay increase to receive O2 equal to s22 : All other requests remain unaffected. Therefore, the increase of the mean aggregate delay is: ½2p2 ðs21 2 l12 Þl12 þ p2 l12 s22 =T The total increase of the mean aggregate delay DD is: DD ¼ ½p1 s1 l2 2 p2 ðs21 2 l12 Þl12 þ p2 l12 s22 =T ¼ ½p1 s1 l2 þ p2 l212 2 p2 ðs21 2 s22 Þl12 =T: Clearly, the delay increase depends on l12 : If we set dDD=dl12 ¼ 0; we find that DD takes a minimum value, if l12 ¼ ðs21 2 s22 Þ=2: This minimum value is: DDmin ¼ ½p1 s1 l2 2 p2 ðs21 2 s22 Þ2 =4T DDmin is negative (equal to maximum delay decrease) if ðs21 2 s22 Þ2 . 4p1 s1 l2 =p2 Thus, if the above inequality is true, then we can reduce the mean aggregate delay, if we interrupt the transmission of O1 after l1 2 l12 ¼ l1 2 ðs21 2 s22 Þ=2 radio units, in order to transmit O2 : Case 3. : Switching order of object transmission In this case, we transmit O2 ahead of schedule and O1 behind schedule, as shown in Fig. 5. The average delay to receive O1 is affected as follows. Requests for O1 appearing in the interval ½t0 ; t2 will
Fig. 5. Transmission order switch of O1 ; O2 :
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
experience a delay increase equal to l2 ; while requests in the interval ½t2 ; t3 will experience a delay decrease equal to ðs1 2 l2 Þ: The remaining requests are unaffected. Thus, the increase of the mean aggregate delay to receive O1 is equal to: ½p1 s1 l2 2 p1 l2 ðs1 2 l2 Þ=T In regard to O2 ; requests for the object that appear in the interval ½t1 ; t2 will experience a delay decrease to receive O2 equal to l1 : Requests in the interval ½t2 ; t4 will experience a delay increase equal to s22 ; while all remaining requests remain unaffected. Thus, the increase of the mean aggregate delay is equal to: ½2p2 l1 ðs21 2 l1 Þ þ p2 l1 s22 =T
1041
Similarly to the case of O1 ; we calculate DD2 : Clients requesting O2 and arriving in the interval ½t1 ; t3 will have a decrease of their tuning time by l12 : Requests during the interval ½t3 ; t03 will have a mean increase of their tuning time by ðs22 2 l2 =2Þ; requests during the interval ½t03 ; t4 an increase by s22 ; and requests during the interval ½t4 ; t5 a mean increase by l2 =2: Thus, the mean increase of tuning time for O2 is: p2 l DD2 ¼ 2ðs21 2 l12 Þl12 þ ðs22 2 2 Þl2 T 2 # 2 l þðl12 2 l2 Þs22 þ 2 ¼ p2 l12 ðs22 2 s21 þ l12 Þ=T 2
The total delay increase is:
Thus, the condition under which the preemptive transmission reduces the total tuning time is:
DD ¼ ½p1 s1 l2 2 p1 l2 ðs1 2 l2 Þ 2 p2 l1 ðs21 2 l1 Þ þ p2 l1 s22 =T
ðs21 2 s22 Þ2 . 4p1 ðs1 2 l1 Þl2 =p2
¼ ½p1 l22 þ p2 l21 2 p2 l1 ðs21 2 s22 Þ=T This expression becomes negative (i.e. it is a delay decrease) if ðs21 2 s22 Þ . ½l1 þ ðp1 l22 Þ=p2 l1 : Since ðs21 2 s22 Þ ¼ 2l12 (Fig. 5), we obtain: l12 . {½l1 þ ðp1 l22 Þ=p2 l1 }=2: If this inequality is true, then the mean aggregate delay is actually reduced when transmission order of O1 and O2 is interchanged.
5.2. Caching clients In this section, we present the analysis of preemptive transmission only for caching clients, in order to demonstrate the analogy of results with cache-less clients. The analysis of all other interventions is similar. Consider the preemptive transmission of object O2 ; shown in Fig. 4, for the case of caching clients. We derive the conditions under which this preemptive transmission reduces the overall tuning time. For the derivation, we calculate DD1 and DD2 ; the increase of the average tuning time for O1 and O2 ; respectively. DD1 is calculated as follows. Clients requesting O1 and appearing in the interval ½t00 ; t2 will suffer an increase of the tuning time by l2 : Requests during the interval ½t2 ; t3 will not be affected, while requests during ½t3 ; t03 will experience a mean decrease of their tuning time by l2 =2: Similarly, requests during ½t03 t4 will have a decrease of their tuning time by l2 ; while requests during ½t4 ; t5 will have a mean decrease by l2 =2: Considering all these delay changes, the mean increase of the tuning time for O1 due to the preemptive transmission of O2 is calculated as: p1 l2 l2 DD1 ¼ l ðs 2 l1 þ l12 Þ 2 2 ðl12 2 l2 Þ 2 T 2 1 2 2 ¼
p1 l2 ðs1 2 l1 Þ T
The maximum reduction of the tuning time is achieved when: l12 ¼ ðs21 2 s22 Þ=2:
6. Conclusions Energy consumption is a significant operational parameter to embedded, mobile systems. Such systems are an important class of clients in emerging broadcast systems that provide demanding services to users. Energy consumption is influenced strongly by the transmission order of objects in push-based broadcast systems, which constitute one of the most popular and widely deployed broadcast system technology. Thus, object transmission scheduling algorithms need to take into account all client parameters and schedule objects appropriately, so that they maximize system performance (i.e. reduce average waiting time of clients) while minimizing energy consumption at the same time: the longer clients wait, the more energy they consume. This scheduling problem has been solved for the case of memory-less clients. In this article, we analyzed broadcast systems for caching clients. We changed the scheduling optimization criterion to use tuning time, providing a more realistic model for estimation of actual client power consumption. In general, caching clients experience shorter reception delays and lead to improved energy consumption. We presented a simple proof that perfect periodicity in object transmission within scheduling cycles is necessary for optimal schedules, and we calculated the conditions that optimal schedulers must satisfy. Furthermore, we analyzed heuristic modifications of broadcast schedules in order to achieve perfect periodicity for the more popular objects, since perfect periodicity is practically impossible to achieve. As we showed, preemptive transmission and exchange of object transmission order can be beneficial for either cache-less or caching clients, under some
1042
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
conditions, which we calculated. Furthermore, in contrast to these heuristics, we proved that object transmission interleaving is always penalizing.
References [1] D. Aksoy, et al. Research in Data Broadcast and Dissemination, in: Proceedings of the First International Conference on Advanced Multimedia Content Processing, AMCP’98, LNCS, Springer, New York, 1999, pp. 194–207. [2] Pointcast, http://www.pointcast.com [3] S. Shekhar, D. Liu, Genesis and Advanced Traveler Information Systems ATIS: Killer Applications for Mobile Computing, MOBIDATA: An Interactive Journal of Mobile Computing, 1 (1) November 1994. [4] DirecPc, http://www.direcpc.com/index.html [5] Intracom, Internet-over-Satellite (IoS), http://www.intracom.gr/pdf/ products/internet_sys/ios.pdf
[6] R. Jain, J. Werth, Airdisks and airRAID: Modeling and scheduling periodic wireless data broadcast, Tech. Rep. 95-11, DIMACS, May, 1995. [7] N. Vaidya, S. Hameed, Scheduling data broadcast in asymmetric communication environments, ACM/Baltzer Wireless Networks 5 (3) (1999) 171 –182. [8] N. Vaidya, S. Hameed, Data broadcast in asymmetric wireless environments, in: Proceedings of Workshop on Satellite-based Information Services (WOSBIS), New York, 1996. [9] G. Heine, GPRS from A to Z, Artech House Inc., 2000. [10] ETS 300 744 rev 1.2.1, (1999–01), Digital broadcasting systems for television, sound and data services (dvb-t); framing structure, channel coding and modulation for digital terrestrial. [11] S. Hameed, N.H. Vaidya, Efficient algorithms for scheduling data broadcast, ACM/Baltzer Wireless Networks 5 (3) (1999) 183–193. [12] J.N.A. Bar-Noy, B. Randeep, B. Schieber, Minimizing service and operation cost of periodic scheduling, in: Proceedings of the Ninth Annual ACM– SIAM Symposium on Discrete Algorithms, 1998, pp. 11– 20.
Scheduling objects in broadcast systems with energy-limited clients D.N. Serpanosa,*, A.P. Traganitisb a
Department of Electrical and Computer Engineering, University of Patras, Patras GR-26500, Greece b Department of Computer Science, University of Crete, GR-71110 Heraklion, Crete, Greece
Abstract Broadcast systems are the popular infrastructure in push-based information distribution environments, where information objects are transmitted to subscribing clients, who are randomly switched on. The main problem in these systems is the construction of a periodic schedule (in cycles), where in every cycle each information object is transmitted several times; the number of appearances of each object is a function of its size and popularity. Existing algorithms try to transmit each object in a cycle with perfect periodicity, i.e. all object instances (appearances) in the cycle are equally spaced in time. These algorithms construct optimal schedules for environments with memory-less clients, optimizing the aggregate access delay for objects, and thus, minimizing client energy consumption accordingly. In this article, we provide an analysis of broadcast systems for memory-equipped (caching) clients. We change the scheduling optimization criterion to include actual object reception time in addition to access time, and thus, we provide a more realistic model for estimation of actual client power consumption. We prove that memory-equipped clients change the system model significantly, and allow for reduced object reception time, leading to improved energy consumption by clients. We give a simple proof of the fact that perfect periodicity in object transmission within scheduling cycles is necessary for optimal schedule, and calculate the conditions that optimal schedulers must satisfy. Since perfect periodicity is practically impossible to achieve (the problem is NP-hard), we analyze heuristic modifications of the broadcast schedule in order to achieve perfect periodicity for the more popular objects; heuristics include object transmission interleaving, preemptive transmission and exchange of object transmission order. We prove that interleaving should always be avoided and we calculate the conditions under which preemptive transmission and exchange of object transmission order result in reduced aggregate object reception delay, in systems with memory-less and memory-equipped clients. q 2004 Elsevier B.V. All rights reserved. Keywords: Broadcast systems; Broadcast scheduling; Energy-limited clients; Memoryless clients; Memory equipped clients
1. Introduction Broadcast systems are becoming increasingly important and popular for the delivery of a wide range of services to an increasing population of mobile and wireless users; importantly, many of the offered services are demanding in terms of bandwidth and user processing power, such as multimedia applications. Broadcast systems are attractive to deliver services to large client populations, because they are scalable in terms of user population and end-user bandwidth, while accommodating heterogeneous user technologies, e.g. mobile and wireless [1]. Their importance is increasing, due to the continuous development and deployment of thin, personalized user terminals that are capable to store, process, transmit and receive information; typical examples of such terminals include mobile phones * Corresponding author. E-mail addresses: [email protected] (D.N. Serpanos); tragani@ csd.uoc.gr (A.P. Traganitis). 0140-3664/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2004.01.022
and personal digital assistants (PDA). The powerful processing and communication characteristics of these client devices enable a wide range of services and applications, which deliver information to users efficiently. However, these terminals have limited resources, and one of the most important ones is energy: considering their reliance on batteries, it is necessary to optimize their operation taking into account their limited amount of power. There exist several architectures for broadcast systems. We focus on a powerful and popular system architecture, which is an asymmetric, push-based system with receiveonly clients (i.e. without uplink bandwidth), who are mobile, or in general, connected occasionally. This architecture has been used for several deployed systems, such as Pointcast [2], Traffic Information Systems [3], Stock, Weather and News dissemination systems, DirecPc by Hughes [4] and Internet-over-Satellite (IoS) by Intracom [5]. In our work, we consider this architecture and in analogy to a web-type environment, we assume that application information is composed of logical objects
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
(e.g. pages, pictures, text segments), where each object is characterized by a different popularity, depending on the number of clients that need to receive it. The system transmits objects, so that their mean aggregate reception delay is minimized for all users. This criterion, the minimized mean aggregate reception delay, is important not only for performance, but for energy consumption as well: minimized delay implies that client devices are switched on for a minimized time, and thus, energy consumption of client devices is minimized. In push-based broadcast systems servers transmit objects in periodic cycles [1,6,7]. The main problem in these systems is the construction of the schedule in every cycle, i.e. the order in which the objects are transmitted. Several scheduling algorithms exist to construct the exact transmission schedule in a cycle, given as optimization criterion the minimization of the mean aggregate access time, where access time is defined as the delay to start receiving an object. The schedule of a cycle (period) may include multiple transmissions of each object, depending on its popularity [6 –8]. Importantly, it has been proven that, when an object is transmitted several times within a cycle, the optimal mean access time is achieved when all transmissions (appearances) of an object are equally distanced within the cycle [6]. Most of the broadcast systems analyzed up to date, consider clients who are memory-less, i.e. they are not equipped with any storage. However, the recent advances in consumer electronics and embedded systems have resulted to the availability of clients with memory today. Importantly, their memory sizes are continuously increasing; in our work, we call this memory a cache, because it is not used for long-term storage. The existence of caching clients changes the model of a broadcast system, because such a client can start collecting an object even if it is turned on during the transmission of the desired object; this can be achieved provided that each object transmission is done using packets that have headers with the appropriate packet information. This provision is the state of the art though, because in modern wireless communications systems, such as GPRS [9] and DVB [10], transmission is performed with fixed size cells (also called radio units), where each cell has an appropriate header that includes sequence information. In this article, we present three main results. First, we provide a simple proof that the instances (appearances) of an object within a cycle must be equally distanced (periodic within the cycle). We provide this proof, because the existing one [6] uses arguments that are quite complex. Furthermore, in the course of our proof we show that, for the same broadcast schedule, caching clients have shorter waiting times for the complete reception of an object. Our second result is a solution to the scheduling problem for caching clients. As we show, the existence of cache not only reduces the reception time of an object, but leads to an optimal broadcast schedule that is different from the optimal
1037
schedule for traditional, cache-less clients. Specifically, we calculate the reduced reception delay and we derive the property that the optimal schedule for caching clients is based on, which is different from the one for cache-less clients. In our work, we use as optimization parameter the mean aggregate reception delay, i.e. the sum of the access delay and the actual object reception time, which is also known as tuning time [7]. Prior analyses [7,8,11] used models similar to teletext, and did not consider caching clients, because it was assumed that access time is significantly larger than actual object reception delay. However, this assumption is not valid in modern systems, because it is often necessary to transmit large objects often, i.e. objects with high popularity. It is important to note that, our optimization parameter reflects power consumption more realistically than existing models and calculations. This occurs because objects have variable sizes and thus, reception delay is not equal for all objects; however, prior work does not include this extra delay (and corresponding power consumption), since they consider cache-less clients for whom the access time is the only variable time. Finally, our third result is the analysis of a number of techniques to improve the scheduling for broadcast systems with or without caching clients. Considering that achievement of perfect periodicity of the multiple transmissions for all objects within a cycle is an NP-hard problem [12], we prove that the mean aggregate tuning time of an object may be further reduced, if we allow pre-emption (interruption) of an object’s transmission in order to transmit on schedule another more popular one; this is opposite to the usual practice to transmit objects without interruption. We describe the conditions under which pre-emption is advantageous and we show that switching the transmission order of two neighboring objects is also beneficial in some cases. Finally, we prove that interleaving the transmission of two neighboring objects is not advantageous under any conditions; such interleaving seems like a reasonable method, considering object packetization and the popularity of interleaving in ATM networks. The article is organized as follows. Section 2 presents our system model and notation, while Section 3 presents the simple proof of the perfect periodicity requirement. Section 4 describes the analysis for clients with caches, including the calculation of the optimum separation between instances of an object for the minimization of the aggregate reception delay. Finally, Section 5 describes the conditions under which pre-emptive scheduling and switching object transmission order provide improved results; furthermore, it presents a proof that interleaving is not beneficial under any conditions.
2. Model We consider an environment where a server S broadcasts information objects to a set of clients (users); clients are
1038
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
considered to have subscribed for reception of specific objects off-line. The server transmits objects from a set O ¼ {O1 ; O2 ; …; OM ; } where each object Oi is characterized by a popularity pi ; popularity is a probability that quantifies the number of clients expecting a specific object. Objects have arbitrary lengths, and transmission is done using fixed size cells, also called radio units. The length li of an object Oi is measured in radio units. In broadcast systems, transmission is performed in periodic cycles with period T: Each object is transmitted at least once in every cycle, and possibly multiple times, depending on its popularity and length. Appearance of an object Oi in a cycle is denoted as an instance of Oi : The spacing sij between two consecutive instances of Oi is the time between the beginning of its jth and its ðj þ 1Þth instance. Clients are devices that are switched on arbitrarily in time. When a client is switched on, it remains on until it receives the desired object(s), and then it is turned off. The activity to switch on a client can be modeled as posting a request for the object, which the client is waiting for. Hence, in the following, we will refer to the client switchon as a request. We assume that client requests for an object Oi appear uniformly distributed during a broadcast cycle T: We consider two different types of clients in our system: caching and cache-less. Cache-less clients have no storage capability and need to receive an object from beginning to end, in order to consume it. In contrast, caching clients have storage and can store partial object information; so, they can store the last part of an object first and then wait for the reception of its beginning later. We define as tuning time for an object the access time of the object (the time until the beginning of its reception) plus its actual reception time [8]. Thus, we define waiting time differently from others, who consider the waiting time equal to the access time. Our definition does not affect the construction of the optimal schedule for cacheless clients, who are considered in prior work. We use for optimization the mean aggregate tuning time, which is defined as the average of the mean tuning times of all P users, i.e. D ¼ M p D i¼1 i i ; where Di is the mean tuning time for Oi :
3. Periodic object transmission in a cycle In regard to broadcast systems with cache-less clients, there exists a proof that, in an optimal schedule all instances of an object must be equally spaced within a cycle T; an optimal schedule is one that minimizes the mean aggregate access delay [6]. In Lemma 1 below, we provide a simple proof of this requirement, in contrast to the existing proof [1] which is quite obscure. Furthermore, we present a proof that the same requirement exists for optimal scheduling with caching clients.
Fig. 1. Calculating mean tuning delay of Oi :
Lemma 1. The spacing si between any two consecutive instances of the same object Oi should be equal in a transmission cycle T. Proof. : We consider cache-less and caching clients separately. Cache-less clients: assume that object Oi must be broadcast fi times in a broadcast cycle T: Then, the instances of Oi within P1 a T are at spacings si1 ; si2 ; …; sifi ; where fj¼1 sij ¼ T; Fig. 1 shows the jth spacing, sij ; between the jth and ðj þ 1Þth transmission of Oi (with length li ). If there is one request for Oi during T; then the probability that it will appear during the spacing si1 is sTi1 ; and the mean delay to receive all of Oi is ðli þ ðsi1 =2ÞÞ: The same calculation holds true for every spacing sij : Thus, the average delay Di to receive Oi during a broadcast cycle T is: sif s Di ¼ ð1=TÞ si1 li þ i1 þ · · · þ sifi li þ i 2 2 ¼ ð1=TÞ{ðs2i1 =2Þ þ ðs2i2 =2Þ þ · · · þ ðs2ifi =2Þ þ li ðsi1 þ si2 þ · · · þ sifi } ¼ ð1=TÞ{ðs2i1 =2Þ þ · · · þ ðs2ifi =2Þ þ li T} Given that the sum of squares of numbers with a constant sum is minimized when the numbers are equal, it follows that Di is minimum for si1 ¼ · · · ¼ sif1 ¼ si ¼ T=fi : So: Di min
f ¼ i 2T
T fi
2
þli ¼
si þ li 2
Caching clients: under the same assumptions as above, the average delay to receive Oi in a cycle T is: 1 {l s þ ðsi1 2 li Þ½ðsi1 2 li Þ=2 þ li þ li si2 T i i1 þ ðsi2 2 li Þ½ðsi2 2 li Þ=2 þ li þ · · · þ li sifi 8 f1 1< X s2 li sij þ i1 þ ðsifi 2 li Þ½ðsifi 2 li Þ=2 þ li } ¼ T : j¼1 2
Dcachei ¼
9 f1 f1 = X X s2ifi s2i2 l2i 2 þ ··· þ 2 li sij þ fi þ li sij 2 li fi þ ; 2 2 2 j¼1 j¼1 8 9 f1 X s2if 1 < s2i1 l2 = ¼ þ · · · þ i þ li sij 2 i fi T: 2 2 2 ; j¼1 ( ) s2if 1 s2i1 l2 ¼ þ · · · þ i þ li T 2 i fi T 2 2 2
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
This expression is minimized when si1 ¼ · · · ¼ sifi ¼ si ¼ T=fi ; and so: Dcachei
min
¼ fi ðT=fi Þ2 =ð2TÞ þ li 2 ðl2i =2Þðfi =TÞ
¼ pi li þ pi s2i =li ¼ constant A
The last expression shows that, for the same broadcast schedule (same si ), local caching at a client leads to shorter delay to receive object Oi : The scheduling algorithm presented in Refs. [7,8], which requires that s2i pi =li ¼ constant does not take cache into account. If one wants to take advantage of a client’s cache, one must modify accordingly the condition on which the scheduling algorithm is based. As we prove in Section 4, the optimal broadcast schedule for caching clients requires that s2i pi =li þ li pi ¼ constant. The difference is significant, if there exist large (lengthy) objects with high popularities in the system.
4. Broadcast system with caching clients Theorem 1. The optimum broadcast schedule in a system with cache-enabled clients requires that: s2i pi þ li pi ¼ constant li Proof. : our proof is analogous to the one for cache-less clients [8]. Let us assume that, the requirement for equalspacing between successive transmissions of the same object within a cycle T is satisfied. Thus, the spacing si between all pairs of consecutive transmissions of the same object Oi within a cycle T is equal (as we will see this is not always possible, but this is approximately true for large broadcast cycles). Then, the average delay to receive object Oi is: Dcachei ¼ ðsi =2Þ þ li 2 ðl2i fi =ð2TÞÞ. Hence, the mean aggregate delay for all objects is: Dcachei pi ¼
i¼1
¼
M X
i¼1
li pi þ ð1=2Þ
i¼1
¼
M X
M X
M X
ðsi pi =2Þ þ
M X
li pi 2
i¼1
M X
ðl2i fi pi =ð2TÞÞ
i¼1
pi ½si 2 ðl2i fi =TÞ
i¼1
li pi þ ð1=2Þ
i¼1
M X
pi li ½ðsi =li Þ 2 ðli fi =TÞ
i¼1
We denote qi ¼ li fi =T: Clearly: P PMwith qi the 21quantity PM M i¼1 qi ¼ i¼1 ðli fi =TÞ ¼ T i¼1 li fi ¼ 1: Then, si =li ¼ si fi =li fi ¼ T=li fi ¼ q21 i : Thus: Dcache
M X i¼1
pi li þ ð1=2Þ
M X
22 p1 l1 ð1 þ q22 1 Þ ¼ · · · ¼ pM lM ð1 þ qM Þ ¼ constant
2 pi li ð1 þ q22 i Þ ¼ pi li þ pi li ðsi =li Þ
¼ Di min 2 ðl2i fi =ð2TÞÞ A
M X
DM : So, we find that the following relation must hold:
This leads to:
¼ ðsi =2Þ þ li 2 ðl2i fi =ð2TÞÞ
Dcache ¼
1039
pi li ½q21 i 2 qi
i¼1
We set ð›Dcache =›qi Þ ¼ 0; for i ¼ 1; 2; …; N; in order to identify the qi that minimize the mean aggregate delay
Considering the condition of the theorem, one can easily construct a scheduling algorithm that calculates the optimal schedule for caching clients following the steps in Ref. [11].
5. Pre-emptive object scheduling The optimum schedule requires that, for every object Oi ; its consecutive instances must be equally spaced within a broadcast cycle. This property is highly desirable, especially for energy-limited users, because it reduces busy waiting1 for a desired object. The design of an optimum schedule for several objects is an NP-hard problem [12], and this leads to adoption of heuristics which achieve sub-optimal (‘nonperfect’) schedules. Thus, it is very probable, as shown in Fig. 2, that an instance of an object (O2 with probability p2 and length l2 ) will be broadcast after the appropriate time (for perfect periodicity, in order to achieve the optimal schedule), because the transmission of another lengthy object (O1 in the figure) is taking place and must be completed first; even worse, this length object may have low popularity. One can attempt to improve the schedule with the following interventions: 1. Interleave transmission of O1 and O2 ; i.e. interrupt transmission of O1 ; transmit part of O2 ; complete the transmission of O1 and then of O2 (Fig. 3). As we prove below, this approach is ineffective, because it always increases total delay. Clearly, the same conclusion holds true, if we attempt a finer interleaving. 2. Interrupt transmission of O1 ; transmit O2 completely, and then resume and complete transmission of O1 (Fig. 4); this is an example of pre-emptive transmission. Below, we prove that this approach may decrease the total delay under the right circumstances. 3. Transmit O2 first (ahead of schedule) and then O1 (behind schedule), as shown in Fig. 5). As we prove below, this approach decreases the total delay under certain conditions. The results for each one of the above interventions hold for both client models, cache-less or caching. In Section 5.1, we present the complete analysis for cache-less clients, because it is much easier to present. In Section 5.2 we provide, for comparison, the analysis of intervention #2 1
As opposed to stand-by or doze waiting.
1040
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
Fig. 2. Impossibility of perfect periodicity.
(preemptive transmission) for the case of clients with caches and derive the condition under which the total delay is reduced. 5.1. Cache-less clients We analyze the effect of all three interventions for cacheless clients. In all cases analyzed below, only the delays of clients waiting for O1 and O2 are influenced. Remaining clients do not experience any difference in performance. Case 1. : Interleaving First, we calculate the increase of the average delay for O1 ; as shown in Fig. 3: requests for object O1 appearing in the interval ½t0 ; t2 will experience a delay increase to receive O1 ; this increase is equal to l21 : Requests in all other time intervals are unaffected. Therefore, the increase of the mean aggregate delay is equal to: p1 s1 l21 =T: Second, we calculate the increase of the average delay for O2 : requests for O2 that appear in the interval ½t3 ; t4 will experience a delay increase to receive O2 equal to: t6 2 t5 ¼ s22 ( ¼ spacing between instances #2 and #3). All other requests are unaffected. Given that the interval ½t3 ; t4 has length l12 ; the increase of the mean aggregate delay is equal to: p2 s22 l12 =T: Combining the two results above, we see that it is not beneficial to interrupt the broadcast of O1 in order to transmit part of O2 ; because this choice always increases the mean aggregate delay by the amount: Total delay increase ¼ ½p1 s1 l21 þ p2 s22 l12 =T . 0 Case 2. : Pre-emption In this case, transmission pre-emption means that the transmission of O1 is interrupted, O2 is transmitted completely and then, the transmission of O1 is concluded,
Fig. 3. Interleaving objects O1 and O2 :
Fig. 4. Pre-emptive transmission of O2 :
as depicted in Fig. 4. In order to calculate the effect of preemption, we need to calculate the increase of the average delay for O1 and for O2 : We calculate the average delay increase for O1 as follows. Requests for O1 appearing in the interval ½t0 ; t2 will experience a delay increase equal to l2 ; in order to receive O1 : All other requests remain unaffected. Therefore, the increase of the mean aggregate delay is equal to: p1 s1 l2 =T: In regard to O2 ; requests for the object that appear in the interval ½t1 ; t3 (with length s21 2 l12 ) will experience a delay decrease to receive O2 equal to l12 : Requests made during ½t3 ; t4 will experience a delay increase to receive O2 equal to s22 : All other requests remain unaffected. Therefore, the increase of the mean aggregate delay is: ½2p2 ðs21 2 l12 Þl12 þ p2 l12 s22 =T The total increase of the mean aggregate delay DD is: DD ¼ ½p1 s1 l2 2 p2 ðs21 2 l12 Þl12 þ p2 l12 s22 =T ¼ ½p1 s1 l2 þ p2 l212 2 p2 ðs21 2 s22 Þl12 =T: Clearly, the delay increase depends on l12 : If we set dDD=dl12 ¼ 0; we find that DD takes a minimum value, if l12 ¼ ðs21 2 s22 Þ=2: This minimum value is: DDmin ¼ ½p1 s1 l2 2 p2 ðs21 2 s22 Þ2 =4T DDmin is negative (equal to maximum delay decrease) if ðs21 2 s22 Þ2 . 4p1 s1 l2 =p2 Thus, if the above inequality is true, then we can reduce the mean aggregate delay, if we interrupt the transmission of O1 after l1 2 l12 ¼ l1 2 ðs21 2 s22 Þ=2 radio units, in order to transmit O2 : Case 3. : Switching order of object transmission In this case, we transmit O2 ahead of schedule and O1 behind schedule, as shown in Fig. 5. The average delay to receive O1 is affected as follows. Requests for O1 appearing in the interval ½t0 ; t2 will
Fig. 5. Transmission order switch of O1 ; O2 :
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
experience a delay increase equal to l2 ; while requests in the interval ½t2 ; t3 will experience a delay decrease equal to ðs1 2 l2 Þ: The remaining requests are unaffected. Thus, the increase of the mean aggregate delay to receive O1 is equal to: ½p1 s1 l2 2 p1 l2 ðs1 2 l2 Þ=T In regard to O2 ; requests for the object that appear in the interval ½t1 ; t2 will experience a delay decrease to receive O2 equal to l1 : Requests in the interval ½t2 ; t4 will experience a delay increase equal to s22 ; while all remaining requests remain unaffected. Thus, the increase of the mean aggregate delay is equal to: ½2p2 l1 ðs21 2 l1 Þ þ p2 l1 s22 =T
1041
Similarly to the case of O1 ; we calculate DD2 : Clients requesting O2 and arriving in the interval ½t1 ; t3 will have a decrease of their tuning time by l12 : Requests during the interval ½t3 ; t03 will have a mean increase of their tuning time by ðs22 2 l2 =2Þ; requests during the interval ½t03 ; t4 an increase by s22 ; and requests during the interval ½t4 ; t5 a mean increase by l2 =2: Thus, the mean increase of tuning time for O2 is: p2 l DD2 ¼ 2ðs21 2 l12 Þl12 þ ðs22 2 2 Þl2 T 2 # 2 l þðl12 2 l2 Þs22 þ 2 ¼ p2 l12 ðs22 2 s21 þ l12 Þ=T 2
The total delay increase is:
Thus, the condition under which the preemptive transmission reduces the total tuning time is:
DD ¼ ½p1 s1 l2 2 p1 l2 ðs1 2 l2 Þ 2 p2 l1 ðs21 2 l1 Þ þ p2 l1 s22 =T
ðs21 2 s22 Þ2 . 4p1 ðs1 2 l1 Þl2 =p2
¼ ½p1 l22 þ p2 l21 2 p2 l1 ðs21 2 s22 Þ=T This expression becomes negative (i.e. it is a delay decrease) if ðs21 2 s22 Þ . ½l1 þ ðp1 l22 Þ=p2 l1 : Since ðs21 2 s22 Þ ¼ 2l12 (Fig. 5), we obtain: l12 . {½l1 þ ðp1 l22 Þ=p2 l1 }=2: If this inequality is true, then the mean aggregate delay is actually reduced when transmission order of O1 and O2 is interchanged.
5.2. Caching clients In this section, we present the analysis of preemptive transmission only for caching clients, in order to demonstrate the analogy of results with cache-less clients. The analysis of all other interventions is similar. Consider the preemptive transmission of object O2 ; shown in Fig. 4, for the case of caching clients. We derive the conditions under which this preemptive transmission reduces the overall tuning time. For the derivation, we calculate DD1 and DD2 ; the increase of the average tuning time for O1 and O2 ; respectively. DD1 is calculated as follows. Clients requesting O1 and appearing in the interval ½t00 ; t2 will suffer an increase of the tuning time by l2 : Requests during the interval ½t2 ; t3 will not be affected, while requests during ½t3 ; t03 will experience a mean decrease of their tuning time by l2 =2: Similarly, requests during ½t03 t4 will have a decrease of their tuning time by l2 ; while requests during ½t4 ; t5 will have a mean decrease by l2 =2: Considering all these delay changes, the mean increase of the tuning time for O1 due to the preemptive transmission of O2 is calculated as: p1 l2 l2 DD1 ¼ l ðs 2 l1 þ l12 Þ 2 2 ðl12 2 l2 Þ 2 T 2 1 2 2 ¼
p1 l2 ðs1 2 l1 Þ T
The maximum reduction of the tuning time is achieved when: l12 ¼ ðs21 2 s22 Þ=2:
6. Conclusions Energy consumption is a significant operational parameter to embedded, mobile systems. Such systems are an important class of clients in emerging broadcast systems that provide demanding services to users. Energy consumption is influenced strongly by the transmission order of objects in push-based broadcast systems, which constitute one of the most popular and widely deployed broadcast system technology. Thus, object transmission scheduling algorithms need to take into account all client parameters and schedule objects appropriately, so that they maximize system performance (i.e. reduce average waiting time of clients) while minimizing energy consumption at the same time: the longer clients wait, the more energy they consume. This scheduling problem has been solved for the case of memory-less clients. In this article, we analyzed broadcast systems for caching clients. We changed the scheduling optimization criterion to use tuning time, providing a more realistic model for estimation of actual client power consumption. In general, caching clients experience shorter reception delays and lead to improved energy consumption. We presented a simple proof that perfect periodicity in object transmission within scheduling cycles is necessary for optimal schedules, and we calculated the conditions that optimal schedulers must satisfy. Furthermore, we analyzed heuristic modifications of broadcast schedules in order to achieve perfect periodicity for the more popular objects, since perfect periodicity is practically impossible to achieve. As we showed, preemptive transmission and exchange of object transmission order can be beneficial for either cache-less or caching clients, under some
1042
D.N. Serpanos, A.P. Traganitis / Computer Communications 27 (2004) 1036–1042
conditions, which we calculated. Furthermore, in contrast to these heuristics, we proved that object transmission interleaving is always penalizing.
References [1] D. Aksoy, et al. Research in Data Broadcast and Dissemination, in: Proceedings of the First International Conference on Advanced Multimedia Content Processing, AMCP’98, LNCS, Springer, New York, 1999, pp. 194–207. [2] Pointcast, http://www.pointcast.com [3] S. Shekhar, D. Liu, Genesis and Advanced Traveler Information Systems ATIS: Killer Applications for Mobile Computing, MOBIDATA: An Interactive Journal of Mobile Computing, 1 (1) November 1994. [4] DirecPc, http://www.direcpc.com/index.html [5] Intracom, Internet-over-Satellite (IoS), http://www.intracom.gr/pdf/ products/internet_sys/ios.pdf
[6] R. Jain, J. Werth, Airdisks and airRAID: Modeling and scheduling periodic wireless data broadcast, Tech. Rep. 95-11, DIMACS, May, 1995. [7] N. Vaidya, S. Hameed, Scheduling data broadcast in asymmetric communication environments, ACM/Baltzer Wireless Networks 5 (3) (1999) 171 –182. [8] N. Vaidya, S. Hameed, Data broadcast in asymmetric wireless environments, in: Proceedings of Workshop on Satellite-based Information Services (WOSBIS), New York, 1996. [9] G. Heine, GPRS from A to Z, Artech House Inc., 2000. [10] ETS 300 744 rev 1.2.1, (1999–01), Digital broadcasting systems for television, sound and data services (dvb-t); framing structure, channel coding and modulation for digital terrestrial. [11] S. Hameed, N.H. Vaidya, Efficient algorithms for scheduling data broadcast, ACM/Baltzer Wireless Networks 5 (3) (1999) 183–193. [12] J.N.A. Bar-Noy, B. Randeep, B. Schieber, Minimizing service and operation cost of periodic scheduling, in: Proceedings of the Ninth Annual ACM– SIAM Symposium on Discrete Algorithms, 1998, pp. 11– 20.