1224
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 4, JULY 2004
Performance of Proactive Earliest Due Date Packet Scheduling in Wireless Networks Peng-Yong Kong, Member, IEEE, and Keng-Hoe Teh
Abstract—With the convergence of multimedia applications and wireless communications, there is a need to support real-time traffic in wireless networks. In general, real-time packets must be delivered before a certain delay upper bound. In the literature, feasible earliest due date (FEDD) is one of the scheduling algorithms proposed to provide packet delay upper bound guarantees over a time-varying wireless channel. However, FEDD is reactive with respect to changes in the wireless channel. In view of this, we propose a novel deadline-based scheduling algorithm called proactive earliest due date (PEDD), which dynamically adjusts a packet’s deadline in anticipation of an upcoming change in the channel condition. Similar to FEDD, PEDD is idealistic, as they both assume the availability of the exact channel knowledge. This is not implementable and, thus, this paper further proposes a realistic version of PEDD, called R-PEDD. R-PEDD uses a probing mechanism to acquire the channel knowledge, which in turn is used for the packet deadline adjustment. Since probe packets consume bandwidth, a modified version of R-PEDD, called R-PEDD is proposed to derive the required channel information from recent packet transmissions. We have performed extensive simulations using OPNET to evaluate the performance of these proposed algorithms. In short, PEDD always outperforms a couple of existing algorithms in the literature. R-PEDD and R-PEDD are both capable of approximating the performance of the idealistic PEDD in a realistic wireless channel. However, their performance deteriorates with more rapid changes in the channel condition.
+
+
Index Terms—Channel-knowledge acquisition, delay upper bound, earliest due date (EDD), packet scheduling, wireless networks.
I. INTRODUCTION
T
HE RAPID growth in wireless technologies has spurred interest in providing wireless networks a wide range of multimedia applications. Some of these applications, such as multimedia conferencing and video telephony, produce real-time packets that must be delivered to the destination before certain delay upper bounds. One of the most important mechanisms to ensure the timely delivery is packet scheduling. Packet scheduling in wireless networks faces a few unique challenges as a result of the radio characteristics [1]. More specifically, these scheduling algorithms need to consider the time-varying connectivity and channel quality while allocating
Manuscript received July 13, 2003; revised November 12, 2003 and March 16, 2004. The work of K.-H. Teh was supported by the Networking Department, Institute for Infocomm Research. P.-Y. Kong is with the Networking Department, Institute for Infocomm Research, Singapore 119613, Singapore (e-mail:
[email protected]). K.-H. Teh was with the Electrical and Computer Engineering Department, National University of Singapore 119260, Singapore. Digital Object Identifier 10.1109/TVT.2004.830942
the shared wireless bandwidth on a packet-by-packet basis. For example, when the channel is not connected or is of low quality, the user should not be scheduled to transmit its packet because, if transmitted, the packet may be erroneous. Thus, the existing results for scheduling algorithm in wireline networks cannot be directly carried over to wireless networks [2]. In the literature, a majority of the wireless scheduling algorithms that are capable of providing delay upper bounds can be broadly classified into two groups based on the two basic disciplines, namely generalized processor sharing (GPS) and earliest due date (EDD). In GPS [3], each user is assigned a weight based on its packet delay upper bound requirement. According to these weights, the wireless bandwidth is partitioned and shared among different users. Given the bandwidth share of a user, the virtual departure time for each packet of the user can be determined. As such, GPS always schedules the packet with the smallest virtual departure time among all the users as long as the time-varying channel condition allows. Different wireless GPS algorithms vary in how they deviate from selecting the packet with the smallest virtual departure time when the channel condition changes from time to time. In EDD [4], different from GPS, each packet is directly assigned a deadline (or due date) that is the packet’s arrival time plus its target delay upper bound. With this, EDD always first schedules the packet with the smallest deadline as long as the time-varying channel condition allows. Compared to the wealthy literature on wireless GPS, to the best of our knowledge, only a few works have been published on wireless EDD. Hence, this paper studies the use of EDD algorithms to deliver real-time packets over the wireless channel to meet their delay upper bound requirements. In [5], EDD has been shown to minimize the packet delay violation ratio in wireline networks that do not suffer from timevarying connectivity. For wireless networks, [6] has proposed feasible earliest due date (FEDD), which is identical to EDD but schedules packets only from the users with good channel quality. FEDD has been analytically proven to minimize packet delay violation ratio in wireless networks when it operates on the snapshots of the users’ queue and does not consider a new arrival to the users between these snapshots. Here, a new snapshot begins only after all the packets in the previous snapshot are scheduled. While the analysis is good, it may not be applicable to the case where the algorithm works on a packet-by-packet basis, but not snapshot-by-snapshot, and there will be packet arrivals between each scheduling instant. Also, we notice that FEDD is idealistic and reactive. It is idealistic because it assumes that the scheduler knows exactly the actual channel conditions for all users at all times. This is not a realistic assumption because the channel knowledge must be acquired through
0018-9545/04$20.00 © 2004 IEEE
KONG AND TEH: PERFORMANCE OF PROACTIVE EARLIEST DUE DATE PACKET SCHEDULING IN WIRELESS NETWORKS
1225
Fig. 1. System model.
a separate mechanism and is not normally exact. FEDD is reactive because it stops scheduling for a user only after its channel quality turns bad. In other words, FEDD is identical to EDD that does not differentiate between the users, but the deadlines of packets as long as the channel quality is good. Later in this paper, we will show that the system throughput can be improved by making EDD proactive as compared to the reactive nature of FEDD. Specifically, it is beneficial to adjust a packet’s deadline to a smaller value in anticipation of an upcoming degradation in channel quality. We call this new variant of wireless EDD algorithm, which proactively adjusts a packet’s deadline, the proactive earliest due date (PEDD) [7]. Similar to FEDD, PEDD is idealistic by definition. In order to consider a more realistic and pragmatic scenario where channel knowledge needs to be explicitly acquired, we develop another variant of EDD algorithm, called realistic PEDD (R-PEDD). R-PEDD is different from PEDD in the sense that it requires a separate channel-knowledge acquisition mechanism. We will develop the mechanism later in this paper. In order to improve the efficiency in channel-knowledge acquisition, a slight modification of R-PEDD, called R-PEDD , is further proposed. This paper contributes in two aspects: 1) defining and developing PEDD, R-PEDD, and R-PEDD packet-scheduling algorithms and 2) studying the performance of the three algorithms. The rest of this paper is organized as follows. Section II describes the system model and wireless channel model. Section III first presents the details of PEDD and R-PEDD. In this section, we also develop the wireless channel-knowledge acquisition mechanism for R-PEDD. In addition, the modification from R-PEDD to R-PEDD is described. Section IV shows the evaluation results obtained from simulations. We conclude this paper by presenting a summary of our results and observations in Section V. II. SYSTEM MODEL AND WIRELESS CHANNEL MODEL We consider a centralized wireless network architecture where a central packet scheduler serving users each has its own queue for downlink transmissions. As illustrated in Fig. 1, queue holds incoming packets heading to user . The scheduler is an implementation of a packet-scheduling
Fig. 2.
Wireless channel model.
algorithm that we study in this paper and is responsible for deciding which of the queues will have its packet transmitted next. There can be only one packet being transmitted at a time. Thus, the process of scheduling a new packet begins as soon as the system becomes idle and not all the queues are empty. As depicted in Fig. 1, the scheduler considers the channel conditions of the users while making scheduling decision. The mechanism to acquire the channel knowledge is illustrated as a dashed line and is relevant to R-PEDD only because PEDD is idealistic, as explained earlier. The channel conditions for different users are assumed to be statistically identical and independent. The time-varying condition of each user is modeled as a stationary stochastic process using a two-state Markov chain that is proposed in [8] and illustrated in Fig. 2. According to the model, the channel condition of a user can only be either good or bad at a time. Specifically, the channel condition alternates between bad and good at the rates and , respectively. As such, when the channel turns good (bad), it is expected to stay in that condition for a duration that is an exponentially distributed random variable with mean before becoming bad (good) again. When the channel is good (bad), its packet error probability is where . For simplicity, but without losing generality, we assume and . This implies that all the packets transmitted during a good (bad) channel will be received without (with) error. Despite being simple, the finite-state Markov chain described above can be generalized to model a Rayleigh-fading wireless channel [9]. Specifically, different characteristics of a Rayleighfading wireless channel are translated into a set of appropriate transition rates (e.g., and ) between two states of the Markov chain. We next describe an example of how these transition rates
1226
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 4, JULY 2004
can be determined. Consider a Rayleigh-fading wireless channel with an average bit-error probability (BEP). This BEP implies a probability density function (pdf) for the signal-to-interference-and-noise-ratio (SINR) of the channel. This pdf will be partitioned into a number of nonoverlapping sections, each of which correspondences to a state of the Markov chain. So, in the two-state Markov chain model, the pdf is partitioned into two sections. The SINR value at which the partitioning is done is called the SINR threshold. With this partitioning, the average BEP of a state is determined by the cumulative probability of the corresponding section. With this average BEP, the average and ) can be determined for packet error probability (e.g., a given packet size. The transition rates between two states depend on the SINR threshold-crossing rates. The crossing rate is a function of the Doppler frequency; as the Doppler frequency increases with a user’s traveling speed at a fixed radio wavelength, transitions between the states become more rapid when a user moves faster.
Fig. 3. Logical comparison between FEDD and PEDD.
III. PEDD PACKET SCHEDULING In this section, we first present the preliminary of FEDD before developing the PEDD and R-PEDD. Describing FEDD is important such that the novelty of the two new algorithms can be seen easily. In FEDD, the deadline of a packet from queue , is computed as where is the arrival time of the packet and is the packet delay upper bound. In each queue, packets are ordered in a nondecreasing manner according to their deadlines such that the first packet has the smallest deadline. Different from EDD that will schedule the packet with the smallest deadline among all the queues, FEDD will schedule the packet with the smallest deadline only among all the queues in good channel condition. Therefore, FEDD essentially is a simple modification from EDD that masks off all the queues with a bad channel condition. In FEDD, after a packet has been scheduled, it is removed from the queue and transmitted over the wireless channel. If the packet has not been scheduled, it stays in the queue and will be considered for service at the next scheduling instant. If a packet is not scheduled before its deadline, it contributes to the packet delay violation ratio. If all these packets with expired deadline are removed from their queues and dropped before scheduling, the packet delay violation ratio equals the packet drop ratio. A. PEDD Similar to FEDD described above, PEDD does not schedule packets from the queues in a bad channel condition. In fact, PEDD differs from FEDD in only one aspect, as illustrated in Fig. 3. The difference is that PEDD proactively adjusts a packet’s deadline in anticipation of an upcoming degradation in its channel quality before scheduling the packet with the smallest adjusted deadline. Although PEDD makes scheduling decisions based on the adjusted deadlines, packets are considered expired only if their original deadlines are exceeded. The packet deadline adjustment in PEDD is performed as illustrated in Fig. 4. In short, if a packet has its deadline that falls within the bad channel duration, it will be assigned a new
Fig. 4. Packet deadline adjustment in PEDD.
deadline that is the time right before its channel turns bad. This packet deadline adjustment is necessary to reflect the actual urgency in the present of time-varying channel conditions, because it is not productive to transmit a packet during a bad channel condition. To explain this further, consider a packet with an original deadline that is after a good to bad transition in the channel condition. According to PEDD, the deadline is adjusted to the transition time. Without the adjustment, the packet may be transmitted right before its original deadline and makes no contribution to the system throughput. With the adjustment, the packet is given a higher priority as compared to other packets that do not experience a bad channel condition. Through this deadline adjustment, PEDD aims at improving the system throughput. We acknowledge the fact that achieving a better fairness is not the objective of PEDD. Thus, as a result of performing the deadline adjustment, packets from users with upcoming good to bad channel transitions may be given extra service opportunities. This leads to unfairness to other users. However, we argue that the unfairness occurs only in the short term. In the long term, PEDD is fair to all users, because a user that receives extra service at one time may experience the opposite at another time. From the above, the adjusted deadline in PEDD should be more accurate compared to that of FEDD, as it takes into consideration not only the delay upper bound requirement but also the time-varying channel condition. Also, the deadline adjustment is the only operational difference between PEDD and FEDD. This difference exists if and only if the packet deadlines fall in the bad channel duration. When there is deadline adjustment, there may be a difference in service order between PEDD and
KONG AND TEH: PERFORMANCE OF PROACTIVE EARLIEST DUE DATE PACKET SCHEDULING IN WIRELESS NETWORKS
FEDD. For instance, consider a packet with the original deadline and the anticipated the next bad channel duration begins . In PEDD but not FEDD, this packet will be schedat uled before another packet with the original deadline , where . When there is a difference between PEDD and FEDD in the service orders, there is a possible difference in their performance. Proposition 1: The number of packets dropped in PEDD will not exceed that of FEDD. Proof: We have learned from the above that there will be a performance difference between the two algorithms only if there is a difference in their service orders. Thus, we need to only examine the case where there is a service-order difference in order to prove the proposition. We do this by case analysis. Consider two queues and that have, respectively, their head packets with original deadlines and . Both queues and are anticipated to experience their next bad channel durations begin at time and , respectively. Let and . There is a difference in service order between the two algorithms when . For this occurrence, there can only be the following two cases in which the packet for queue with the original deadline is scheduled. Case 1) The packet is scheduled before or at in FEDD. Case 2) The packet is scheduled after in FEDD. Case 1: FEDD schedules the queue ’s packet before or at . In this case, FEDD can schedule both the packets before or at their original deadlines because . Thus, no packet delay violation occurs and no packet is dropped. Consequently, drop where drop is the number of packet dropped in FEDD. For this case, PEDD can also schedule both the packets before or at their adjusted deadlines, but with a different service order. In PEDD, the queue ’s packet will be scheduled before that of queue ’s. Since both the packets do not violate their original deadlines in PEDD, no packet is dropped. Therefore, drop drop is proven for the proposition in this case. Case 2: FEDD attempts to schedule the queue ’s packet after . There are two subcases, namely Case 2a and Case 2b for this case: Case 2a: FEDD schedules the queue ’s packet after . In FEDD, since the queue ’s packet must be scheduled before or at its original deadline, the scheduling instant occurs within the time window . According to FEDD, the queue ’s packet can only be scheduled after that of queue . Thus, when the attempt is made to schedule for queue , it will have been in the bad channel duration, which begins at . Since FEDD does not schedule for a queue in the bad channel condition, the queue ’s packet will be dropped. Consequently, drop . For this case, PEDD will schedule the queue ’s packet before that of queue . Let the queue ’s packet take over the resource originally allocated to queue . Then, queue ’s packet will be attempted scheduled in the same time window, which is after . At the time of attempt, queue has already entered a bad channel duration and the packet is not scheduled but dropped. Since the queue ’s packet is dropped and resource is not used, the queue ’s packet can be scheduled next, before its original deadline. . With this, drop drop As a result, drop is proven for the proposition in this subcase.
1227
Case 2b: FEDD schedules the queue ’s packet before or at . In FEDD, since the queue ’s packet is scheduled before , it will not be dropped. For the queue ’s packet, depending on how early the actual scheduling instant takes place before , there is a chance that FEDD may schedule it before . However, in order to make this case exclusive to that of Case 1, we consider only the situation where the queue ’s packet is not scheduled before . With this, the queue ’s packet will violate the delay upper bound and be dropped. Consequently, drop . For this case, PEDD will schedule the queue ’s packet first. Under the same condition for FEDD, where the queue ’s packet can be scheduled, this queue ’s packet can also be scheduled in PEDD before . But, the queue ’s packet may or may not be scheduled in PEDD, depending on the available bandwidth. As a result, drop . Therefore, drop drop is proven for the proposition in this subcase. In the proof for proposition 1, Cases 1 and 2a indicate equal performance in both the algorithms. On the other hand, Case 2b suggests that PEDD can outperform FEDD in terms of fewer packets dropped. This proof has implicitly assumed that there are no multiple transitions in channel condition before a packet deadline expires. We realize that this assumption may not hold at all times. In order to supplement this deficiency, extensive simulation results will be provided later in this paper to show that PEDD performance never drops below that of FEDD. From the simulation results, we will also quantify the actual performance improvement in packet drop ratio and system throughput brought about by PEDD. B. R-PEDD Similar to FEDD, PEDD is idealistic. Specifically, PEDD assumes the channel knowledge is known to the scheduler. To be realistic or pragmatic, in this section we develop R-PEDD that has an explicit mechanism for the scheduler to acquire the required channel knowledge. Recall that the mechanism is illustrated as the dashed line in Fig. 1 and will be explained later. Before developing the channel-knowledge acquisition mechanism, we introduce how the acquired channel information is used in making scheduling decisions. R-PEDD uses the acquired information in the following two ways: 1) to decide when to mask off which queue from receiving service, as illustrated in Fig. 3, and 2) to proactively adjust the packet deadlines, as illustrated in Fig. 4, for all queues that are not masked off. Contrary to PEDD with exact channel knowledge, R-PEDD does not have the exact knowledge through acquisition. Since the information is not completely accurate, R-PEDD needs to estimate or predict the next time instant at which channel conditions turn from good to bad so that the packet deadlines can be adjusted accordingly. R-PEDD suggests to predict the next transition time for queue as
(1) where is the pdf of good channel durations experienced by queue and is the probability threshold. The density function is dynamically constructed for each queue based on the
1228
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 4, JULY 2004
analyzing the difference for either good or bad channel durations, we focus on the good channel duration hereafter. From Fig. 5, the measured good channel duration is given as
(3) where
is the error term. Then (4)
Fig. 5. Measurement of good and bad channel durations through channel probing.
channel knowledge acquired about the queue. On the other hand, is a design parameter and its effect on the R-PEDD performance will be investigated later in this paper. The immediate for queue problem now is to decide how to construct so that can be determined for a given . R-PEDD proposes two strategies: 1) to construct through channel probing and 2) to mask off all queues that are probed to be in a bad channel duration for as long as the run-time-averaged bad channel period. For a queue , the average bad period is computed as (2) where is the previous average value and is the th measured bad channel duration. The value of is measured from the outcomes of channel probing, as illustrated in Fig. 5. In short, the measured bad channel duration is the time elapsed between the first detected bad channel instant and the next first detected good channel instant. Given the above, both the construction of and the masking off of queues rely on channel probing. Other than channel probing, there are existing methods to predict channel conditions through signal-processing techniques. For example, [10] suggests that radio signal strength exhibits self-similar characteristics and proposes a mechanism to predict the upcoming signal strength given the past signal strengths. In general, this signal-processing-based method is not suitable for PEDD due because: 1) this method assumes each user will continuously transmit or receive in all time instants; it is this continuous transmissions assumption that eliminates the need of channel probing and 2) this method needs to know the signal strengths that are available in the physical layer, but PEDD sits in the higher layer. In the literature [11], cross-layer coupling has been introduced so that higher layer protocols, such as the packet-scheduling algorithms, are provided with the signal strengths collected from the lower layer. While this cross-layer coupling may improve the system performance, it is out of the scope of this paper. Conceptually, R-PEDD probes the channel by sending a small probe packet to each user at every probe interval. If the scheduler receives a response from a user with respect to the probe, the user is perceived as in good channel condition. Otherwise, it is in a bad channel duration. With this probing method, as depicted in Fig. 5, there will be a difference between the actual and measured good (or bad) channel durations. Since it is similar in
Let be the probe interval, as depicted in Fig. 5. Then, assume that is uniformly distributed between 0 and . Consequently, the expected value of is given as (5) Substituting the expected value of
into (4), we have (6)
is the expected value of the acwhere tual good channel duration. As such, the error in measuring the mean good channel duration is half of the probe interval. Thus, a smaller probe interval yields a more accurate measurement. Unfortunately, we cannot afford to reduce the probe interval indefinitely, because each probe packet consumes bandwidth and a smaller probe interval implies a lesser remaining bandwidth for data packets [12]. Ideally, to conserve bandwidth for data packets, we would like to make the probe interval as long as possible so long as the acquired channel knowledge is accurate enough. For a queue , let be the inaccuracy in the acquired channel knowledge and be the probe traffic intensity. Then, we combine the two variables to form a single cost function as (7) where and are the fixed weights assigned to the respective variables. Different values of the weights will affect the performance differently. We will study the effect of and in Section IV before suggesting a set of values to use. Defining could be straightforward, as it is the inversion of the probe interval, i.e., . Consider the acquired channel knowledge is used to construct the pdf . We define the inaccuracy of the acquired knowledge as how different the constructed compared to the actual pdf. Formally, this distance between two pdfs can be given by their Kullback–Leibler distance [13], which is also called the relative entropy. Let be the actual pdf for the constructed . Then, the relative entropy (or inaccuracy) for queue , is given as (8) Generally, is a nonnegative value and a smaller value means the closer the two probability functions. Specifically, is zero when .
KONG AND TEH: PERFORMANCE OF PROACTIVE EARLIEST DUE DATE PACKET SCHEDULING IN WIRELESS NETWORKS
1229
Recall from Section II that the good channel durations are ex. Thus, . ponentially distributed with mean Assume that the measured good channel durations are also exponentially distributed with mean , as given in (6). Then, we may re-rite (8) as
(9) By substituting the expressions of and into (7), we may rewrite the final form of the cost function as (10) Hence, given a particular channel characteristic as indicated by , the objective now is to find the value of such that the value of can be minimized. It is reasonable to assume that is known beforehand through offline analysis of historical data. Since this is a single-dimensional optimization problem, we can then find the desired through derivation of . By partial differentiation (11) There are three roots for (11), namely two complex roots and one real root. Since the probe interval can only be positive and real, there is only one possible solution , given as (12) where
is a function of
and is given as
(13) Recall from the beginning of this section that R-PEDD is different from PEDD because it does not assume the availability of perfect channel knowledge. Instead, R-PEDD depends on channel probing to acquire the desired channel knowledge to estimate the occurrence of the next bad channel duration and to mask off queues that are already in a bad channel condition. As such, in (12) is the probe interval suggested for R-PEDD. With all these, Fig. 6 shows the algorithmic implementation of R-PEDD that estimates the upcoming bad channel duration to happen at as given in (1). In fact, Fig. 6 is the detailed version of Fig. 3. According to the implementation, the operation to mask off queue depends on the value of , which is being updated by the channel probing process illustrated in Fig. 7. Practically, is the current bad channel duration and the queue will be masked off if is greater than a threshold that is equal to the suggested probe interval . Setting the threshold prevents the scheduling algorithm to wrongfully mask off queues that are experiencing transient errors.
Fig. 6.
Algorithmic implementation of R-PEDD.
In Fig. 7, is the probe interval. One of the criteria in selecting according to (12) is to reduce the probe traffic intensity so that bandwidth can be saved for data packets. In the same line of thought, we propose a variant of the channel-probing algorithm that will not probe a user if its channel knowledge can be derived from a recent data packet communication. We call the R-PEDD that uses this modified probing algorithm R-PEDD . In R-PEDD , when the channel condition of a user needs to be acquired at the probe interval , a probe packet will not be sent to the user if a data packet has been transmitted to the same user in the interval. Specifically, if the past data packet transmission has been completed without (with) error, then R-PEDD assumes the channel condition is good (bad) at the time the user supposed to be probed. IV. SIMULATION RESULTS In this section, we benchmark and evaluate the various versions of the proposed proactive scheduling algorithms through random event simulations using OPNET [14]. For the purpose of simulation, the packet-generation behavior of each queue is modeled as a statistically independent ON–OFF process. The process has alternating ON and OFF periods. The duration of an OFF period is an exponentially distributed random variable with mean s. During an ON period, fixed length packets of 4800 bits are generated back-to-back at the peak rate 4.8 Kb/s. The number of generated packets during an ON period is a geometrically distributed random variable with a mean value of 5. All the users have an identical packet delay upper bound, i.e.,
1230
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 4, JULY 2004
Fig. 8. Packet drop ratio with the transition rate from a good to bad channel condition = 0:05.
Fig. 7. Channel-probing algorithm for R-PEDD.
s. Given the value of , the number of admissible users (refer to Fig. 1 for ) are determined as , where is the system bit rate and is the user peak rate. Unless it is stated otherwise, we let Kb/s. Since Kb/s as introduced above, equals 25.
Fig. 9. Packet drop ratio with different transition rates from a good to bad channel condition and bad to good channel conditions .
A. Benchmark PEDD Against FEDD and EDD We first compare the performance of PEDD against that of FEDD and EDD. Fig. 8 shows the packet drop ratio when the average traffic source off period is set to 2 s and the transition rate from a good to bad channel condition is set to 0.05. Recall from Section III that packets are dropped before being scheduled, so that all the transmitted packets can meet the delay upper bound . As such, EDD should have zero packets dropped after using the method presented above to admit only 25 users. Since there is no packet dropped, EDD is not shown in Fig. 8. The same figure also indicates that PEDD always has a lower packet drop ratio compared to FEDD (this has been explained earlier in proposition 1). The performance difference increases with increasing , i.e., the transition rate from the bad to good channel condition. This performance trend can be explained by first assuming that all packets with deadlines that fall within a bad channel duration are dropped in FEDD. Then, the performance improvement from FEDD to PEDD can be approx-
Fig. 10. System throughput with different transition rates from a good to bad channel condition and bad to good channel conditions .
imated as the fraction of these packets that can be scheduled before the bad channel duration begins and the improved schedulability is a result of changes in the service order introduced by
KONG AND TEH: PERFORMANCE OF PROACTIVE EARLIEST DUE DATE PACKET SCHEDULING IN WIRELESS NETWORKS
Fig. 11.
System throughput with different off periods for traffic sources.
Fig. 12.
Packet drop ratio with different off periods for traffic sources.
PEDD. Since the service order will be changed for a user only if it is in the good channel condition, this fraction equals the stationary probability for the channel being in good condition multiplies the ratio of delay bound to average bad channel duration. Hence, the packet drop ratio improvement from FEDD to PEDD is given as drop drop
(14)
where means is approximately proportional to . The symbol is introduced to reflect the use of approximation in the heuristic leading to the relationship. From (14), we may predict that the performance difference between PEDD and FEDD becomes larger at a smaller value of , i.e., the transition rate from a good to bad channel condition. This prediction is correct as indicated in Fig. 9 and, thus, our heuristic reasoning on the performance improvement brought about by PEDD should be acceptable. Generally, a lower packet drop ratio alone is not enough to indicate a better performance, because transmitted packets may be received with error and do not contribute to system throughput. Hence, a lower packet drop ratio must come together with a
1231
higher throughput to indicate a better scheduling algorithm. In addition to the better packet drop ratio in Figs. 8 and 9, PEDD can achieve a higher throughput compared to both FEDD and EDD, as illustrated in Fig. 10. Since FEDD and PEDD are idealistic, the higher throughput in PEDD is a direct result of a fewer packets dropped. Contrary to the packet drop ratio (cross-refer Fig. 9 to Fig. 10), the difference in throughput between PEDD and FEDD is greater with a larger value of . This can be explained by assuming that only a fraction of the improvement in packet drop ratio leads to improvement in throughput. This fraction can be approximated as the ratio of packet transmission time to average good channel duration. Therefore, by extending the heuristic in (14), the throughput improvement from FEDD to PEDD can be written as throughput throughput
(15)
In Fig. 10, despite recording zero packet drop ratio (see Fig. 9), EDD always has the lowest throughput as compared to PEDD and FEDD. This is because EDD does not consider channel condition and may schedule transmission for a user when it is in a bad channel duration. This performance trend is
1232
Fig. 13. w =w values that yield the highest system throughput for R-PEDD with different transition rates from a good to bad channel condition . For these results, is fixed at 0.55.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 4, JULY 2004
Fig. 14.
Suggested probe interval
1
for different while is fixed at 0.55.
also shown when the simulations are repeated using different , which is the average traffic source off period. In Fig. 11, EDD and PEDD have the lowest and highest system throughput, respectively. Comparing Fig. 11 to Fig. 10, we notice that the performance difference between PEDD and FEDD reduces at a larger value of . This is because a larger implies a lower traffic intensity and, thus, fewer packets available for service reordering in PEDD. With fewer chances to change the service order, PEDD performs more similarly to FEDD. This trend of a smaller performance difference at a larger is also noticeable in terms of packet drop ratio, as illustrated in Fig. 12 (cross-refer to Fig. 9). B. Benchmark R-PEDD Against PEDD In this section, we benchmark the idealistic PEDD against the realistic R-PEDD. For the ease of presentation, we focus on the results with (cross-refer to Figs. 9 and 10). While PEDD has the accurate channel knowledge by definition, R-PEDD needs to acquire the channel knowledge through probing. We assume the size of each probe packet is fixed at 120 bits, i.e., 40 times smaller than a data packet. These probe packets are sent at each probe interval. The probe interval depends on the weighting factors and . Thus, we begin by studying the effects of the weighting factors on R-PEDD performance. We have observed that the system throughput of R-PEDD depends on the ratio (results are not presented due to space limitation). In general, decreasing leads to an increasing throughput before a certain threshold, after which a further decreasing results in a smaller throughput. The values of this threshold obtained from simulations at different are shown in Fig. 13. Note that these values are very small, i.e., in the range of . This indicates that it is much more beneficial to frequently probe the channel to acquire accurate knowledge than saving the bandwidth for data packet transmissions. We fixed and , respectively, at 1.0 and 0.975 , hereafter.
Fig. 15. Comparison between the actual andestimated pdfs of good channel durations with different and . The estimated functions are constructed from the channel information acquired through probing.
Given that and are fixed, we may directly compute the probe interval for different . An example of the computed is shown in Fig. 14. From the figure, the probe interval is much longer with a smaller . This is reasonable because a smaller implies a longer average good channel duration, i.e., and, thus, a longer interval between changes in channel condition. Using the probe interval from Figs. 14 and 15 compares the actual and estimated pdfs of good channel durations with different and . Recall that the estimated functions are constructed from the channel information acquired through probing. The figure indicates that, with the , the estimated pdf is close to the actual one. This observation suggests the correctness of and the efficiency of the proposed channel-probing mechanism. Given that the correctness of the probing mechanism is verified as above, we now proceed to benchmark the performance of R-PEDD against that of PEDD. Recall from (1) that the estimated pdf is used by R-PEDD to predict the upcoming change in
KONG AND TEH: PERFORMANCE OF PROACTIVE EARLIEST DUE DATE PACKET SCHEDULING IN WIRELESS NETWORKS
1233
Fig. 16. 2.0
Comparison between PEDD and R-PEDD in terms of system throughput and packet drop ratio. The average off period for all the 25 traffic sources is s.
Fig. 17. is 2.0
Comparison between R-PEDD and R-PEDD s.
210
210
+ in terms of system throughput and packet drop ratio. The average off period for all the 25 traffic sources
channel condition. The prediction requires presetting of a probability threshold . We have observed that the selection of affects the system performance (results are not presented due to space limitation). In general, should be selected within the 0–0.5 range, depending of the channel characteristics, but there is no intuitive rule for such a selection. For a particular channel characteristics as described by and , we have performed extensive simulations to identify the value of that yields the highest system throughput. At these best values of , the respective system throughput and packet drop ratio are shown in Fig. 16. In Fig. 16, the performance of R-PEDD is not always as good as the performance of PEDD. The lower performance level of R-PEDD is observed although Fig. 15 has indicated that R-PEDD is capable of acquiring accurate channel knowledge. This observation suggests that the current channel predication accuracy provided by R-PEDD is not good enough and that a better channel-knowledge acquisition mechanism is necessary to approximate the performance level of PEDD. The current mechanism is not perfect because it is not sensitive enough at larger values of , as indicated in Fig. 14. In this figure, the
becomes invariant to for but, probe interval intuitively, should not converge. The convergence of is a direct result of the way we formulate the cost function . The cost function leads to a compromise in the accuracy for system throughput, but the impact of inaccuracy on throughput is not well captured. The less than perfect performance of R-PEDD at larger values of is clearly illustrated in Fig. 16, where the performance difference between PEDD and R-PEDD grows with in both system throughput and packet drop ratio. This observation implies that R-PEDD is suitable only for small values of . C. Comparison Between R-PEDD and R-PEDD In this section, we compare the performance of R-PEDD to that of R-PEDD . Recall that R-PEDD is identical to R-PEDD, but does not send a probe when the required information can be derived from a recent data packet transmission. Logically, R-PEDD should save some bandwidth due to the lower probe traffic intensity, but it may result in a lower accuracy in the acquired channel knowledge. With reference to Fig. 17, the R-PEDD outperforms R-PEDD in both system
1234
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 4, JULY 2004
throughput and packet drop ratio. This implies that the effect of reduced channel-knowledge accuracy is not as critical as the gain in bandwidth. As a matter of fact, compared to R-PEDD, the performance of R-PEDD is closer to that of PEDD. This is again an indicator that a better channel-knowledge acquisition mechanism is required after reformulating the cost such that the impact of prediction inaccuracy on function system throughput can be captured. Nevertheless, R-PEDD and R-PEDD are both capable of approximating the performance of the idealistic PEDD in a realistic wireless channel. V. CONCLUSION Compared to FEDD, which is reactive with respect to changes in a wireless channel, we have proposed PEDD, which will dynamically adjust a packet’s deadline in anticipation of an upcoming change in its channel condition. PEDD will outperform FEDD. Specifically, the performance difference in terms of system throughput will grow with larger values of and . On the other hand, the performance difference in terms of packet drop ratio will grow only with respect to a larger value of , but not . These performance trends have been validated through extensive simulations. Similar to FEDD, PEDD is idealistic as it assumes availability of the exact channel knowledge. To be pragmatic, we have further proposed two realistic versions of PEDD, called R-PEDD and R-PEDD , respectively. The two realistic algorithms use the probing mechanism to acquire the channel knowledge, which in turn is used for packet deadline adjustment. In order to decide the probe interval, i.e., how frequent to send a probe packet, we have formulated a cost function that takes into account both the channel-knowledge inaccuracy and the probe traffic intensity. By solving for the cost function, we have suggested the best probe interval. With the suggested probe interval, simulation results have indicated that the performance of R-PEDD and R-PEDD is not as good as that of PEDD. The performance difference is greater at larger . This is because the formulation of the cost function have not taken into account the direct impact of channel-knowledge inaccuracy on system throughput. As a result, both R-PEDD and R-PEDD should be suitable only for small values of . Also, compared to R-PEDD, the performance of R-PEDD is closer to that of PEDD. Last but not least, all simulations results have indicated that R-PEDD and R-PEDD are both capable of approximating the performance of the idealistic PEDD in a realistic wireless channel. For future work, we consider studying the performance of the proposed algorithms using real audio, video, or any other multimedia traces. Also, we consider proposing a better channel-knowledge acquisition mechanism to improve R-PEDD performance. ACKNOWLEDGMENT The authors would like to thank S. Jiang for his invaluable advice during the initial state of the research and the anonymous reviewers and the editor for their helpful comments.
REFERENCES [1] D. Eckhardt and P. Steenkiste, “Measurement and analysis of the error characteristics of an in-building wireless network,” in Proc. Association for Computing Machinery Special Interest Group on Data Communications (SIGCOMM), Aug. 1996, pp. 243–254. [2] Y. Cao and V. O. K. Li, “Scheduling algorithms in broadband wireless networks,” Proc. IEEE, vol. 89, pp. 76–87, Jan. 2001. [3] V. Bharghavan, S. Lu, and T. Nandagopal, “Fair queuing in wireless networks: Issues and approaches,” IEEE Pers. Commun., pp. 44–53, Feb. 1999. [4] D. Ferrari and D. Verma, “A scheme for real-time channel establishment in wide area networks,” IEEE J. Select. Areas Commun., vol. 8, pp. 368–379, Apr. 1990. [5] S. Panwar, D. Towsley, and J. Wolf, “Optimal scheduling policies for a class of queue with customer deadlines to the beginning of services,” J. Assoc. Comput. Mach., vol. 35, no. 4, pp. 832–844, 1988. [6] S. Shakkottai and R. Srikant, “Scheduling real-time traffic with deadlines over a wireless channel,” ACM/Baltzer Wireless Networks, vol. 8, no. 1, pp. 13–26, 2002. [7] K.-H. Teh, P.-Y. Kong, and S. Jiang, “Proactive earliest due date scheduling in wireless packet networks,” in Proc. Int. Conf. Communication Technology, Apr. 2003, pp. 816–820. [8] E. N. Gilbert, “Capacity of a burst-noise channel,” Bell Syst. Tech. J., vol. 39, pp. 1253–1265, 1960. [9] H. S. Wang and N. Maoyeri, “Finite-state markov chain—A useful model for radio communication channels,” IEEE Trans. Veh. Technol., vol. 44, pp. 163–171, Feb. 1995. [10] Y. Zhou, P. C. Yip, and H. Leung, “On the efficient prediction of fractal signals,” IEEE Trans. Signal Processing, vol. 45, pp. 1865–1868, July 1997. [11] I. Koutsopoulos and L. Tassiulas, “Channel state-adaptive techniques for throughput enhancement in wireless broadband networks,” in Proc. IEEE INFOCOM’01, Apr. 2001, pp. 757–766. [12] S. Choi and K. Shin, “A unified wireless LAN architecture for real-time and nonreal-time communication services,” IEEE/ACM Trans. Networking, vol. 8, pp. 44–59, Feb. 2000. [13] T. A. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [14] Optimum Network Performance (OPNET). [Online]. Available: http://www.opnet.com
Peng-Yong Kong (S’00–A’03–M’03) received the B.Eng. degree in electrical and electronic engineering (first-class honors) from the Universiti Sains Malaysia, Perak, in 1995, and the Ph.D. degree in electrical and computer engineering from the National University of Singapore, Singapore, in 2002. He was with Intel Malaysia as an Engineer from 1995 to 1998 After that, he was a Research Scholar with the Center for Wireless Communications, Singapore (currently the Institute for Infocomm Research), working toward the Ph.D. degree. Since 2001, he has been with the Institute for Infocomm Research, where he currently is a Scientist. His research interests are primarily in medium-access control protocols, traffic scheduling, traffic control, and quality of service provisioning for both wired and wireless networks.
Keng-Hoe Teh received the B.Eng. degree (first-class honors) from the Multimedia University, Cyberjaya, Malaysia, in 2001 and was working toward the M.Eng. degree in the Electrical and Computer Engineering Department, National University of Singapore. He was a Research Scholar with the Institute for Infocomm Research, Singapore. His research interests include the quality-of-service provision in wireless networks, with a focus on real-time packet scheduling and traffic control.
