Scheduling Algorithms for Efficient. Delivery of Synchronous Traffic in. Wireless Access Networks

Transcription

1 Scheduling Algorithms for Efficient Delivery of Synchronous Traffic in Wireless Access Networks Liran Katzir

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3 Scheduling Algorithms for Efficient Delivery of Synchronous Traffic in Wireless Access Networks Research Thesis Submitted in Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy Liran Katzir Submitted to the Senate of the Technion Israel Institute of Technology AV 5768 HAIFA AUGUST 2008

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5 The Research Thesis Was Done Under The Supervision of Assoc. Prof. Reuven Cohen in Computer Science Department I wish to express gratitude to my mother, Nava Katzir, for years of dedication and devotion and mainly for bestowing me with education as a key principle since early age. I attribute my success to her. I wish to express gratitude to my immediate family, and especially to my grandpa and grandma, Reuven and Hana Tahan, for their help during my younger years. I wish to express gratitude to Assoc. Prof. Reuven Cohen for introducing me to the world of research during my undergraduate studies and for supervising my Ph.D. I thank him for the potential he saw in me, the faith in my ability to succeed, and for the persistent strive for excellence. I could not have asked for a better supervisor. The generous financial help of the Technion, the REMON Consortium, the Jacobs-Qualcomm fellowship, and the Gutwirth fund is gratefully acknowledged

6 Dedicated to the memory of my father, Dan Katzir (RIP).

7 Contents Abstract xi 1 Introduction 1 2 A Generic Quantitative Approach to the Scheduling of Synchronous Packets in a Shared Uplink Wireless Channel Introduction Related Work The Quantitative-Based Framework Computing the Probability for Successful Transmissions and Retransmissions Transmissions by Stationary and Mobile Hosts Retransmission Scheduling Algorithms for Finding the Optimal Schedule in µ and Some Practical Considerations Scheduling Algorithms Tolerated Jitter vs. Packetization Time Simulation Results Computational Analysis and Efficient Algorithms for Micro and Macro OFDMA Scheduling Introduction Related work Problem Formulation and Complexity Classification The Macro Scheduling Problem (MaSP) The Micro Scheduling Problem (MiSP) Computational analysis of MiSP A Linear Time Dual Approximation for MiSP A Dual PTAS for MiSP Heuristics for MiSP The Extended Macro Scheduling Problem (E-MaSP) Multi frame optimization v

8 3.8 Simulation Study Scheduling of Time-Dependent Multicast Packets Over a Broadcast Channel Introduction Related Work Fixed-size items Greedy approaches An optimal algorithm Variable-size items Hardness Greedy approaches A 2-approximation algorithm Macro-scheduling Simulation Study The Benefit Matrices Micro-Scheduling Simulations Macro-Scheduling Simulations An Efficient Approximation for the Generalized Assignment Problem Introduction The new algorithm Implementation Notes Extensions The Generalized Maximum Coverage Problem Introduction An Efficient Greedy-Based Approximation for GMC Run-Time Complexity Analysis Improving the Performance Guarantee Extensions Conclusions 97 References 99 Hebrew Abstract ä vi

9 List of Figures 1.1 The thesis s structure The three tasks of the proposed scheduling algorithm The Structure of an OFDMA Downlink Frame The two approaches studied in Chapter A scheduling period The variations of MC and their upper bounds and restrictions The loss rate vs. jitter and load for VoIP packets A 3D (2x3x8) expected profit matrix µ with 2 calls, 3 PHY profiles and 8 slots Gilbert Model Possible relationships between t 0 and Release(P) The two cases of Eq The function λ 2 (c, t) Tolerated jitter vs. packetization time The reduction in the bandwidth consumed by the quantitative algorithm vs. the reference algorithm when tolerated jitter is 1 sec The performance as a function of and lookahead ratio T /T when synchronous load is 30% The performance as a function of and lookahead ratio T /T when synchronous load is 60% The performance as a function of and BER for Scenario C Computational Complexity and Running Time for the Various Scheduling Problems The Two Approaches for Micro Scheduling An example of Algorithm 3.8, with k = The Performance of the Algorithms as a Function of Various Parameters The Relative Performance Improvement of E-MaSP vs MaSP Example of one iteration of Algorithm The proof that S contains at most one transmission instance from B(I i ).. 65 vii

10 4.3 An example for a profit function The performance of the various algorithms for the fixed size case as a function of the number of end users The performance of the various algorithms for the fixed size case as a function of N/T The performance of the various algorithms for the variable size case as a function of N/T The performance of the various algorithms for the variable size case as a function of the maximum item size The performance of the various algorithms for the variable size case as a function N/T Algorithm 5.3 running example for 4 items viii

11 List of Tables ix

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13 Abstract Scheduling algorithms are an integral part of network design. They are essential for reducing the congestion in the network, for enforcing fairness, for controlling the delay and jitter, and for saving scarce resources like energy and bandwidth. In wireless networks, scheduling algorithms are critical, because bandwidth is usually limited and it is shared by multiple nodes, transmission errors are likely to take place, and energy saving is very often the main factor for a successful operation of the network. In this research we identify issues in wireless networks that must be addressed using efficient scheduling algorithms. We develop such algorithms, and test the schedulability of various applications under different network conditions. In wireless networks many factors come into play when designing an efficient scheduler, including channel condition, channel load, and information due date. The increase in the number of factors that should be considered makes these schedulers more and more complicated. We proposed a new approach that is quantitative rather than qualitative. All relevant factors are translated into a merit function, and a schedule that maximizes the aggregated profit is then chosen. This approach is applied to a generic uplink channel. The IEEE (WiMax) and IEEE (WiFi) are new emerging future generation standards in wireless communication. The modulation of choice for this system is Orthogonal Frequency Division Multiple Access (OFDMA). In these system the downlink frame has a complex 2D structure. This structure incurs an overhead for a set of packets that share the transmission parameters. Therefore, a scheduler should take all previous consideration that deal with a wireless link and these new OFDMA constraints. We developed efficient algorithms that address the OFDMA scheduling problem that use the quantitative approach. The broadcast push-based system for mobile users is a typical example of a system where time-critical data has to be sent to multiple receivers over a broadcast channel. In these system a sent packet is shared by many users, unlike typical voice communication. We developed a quantitative scheduler that addresses the broadcast nature of these systems. As a part of developing and applying the quantitative approach, we found some generic context-free theoretical computer science results. We found an efficient algorithm for a well-established theoretical problem in computer science known as the generalized assignment problem. Under certain assumptions, the technique used for the solution can be used to translate any single frame scheduling algorithm into a multiframe scheduling algorithm xi

14 with almost the same performance guarantee, at the cost of extra linear running time. This generic result implies that schedulers using the quantitative approach can focus mainly on single frame algorithms. Another theoretical result deals with an extension of the well-known maximum coverage problem. We define a new theoretical problem called generalized maximum coverage. This problem is an extension of the budgeted maximum coverage problem, which in turn is an extension of the maximum coverage problem. We provide an algorithm for this problem with a tight performance guarantee. xii

15 Chapter 1 Introduction In this thesis we consider a wireless access network with a set of nodes connected to a shared channel. The channel is controlled by a centralized node, referred to as a basestation. The upstream channel carries information from the end nodes to the base-station. This information is then forwarded from the base-station to the rest-of-the-world. The downstream channel carries information in the reverse direction: through the base-station to the end nodes. The downstream channel is accessed by the base-station only, and no MAC protocol is therefore needed. In Chapter 2, we propose a new quantitative approach for solving scheduling problems in wireless networks. In this approach the relevant factors are translated into a merit function. Then, the scheduler finds a solution that maximizes the aggregated profit. In Chapter 3, the approach of Chapter 2 is used for the specific constraints of an OFDMA system. In Chapter 4, the quantitative approach is used for solving scheduling problems in a broadcast push-based system for mobile users. In Chapter 5, we present a method to translate any single frame quantitative algorithm into a multi-frame quantitative algorithm. In Chapter 6, we present a detailed solution for one of the mathematical problems presented in Chapter 3. The structure of this thesis chapters flow is summarized in Figure 1.1. A synchronous application like streaming (one-way voice/video) or telephony (twoway voice) is one that demands from the network guaranteed bandwidth, guaranteed maximum delay, and guaranteed loss rate. In wireless access networks there is a common channel that needs to be shared by many stations using a MAC (Medium Access Control) protocol. In Chapter 2 we propose a MAC layer uplink scheduling for synchronous traffic in wireless networks where the transmission of such traffic is governed by explicit grants allocated by the base station. Examples for such networks are IEEE [37, 31] or IEEE (in infrastructure mode). In the proposed scheme the base station allocates transmission grants to the end host of each synchronous call while taking into account the following scheduling considerations (SCs): SC1. The specific QoS requirements of each call: the grants should meet the negotiated grant size, grant periodicity, and tolerated grant jitter. 1

16 Chapter 4 Chapter 5 Chapter 2 Chapter 3 Chapter 6 Figure 1.1: The thesis s structure SC2. The specific conditions of each uplink channel: basically, if a channel experiences bad SNR, the scheduler will try to delay the grant as much as possible. SC3. The specific Application layer loss recovery mechanism employed by each synchronous call codec. Several researchers have shown that some synchronous packets are more sensitive to loss than others [54, 55]. The quality of a synchronous call can therefore be improved by assigning a higher drop priority to the more important packets. For example, when media-dependent FEC is employed and a packet is lost due to a bad channel, the scheduler should increase the priority of the next packet from the same synchronous call, in order to increase the probability that this packet will be received on time. SC4. The specific MAC layer loss recovery mechanisms employed by the network, and in particular, whether ARQ is employed. SC5. Adaptive Modulation and Coding (AMC), along with power control. We describe the proposed scheme in the context of a single upstream channel. In terms of the standard [37], such a channel is provided by the single carrier PHY and by the OFDM PHY. We also extend the scheme to address multiple simultaneous transmissions on different sub-channels when OFDMA (or OFDM with sub-channelization) PHY is used. The main idea behind the proposed scheme is to assign a profit to the transmission of each synchronous packet at every time slot, while taking into account all the relevant scheduling aspects. For example, the profit is proportional to the priority of the packet, to the distance to the packet due date, and to the probability of a successful transmission. In the proposed scheme the scheduler has three tasks, as described in the following and summarized in Figure 1.2. The first (Scenario A in Figure 1.2) is to determine which of the 2

17 Scenario Synch. Tolerated Scheduler Benefit gained from load jitter challenge efficient scheduling high short selecting the most on-time transmission of A normalized important packets the most important jitter for transmission synchronous packets longer than selecting the best successful transmission B irrelevant error burst time and PHY of more synchronous length profile for packets using each packet less bandwidth (a) more available bandwidth for minimizing non-synchronous shorter than the number of applications C irrelevant error burst bad synchronous (b) successful length transmissions transmission of more synchronous packets using less bandwidth Figure 1.2: The three tasks of the proposed scheduling algorithm synchronous packets should be dropped and which transmitted. This decision is important only if the channel bandwidth and the tolerated jitter cannot accommodate the demand of the synchronous applications. This is shown clearly in Figure 2.1, which depicts the loss rate of VoIP packets as a function of the normalized jitter, i.e., tolerated grant jitter packet transmission time for various loads. These graphs were obtained through simulations, assuming an errorfree channel, 1-slot packets, and EDF (Earliest Deadline First) scheduling, which is known to be optimal for minimizing the number of packets that miss their deadlines under these conditions. When the normalized jitter is higher than 10, losses due to scheduling conflicts are not likely even if the load of the synchronous traffic is very close to 100%. Note that non-synchronous (best-effort) traffic has no effect on the graph because it is accommodated only when there is no synchronous traffic. In a bad channel, the scheduler has an important task even if the load imposed by the synchronous traffic is low compared to the channel bandwidth. If the tolerated jitter is long enough compared to the average length of an error burst (Scenario B in Figure 1.2), as might be the case for video streaming, the second task of the scheduler is to determine the best combination of transmission time and AMC (Adaptive Modulation and Control) for each packet, in order to maximize the number of synchronous packets that are received on time, with no error using minimal bandwidth. When the tolerated jitter is not long enough 3

18 (Scenario C in Figure 1.2), as in the case of packetized telephony, the scheduler does not have enough flexibility to wait until an error burst is likely to end. In that case, the third task of the scheduler is to determine which packets should be ignored, in order to minimize bandwidth waste. This can increase the available bandwidth for non-synchronous (besteffort) applications that experience a good channel. The proposed quantitative-based approach is said to be generic because it is applicable for all the scenarios described in Figure 1.2, and for any combination of scheduling considerations SC1-SC5 discussed earlier. This result was also reported in [22]. Orthogonal Frequency Division Multiple Access (OFDMA) is one of the most important modulation and multiple access methods for the future mobile networks. It is an extension of OFDM, which is today the modulation of choice for non-mobile wireless access systems such as IEEE (WiFi) and IEEE (WiMax). OFDM divides a single frequency band into dozens of sub-carriers for parallel transmissions by the same user. This division increases the tolerance to noise and multi-path, while enabling more efficient use of bandwidth allocation. OFDMA extends OFDM by dividing the original band into many subchannels, each comprising a group of orthogonal carriers. Various modulations and FEC (Forward Error Correction) techniques are used for each subchannel, in order to provide improved coverage and throughput. Unlike OFDM, OFDMA has many intricate constraints on its frame structure. The structure of an OFDMA downlink frame is depicted in Figure 1.3. The frame can be viewed as a 2-dimensional matrix, with the y-axis indicating the number of subchannels, each consisting of several orthogonal and not necessarily adjacent frequencies, and the x-axis indicating the time. The frame starts with a Frame Control Header (FCH), which contains information about the code rate, modulation level, and the length of the downlink (DL) and uplink (UL) maps. This part of the frame is always coded using a pre-determined Phy-profile. The data messages (PDUs) are transmitted in multiple bursts. There are 7 bursts shown in Figure 1.3. Each burst is transmitted by the base station using a specific Phy-profile. Due to Physical layer considerations, each data burst must be allocated a rectangular region within the frame. A burst may contain one or more PDUs destined for one or more receivers. To ensure correct decoding of the received signals, certain Phyprofile information about every burst is required. This information is included as overhead in the DL map burst. Chapter 3 studies combinational aspects of scheduling on an OFDMA channel. Before transmitting a downstream frame, typically once every few ms., the base station has to invoke a scheduling algorithm that will generate the frame matrix as shown in Figure 1.3. To this end, the scheduler needs to address the following decision problems: To decide which Phy-profile will be used for each PDU. There are several dozen potential profiles, each with its own bandwidth cost and robustness against transmission errors. It is not possible to transmit all the profiles in the same frame. Moreover, each profile introduces a fixed significant overhead in the DL map field of the frame, and 4

19 time (OFDMA symbol number) subchannel logical number p r e a m b l e downlink & uplink map FCH downlink & uplink map downlink burst #2 downlink burst #7 downlink downlink burst #4 burst #3 downlink burst #5 downlink burst #1 downlink burst #6 Figure 1.3: The Structure of an OFDMA Downlink Frame therefore, because of throughput considerations, the scheduler should try to minimize the number of profiles accommodated in every frame. To determine which PDU will be transmitted in the next frame. This decision has to take into account many factors, such as: (a) Quality of Service, since some of the PDUs have a guaranteed upper bound on their maximum delay; (b) total throughput maximization, since transmission to some of the hosts is difficult and requires more bandwidth for reliable delivery, and (c) fairness. To decide where exactly in the frame every burst will be located. Here there are also several constraints, some of which are: (a) power boosting, namely the ability of the base station to increase the transmission power used for some burst while decreasing the power used for other bursts transmitted at the same time on different subchannels; (b) efficiency, since the requirement that every Phy-profile will be represented as a rectangle in the frame matrix may leave some unused space in each rectangle. There are two kinds of difficulties associated with addressing these problems. The first is that a solution for each problem depends on the solutions for the other two. This leads to a circular dependency, which has to be broken somehow. The second difficulty is that each of these decision problems is NP-complete, which means that even if the circular dependency is broken, finding an efficient scheduling algorithm is difficult. In Chapter 3 we address the OFDMA scheduling problem in two approaches. In the first approach (Section 3.4 and Section 3.5) we assume that the base station determines in advance the Phy-profile to be used for each PDU, and the profit gained from transmitting 5

20 A list of PDUs and Input the association between each PDU and its A list of PDUs Input selected Phy profile Macro Sched. Which PDUs will be transmitted Which PDUs will be transmitted and in which Phy profile Extended Macro Sched. Micro Sched. How to build the matrix How to build the matrix Micro Sched. (a) the first approach (b) the second approach Figure 1.4: The two approaches studied in Chapter 3 this PDU using its selected Phy-profile. Then, there are two phases. In the first phase, referred to as macro scheduling(masp), the base station determines which of the PDUs will actually be selected for transmission. In the second phase, referred to as micro scheduling(misp), the scheduler determines how to build the OFDMA matrix from the selected PDUs. In the second approach we extend the macro scheduler such that it also determines the Phy-profile for each PDU. This is referred to as extended macro scheduling(emasp). Figure 1.4 summarizes these two approaches. This result was also reported in [23]. The assignment of profit to PDUs, which is based on various considerations such as QoS, throughput maximization, fairness, and channel state information (CSI), is discussed in Chapter 2 in the context of uplink transmission. Similar ideas can be used for the downlink as well. The main contributions of Chapter 3 are: Breaking the OFDMA scheduling problem into two more tractable problems, referred to as macro scheduling and micro scheduling. 6

21 Analyzing the computational complexity of the macro and micro scheduling problems, and providing the best possible approximations. Developing efficient and practical algorithms for these NP-hard scheduling problems. Specifically, we provide an optimal macro scheduling problem, and an algorithm with only 2.5% overhead for the micro scheduling problem. Introducing a new concept in which the transmission profile of a PDU is not only determined by the channel condition of the intended receiver, but on the entire system conditions. To this end, we extend the macro scheduling problem to include the selection of transmission parameters for each PDU, and provide an efficient algorithm for this extended macro scheduling problem. This algorithm has the best possible approximation guarantee up to a small constant. It provides a significant improvement over the algorithm for the macro scheduling problem, at the cost of higher running time. One of the most important issues in a broadcast push-based system for mobile users is determining what data-items should be broadcast and according to which transmission program. A practical solution is to ask the clients to subscribe to content channels while providing profiles of their interest [5, 17, 51], e.g. a sports channel, a news channel, a traffic reports channel etc. Cyclic Data Broadcast (CDB), also known as broadcast disks [2] or data carousel, is a well accepted approach used to decide how often should every data-item be broadcast. The idea is that the broadcasting station, namely the wireless base-station, broadcasts all of the required data-items in a cyclic manner, according to some pre-determined program. Several models have been proposed in this context [2, 12, 42, 48]. In [20] a similar system is considered, but the issue of broadcast reliability is addressed. A solution called digital fountain is proposed, where the broadcasting station transmits a stream of distinct encoded packets. When the concept of cyclic data broadcast (CDB) is employed, the broadcast frequency of each data-item is determined based on several parameters. The most important of which are the popularity of the data-item and its expiration time: popular data-items should be broadcast more frequently than less popular items, and data items whose content is valid only for a short time period like stock quotes should be broadcast more frequently than data-items whose content is valid for a longer time period. Consider a set of N data-items I 1 I N. Let the broadcast frequency of data-item I i be φ(i i ) (broadcasts in a second). In order to guarantee that the broadcast channel has enough capacity, the following condition must be fulfilled: N φ(i i ) σ(i i ) B, (1.1) i=1 7

22 Frame 1 slot 1 slot 2 slot 3 a scheduling period=t slots Frame 2 slot 1 slot 2 slot 3 Frame L slot 1 slot 2 slot 3 time Figure 1.5: A scheduling period where σ(i i ) is the size of data-item i and B is the long time average of the bandwidth available for the CDB services. The value of B can be determined in several ways. If the CDB services are of first priority, then the value of B can be determined in advance based on the actual requirement of the supported data channels. Alternatively, if the CDB services are configured to use only the bandwidth not used by the other (unicast) services, then B is equal to the available bandwidth minus the average bandwidth used by the other services. Another option is a combination of these two approaches: to give the CDB services the bandwidth left by the other services, but with some minimum guarantee. Regardless of the way B is computed, its value may be modified from time to time, e.g. once a hour if the first approach is used, or even more often if the second or third approaches are used. Whenever this happens, the value of φ(i i ) for every data-item I i might have to be adjusted in order to guarantee Eq In Chapter 4 we consider the problem of determining when exactly should every dataitem I i be broadcast, given that the value of φ(i i ) has already been determined for every data-item. We assume that this scheduling decision is made once for every scheduling period of T slots, and that each item should be transmitted either 0 or 1 time during a scheduling period. A typical value of T could the number of slots in a period of seconds. Decreasing the value of T imposes a larger scheduling processing burden on the broadcasting station, while increasing this value reduces the scheduling efficiency because changes in the short-time average available bandwidth cannot be taken into consideration by the scheduler. Each scheduling period consists of L (possibly hundreds) layer-2 downstream frames. For each scheduling period, the scheduling logic at the broadcasting station has to determine which of the N data-items will have to be broadcast in each time slot of every frame (see Figure 1.5). We develop a model and algorithms that address this scheduling problem. The approach we adopt is based on giving a profit to the transmission of every data-item I in every time slot, based on the various scheduling considerations. We then translate the problem into a theoretical scheduling problem called STDMP ( Scheduling of Time-Dependent Multicast Packets ), and present algorithms for solving this problem. For the case where all the dataitems have the same size, a polynomial time optimal algorithm is presented. In the more practical case, where each item has a different length, STDMP is NP-complete, and several approximation algorithms are introduced. 8

23 We then turn to a more practical scenario where the granularity of the scheduling is per frame. That is, we are considering the macro-scheduling problem in which we only want to define the data items to be broadcast in each frame, and we let a lower layer (PHY) scheduler to decide how exactly to spread these data items inside the frame. Our algorithms for STDMP could be used here as well. However, since their complexity depends on the actual benefit for each data item in each slot, whereas the macro-scheduling problem associates with each data item only one benefit value for the entire frame, these algorithms are no longer polynomial. To address this problem we develop a polynomial approximation algorithm for this case. In order to study the practical performance of the proposed algorithms we conducted a thorough simulation study in which the different algorithms are compared to various greedy heuristics. In order to perform this study and simulate realistic broadcast pushbased scenarios, we also have to deal with the definition of the exact benefit one can get from broadcasting a given data item in a given time slot, or in a given frame for the macroscheduling case. This result was also reported in [25]. In Chapter 5 we present a simple family of algorithms for solving the Generalized Assignment Problem (GAP). Our technique is based on a novel combinatorial translation of any algorithm for the knapsack problem into an approximation algorithm for GAP. If the approximation ratio of the knapsack algorithm is α and its running time is O(f(N)), our algorithm guarantees a (1 + α) approximation ratio, and it runs in O(M f(n) + M N), where N is the number of items and M is the number of bins. Our technique comprises a general interesting framework for the GAP problem; and it also matches the best combinatorial approximation for this problem, using a much simpler algorithm with a better running time. This approach can translate any quantitative scheduler for a single frame to a quantitative scheduler for multiple frames. This result was reported in [26]. In Chapter 6 we define a new problem, called the Generalized Maximum Coverage Problem (GMC). GMC is an extension of the Budgeted Maximum Coverage Problem, and it has important applications in wireless OFDMA scheduling. In the GMC problem there is a budget, a set of bins, and a set of items. Each item has a size and a profit for each of the bins, and each bin has a size. The objective is find an assignment of the items to the bins such that the total size does not exceed the budget and the total profit is maximized. In the OFDMA context each bin is a burst with specific transmission parameters, the bins size is the overhead for each burst, the budget is the downlink frame s size, and the items are the packets awaiting transmission, where each such a packet has a different size and profit according to the transmission parameters of each burst (bin). We use a variation of the greedy algorithm to produce a ( 2e 1 + ɛ)-approximation for every ɛ > 0, and then use e 1 partial enumeration to reduce the approximation ratio to in [24]. e e 1 +ɛ. This result is also reported Formally, the new maximization problem, referred to as the Generalized Maximum Coverage Problem (GMC) is defined as follows: 9

24 Instance: A budget L, a set E of elements, and a collection B of bins. For each bin b B, each element e E has a positive profit and a non-negative weight, denoted by P (b, e) and W (b, e) respectively. In addition, each bin b has a weight W (b), which can be considered as the overhead for using this bin. Objective: Find a feasible selection with maximum profit. A selection is a triple S = (β, η, f) where β B, η E, and f is an assignment function from η to β. The selection s weight is W (S) = b β W (b) + e η W (f(e), e). A feasible selection is a selection such that W (S) L. The selection s profit is denoted by P (S), where P (S) = e η P (f(e), e). Note that function f guarantees that an element e can only be chosen for one bin. Note that the bins have no individual capacity. Instead, there is an overall budget L that limits the usage of bins and elements within the packing. The new problem is related to three known problems: the Maximum Coverage Problem (MC) [36], the Budgeted Maximum Coverage Problem (BMC) [43], and the Multiple Choice Knapsack Problem (MCKP) [41] (see Figure 1.6). In MC, the input is a number L, a set of elements E, and a collection C of subsets of E. The goal is to find a subset of C whose size is L and which maximizes the number of elements covered. MC can be captured by the GMC formulation in the following way: (1) the bins in GMC represent the collection of subsets; (2) the weights of all bins are exactly 1; (3) the profit of every element is the same for all bins, and it is equal to 1; and (4) if an element e is a member of a collection C, then in its corresponding bin b we have W (b, e) = 0 and W (b, e) = otherwise. By setting W (b, e) = we prevent the selection of element e for bin b, while by setting W (b, e) = 0 we allow the selection of e for bin b with no additional cost. BMC is a generalization of MC in which the weight of the bins is not necessarily equal. GMC, defined in this chapter, is a generalization of BMC, in which elements have weights and an element s weight/profit may vary from bin to bin. MC is NP-hard [36], and therefore both BMC and GMC are also NP-hard. A simple greedy algorithm yields an ( e )-approximation for MC [36]. The e 1 special case where an element can be selected for no more than ρ bins can be approximated within [6]. For small values of ρ, this approximation ratio is better 1 than 1 (1 1 ρ) ρ e. Several special cases where the bins are vertices of a graph and the elements are its e 1 e edges are discussed in [45]. In [43], the authors prove a lower bound of ɛ, for every e 1 ɛ > 0, on the approximation ratio of MC. This lower bound also applies to BMC and GMC since they hold MC as a special case. The authors of [43] also present two greedy-based algorithms for BMC. The first achieves a ( e e 1 )-approximation and the second a ( e )- e 1 approximation (the best possible). If the bins have no weight, i.e., b B W (b) = 0, GMC reduces to MCKP [41], which is known to have an FPTAS [41]. Therefore, the hardness of GMC stems from the bins weights. Another closely related problem is the generalized assignment problem discussed in Chapter 5. In this problem, each bin has no weight, but rather its own budget. The first 10

25 Problem Restrictions Best Known Approx e MC b B W (b) = 1, e E,b B P (b, e) = 1, [36] e 1 Elements can be selected only for a subset of the bins e BMC e E,b B P (b, e) = 1, [43] e 1 Elements can be selected only for a subset of the bins MCKP b B W (b) = 0 FPTAS [41] e GMC none + ɛ, ɛ > 0 e 1 (this work) Figure 1.6: The variations of MC and their upper bounds and restrictions. known approximation algorithm for GAP is an LP-based 2-approximation algorithm, presented implicitly in [58]. an LP-based ( e + ɛ)-approximation algorithm is presented e 1 in [32]. In Chapter 5, we presented a combinatorial linear time (2 + ɛ)-approximation. Both BMC and MC are well established and have many applications [36]. An important application of GMC is in the area of OFDMA wireless scheduling. Consider a wireless base station with a set of packets waiting for transmission to different users. Each packet can be transmitted using several different Phy-profiles, each of which includes a modulation technique, code rate, and a scheme of forward error correcting code (FEC). There is a profit/cost tradeoff associated with selecting a Phy-profile for each packet. Loosely speaking, the more robust Phy-profiles require more bandwidth than the less robust ones. This added cost can be justified when the quality of a wireless link is bad and the robustness of the Phy-profile significantly increases the likelihood of a correct transmission. For each frame transmitted by the base station, the scheduler needs to decide which of the packets will be transmitted, and what Phy-profile to use for each packet. All the packets transmitted using the same Phy-profile are grouped together and are considered as a single burst. In addition to the bandwidth cost associated with the transmission of every packet, there is also a fixed cost for every burst, because the base station needs to inform all the receiving nodes about the exact location of each burst within the frame. Due to the importance of this control information, it is transmitted using the most expensive and robust Phy-profile. Hence, the cost associated with each Phy-profile cannot be ignored. In terms of GMC formulation, the frame s size is the budget L, each packet is an element, and each Phy-profile is a bin. The cost associated with the transmission of each packet using each Phy-profile is equivalent to the weight of the corresponding element in the corresponding bin. The profit associated with each packet indicates the likelihood that this packet will be transmitted correctly, multiplied by a factor that indicates the packet s priority at the considered time. The latter factor depends not on the Phy-profile, but only on the content of the packet. 11

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27 Chapter 2 A Generic Quantitative Approach to the Scheduling of Synchronous Packets in a Shared Uplink Wireless Channel 2.1 Introduction The scheduling logic at the base station of a shared wireless medium supports time-dependent (synchronous) applications by allocating timely transmission grants. To this end it must take into account not only the deadlines of the pending packets, but also the channel conditions for each potential sender, the requirements of non-synchronous applications, and the packet retransmission strategy. With respect to these factors, in this chapter we identify three scheduling scenarios and show that the scheduler logic faces a different challenge in addressing each of them. We then present a generic scheduling algorithm that translates all the factors relevant to each scenario into a common profit parameter, and selects the most profitable transmission instances. This result was reported in [22]. This chapter is organized as follows. In Section 2.2 we discuss related work. In Section 2.3 we present the proposed scheme. In Section 2.4 we show how to compute the profit of transmitting a packet in every slot, with or without MAC layer retransmission support. In Section 2.5 we address some practical considerations related to the proposed scheme. Section 2.6 presents a simulation study of the proposed scheme. 2.2 Related Work We are not aware of previous works that address the scheduling problem while taking into account the aforementioned scheduling considerations. Ref. [4] proposes interesting scheduling algorithms that do take into consideration packet loss due to transmission errors. It also provides a general guideline for addressing SC2 and SC4 using the notion of 13

28 % load 80% load 60% load 30 Packet loss rate Normalized Jitter Figure 2.1: The loss rate vs. jitter and load for VoIP packets backoff time. The idea is that if a call has experienced a recent loss due to transmission errors, then a new packet generated by this call at time t 0 and having a deadline of t 1 will be scheduled during [(t 0 + t 1 )/2, t 1 ] rather than during [t 0, t 1 ], in order to allow the channel to recover. However, this scheme is only mentioned as a possible strategy and its performance is not discussed. In [49], a model for wireless fair scheduling based on the adaptation of fluid fair queuing is proposed. This model is different from the one proposed in this chapter. While we assume that the base station knows exactly when each host has a new packet awaiting transmission, this assumption is not made in [49]. Moreover, [49] does not take into account issues related to SC3-SC5. In [57] it is shown that when EDF (Earliest Deadline First) is implemented over channels that are in good condition, the number of packets lost due to deadline expiration is minimized. This result has a bearing on congested channels, for which it is important to determine which of the packets should be transmitted and which should be dropped. This is in contrast to the problem considered in this chapter where we determine the optimal time for transmitting each packet under not necessarily congested conditions. In [9], a scheduling algorithm that uses an N-state Markov model, where N > 2, to characterize the channel is presented. This algorithm supports AMC in order to adjust the modulation and FEC to the forecasted channel state. A common way to address SC2 and SC5 is by assigning higher data rates to hosts with a better channel, in order to maximize throughput while ensuring acceptable bit-error rate 14

29 (BER) [34, 52]. In the uplink channel of an OFDM network, multiple hosts can transmit simultaneously over different sub-carriers. Since the channel characteristics for different users may be independent, dynamic assignment of sub-carriers to hosts can significantly improve the throughput [63, 62]. However, this water filling approach for maximizing instantaneous throughput does not take into account the QoS requirements of the calls originating these packets, and it is therefore unsuitable for synchronous traffic. The authors of [44] address this problem, in the context of OFDMA, by asking how, given a set of hosts and a set of sub-carriers, the sub-carriers should be allocated to the hosts in a way that satisfies the rate requirements of each host while using minimum power. In [60] utility-based cross-layer optimization problems are defined using the channel model, utility functions, adaptive modulation and frequency power allocation. 2.3 The Quantitative-Based Framework The scheme proposed in this chapter is based upon quantitative rather than qualitative considerations. For each scheduling interval, namely, an interval of time for which scheduling decisions are made by the base station for all the packets available for transmission, the scheduler determines the expected profit for scheduling each packet in every time slot. Then, the schedule with maximum expected profit is chosen. The most difficult part of this scheme is finding a good profit function that takes into account the various considerations. To simplify the presentation of our quantitative framework, we present it incrementally. We start with the basic model, referred to as Model 2.1. In this model the channel is assumed to be clean, and only the QoS requirements of each call (SC1) and the profit of each packet in the context of its specific call (SC3) are taken into consideration. We then consider Model 2.2, where the channel conditions for each host (SC2) are also taken into account. Model 2.3 extends the framework to accommodate multiple PHY profiles (SC5), and Model 2.4 addresses MAC layer retransmissions (SC4) as well. Model 2.1 Assume that the head-end executes the scheduling algorithm every scheduling interval of T uplink slots. T is usually equal to the length of the uplink frame (a few milliseconds). However, its value can be dynamically adjusted by the base station. For example, in order to reduce the load imposed on the base station, this value can be extended to several frame times. The penalty in such a case is slower response to the first packet of a new flow or to the first packet of a new talkspurt when silence suppression is used. The base station maintains a profit matrix φ. Entry φ[c, t] in this matrix indicates the profit if the first pending packet of call c is transmitted starting from slot t and is correctly received by the base station. If the transmission starting at slot t cannot be completed before the deadline of the packet, the profit is 0; otherwise, the profit is 1. In this case, by finding a schedule that maximizes the profit, we maximize the number of packets that meet their QoS requirement, thereby addressing SC1. However, in order to take into consideration not only SC1 but also SC3, φ can be viewed as a non-binary profit function that takes into account the priority of 15

30 the packet, as determined by upper layer information. This priority can be modified by the scheduler dynamically. For instance, it can be increased if a previous packet of the same call was not received on time. It will be convenient to assume in the meantime that each synchronous call has only one pending packet during each scheduling interval. A pending packet is a packet that (1) was released, (2) has not yet been successfully transmitted, and (3) whose due date has not yet expired. This assumption holds only when the tolerated jitter for a call is shorter than the packetization time (i.e., packet inter-arrival time). Model 2.2 We assume that at slot t there is a probability λ(c, t) 0 that the transmission from the host of call c over the shared channel will not be lost due to a transmission error. The optimization criterion is to maximize the sum of the profits of packets that are transmitted on time and experience no transmission error. Suppose that the base station is able to compute the value of λ(c, t) for every time slot t and for every synchronous call c. The profit matrix φ used for the previous model is replaced by a new expected profit matrix µ, where µ[c, t] = φ[c, t] λ(c, t). We later show how λ(c, t) can be computed for fixed and for mobile hosts. A schedule σ is a transmission vector that indicates which packet should start being transmitted in which slot. If σ(t) = c, then at time slot t the transmission of the current packet of call c should start. The overall profit gained from a schedule σ is Profit(σ) = T t=1 µ[σ(t), t], where [1 T ] is the scheduling interval. We seek a schedule σ for which Profit(σ) is maximum. Algorithms for finding the best schedule for a given profit matrix µ are discussed in Section 2.5. There is one problem with employing the profit-based framework in Model 2.2. Suppose that the channel of call c is in bad condition when a packet is released. If the shared channel is not heavily loaded, an algorithm that seeks to maximize the expected profit will choose to transmit the packet at a slot where the error probability is the smallest, even if this probability is still very high, just because there are no other waiting synchronous packets. However, a better decision is not to schedule this packet during the current scheduling interval, but to wait for one of the subsequent scheduling intervals, where the probability for an error might be smaller. The vacant slots can then be used for the transmission of some best-effort packets. There are several possible approaches for addressing this problem: 1. To change the optimization criterion from maximizing the aggregated expected profit to maximizing the average expected profit per slot. This criterion penalizes the scheduler for transmitting packets in bad slots. The drawback of this approach is that the scheduler will avoid transmitting a packet in a bad channel even if this packet is very close to its deadline. 2. To determine a minimum threshold for the probability of a successful transmission. When this probability is smaller than, the expected profit is set to 0, and 16

31 slot 1 slot 2 slot 3 slot 4 slot 5 slot 6 slot 7 slot 8 profile 1 call 1 profile 2 profile 3 slot 1 slot 2 slot 3 slot 4 slot 5 slot 6 slot 7 slot 8 profile 1 call 2 profile 2 profile 3 Figure 2.2: A 3D (2x3x8) expected profit matrix µ with 2 calls, 3 PHY profiles and 8 slots the scheduler will not select the packet for transmission. The value of can be dynamically adjusted according to the load of the non-synchronous traffic. 3. To run the algorithm for a period of T slots, longer than the scheduling interval T, while implementing only its short-term decisions. Namely, we find the best packet to be transmitted during each of the next T slots, but take into consideration only the decisions made for the first T slots, while ignoring those made for slots [T + 1 T ]. The next time the scheduler is invoked, after a time equivalent to T slots, the matrix will contain all the packets not transmitted before, including those scheduled to be transmitted during slots [T + 1 T ]. The fraction T /T is considered as the lookahead ratio. The main drawback of this approach is that it increases the running time of the scheduler by a factor of T /T (see Section 2.5). Model 2.3 The various PHY layer FEC options are paired with modulation schemes, like QPSK, 16-QAM and 64-QAM, to form a pre-defined set of PHY profiles with varying robustness and efficiency. To accommodate this model, the expected profit matrix µ is extended to 3 dimensions: calls, PHY profiles, and slots. Entry µ[c, t, m] in this matrix is set to φ[c, t] λ(c, t, m), where λ(c, t, m) is the probability that the packet of call c will be transmitted correctly starting from slot t using PHY profile m. Figure 2.2 shows the expected profit matrix µ with 2 calls and 3 PHY profiles. The packet of call-1 is available for transmission in slot 1. It must be transmitted not later than slot 6 in order to meet its tolerated jitter. This packet requires a bandwidth of 4 slots for the most robust PHY profile (profile-1), 2 slots for the less robust profile, and only one slot for the least robust profile. As the figure shows, there are 3 possible transmission instances for this packet using profile-1, 5 possible transmission instances using profile-2, and 6 possible transmission instances using profile-3. From these 17