Wireless systems, cross-layer design, packet scheduling, multimedia communication ..... discuss extensions to a system with more than two users in Section V-B.
Downlink Wireless Packet Scheduling with Deadlines TO APPEAR IN THE IEEE TRANSACTIONS ON MOBILE COMPUTING
Aditya Dua and Nicholas Bambos Department of Electrical Engineering Stanford University 350 Serra Mall, Packard Building Stanford CA 94305 {dua,bambos}@stanford.edu Abstract Next generation cellular wireless communication networks aim to provide a variety of quality-ofservice (QoS) sensitive packet based services to downlink users. Included amongst these are real-time multimedia services, which have stringent delay requirements. Downlink packet scheduling at the base station plays a key role in efficiently allocating system resources to meet the desired level of QoS for various users. In this paper, we employ dynamic programming (DP) to study the design of a downlink packet scheduler capable of supporting real-time multimedia applications. Under well justified modeling reductions, we extensively characterize structural properties of the optimal control associated with the DP problem. We leverage intuition gained from these properties to propose a heuristic scheduling policy, namely CA-EDD (Channel Aware Earliest Due Date), which is based on a “quasi-static” approach to scheduling. The per time-slot implementation complexity of CA-EDD is only O(K) for a system with K downlink users. Our simulation results demonstrate that CA-EDD comfortably outperforms benchmark schedulers, both in terms of overall system performance and fairness. CA-EDD achieves these performance gains by using channel and deadline information in conjunction with application layer information (relative importance of packets) in a systematic and unified way for scheduling. Index Terms Wireless systems, cross-layer design, packet scheduling, multimedia communication, quality of service (QoS), dynamic programming (DP).
I. I NTRODUCTION Next generation wireless communication systems aim to support quality-of-service (QoS) sensitive services like media streaming and jitter-sensitive high speed data for downlink subscribers [1]. However, bandwidth and power constraints, as well as the unpredictable nature of wireless channels, renders the design of such systems a challenging task. While advances in digital communication system design principles at the physical layer [2] have played a key role in the evolution of first and second generation wireless networks, crosslayer design methodologies have attracted much interest in recent years for the design of next generation systems [3]. The cross-layer design approach leverages the synergy existing between different layers of the communication protocol stack to achieve more efficient designs, instead of treating each layer as an individual entity. QoS and channel-aware packet scheduling is an important illustration of the cross-layer design approach, which exploits interactions between
the physical, network and application layers. Cross-layer design for packet scheduling plays an important role in enhancing the performance of high-speed downlink packet access (HSDPA), an architecture which has been introduced in the 3GPP Release 5 specifications to provide high data rates and QoS sensitive services to downlink users [4]. Starting primarily with the work in [5], packet scheduling for QoS provisioning and fairness in wired networks has been studied extensively. However, scheduling algorithms designed for wired networks are not directly applicable to the wireless scenario, because radio channels exhibit unique characteristics like bursty errors, and temporally and spatially varying capacity. Owing to their simplicity, channel-aware scheduling algorithms like Maximum SNR (currently best channel), Proportional-Fair (PF) [6], and numerous variants have been the subject of much attention. While these schedulers possess several desirable properties for supporting delay tolerant packet traffic, they are less suitable for delay sensitive real-time applications. The QoS requirements of the latter are very different from those of non-real-time applications. For instance, video-conferencing (a real-time variable-bit-rate application) has a delay bound of 40-90 ms and an acceptable loss rate of 10−3 . In contrast, non-real-time applications like web browsing and file transfer have a stringent acceptable loss rate requirement of 10−8 , but can withstand large delays [7]. This disparity between QoS requirements motivates the design of scheduling algorithms adapted to supporting real-time traffic over radio channels. Our goal in this paper is to develop low-complexity downlink packet scheduling algorithms, which are capable of supporting delay-sensitive multimedia applications, and provide a desirable trade-off between overall system performance and user-level fairness. We substantially extend our work on deadline constrained packet scheduling initiated in [8]. Our primary methodological tool here for studying the scheduling problem is dynamic programming (DP) [9]. While conceptually powerful, DP can often become practically intractable if the underlying statespace is not chosen judiciously. We characterize real-time traffic using differential deadlines between packets, or inter-packet deadlines (IPD), which helps us circumvent the state-space explosion problem. Moreover, another drawback of DP-based solutions is that they can often only be computed numerically. Such computations are prohibitive in wireless systems on-line for both time and/or power reasons. We mitigate this issue by developing approximations to the optimal scheduling policy which can be computed analytically without explicitly solving any DP equations. The proposed approximations result in heuristic scheduling policies which have low (linear) complexity in the number of users. A. Previous work The earliest-deadline-first (EDF) scheduling policy and its variants - also known in literature as the earliest-due-date (EDD) or shortest time-to-extinction (STE) - have been shown to be optimal for deadline constrained scheduling over wired (error-free) channels [10]. However, EDD is not well-suited to the wireless scenario, owing to its disregard for channel variations. Shakkottai et. al. [11] model the wireless channel as a two-state ON-OFF Markov chain and demonstrate that a feasible earliest due date (FEDD) policy is nearly optimal. Their work is the first attempt at studying a channel-aware version of EDD. Kong et. al. [12] extend the work in [11] and propose a proactive EDD policy which adaptively adjusts packet deadlines in anticipation of upcoming bad channel conditions. Khattab et. al. [13] study a heuristic channel-dependent EDD policy and demonstrate its efficacy via simulations. They also propose a fairness-aware version of the scheduler. On a different strand of research, Ren et. al. [14] employ dynamic programming to study scheduling of constant bit rate (CBR) traffic over wireless channels (modeled as Markov
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Fig. 1. (a) Schematic of the wireless downlink with two users and a single time-multiplexed scheduler at the base station, and (b) The system model under Reduction 1 and 2, with one packet per queue and static channel conditions.
chains). The simulation results presented in the paper point to the need for designing deadline and channel-aware scheduling policies for supporting real-time traffic. Johnsson et. al. [15] propose a heuristic cost function and demonstrate via simulations that a policy minimizing it performs well with respect to “normalized packet delays” and “missed packet deadline penalties”. Agarwal et. al. [16] adopt a different approach. They use a revenue based model to analyze the deadline constrained scheduling problem and show that a greedy online algorithm achieves a competitive ratio of 12 . However, they impose deadline requirements on a per-request (bursts of packets) basis, rather than a per-packet basis. With the exception of [16] and to an extent [11], [12], scheduler design for deadline-constrained wireless traffic has primarily been driven by simulations and heuristics, due to inherent system complexities. However, our strategy here is to focus on 1) analyzing parsimonious mathematical models that capture well the essence of the scheduling problem and then 2) leverage provable properties of the optimal policies to design practical scheduling algorithms. B. Paper outline We state the deadline-constrained scheduling problem for wireless traffic in Section II, and present key ingredients of our system model. In particular, we discuss the merits of using interpacket deadlines (IPD) as a modeling tool, and describe the finite-state Markov chain model (FSMC) for wireless channels. We consider two reductions of our system model in Section III. The motivation is to construct a system model which captures the fundamental scheduling tradeoffs, and yet, is amenable to analysis, and leads to implementation friendly scheduling policies. We then formulate the scheduling problem under the reduced model in a dynamic programming framework. Based on the optimal solution to the reduced problem, we propose our scheduling algorithm, namely CA-EDD (Channel-Aware Earliest Due Date). In Section IV, we establish numerous properties of the optimal solution to the DP problem formulated in Section III, for a system with two users. Key amongst these is the optimality of a switch-over policy. Based on these structural properties, we develop approximations in Section V, which enable a lowcomplexity implementation of CA-EDD. We also demonstrate that for a system with more than two users, a pairwise scheduling approach based on the optimal solution to the two-user problem, is optimal. We show the efficacy of CA-EDD relative to benchmark schedulers like EDD via simulations in Section VI. We provide concluding remarks in Section VII.
II. P ROBLEM S TATEMENT AND M ODEL C ONSTRUCTION A. The scheduling problem We consider a time-division multiplexed (TDM) wireless communication system, with one transmitter and multiple mobile downlink receivers/users (used interchangeably throughout the paper). A schematic diagram of a two-user system is depicted in Fig. 1(a). Time is divided into fixed size transmission time-slots. There is a queue corresponding to each downlink user at the base-station (BS), which contains the packets destined for that user. At the beginning of every time-slot, the BS schedules the head-of-line (HOL) packet of a non-empty queue for transmission according to some scheduling policy. Packets have associated deadlines, i.e., if they are not successfully transmitted prior to the expiration of their deadline, they are rendered worthless to the receiver and are discarded. Dropped packets result in QoS degradation at the receiver. For instance, in the case of video streaming, dropped packets lead to playout freezes and hence user annoyance. Instantaneous and error-free feedback indicating the success/failure of a transmission (ACK/NAK) is made available to the BS by each receiver via an uplink channel. The BS has the option to re-schedule failed transmissions subject to deadline constraints. Our goal is to design a scheduling policy which minimizes the aggregate packet drop rate (due to missed deadlines) and simultaneously respects certain fairness guarantees. In addition, we would like the scheduler to possess desirable properties such as low computational complexity and a parsimonious characterization. Given the multiplicity of problem dimensions and performance criteria, modeling and solving the scheduling problem in its entirety is a mammoth task. Toward this end, we explore a class of mathematical models that capture the essence of the scheduling problem, and yet, remain amenable to analysis. B. Inter-packet deadlines For multimedia applications, packets reside in a queue in non-decreasing order of their respective deadlines. It is important for a packet to satisfy “differential” deadline requirements relative to its predecessors, rather than an “absolute” deadline constraint. We refer to these differential deadlines as inter-packet deadlines (IPD). For example, suppose the HOL packet (call it P) of a queue is successfully transmitted 2 time-slots prior to the expiration of its deadline, and the receiver expects the next packet (call it P’) 4 time-slots after it has “consumed” P. These packets could be video frames being played out at the receiver at the rate of 0.25 frames/time-slot. Then, the IPD between P and P’ is 4 time-slots. If P’ is available at the BS at the time of departure of P, it acquires a deadline of 4+2=6 time-slots. This would be the case for applications like streaming of pre-cached multimedia. Alternatively, if the packets are being generated by a realtime periodic source (with a period of 4 time-slots), P’ arrives to the queue only 2 time-slots after the departure of P and acquires a deadline of 4 time-slots. This would be the case for applications like transmission of live multimedia or two-way media communication. The IPDs for a user’s traffic stream are a measure of the “stress” the user imposes on the system. The tighter or smaller the IPDs, the more the resources required by the user, and viceversa. IPDs can be modulated by adaptive buffering and playout strategies employed at the receiver, or variable rate encoding of multimedia at the transmitter. For instance, if a receiver is experiencing poor channel conditions, it could slow down the playout rate from 0.25 frames/timeslot to 0.2 frames/time-slot without perceptible degradation in video quality, and in the process stretch out the IPDs from 4 to 5 time-slots. Alternatively, the transmitter could choose to encode the video stream with lower resolution and thereby reduce the rate requirements of the user. Such considerations make IPDs an attractive modeling tool.
C. Wireless channel model Wireless channels exhibit temporal and spatial fluctuations which are attributed to user mobility, interference from neighboring cells, and signal attenuation due to physical phenomena. Different models have been used in the literature to abstract the behavior of wireless channels. While some authors model the wireless channel as a reliable “bit-pipe” with time-varying capacity, others model it as a fixed-size “bit-pipe” with time-varying reliability. We adopt the latter approach and quantify the wireless channel quality in a time-slot by the probability of successful transmission of a packet over the channel, if the channel is used in that time-slot. We model variations in wireless channel reliability as a time homogeneous finite-state Markov chain (FSMC). This model has been widely employed in the wireless literature (see [17] and references therein). The FSMC for the downlink channel corresponding to the ith user is characterized by the state-space Si = {1, . . . , Mi } and a corresponding Mi × Mi state-transition matrix Πi . State transitions are assumed to occur at the beginning of each time-slot. With the FSMC for the ith user, we associate a mapping si : Si 7→ (0, 1], such that si (ji ) denotes the probability of successful packet transmission if the HOL packet of the ith user is scheduled when his/her channel state is ji ∈ Si . We assume that the FSMCs for all users evolve independently of each other and also independently of the scheduling decisions made at the BS. D. Time-to-expiry and packet dropping costs With an IPD based characterization, it suffices to keep track of only the time-to-expiry (TTE) of the deadline associated with the HOL packet of each user as part of the system description. To this end, we associate with the ith user a counter ni ∈ {0, 1, 2 . . .}, which keeps track of the TTE of the deadline of his/her HOL packet. If the ith user is not scheduled in the current time-slot, or is scheduled and the transmission fails, his/her TTE counter decrements by 1. If the ith user is scheduled and the transmission is successful, his/her TTE counter is reset, depending on availability of another packet in the queue and the IPD between the just departed packet and the new HOL packet (if available). Note that a single time-slot is sufficient to deliver a packet. An unsuccessfully transmitted packet can be re-scheduled in a later time-slot subject to its deadline constraint. However, if a packet misses its deadline (TTE decrements to 0 before successful transmission), it is rendered worthless to its intended receiver and is dropped from the queue. The scheduler incurs a cost or penalty in this case, which reflects the “importance” of the packet to the receiver. We denote this cost by λi > 0 for the HOL packet of the ith user. These costs can vary from packet to packet, depending upon the information content of the packet. For example, in the case of video, an I-frame is more important than a P-frame and hence will be associated with a higher cost. Packet dropping costs also equip the scheduler with a means to enforce “fairness” between different users. For example, consider a scenario where the packets of two users are of identical importance and have the same deadlines, but the users are experiencing very disparate channel conditions (one could be at the edge of the cell and the other very close to the BS). In this case, the packet dropping cost associated with the weaker user could be increased “artificially” by the scheduler to offset the imbalance caused by channel disparity. We will illustrate this feature in our simulation results in Section VI. E. Packet holding costs The scheduler can be equipped with an additional control knob by associating a packet holding cost with the HOL packet of each user. In particular, let ci > 0 denote the holding cost incurred
by the HOL packet of Qi for every time-slot it spends in the queue. With a non-zero holding cost, there is an incentive to not merely transmit a packet prior to its deadline expiration, but to transmit it as soon as possible. One application scenario where such a notion is useful is streaming of progressively encoded video, where enhancement layer data can be transmitted in addition to base layer data to refine video quality, if the latter has been transmitted before deadline expiration. We will initially study the deadline constrained scheduling problem with zero holding costs, and incorporate them into our framework at a later stage (in Section III-C). III. M ODEL R EDUCTION AND DYNAMIC P ROGRAMMING F ORMULATION In general, the scheduling problem can be formulated as a sequential decision making problem and solved optimally using numerical techniques, given statistical characterization or actual realizations of the channel conditions and IPDs for all users. Such an approach is, however, computationally prohibitive from an implementation perspective and is also unlikely to shed much light on the fundamental trade-offs inherent in the scheduling problem. Moreover, detailed knowledge of traffic and channel conditions is bound to be unavailable in advance to the scheduler in real wireless systems. We would like to formulate the scheduling problem as an optimal control problem which (a) encapsulates the fundamental scheduling trade-offs, (b) is amenable to analysis, and (c) leads to implementation friendly scheduling policies. To this end, we will consider the following two reductions of the scheduling problem (see Fig. 1(b)). Reduction 1: To formulate our optimal control problem, we will assume that each queue contains only one packet. Reduction 2: We will assume static (in a probabilistic sense) channel conditions in the optimal control problem formulated on the basis of Reduction 1. The ultimate test to determine whether the modeling reductions are justified is the “test against nature”, i.e., whether a scheduling policy based on the properties of the solution to the reduced problem perform better than benchmark policies or not, in a real-world scenario. Experimental evaluation of our proposed policy in Section VI shows that this is indeed the case. A. Problem formulation under reduction 1 — problem P1 We first formulate our scheduling problem as an optimal control problem under Reduction 1 only. We will call this problem P1 . In accordance with Reduction 1, we will assume that each queue at the BS contains only one packet (its HOL packet). This assumption is reasonable if packets are being generated periodically by a real-time media source, so that a new packet arrives to a queue only after the current packet has been successfully transmitted, or dropped. For ease of exposition, we formulate problem P1 for a canonical two-user system. The description of P1 for a two-user system is as follows: There are two queues (denoted Q1 and Q2 ) at the BS corresponding to the two downlink users, with one packet each, and TTE of their deadlines n1 and n2 . The packets are associated with dropping costs λ1 and λ2 (Section II-D). The downlink channel for each user is characterized by an FSMC (Section II-C), with channel states denoted by j1 and j2 . We denote the state of the system by the four-tuple (n1 , n2 , j1 , j2 ). Our goal is to find the optimal scheduling policy P1? which minimizes the total expected cost of transmitting both packets, starting in any state. We only admit non-idling, non-anticipative, and stationary scheduling policies and denote the set of such policies by P. We adopt the methodology of dynamic programming (DP) to compute the optimal policy ? P1 . Before doing so, let us examine the system dynamics under a candidate scheduling policy P ∈ P. The states with n1 = 0 or n2 = 0 correspond to Q1 and Q2 respectively being empty, so
no scheduling decision needs to be made in these states. If P schedules Q1 in state (n1 , n2 , j1 , j2 ) when n1 , n2 > 0, one of the following state transitions will occur: • n1 , n2 > 1: The transmission is successful with probability (w.p.) s1 (j1 ) and the new state is (0, n2 − 1, j10 , j20 ). The transmission fails w.p. (1 − s1 (j1 )) and the new state is (n1 − 1, n2 − 1, j10 , j20 ). 0 0 • n1 = 1 and n2 > 1: The new state is (0, n2 − 1, j1 , j2 ). However, the transmitted packet gets dropped w.p. (1 − s1 (j1 )) and a dropping cost of λ1 is incurred. • n1 > 1 and n2 = 1: The transmission is successful w.p. s1 (j1 ) and the new state is (0, 0, j10 , j20 ). The transmission fails w.p. (1 − s1 (j1 )) and the new state is (n1 − 1, 0, j10 , j20 ). The HOL packet of Q2 is dropped in both cases and a dropping cost of λ2 is incurred. The system evolution can be similarly described if P schedules Q2 in state (n1 , n2 , j1 , j2 ). For any initial state (n1 , n2 , j1 , j2 ), the system reaches the state (0, 0, j10 , j20 ) for some j10 ∈ S1 , j20 ∈ S2 within max(n1 , n2 ) time-slots (since the deadlines on both HOL packets expire), whereupon no further cost is incurred. Since the system dynamics are stationary (depend on the state only) and the total expected cost is upper-bounded by λ1 + λ2 , P1 is a stochastic shortestpath problem for which an optimal stationary policy P1? exists. The associated value function V (n1 , n2 , j1 , j2 ) satisfies the following Bellman’s equations ∀ n1 , n2 ∈ N∗ and ∀ j1 ∈ S1 , j2 ∈ S2 : V (n1 , n2 , j1 , j2 ) = min{ s1 (j1 )Λ(0, n2 − 1, j1 , j2 ) + [1 − s1 (j1 )]Λ(n1 − 1, n2 − 1, j1 , j2 ), {z } | {z } | Q1 scheduled, transmission successful
Q1 scheduled, transmission failed
s2 (j2 )Λ(n1 − 1, 0, j1, j2 ) + [1 − s2 (j2 )]Λ(n1 − 1, n2 − 1, j1 , j2 )}, | {z } | {z }
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n1 , n2 > 1, (1)
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V (1, n2 , j1 , j2 ) = min{Λ(0, n2 − 1, j1 , j2 ) + [1 − s1 (j1 )]λ1 , [1 − s2 (j2 )]Λ(0, n2 − 1, j1 , j2 ) + λ1 } V (n1 , 1, j1 , j2 ) = min{[1 − s1 (j1 )]Λ(n1 − 1, 0, j1 , j2 ) + λ2 , Λ(n1 − 1, 0, j1, j2 ) + [1 − s2 (j2 )]λ2 } V (1, 1, j1, j2 ) = min{[1 − s1 (j1 )]λ1 + λ2 , λ1 + [1 − s2 (j2 )]λ2 }, (2) where Λ(n1 , n2 , j1 , j2 ) is V (n1 , n2 , j1 , j2 ) averaged over all possible channel state transitions, X X Π1 (j1 , j10 )Π2 (j2 , j20 )V (n1 , n2 , j10 , j20 ), (3) Λ(n1 , n2 , j1 , j2 ) , j10 ∈S1 j20 ∈S2
and Si and Πi respectively denote the state-space and state-transition matrix associated with the downlink channel of the ith user. All terms in (2) have an interpretation similar to corresponding terms in (1). We can compactly represent the DP equations as V (n1 , n2 , j1 , j2 ) = min{s1 (j1 )α(n1 , n2 , j1 , j2 ), s2 (j2 )β(n1 , n2 , j1 , j2 )}+ Λ(n1 − 1, n2 − 1, j1 , j2 ) + λ1 I{n1 =1} + λ2 I{n2 =1} , (4) where we define α(n1 , n2 , j1 , j2 ) , Λ(0, n2 − 1, j1 , j2 ) − Λ(n1 − 1, n2 − 1, j1 , j2 ) − λ1 I{n1 =1} β(n1 , n2 , j1 , j2 ) , Λ(n1 − 1, 0, j1 , j2 ) − Λ(n1 − 1, n2 − 1, j1 , j2 ) − λ2 I{n2 =1} , ∗
Throughout the paper, N = {1, 2, . . .}.
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We also have the following boundary conditions ∀ n1 , n2 ∈ N and ∀ j1 ∈ S1 , j2 ∈ S2 : V (n1 , 0, j1 , j2 ) = [1 − s1 (j1 )]Λ(n1 − 1, 0, j1 , j2 ) + λ1 I{n1 =1} V (0, n2 , j1 , j2 ) = [1 − s2 (j2 )]Λ(0, n2 − 1, j1 , j2 ) + λ2 I{n2 =1} . Finally, we set V (0, 0, j1 , j2 ) = 0 ∀ j1 ∈ S1 , j2 ∈ S2 . It follows from (4) that the behavior of P1? is completely governed by the sign of the decision function γ(n1 , n2 , j1 , j2 ), defined as γ(n1 , n2 , j1 , j2 ) , s1 (j1 )α(n1 , n2 , j1 , j2 ) − s2 (j2 )β(n1 , n2 , j1 , j2 ).
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P1? schedules Q1 in state (n1 , n2 , j1 , j2 ) if γ(n1 , n2 , j1 , j2 ) ≤ 0, and Q2 else. The DP formulation for the two-user case extends in straightforward fashion to a system with K > 2 users, with a2K-dimensional state-space and K possible actions in each state. B. A quasi-stationary scheduler based on P1? — QS Now that we have the means to compute the optimal policy P1? for the scheduling problem under Reduction 1, the following question arises naturally: How does P1? translate into a scheduling policy which can be implemented in a real wireless system? If an FSMC characterization of all downlink channels is available to the BS, it can solve the DP equations which describe P1? offline, and store the optimal action associated with every possible state in a look-up table (LuT)† . Then, in the real system, the BS simply needs to look at the TTE of the HOL packet of each user (ni ) and the associated channel state (ji ), and extract the optimal scheduling decision from the LuT. Recall that P1 was formulated under the assumption of stationary channel statistics. Thus, the LuT is regenerated afresh every time the underlying channel statistics change. We call this policy QS. The scheduling decisions of QS are based on the assumption that channel statistics are stationary; however, the stationary operating point is updated continuously as channel statistics vary over time. Thus, QS exemplifies a quasi-stationary approach to scheduling. It is evident that implementing QS in a real wireless system is fraught with difficulties. The BS will seldom possess detailed knowledge of channel statistics in a real wireless system. The BS can potentially estimate the FSMC parameters based on receiver feedback. However, channel statistics are non-stationary due to user mobility, implying the need for frequent re-computation of the LuT. Moreover, every time the packet dropping cost associated with a user changes, the LuT would have to be re-computed. This will occur frequently in multimedia transmission, where high priority packets are inter-laced with low priority packets (see Fig. 8 in Section VI for an illustration). Such considerations render QS impractical from an implementation perspective. f2 C. Problem formulation under reduction 1 and 2 — problems P2 and P To overcome the difficulties discussed in Section III-B, we now incorporate Reduction 2 in addition to Reduction 1 into our problem formulation and construct problem P2 . Recall that incorporating Reduction 2 is tantamount to assuming static channel conditions in the formulation of P1 . i.e., P2 is gotten from P1 by assuming that the FSMCs used to model the downlink channels have only one state each. In particular, we set Si = {1} and Πi = [1] for all users. Equivalently, †
For a K-user system, with the deadline on all HOL packets upper-bounded by D, and an M -state FSMC channel model for every user, the size of the LuT is (DM )K .
the success probability associated with every user’s channel is fixed (static) over the duration of his/her HOL packet’s stay at the BS. We study a two-user system for ease of exposition and discuss extensions to a system with more than two users in Section V-B. The description of P2 is very similar to that of P1 . Queues Q1 and Q2 at the BS have one packet each, with TTE on their respective deadlines n1 and n2 , and dropping costs λ1 and λ2 . The channel for each user is now described by a single parameter — the static success probability associated with that channel. We will denote these success probabilities for the two users by s1 and s2 , respectively. Our goal is to find a scheduling policy which minimizes the total expected cost of transmitting both packets, starting in any state. Since the channel state is now degenerate for both users, the system state reduces to the two-tuple (n1 , n2 ). We denote the optimal policy in this case by P2? , which is indeed a special case of P1? computed under Reduction 2. Before we delve deeper into the computation of P2? , we incorporate packet holding costs into our model. As pointed out in Section II-E, packet holding costs provide the scheduler an extra f2 , and dimension of control. We will call this modified version of P2 with packet holding costs P ? f2 . Note that all other modeling assumptions stay unchanged. the corresponding optimal policy P Unfortunately, problem P1 does not lend itself well to analysis under the assumption of non-zero holding costs. However, we can establish interesting structural properties of the optimal policy f2 (which is P2 with non-zero holding costs), as we will show in Section IV. for problem P f2? via dynamic programming. The Bellman’s equations Similar to P1? , we can compute P2? and P ? f can be compactly written as associated with P 2 V (n1 , n2 ) = min{s1 α(n1 , n2 ), s2 β(n1 , n2 )} + V (n1 − 1, n2 − 1)+ c1 + c2 + λ1 I{n1 =1} + λ2 I{n2 =1} ,
∀ n1 , n2 ∈ N, (8)
where α(n1 , n2 ) = V (0, n2 − 1) − V (n1 − 1, n2 − 1) − λ1 In1 =1 and β(n1 , n2 ) = V (n1 − 1, 0) − V (n1 −1, n2 −1)−λ2 In2 =1 are obtained by setting j1 = j2 = 1 in (5). We also have the boundary conditions: V (n1 , 0) = (1 − s1 )V (n1 − 1, 0) + c1 + λ1 I{n1 =1} and V (0, n2 ) = (1 − s2 )V (0, n2 − 1) + c2 + λ2 I{n2 =1} . The decision function γ(n1 , n2 ) is obtained by setting j1 = j2 = 1 in (7) γ(n1 , n2 ) = s1 α(n1 , n2 ) − s2 β(n1 , n2 ).
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f? schedule Q1 Corresponding equations for P2? are obtained by setting c1 = c2 = 0. P2? and P 2 in state (n1 , n2 ) if γ(n1 , n2 ) ≤ 0, and Q2 else. f? — CA-EDD D. A quasi-static scheduler based on P2? and P 2
f? translate into implementable Once again, a natural question to ask is: How do P2? and P 2 scheduling policies? In line with our quasi-stationary approach in Section III-B, we now propose a quasi-static approach to scheduling. We call our proposed policy CA-EDD (Channel Aware Earliest Due Date). We will provide a mathematical interpretation of CA-EDD as a channel aware version of the basic EDD policy in Section V. The key steps involved in CA-EDD are: 1) Given the channel conditions, HOL packet deadlines, and packet dropping and holding f? ) for the costs for all users in the current time-slot, compute the optimal policy P2? (or P 2 scheduling problem formulated under Reduction 1 and 2 by solving the DP equations. f? ), computed in Step 1 above. 2) Schedule a packet in the real system based on P2? (or P 2 3) Update the system state and other parameters based on the outcome of Step 2. 4) Repeat steps 1-3 in every time-slot.
Policy P1? P2? f? P 2 QS CA-EDD
Description Optimal solution to P1 , formulated under Reduction 1 Optimal solution to P2 , formulated under Reduction 1 and 2 f2 , formulated under Reduction 1 and 2, with holding costs Optimal solution to P Quasi-stationary policy, based on P1? f2? Quasi-static policy, based on P2? or P TABLE I
A BRIEF DESCRIPTION OF DIFFERENT SCHEDULING POLICIES DEFINED IN THE PAPER .
Thus, the scheduling decision of CA-EDD in every time-slot is based on the assumption of static channel conditions. However, the static operating point is updated in every time-slot as the wireless channels evolve over time. This justifies the nomenclature quasi-static. f? ) are distinct from CA-EDD. While the former are optimal policies for Note that P2? (and P 2 the scheduling problem formulated under modeling reductions 1 and 2 (viz. only HOL packet in queue and static channel conditions), the latter is a heuristic quasi-static scheduling policy f2? ) to compute its which does not impose any assumptions on the system but uses P2? (or P ? scheduling decision in each time-slot. Similarly, P1 is the optimal policy for the scheduling problem formulated under modeling reduction 1 (viz. only HOL packet in queue) and QS is the corresponding quasi-stationary heuristic policy which imposes no assumptions on the system but uses P1? in each time-slot to compute its scheduling decision. For clarity, we provide a summary of different scheduling policies introduced in the paper in Table I. In contrast to QS, which requires a detailed statistical channel characterization, CA-EDD only requires instantaneous channel knowledge. Moreover, based on the approximations we will develop in Section V, CA-EDD can be implemented without actually solving the DP equations to f? ) in every time-slot. We conclude this section by re-iterating that the performance obtain P2? (or P 2 gains provided by CA-EDD vis-`a-vis benchmark policies (Section VI) provide ample justification for our modeling reductions. IV. S TRUCTURAL P ROPERTIES
OF
P1? , P2?
AND
f2? P
f? for the twoIn this section, we present structural properties of the policies P1? , P2? and P 2 user case. These properties provide key insights into the fundamental trade-offs inherent in the scheduling problem and also help us develop a low-complexity implementation of CA-EDD. A. Properties of P1? The value function associated with the DP formulation of P1 satisfies the following: Lemma 1 (Monotonicity of V ): V (n1 , 0, j1 , j2 ) and V (0, n2 , j1 , j2 ) are non-increasing functions of n1 and n2 respectively, ∀ j1 ∈ S1 , j2 ∈ S2 . Proof: See Appendix VIII-A. Lemma 2 (Monotonicity of V ): V (n1 , n2 , j1 , j2 ) is a non-increasing function of n1 for fixed n2 > 0 and a non-increasing function of n2 for fixed n1 > 0, ∀ j1 ∈ S1 , j2 ∈ S2 . Proof: See Appendix VIII-B The following corollary is a direct consequence of Lemma 2.
Corollary 1 (Monotonicity of Λ): Λ(n1 , n2 , j1 , j2 ) is a non-increasing function of n1 for fixed n2 and a non-increasing function of n2 for fixed n1 , ∀j1 ∈ S1 , j2 ∈ S2 . Equipped with Lemma 2 and Corollary 1, we are ready to establish the most important structural property of P1? . First, we need a definition. Definition 1 (Switch-over Policy): We say that a scheduling policy P ∈ P is a switch-over policy if ∀j1 ∈ S1 , j2 ∈ S2 there exist non-decreasing switch-over curves φj1 ,j2 : N → N ∪ {0}, such that P schedules Q1 in state (n1 , n2 , j1 , j2 ) if n2 ≥ φj1 ,j2 (n1 ), and Q2 else. For example, EDD is a switch-over policy with φj1 ,j2 (n1 ) = φ(n1 ) = n1 ∀ j1 ∈ S1 , j2 ∈ S2 . Theorem 1 (Optimality of Switch-over Policy for P1 ): The optimal scheduling policy for P1 , namely P1? , is a switch-over policy. Proof: See Appendix VIII-C. Thus, solving Bellman’s equations given by (4) to compute P1? is equivalent to computing the optimal switch-over curves φj1 ,j2 , for every (j1 , j2 ) pair. Unlike EDD, however, the switch-over curves for P1? depend on the channel states. This corroborates our belief that a “good” scheduler for real-time wireless traffic ought to utilize both deadline and channel-state information. f? B. Properties of P2? and P 2 The decision function γ(n1 , n2 ) defined in (9) satisfies the following important property: Lemma 3: γ(n1 , n2 ) is a non-decreasing function of n1 for fixed n2 and a non-increasing function of n2 for fixed n1 . Proof: See Appendix VIII-D. f? is a switch-over policy (under qualifying Based on Lemma 3, we can show that the policy P 2 conditions). As a special case (zero holding costs), the conclusion applies to policy P2? as well. Note that the switch-over behavior of P2? also follows as a special case of Theorem 1, since P2 is a special case of P1 (under Reduction 2). However, we cannot deduce such a relation between f? and P ? , because P1 does not incorporate packet holding costs, while P f2 does. More formally, P 2 1 f2 ): If c1 ≤ s1 λ1 and c2 ≤ s2 λ2 , the optiTheorem 2 (Optimality of Switch-over Policy for P f2 , namely P f? , is a switch-over policy. mal scheduling policy for P 2 Proof: See Appendix VIII-E. Corollary 2 (Optimality of Switch-over Policy for P2 ): The optimal scheduling policy for P2 , namely P2? , is a switch-over policy. The conditions of Theorem 2 are trivially satisfied for P2 , because c1 = c2 = 0. f2? C. Asymptotic behavior of P2? and P
f2? . Asymptotic analysis We will now investigate the “asymptotic” behavior of policies P2? and P refers to the regime n1 , n2 → ∞, which is representative of non-real-time traffic, i.e., packets without deadline constraints. The analysis in this section helps us develop an approximation to f? . It also helps us establish an equivalence with a well known the optimal policies P2? and P 2 result for scheduling of non-real-time traffic (the cµ rule [18]). c1 c1 n1 λ1 − , V (0, n2 ) = + (1 − s1 ) It is easy to show that ∀ n1 , n2 ∈ N V (n1 , 0) = s1 s1 c2 c1 c2 c2 +(1−s2 )n2 λ2 − . Consequently, lim V (n1 , 0) = , lim V (0, n2 ) = . Our next n1 →∞ n2 →∞ s2 s2 s1 s2 result states that both successive limits of γ(n1 , n2 ) exist, and are equal.
Lemma 4 (Limits of γ): lim
lim γ(n1 , n2 ) = lim
n1 →∞ n2 →∞
lim γ(n1 , n2 ) , γ ? .
n2 →∞ n1 →∞
Proof: See Appendix VIII-F. Now, from the definition of γ(n1 , n2 ) and the limits of V computed above, s 1 c2 s2 c1 lim lim γ(n1 , n2 ) = − + (s2 − s1 ) lim lim V (n1 − 1, n2 − 1) n1 →∞ n2 →∞ n1 →∞ n2 →∞ s2 s1 s 1 c2 s2 c1 lim lim γ(n1 , n2 ) = − + (s2 − s1 ) lim lim V (n1 − 1, n2 − 1). n2 →∞ n1 →∞ n2 →∞ n1 →∞ s2 s1
(10)
From (10) and Lemma 4, we conclude that the successive limits of V (n1 , n2 ) exist and are equal, i.e., lim lim V (n1 , n2 ) = lim lim V (n1 , n2 ) , V ? . As a direct consequence, α(n1 , n2 ) and n1 →∞ n2 →∞
n2 →∞ n1 →∞
β(n1 , n2 ) also have unique successive limits, denoted by α? and β ? , respectively. In particular, c2 c1 the limits are related by α? = − V ?, β ? = − V ? . From the definition of V (n1 , n2 ), we s2 s1 note that V ? must satisfy‡ c1 c2 ? ? ? ? ? ? + V ? + c1 + c2 . (11) V = min(α , β ) + V + c1 + c2 = min s1 − s1 V , s2 − s2 V s2 s1 We have two cases: c2 c1 f2? schedules Q1 as n1 , n2 → ∞. We use (11) 1) s1 − s1 V ? ≤ s2 − s2 V ? : In this case, P s2 s1 c1 c1 + c2 + . Upon substitution, we obtain the condition s1 c1 ≥ s2 c2 . to get V ? = s1 s2 c1 c2 f2? schedules Q2 as n1 , n2 → ∞. We use (11) 2) s1 − s1 V ? > s2 − s2 V ? : In this case, P s2 s1 c1 + c2 c2 to get V ? = + . Upon substitution, we obtain the condition s2 c2 > s1 c1 . s1 s2 f? schedules Q1 if s1 c1 ≥ s2 c2 , and Q2 else. Also, V ? = To summarize, as n1 , n2 → ∞, P 2 c1 c2 c1 c2 + + min , . From (10), we get s1 s2 s2 s1 s2 c2 − c1 ; s 1 c1 ≥ s 2 c2 s1 γ? = (12) c 2 − c 1 s 1 ; s 1 c 1 < s2 c 2 . s2
The optimal policy in the asymptotic regime is of index-type. The indices agree with those obtained from the cµ scheduling rule [18], which corresponds to scheduling of packets (without deadlines) with geometrically distributed service times and fixed packet holding cost rates. Inf2 , the service times have a truncated geometric distribution§. deed, in our formulation of P2 and P f2? . We conclude this section with another interesting structural property of P Theorem 3 (Saturation of Switch-over Curve): If c1 , c2 > 0, 1) If s1 c1 ≥ s2 c2 ∃ n?1 , n?2 ∈ N such that φ(n1 ) = n?2 ∀ n1 ≥ n?1 . 2) If s1 c1 < s2 c2 ∃ n?1 ∈ N such that φ(n1 ) = ∞, ∀ n1 > n?1 , f2? . where φ is the optimal switch-over curve for P Proof: See Appendix VIII-G. ‡ §
Note that interchanging min with lim
lim is valid because V ? , and hence α? and β ? are finite.
n1 →∞ n2 →∞
The service times are of type min(G, n), where G is a geometric random variable and n is the TTE of the packet deadline.
f? , and not P ? . Note that the hypothesis of Theorem 3 applies only to P 2 2
f? , AND PAIRWISE S CHEDULING P 2 ? ? f2 A. Piecewise linear approximations for P2 and P Despite the simplifications introduced so far, CA-EDD still needs to compute the optimal switch-over curve in each time-slot by numerically solving the DP equations associated with P2 f2 . This is not an attractive proposition from an implementation perspective. Thus, it is natural or P to seek approximations to switch-over curves which eliminate the need to repeatedly solve the DP equations. In this section, we develop piecewise linear approximations to the switch-curves f? , hinging on the analysis of Section V and the following result: for P2? and P 2 Lemma 5 (Straight Line Lemma): The decision function γ(n1 , n2 ) associated with P2? satisfies sgn{γ(n1 , n2 )} = sgn{γ(n1 − 1, n2 − 1)} ∀ n1 , n2 > 1. Proof: See Appendix VIII-H. V. A PPROXIMATIONS
FOR
P2?
AND
Switchover Curve
(a)
(b)
K K O
n2
O
K K O
O
(d)
(c) n1
f
Fig. 2. Possible shapes (approximate) of the switch-over curve for policy P2? - Clockwise: (a) s1 c1 ≥ s2 c2 , s1 λ1 < s2 λ2 , (b) s1 c1 ≥ s2 c2 , s1 λ1 ≥ s2 λ2 , (c) s1 c1 < s2 c2 , s1 λ1 ≥ s2 λ2 , and (d) s1 c1 < s2 c2 , s1 λ1 < s2 λ2 ; • and ◦ denote the states in which P2? schedules Q1 and Q2 , respectively. O and K respectively denote the location of the offset and the knee.
f
In Lemma 5, sgn(x) = 1 if x ≥ 0, and sgn(x) = −1 if x < 0. Since P2? is completely determined by the sign of γ(n1 , n2 ), Lemma 5 implies that the switch-over curve for P2? is f? , we exactly a straight line of slope π/4. While an equivalent result cannot be established for P 2 empirically observed that the switch-over curve is well approximated by a piecewise linear curve with two components, one with slope 0 or π/2 and the other with slope π/4 (see Fig. 2). The piecewise linear approximation is characterized by two points: the knee and the offset, which can be computed analytically in terms of the problem parameters, namely, si , ci , λi (i = 1, 2). For P2? , the straight line switch-over curve is characterized by a solitary point, the offset. 1) Offset computation: We have two distinct cases — s1 λ1 ≥ s2 λ2 and s1 λ < s2 λ2 . The offset for the two cases is determined by the zero-crossings of γ(n1 , 1) and γ(1, n2 ), respectively. It follows from Lemma 3 that the zero-crossings (if any) are unique. s2 λ2 −c1 log s1 λ1 −c1 . 1) s1 λ1 ≥ s2 λ2 : The offset is at the point (o?1 , 1), where o?1 = 1 + log(1 − s1 )
TTE of deadline on HOL packet of Q2 (n2)
7
6
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Knee (o1*+k2*,k2*)
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Optimal Approximate EDD 0
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Fig. 3. Optimal (from DP), approximate (piecewise linear), and EDD switch-over curves for a typical choice of parameters: λ1 = 8, λ2 = 1, c1 = 2, c2 = 1, s1 = 0.5 and s2 = 0.2.
log
s1 λ1 −c2 s2 λ2 −c2
2) s1 λ1 < s2 λ2 : The offset is at the point (1, o?2 ), where o?2 = 1 + log(1 − s2 ) . 2) Knee computation: We have two distinct cases — s1 c1 ≥ s2 c2 and s1 c1 < s2 c2 . The knee for the two cases is determined by the zero-crossings of lim γ(n1 , n2 ) and lim γ(n1 , n2 ), n1 →∞ n2 →∞ respectively. It follows from Lemma 3 that the zero-crossings (if any) are unique. 1) log 1 + s2s1(sc21λ−s2 −c 2 c2 . 1) s1 c1 ≥ s2 c2 : The knee is located on the line n2 = k2? , k2? = 1 + 1 log 1−s2 1 −c2 ) log 1 + ss12(sc12λ−s 1 c1 . 2) s1 c1 < s2 c2 : The knee is located on the line n1 = k1? , k1? = 1 + 1 log 1−s 1
The precise location of the knee is determined in conjunction with the offset. For instance, if the offset is at the point (o?1 , 1) and the knee is located on the line n2 = k2? , then the location of the knee is given by (o?1 + k2? − 1, k2? ). Summarizing, we have four mutually exhaustive cases, depending on whether the ratios s1 λ1 /s2 λ2 and s1 c1 /s2 c2 are greater or less than 1. For P2? , the switch-over curve does not obey Theorem 3 and the knee is always located at infinity. Thus, we have only two distinct cases, determined by the ratio s1 λ1 /s2 λ2 . The offset is computed as outlined above, with c1 and c2 set to zero. Recall that the switch-over curve for EDD is a straight line of slope π/4 passing through the origin. The switch-over curve for P2? is a straight line of slope π/4, but is offset from the origin, with the offset being a function of the channel conditions s1 , s2 and packet dropping costs λ1 , λ2 . Thus, the switch-over curve for P2? is a channel-aware version of the switch-over curve for EDD. This justifies the choice of the name CA-EDD for our proposed policy, because CA-EDD has P2? at its core. We illustrate the computations in the section with a numerical example. The approximate (piecewise linear), optimal (obtained by solving the DP equations), and EDD switch-over curves computed for a particular choice of parameters are depicted in Fig. 3. Observe that the piecewise linear curve provides a good approximation to the optimal switch-over curve.
B. Optimality of pairwise comparisons While our DP formulation for the two-user case extends in natural fashion to more than f? two users, we do not yet have a means of (approximately or exactly) computing P2? and P 2 for the latter case without solving the DP equations. Being able to do so is desirable from f? (in Step 1) to an implementation perspective because CA-EDD uses the solution of P2? or P 2 make a scheduling decision in every time-slot. To this end, we consider the following pairwise ? comparison based implementation (called P2,PW ) of P2? for a system with two or more users: 1) Users are arbitrarily grouped into pairs. 2) Users in each pair are compared using the optimal two-user policy, P2? , which can be computed without directly solving the DP equations (as outlined in Section V-A). 3) The winner in each pair is promoted to the next round, while the losers are eliminated. 4) Steps 1-3 are repeated until only one user survives; The HOL packet of this user is scheduled for transmission in the current time-slot. ? Steps 1-4 above are repeated in every time-slot. For a K-user system, P2,PW needs to perform ¶ exactly K − 1 comparisons to arrive at its scheduling decision . Each pairwise comparison has computational complexity O(1) (since only one point, the offset, needs to be computed). ? It follows that an implementation of CA-EDD based on P2,PW has a complexity of O(K) per time-slot (linear in K). In contrast, solving the DP equations directly to compute P2? has a complexity of O(nK ) (exponential in K) per time-slot if ni = O(n). ? Our next result says us that the pairwise comparison strategy is optimal, i.e., P2? = P2,PW . ? This implies that the optimal policy P2 for a K-user system can be obtained by solving K − 1 two-user problems and combining their results in pairwise fashion. Theorem 4 (Optimality of pairwise Comparisons): For the scheduling problem P2 for a K? ? user system, the pairwise comparison rule P2,PW is optimal, i.e., P2? = P2,PW . Proof: See Appendix VIII-I. Theorem 4 provides solid justification for a pairwise comparison based implementation of CA-EDD. All other steps in CA-EDD remain the same (as outlined in Section III-D). In Step ? 1, however, P2? is replaced by the pairwise strategy P2,PW . Theorem 4 tells us that the two approaches are equivalent; however, the latter has a much lower computational complexity. f2? . Finally, we remark that the pairwise comparison approach is also optimal for P VI. P ERFORMANCE E VALUATION In this section, we present a comparison between CA-EDD and benchmark scheduling policies based on experimental results. We considered three representative benchmarks: (a) Round-Robin (RR) — Schedules users in cyclic fashion, (b) Best Channel First (BCF) — Schedules the user with the highest successful transmission probability, and (c) Earliest Due Date (EDD) — Schedules the user with the smallest TTE on the deadline of his/her HOL packet. While other QoS aware schedulers have been proposed in the literature, their performance depends on the choice of associated weights/parameters, which will have be to tuned to guarantee a fair comparison with CA-EDD. We consider this beyond the scope of our work and thereby contrast CA-EDD to universal benchmarks only. We simulated the performance of a system with eight downlink users (except the results in Section VI-A, which are for a two-user system). All users were assumed to be mobile at a ¶
Let mk be the number of pairwise comparisons required for a k-user system. Then, mk = k/2 + mk/2 if k is even and mk = (k − 1)/2 + m(k+1)/2 if k is odd. It is easy to see that these equations are satisfied by mk = (k − 1) ∀ k ∈ N.
Packet drop rate for both users
QS (N=2) QS (N=8) CA−EDD
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Average SNR for both users (dB)
Fig. 4. Packet drop rate averaged over both users as a function of γ ave for a two user system with periodic arrivals and identical packets
speed of 3kmph. Each downlink channel was modeled as a single-path channel with a Rayleigh faded envelope and Doppler spectrum. The channel realizations were generated by filtering white Gaussian noise through a one pole IIR filter [19]. We denote the average downlink SNR for the ith user by γiave . The instantaneous channel SNR (γ) was mapped to packet success probability (s) via a monotonically increasing function of the form s = 1 − exp(−δx), where δ > 0 is a system dependent parameter which captures the effect of the modulation and coding scheme, receiver structure etc. We set δ = 0.6 for all our simulations. In a more elaborate link-level simulation, s(·) would be specified via actual value interface (AVI) tables [20]. We also tried other functional forms for s(·) (for example, sigmoidal) but did not observe any significant differences in the relative performance of different schedulers. A time-slot length of 2ms (HSDPA) was assumed. Ideal and instantaneous channel knowledge (in terms of successful transmission probabilities), and error-free and instantaneous ACK/NAK feedback were assumed to be available at the BS. Each point on all the performance curves depicted here was generated by simulating the system for 105 time-slots (200sec of real time). We considered two traffic models, corresponding to the two scenarios discussed in Section II-B. (a) Periodic Arrivals — Under this model, one packet arrives to Qi every Di > 0 timeslots and acquires a deadline Di upon its arrival. i.e., Qi stays empty until the arrival of the next packet if the current HOL packet departs before its due deadline. Different streams may have equal periods but have random offsets/phases with respect to each other. (b) Infinite Backlogs — Under this model, traffic streams for all users are pre-cached at the BS. The ith user expects to receive a packet every Di time-slots, i.e., the inter-packet deadline between successive packets in Qi is Di . Thus, if the HOL packet of Qi departs Ti time-slots before its deadline expires, the new HOL packet acquires a deadline of Di + Ti . The “load” imposed on the system by the ith user in both cases was varied by changing Di . We also considered two models for the packet dropping costs. (a) Identical Packets — Under this model, all packets are of equal importance and packet dropping costs are artificially imposed by the BS to ensure fairness amongst users. (b) Non-identical packets — Under this model, the dropping cost of a packet reflects its application layer importance. We used real video test sequences to assign costs to packets.
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Packet drop rate for weak users
RR BCF EDD CA−EDD
2.2 dB
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1.8 dB
−2
−1
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Average SNR for weak users (dB)
ave Fig. 5. Packet drop rate averaged over four weak users as a function of γweak for an eight user system with periodic arrivals (equal periods) and identical packets
A. QS versus CA-EDD (periodic arrivals, identical packets) Here, we contrast the performance of QS and CA-EDD for a two-user system. To implement QS, the transition probabilities for an N-state FSMC channel model were estimated by partitioning the interval [0, 1] into N equal bins and empirically computing the transition rates between different bins from a sufficiently long trace of the channel realization. The periods for both users were fixed at D = 4 and the average downlink SNR (assumed equal for both users) γ ave was varied from 6dB to 12dB. The packet drop rate (averaged over both users) for QS (for N = 2, 8) and CA-EDD are depicted in Fig. 4. Increasing the number of states in the FSMC model offers no tangible benefit. In fact, CA-EDD performs as well as QS. Recall that QS is based on P1 (formulated under Reduction 1), while CA-EDD is based on P2 (formulated under Reduction 1 and 2). The results in Fig. 4 indicate that their is no loss in performance by incorporating the static channel assumption (Reduction 2) into our problem formulation. B. CA-EDD versus benchmarks (periodic arrivals, identical packets) Equal periods: For the results in this section, the eight downlink users were split into two ave groups of four each. The average downlink SNR for users in the “strong” group was fixed at γstrong ave 4.7dB and the average downlink SNR for users in the “weak” group was varied (γweak ) was varied from -1.3dB to 4.7dB. The period for all eight users was fixed at D = 12. The packet dropping costs for CA-EDD were experimentally chosen to ensure fairness amongst users (comparable packet drop rates). Fig. 5 depicts the average drop rate (averaged over the weak users) as a ave function of γweak . CA-EDD requires ∼2dB lower SNR than its nearest competitor BCF on an average to achieve drop rates below 10%. The performance of CA-EDD and BCF was similar with respect to the strong users. They both comfortably outperformed RR and EDD. Unequal periods: To stress the system further, the period corresponding to the weak users was ave ave reduced to 10. In this case, γstrong was fixed at 10dB and γweak was varied from 0dB to 6dB. The ave packet drop rate (averaged over weak users) as a function of γweak is depicted in Fig. 6. Once again, CA-EDD delivers a performance gain of nearly 2dB over BCF. Note the improvement in performance of EDD. The gap between EDD and BCF closes further as the disparity between the periods of different streams increases.
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RR BCF EDD CA−EDD
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ave Fig. 6. Packet drop rate averaged over four weak users as a function of γweak for an eight user system with periodic arrivals (unequal periods) and identical packets
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Packet drop rate for weak users
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ave Fig. 7. Packet drop rate averaged over four weak users as a function of γweak for an eight user system with infinite backlogs (equal inter-packet deadlines) and identical packets
C. CA-EDD versus benchmarks (infinite backlogs, identical packets) Once again, the users were split into two equal groups, as in Section VI-B. The IPDs were set equal to 12 for all streams. However, the deadline acquired by the HOL packet upon its arrival to the head of the queue was determined by the departure time of its predecessor. For example, if the HOL packet of a queue departed 6 time-slots prior to the expiration of its deadline, the next packet acquired a deadline of 6+12=18 time-slots. While making their scheduling decisions, ¯ schedulers were only allowed to consider queues whose HOL packets had a TTE less than D (a design parameter). The constraint arises from the practical consideration of small receiver buffers, so that a receiver cannot be flooded with packets when it is experiencing good channel ¯ = 24 for our conditions. This modeling assumption was also employed in [10]. We set D ave ave simulations. For this experiment, γstrong was fixed at 4.7dB and γweak was varied from -1.3dB to ave 4.7dB. The packet drop (averaged over weak users) as a function of γweak for different policies is depicted in Fig. 7. CA-EDD yields gains of nearly 3dB over BCF in this case. This set of results illustrates the utility of IPDs as a modeling tool (see Section II-B).
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Fig. 8.
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Fig. 9. Cumulative dropping cost (as a fraction of the maximum) averaged over all users as a function of γ ave for an eight user system with periodic arrivals and non-identical packets
D. CA-EDD versus benchmarks (periodic arrivals, non-identical packets) For this experiment, we assumed equal average downlink SNR for all eight users (γ ave ) and equal period (D = 12). However, packet dropping costs were derived from real video test sequences. In particular, 325 frames of the carphone sequence (a standard test sequence used by the video research community) were transmitted to each user at 176 × 144 pixels/frame, 30 frames/sec (QCIF format), encoded using the H.264/MPEG-4 AVC standard [21] with a leading I-frame followed by 324 P-frames. The video was encoded using the H.264 reference software version JM10.2 [22]. Each frame was encoded as an individual network packet. The dropping cost associated with each packet/frame was determined by dropping that packet alone from the video stream, decoding the resultant stream and computing the “distortion” in video quality (offline computation). The distortion was measured in terms of mean squared error, a standard metric employed by the video community. The packet dropping costs for various packets/frames (normalized by the maximum) are depicted in Fig. 8. The frames with large cost are typically the ones whose temporal location corresponds to significant motion in the video sequence. The total normalized cost over 325 frames of the carphone sequence sums to 32.68. The sum cost incurred by different policies (as a fraction of the maximum) as a function of γ ave is depicted in Fig. 9.
The cost incurred by CA-EDD is less than 1/2 the cost incurred by BCF, and a much smaller fraction of the cost incurred by EDD and RR. Since only 325 frames of the video sequence are available for experimentation, the sequence was repeatedly transmitted in a loop. We repeated this experiment with the coastguard sequence and observed similar performance results. To summarize our observations, CA-EDD comfortably outperforms benchmark scheduling policies under a variety of scenarios. BCF generally delivers the best performance amongst the benchmark schedulers. However, BCF can be arbitrarily unfair to weak users when the channel conditions are very disparate. On the other hand, CA-EDD can be tuned to ensure fairness between users. The performance advantages of CA-EDD are further substantiated by visual inspection of received video streams for the trace-driven simulation scenario of Section VI-D. VII. C ONCLUSIONS We studied the problem of deadline-constrained packet scheduling for supporting real-time multimedia applications on the downlink of a wireless communication system. We formulated the scheduling problem in a dynamic programming framework and proposed the CA-EDD scheduling policy, which was interpreted as a channel and QoS aware version of the basic EDD scheduling policy. CA-EDD considers wireless channel conditions, packet deadlines and application layer importance of packets in a unified and systematic way to enhance system performance. Experimental results show that CA-EDD yields significant gains vis-`a-vis benchmark schedulers, both in terms of overall system performance and fairness amongst users. We expect the gains of CA-EDD to improve further as the number of users in the system increases. VIII. A PPENDIX A. Proof of Lemma 1 The proof is based on an inductive argument. 1) Base case: By definition, V (2, 0, j1 , j2 ) = [1 − s1 (j1 )]Λ(1, 0, j1X , j2 ) and V (1, 0, j1, j2 ) = [1 − s1 (j1 )]λ1 . Thus, V (1, 0, j1 , j2 ) − V (2, 0, j1 , j2 ) = V (1, 0, j1, j2 ) s1 (j10 )Π1 (j1 , j10 ) ≥ 0. j10 ∈S1
Since this holds for any j1 ∈ S1 , j2 ∈ S2 , the base case is proved. 2) Inductive step: Now, let us assume V (n1 , 0, j10 , j20 ) ≤ V (n1 −1, 0, j10 , j20 ) ∀ j10 ∈ S1 , j20 ∈ S2 . This implies Λ(n1 , 0, j1 , j2 ) ≤ Λ(n1 − 1, 0, j1 , j2 ). By definition, V (n1 + 1, 0, j1 , j2 ) = [1 − s1 (j1 )]Λ(n1 , 0, j1 , j2 ) and V (n1 , 0, j1 , j2 ) = [1 − s1 (j1 )]Λ(n1 − 1, 0, j1 , j2 ). Thus, V (n1 , 0, j1 , j2 ) − V (n1 + 1, 0, j1 , j2 ) = [1 − s1 (j1 )] [Λ(n1 − 1, 0, i, j) − Λ(n1 , 0, i, j)] ≥ 0. It follows that V (n1 , 0, j1 , j2 ) is non-increasing in n1 for any j1 ∈ S1 , j2 ∈ S2 . The proof for monotonicity of V (0, n2 , j1 , j2 ) as a function of n2 is symmetric. B. Proof of Lemma 2 The proof is by induction. The base case follows from Lemma 1. We focus on the inductive step here. Let us fix n2 and assume V (n1 , n2 − 1, j10 , j20 ) ≤ V (n1 − 1, n2 − 1, j10 , j20 ) ∀j1 ∈ S1 , j2 ∈ S2 . It follows that Λ(n1 , n2 − 1, j1 , j2 ) ≤ Λ(n1 − 1, n2 − 1, j1 , j2 ). For convenience, we denote ∆V = V (n1 , n2 , j1 , j2 ) − V (n1 + 1, n2 , j1 , j2 ), ∆Λ = Λ(n1 − 1, n2 − 1, j1 , j2 ) − Λ(n1 , n2 − 1, j1 , j2 ) ≥ 0. From the inductive assumption, ∆Λ ≥ 0. We split the proof into four cases: 1) γ(n1 , n2 , j1 , j2 ) ≤ 0 and γ(n1 + 1, n2 , j1 , j2 ) ≤ 0: We get, ∆V = [1 − s1 (j1 )]∆Λ ≥ 0. 2) γ(n1 , n2 , j1 , j2 ) > 0 and γ(n1 + 1, n2 , j1 , j2 ) > 0: We invoke Lemma 1 to get, ∆V = [1 − s2 (j2 )]∆Λ + s2 (j2 ) [Λ(n1 − 1, 0, j1 , j2 ) − Λ(n1 , 0, j1 , j2 )] ≥ 0.
3) γ(n1 , n2 , j1 , j2 ) ≤ 0 and γ(n1 + 1, n2, j1 , j2 ) > 0: Since γ(n1 + 1, n2 , j1 , j2 ) > 0, it follows that s1 (j1 )α(n1 + 1, n2 , j1 , j2 ) > s2 (j2 )β(n1 + 1, n2 , j1 , j2 ). We get, ∆V
= s1 (j1 )α(n1 , n2 , j1 , j2 ) − s2 (j2 )β(n1 + 1, n2 , j1 , j2 ) + ∆Λ ≥ s1 (j1 )[α(n1 , n2 , j1 , j2 ) − α(n1 + 1, n2 , j1 , j2 )] + ∆Λ = [1 − s1 (j1 )]∆Λ ≥ 0.
4) γ(n1 , n2 , j1 , j2 ) > 0 and γ(n1 + 1, n2 , j1 , j2 ) ≤ 0: Since γ(n1 + 1, n2 , j1 , j2 ) ≤ 0, it follows that s1 (j1 )α(n1 + 1, n2 , j1 , j2 ) ≤ s2 (j2 )β(n1 + 1, n2 , j1 , j2 ). We get, ∆V
= s2 (j2 )β(n1 , n2 , j1 , j2 ) − s1 (j1 )α(n1 + 1, n2 , j1 , j2 ) + ∆Λ ≥ s2 (j2 )[β(n1 , n2 , j1 , j2 ) − β(n1 + 1, n2 , j1 , j2 )] + ∆Λ = [1 − s2 (j2 )]∆Λ + s2 (j2 ) [Λ(n1 − 1, 0, j1 , j2 ) − Λ(n1 , 0, j1 , j2 )] ≥ 0.
It follows that ∆V ≥ 0, i.e., V (n1 , n2 , j1 , j2 ) ≥ V (n1 + 1, n2 , j1 , j2 ) for any n2 ∈ N ∪ {0}, j1 ∈ S1 and j2 ∈ S2 . The proof for monotonicity of V (n1 , n2 , j1 , j2 ) in n2 is symmetric. C. Proof of Theorem 1 An equivalent characterization of Theorem 1 is as follows: If the optimal policy P1? schedules Q1 in state (n1 , n2 , j1 , j2 ), it necessarily schedules Q1 in states (n1 − 1, n2 , j1 , j2 ) and (n1 , n2 + 1, j1 , j2 ). Similarly, if P1? schedules Q2 in state (n1 , n2 , j1 , j2 ), it necessarily schedules Q2 in states (n1 +1, n2 , j1 , j2 ) and (n1 , n2 −1, j1 , j2 ). We will prove this characterization of the theorem. Let us fix j1 ∈ S1 and j2 ∈ S2 . For the remainder of this section, we suppress the indices j1 , j2 in the interest of readability. In particular, we denote si (ji ) by si , Λ(n1 , n2 , j1 , j2 ) by Λ(n1 , n2 ) etc. We will repeatedly use the following: Policy P1? schedules Q1 in state (n1 , n2 , j1 , j2 ) if (s2 − s1 )Λ(n1 − 1, n2 − 1) ≤ s2 Λ(n1 − 1, 0) − s1 Λ(0, n2 − 1),
(13)
and Q2 else. We will prove the theorem by contradiction. We start with the following four cases: 1) P1? schedules Q1 in state (n1 , n2 , j1 , j2 ) and Q2 in state (n1 − 1, n2 , j1 , j2 ): Using (13) in conjunction with Lemma 1 we get (s2 −s1 )Λ(n1 −2, n2 −1) ≥ s2 Λ(n1 −2, 0)−s1Λ(0, n1 − 1) ≥ s2 Λ(n1 − 1, n2 ) − s1 Λ(0, n2 − 1) ≥ (s2 − s1 )Λ(n1 − 1, n2 − 1). From Lemma 2, Λ(n1 − 2, n2 − 1) ≥ Λ(n1 − 1, n2 − 1). Thus, if s1 ≥ s2 , we get a contradiction. 2) P1? schedules Q1 in state (n1 , n2 , j1 , j2 ) and Q2 in state (n1 , n2 + 1, j1 , j2 ): Using (13) in conjunction with Lemma 1 we get (s2 − s1 )Λ(n1 − 1, n2 ) ≥ s2 Λ(n1 − 1, 0) − s1 Λ(0, n2 ) ≥ s2 Λ(n1 − 1, 0) − s1 Λ(0, n2 − 1) ≥ (s2 − s1 )Λ(n1 − 1, n2 − 1). From Lemma 2, Λ(n1 − 1, n2 − 1) ≥ Λ(n1 − 1, n2 ). Thus, if s2 > s1 , we get a contradiction. 3) P1? schedules Q2 in state (n1 , n2 , j1 , j2 ) and Q1 in state (n1 + 1, n2 , j1 , j2 ): Using (13) in conjunction with Lemma 1 we get (s2 − s1 )Λ(n1 − 1, n2 − 1) ≥ s2 Λ(n1 − 1, 0) − s1 Λ(0, n2 − 1) ≥ s2 Λ(n1 , 0) − s1 Λ(0, n2 − 1) ≥ (s1 − s1 )Λ(n1 , n2 − 1). From Lemma 2, Λ(n1 − 1, n2 − 1) ≥ Λ(n1 , n2 − 1). Thus, if s1 ≥ s2 , we get a contradiction. 4) P1? schedules Q2 in state (n1 , n2 , j1 , j2 ) and Q1 in state (n1 , n2 − 1, j1 , j2 ): Using (13) in conjunction with Lemma 1 we get (s2 − s1 )Λ(n1 − 1, n2 − 1) ≥ s2 Λ(n1 − 1, 0) − s1 Λ(0, n2 − 1) ≥ s2 Λ(n1 − 1, 0) − s1 Λ(0, n2 − 2) ≥ (s2 − s1 )Λ(n1 − 1, n2 − 2). From Lemma 2, Λ(n1 − 1, n2 − 2) ≥ Λ(n1 − 1, n2 − 1). Thus, if s2 > s1 , we get a contradiction. Using the above along with similar computations, the proof can be completed by showing: ? • s1 ≥ s2 : If P1 schedules Q1 in state (n1 , n2 , j1 , j2 ), it schedules Q1 in state (n1 , n2 +1, j1 , j2 ). ? If P1 schedules Q2 in state (n1 , n2 , j1 , j2 ), it schedules Q2 in state (n1 , n2 − 1, j1 , j2 ).
•
s1 < s2 : If P1? schedules Q1 in state (n1 , n2 , j1 , j2 ), it schedules Q1 in state (n1 −1, n2 , j1 , j2 ). If P1? schedules Q2 in state (n1 , n2 , j1 , j2 ), it schedules Q2 in state (n1 + 1, n2 , j1 , j2 ).
D. Proof of Lemma 3 It suffices to show: γ(n1 , n2 + 1) ≤ γ(n1 , n2 ) ≤ γ(n1 + 1, n2 ) ∀ n1 , n2 ∈ N. The proof is based on inductive arguments. We begin with the base case, i.e., n1 = n2 = 1. 1) Base case: γ(1, 2) ≤ γ(1, 1) ≤ γ(2, 1) From definitions, we have γ(1, 2) = −s1 λ1 + s2 V (0, 1), γ(1, 1) = −s1 λ1 + s2 λ2 and γ(2, 1) = −s1 V (1, 0) + s2 λ2 . Also, from the boundary conditions it follows that V (0, 1) = (1 − s2 )λ2 + c2 and V (1, 0) = (1 − s1 )λ1 + c1 . Using our assumptions s1 λ1 ≥ c1 and s2 λ2 ≥ c2 , it is easy to verify that γ(1, 2) ≤ γ(1, 1) ≤ γ(2, 1). The base case is proved. 2) Inductive step: We now assume that for some n1 , n2 > 2, the following holds: γ(n1 − 1, n2 ) ≤ γ(n1 − 1, n2 − 1) ≤ γ(n1 , n2 − 1).
(14)
f2? We will show that (14) implies γ(n1 , n2 + 1) ≤ γ(n1 , n2 ) ≤ γ(n1 + 1, n2 ). We know that P schedules Q1 in state (n1 , n2 ) if γ(n1 , n2 ) ≤ 0, and Q2 else. There are three possible cases: 1) γ(n1 , n2 − 1) ≤ 0 and γ(n1 − 1, n2 − 1) ≤ 0: V (n1 , n2 − 1) = s1 V (0, n2 − 2) + (1 − s1 )V (n1 − 1, n2 − 2) + c1 + c2 V (n1 − 1, n2 − 1) = s1 V (0, n2 − 2) + (1 − s1 )V (n1 − 2, n2 − 2) + c1 + c2 γ(n1 + 1, n2 ) − γ(n1 , n2 ) = (1 − s1 )[γ(n1 , n2 − 1) − γ(n1 − 1, n2 − 1)] ≥ 0. 2) γ(n1 − 1, n2 − 1) > 0 and γ(n1 , n2 − 1) > 0: V (n1 , n2 − 1) = s2 V (n1 − 1, 0) + (1 − s2 )V (n1 − 1, n2 − 2) + c1 + c2 V (n1 − 1, n2 − 1) = s2 V (n1 − 2, 0) + (1 − s2 )V (n1 − 2, n2 − 2) + c1 + c2 γ(n1 + 1, n2 ) − γ(n1 , n2 ) = (1 − s2 )[γ(n1 , n2 − 1) − γ(n1 − 1, n2 − 1)] ≥ 0. 3) γ(n1 − 1, n2 − 1) ≤ 0 and γ(n1 , n2 − 1) > 0: V (n1 , n2 − 1) = s2 V (n1 − 1, 0) + (1 − s2 )V (n1 − 1, n2 − 2) + c1 + c2 V (n1 − 1, n2 − 1) = s1 V (0, n2 − 2) + (1 − s1 )V (n1 − 2, n2 − 2) + c1 + c2 γ(n1 + 1, n2 ) − γ(n1 , n2 ) = (1 − s2 )γ(n1 , n2 − 1) − (s2 − s1 )γ(n1 − 1, n2 − 1) ≥ 0. The final inequality in all three cases follows from (14). Hence, we have established that γ(n1 + 1, n2 ) − γ(n1 , n2 ) ≥ 0. The proof for showing that γ(n1 , n2 ) − γ(n1 , n2 + 1) ≥ 0 is analogous. E. Proof of Theorem 2 It follows from Lemma 3 that for fixed n1 , γ(n1 , n2 ) changes sign at most once as n2 increases, and this sign change is from positive to negative. Equivalently, the optimal policy “switches” from Q2 to Q1 at most once as n2 increases, for each fixed n1 . Thus, for each n1 > 1 we can determine an integer φ(n1 ) such that γ(n1 , n2 ) is positive for all n2 < φ(n1 ), and non-positive else. Since γ(n1 , n2 ) is a non-decreasing sequence in n1 for fixed n2 , φ(n1 ) is a non-decreasing f? is a switch-over policy. sequence for all n2 . By definition, P 2
F. Proof of Lemma 4 Define an2 , lim γ(n1 , n2 ) ∀ n2 and bn1 , lim γ(n1 , n2 ) ∀ n1 . From Lemma 3 we n1 →∞
n2 →∞
conclude that {an2 } is a decreasing sequence and thus has a limit a? , while {bn1 } is an increasing sequence and thus has a limit b? . We want to show that a? = b? . We have the following chain of inequalities: γ(n1 , n2 ) ≥ bn1 =⇒ lim γ(n1 , n2 ) ≥ lim bn1 =⇒ an2 ≥ b? =⇒ lim an2 ≥ n1 →∞
n1 →∞
b? =⇒ a? ≥ b? . Similarly, we can demonstrate b? ≥ a? and conclude a? = b? .
n2 →∞
G. Proof of Theorem 3 1) We will argue by contradiction. Since φ(n1 ) is a non-decreasing sequence, if the claim is false, it follows that lim φ(n1 ) = +∞. This implies lim γ(n1 , n2 ) > 0 ∀ n2 ∈ N. In n1 →∞ n1 →∞ particular, letting n2 → ∞, we get γ ? > 0. However, when s1 c1 ≥ s2 c2 , it follows from (12) that γ ? ≤ 0. This leads to a contradiction, and we conclude that the claim is true. 2) The proof is similar to (a). By assuming the statement to be false, we can demonstrate γ ? ≤ 0. However, when s1 c2 < s2 c2 , it follows from (12) that γ ? > 0, thereby leading to a contradiction and proving the claim. H. Proof of Lemma 5 For convenience, we assume n1 , n2 > 2. The proof when n1 , n2 = 1, 2 is similar, though notationally more burdensome. From (9), γ(n1 , n2 ) = s1 V (0, n2 − 1) − s2 V (n − 1 − 1, 0) + (s2 − s1 )V (n1 − 1, n2 − 1). We have two possible cases, depending on the sign of γ(n1 − 1, n2 − 1). 1) γ(n1 − 1, n2 − 1) ≤ 0: P2? schedules Q1 in state (n1 − 1, n2 − 1). Thus, V (n1 − 1, n2 − 1) = s1 V (0, n2 − 1) + (1 − s1 )V (n1 − 2, n2 − 2) and γ(n1 , n2 ) = (1 − s1 )γ(n1 − 1, n2 − 1). 2) γ(n1 − 1, n2 − 1) > 0: P2? schedules Q2 in state (n1 − 1, n2 − 1). Thus, V (n1 − 1, n2 − 1) = s2 V (n1 − 1, 0) + (1 − s2 )V (n1 − 2, n2 − 2) and γ(n1 , n2 ) = (1 − s2 )γ(n1 − 1, n2 − 1). Since s1 , s2 ≤ 1, it follows that sgn{γ(n1 , n2 )} = sgn{γ(n1 − 1, n2 − 1)}, as desired. I. Proof of Theorem 4 For the K-user problem we define the pairwise decision functions γij (¯ n) , si V (¯ ni ) − j i sj V (¯ n ) + (sj − si )V (¯ n − ¯1), where n ¯ = (n1 , . . . , nK ), n ¯ = (n1 − 1, . . . , ni−1 − 1, 0, ni+1 − ¯ 1, . . . , nK − 1) and 1 is a K-length vector with all unit entries. The optimal policy P2? “prefers” Qi over Qj in state n ¯ if γij (¯ n) ≤ 0, and prefers Qj else. The pairwise decision functions are computed by solving the DP equations associated with P2 . According to the hypothesis of the theorem, a pairwise comparison between Qi and Qj is unaffected by the states of all other queues. In particular, if n ¯ ij , (0, . . . , 0, ni , 0, . . . , 0, nj , 0, . . . , 0), then optimality of pairwise ? comparisons (according to policy P2,PW ) implies that sgn{γij (¯ n)} = sgn{γij (¯ nij )} ∀ i, j, n ¯ . For ease of exposition, we present the proof for K = 3. The proof ideas easily generalize to K > 3. We show that if sgn{γ12 (n1 , n2 , 0)} = sgn{γ12 (n1 , n2 , n3 )}, i.e., the relative ranking of Q1 and Q2 is not affected by the presence of Q3 . The proof is by induction. We assume that sgn{γ12 (n1 −1, n2 −1, n3 −1)} = sgn{γ12 (n1 −1, n2 −1, 0)}. Now, γ12 (n1 , n2 , n3 ) = s1 V (0, n2 − 1, n3 −1)−s2 V (n1 −1, 0, n3 −1)+(s2 −s1 )V (n1 −1, n2 −1, n3 −1). Several cases arise, depending on the decision of P2? in states (0, n2 − 1, n3 − 1), (n1 , n2 , n3 ) and (n1 − 1, n2 − 1, n3 − 1). 1) P2? schedules Q2 in (0, n2 −1, n3 −1), and Q1 in (n1 −1, 0, n3 −1) and (n1 −1, n2 −1, n3 −1): In this case, γ12 (n1 , n2 , n3 ) = (1 − s1 )γ12 (n1 − 1, n2 − 1, n3 − 1) ≤ 0.
2) P2? schedules Q1 in (n1 −1, 0, n3 −1), and Q2 in (0, n2 −1, n3 −1) and (n1 −1, n2 −1, n3 −1): In this case, γ12 (n1 , n2 , n3 ) = (1 − s2 )γ12 (n1 − 1, n2 − 1, n3 − 1) > 0. 3) P2? schedules Q3 in (n1 −1, 0, n3 −1), and Q2 in (0, n2 −1, n3 −1) and (n1 −1, n2 −1, n3 −1): In this case, γ12 (n1 , n2 , n3 ) = s2 γ13 (n1 −1, 0, n3 −1)+(1−s2 )γ12 (n1 −1, n2 −1, n3 −1) > 0. 4) P2? schedules Q3 in (0, n2 −1, n3 −1), and Q1 in (n1 −1, 0, n3 −1) and (n1 −1, n2 −1, n3 −1): In this case, γ12 (n1 , n2 , n3 ) = s1 γ32 (0, n2 −1, n3 −1)+(1−s1 )γ12 (n1 −1, n2 −1, n3 −1) ≤ 0. 5) P2? schedules Q3 in (0, n2 − 1, n3 − 1), (n1 − 1, 0, n3 − 1) and (n1 − 1, n2 − 1, n3 − 1): In this case, γ12 (n1 , n2 , n3 ) = s3 γ12 (n1 − 1, n2 − 1, 0) + (1 − s3 )γ12 (n1 − 1, n2 − 1, n3 − 1). By our inductive assumption, both terms on the right side have the same sign. In all the above cases, we have shown that sgn{γ(n1 , n2 , n3 )} = sgn{γ(n1 − 1, n2 − 1, n3 − 1)}. It follows from our inductive assumption that sgn{γ(n1 , n2 , n3 )} = sgn{γ(n1 − 1, n2 − 1, 0)}. Finally, invoking Lemma 5, we conclude sgn{γ(n1 , n2 , n3 )} = sgn{γ(n1 , n2 , 0)}, as desired. R EFERENCES [1] H. Holma and A. Toskala, Eds., WCDMA for UMTS, 3rd Ed., John Wiley & Sons, 2002. [2] J.G. Proakis, Digital Communications, 4th Ed., New York:McGraw-Hill, 2001. [3] S. Shakkotai, T. Rappaport and P. Karlsson, “Cross-layer design for wireless networks”, IEEE Communications Magazine, vol. 41, no. 10, pp. 74-80, Oct. 2003. [4] 3GPP, “High Speed Downlink Packet Access (HSDPA); overall description”, 3GPP, Sophia Antipolis, France, Technical Specification 25.308, Ver. 5.4.0, Release 5, Mar. 2002. [5] A.K. Parekh and R.G. Gallager, “A generalized processor sharing approach to flow control in integrated services networks: the single node case”, IEEE/ACM Transactions on Networking, vol. 1, no. 3, pp. 344-357, Jun. 1993. [6] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindushayana and A. Viterbi, “CDMA/HDR: A bandwidth-efficient highspeed wireless data service for nomadic users”, IEEE Communications Magazine, vol. 38, no. 7, pp. 70-77, Jul. 2000. [7] H. Fattah and C. Leung, “An overview of scheduling algorithms in wireless multimedia networks”, IEEE Wireless Communications, vol. 9, no. 5, pp. 76-83, Oct. 2002. [8] A. Dua and N. Bambos, “Deadline constrained packet scheduling for wireless networks”, Proceedings of IEEE VTC (Fall), Dallas, TX, vol. 1, pp. 196-200, Sept. 2005. [9] D. Bertsekas, Dynamic Programming and Optimal Control, vol. 1 & 2, 2nd Ed., Athena Scientific, 2000. [10] L. Georgiadis, R. Guerin and A. Parekh, “Optimal multiplexing on a single link: delay and buffer requirements”, IEEE Transactions on Information Theory, pp. 1518-1535, vol. 43, no. 5, Sept. 1997. [11] 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, Jan. 2002. [12] P.Y. Kong and K.H. Teh, “Performance of proactive earliest due date packet scheduling in wireless networks”, IEEE Transactions on Vehicular Technology, vol. 53, no. 4, pp. 1224-1234, Jul. 2004. [13] K.M.F Elsayed and A.K.F Khattab, “Channel-aware earliest deadline due fair scheduling for wireless multimedia networks”, Springer Wireless Personal Communications, vol. 38, no. 2, pp. 233-252, 2006. [14] T. Ren, I. Koutsopolous and L. Tassiulas, “QoS provisioning for real-time traffic in wireless packet networks”, Proceedings of IEEE GLOBECOM, Taipei, Taiwan, pp. 1673-1677, Nov. 2002. [15] K.B. Johnsson and D.C. Cox, “An adaptive cross-layer scheduler for improved QoS support of multi-class data services on wireless systems”, IEEE Journal on Selected Areas in Communication, vol. 23, no. 2, pp. 334-343, Feb. 2005. [16] M. Agarwal and A. Puri, “Base station scheduling of requests with fixed deadlines”, Proceedings of IEEE INFOCOM, New York, NY, pp. 487-496, Jun. 2002. [17] M. Hassan, M.M. Krunz and I. Matta, “Markov-based channel characterization for tractable performance analysis in wireless packet networks”, IEEE Transactions on Wireless Communications, vol. 3. no. 3, pp. 821-831, May 2004. [18] J. Walrand, An Introduction to Queuing Networks, NJ:Prentice Hall, 1988. [19] G.L. Stuber, Principles of Mobile Communication, 2nd Ed., Kluwer Academic Publishers, 2001. [20] S. Hamalainen, P. Slanina, M. Hartman, A. Lappetelainen, H. Holma and O. Salonaho, “A novel interface between link and system level simulations”, ACTS Mobile Telecommunications Summit, Aalborg, Denmark, pp. 599-604, Oct. 1997. [21] “Advanced video coding for general audiovisual services”, ITU-T Recommendation, H.264, Mar. 2005. [22] ITU H.264/MPEG-4 AVC reference software, ver. JM10.2, http://iphome.hhi.de/suehring/tml/.
