Delay-Constrained Video Transmission: Quality-driven Resource Allocation and Scheduling
Amin Abdel Khalek, Constantine Caramanis, Robert W. Heath Jr
aa r X i v : . [ c s . I T ] N ov Delay-Constrained Video Transmission:Quality-driven Resource Allocation and Scheduling
Amin Abdel Khalek,
Student Member, IEEE , Constantine Caramanis,
Member, IEEE ,and Robert W. Heath Jr.,
Fellow, IEEE
Abstract —Real-time video demands quality-of-service (QoS)guarantees such as delay bounds for end-user satisfaction.Furthermore, the tolerable delay varies depending on the usecase such as live streaming or two-way video conferencing. Dueto the inherently stochastic nature of wireless fading channels,deterministic delay bounds are difficult to guarantee. Instead, wepropose providing statistical delay guarantees using the conceptof effective capacity. We consider a multiuser setup wherebydifferent users have (possibly different) delay QoS constraints.We derive the resource allocation policy that maximizes the sumvideo quality and applies to any quality metric with concaverate-quality mapping. We show that the optimal operating pointper user is such that the rate-distortion slope is the inverse of thesupported video source rate per unit bandwidth, a key metricwe refer to as the source spectral efficiency. We also solve thealternative problem of fairness-based resource allocation wherebythe objective is to maximize the minimum video quality acrossusers. Finally, we derive user admission and scheduling policiesthat enable selecting a maximal user subset such that all selectedusers can meet their statistical delay requirement. Results showthat video users with differentiated QoS requirements can achievesimilar video quality with vastly different resource requirements.Thus, QoS-aware scheduling and resource allocation enablesupporting significantly more users under the same resourceconstraints.
I. I
NTRODUCTION
Real-time video transmission requires maintaining stringentdelay bounds to ensure a good user experience. The stringencyof the delay bound is further dependent on the specific usecase. For instance, interactive applications such as video con-ferencing can only tolerate an end-to-end delay on the orderof few hundred milliseconds for a smooth experience whereaswith live streaming, the delay constraint can be relaxed to fewseconds. In bandwidth-limited networks with shared resources,the bottleneck in the end-to-end delay is queuing. Thus, userswith more stringent delay constraints should be allocatedmore physical resources to boost their service rates, reducethe queuing delay, and thus support their QoS requirement.This provides a strong motivation for re-designing resourceallocation taking into account hybrid QoS requirements in thenetwork.Deterministic delay bounds are hard to guarantee overwireless networks due to changing channel conditions [1].Therefore, to provide a realistic and accurate model for qualityof service, statistical guarantees are considered as a design
The authors are with the Wireless Networking & Communications Groupin the Department of Electrical and Computer Engineering at UT AustinWNCG, 2501 Speedway Stop C0806, Austin, Texas 78712-1687. Email: { akhalek,constantine,rheath } @utexas.edu. This work was partially supportedby the Intel-Cisco Video Aware Wireless Networks (VAWN) Program. guideline by defining constraints in terms of the delay-boundviolation probability. The notion of statistical QoS is tied backto the well developed theory of effective bandwidth [2]–[4] andits dual concept of effective capacity [1], [5], [6].Satisfying delay constraints should not come at the expenseof maintaining high perceptual quality. Therefore, we derivea resource allocation policy that maximizes a quality-basedutility such that all users in the network can achieve theirtarget statistical delay bound. We consider two different utilityfunctions: the sum video quality and the minimum videoquality in the network. The only assumption about the qualitymetric is that the rate-quality mapping is concave which isgenerally the case because practical video codecs achievediminishing returns in quality as the source rate increases. Forthe general case where the set of users in the network cannotall be served, we solve the problem of selecting a maximalsubset of users to schedule such that each scheduled user canmeet their target QoS requirement. A. Contributions
The central contribution of the paper is that it partitions thewireless channel resources across real-time video users withhybrid QoS requirements to maximize a video quality-basedutility function. Previous work on delay-constrained videotransmission addresses maximizing rate or throughput [7]–[9], minimizing energy consumption [10], [11], or minimizingresource utilization [11], [12], all of which are not directlyrelevant metrics for the end video quality. Furthermore, asignificant body of literature [7]–[12] is devoted to point-to-point transmission with resource allocation across differ-ent time slots or for different video layers, thus not takinginto account hybrid QoS requirements across different users.Previous work which directly optimizes video quality utilityfunctions across different users is either focused on storedvideo in which content can be buffered ahead opportunisticallyand large delays can be tolerated [13], [14] or considers real-time video with deterministic delay constraints [15], thus notbeing applicable to communication over fast fading channels.Specifically, this paper addresses these issues by answeringthe following three questions:1)
Resource Allocation:
Given a set of users with specifieddelay bounds, target violation probabilities, and ratedistortion characteristics, how should resources be allo-cated across the users and how should the source ratesbe adapted to maximize a video quality-based utilityfunction?2)
Scheduling:
If not all users can meet their delay con-straint simultaneously, what is the subset of users with the largest cardinality such that all scheduled users canmeet their statistical delay bound?3)
User Admission:
Given a network in operation and anew video user requesting a session, what is the useradmission criterion such that the user can meets the QoSrequirement without jeopardizing the QoS of existingusers?In what follows, we summarize the major paper contribu-tions on these three fronts.
1) Quality-driven resource allocation and rate adaptation:
Considering a network with multiple video users with pos-sibly different delay requirements sharing a wireless channelresource, we derive the resource allocation and rate adaptationpolicy that maximizes the sum video quality. Resource alloca-tion adapts the partitioning of the wireless channel resourcesacross the users and rate adaptation adapts the video sourcerate of each user. We show that the optimal operating point peruser is such that the rate-distortion slope is the inverse of thesupported video source rate per unit bandwidth. The maximumsource rate per unit bandwidth is a fundamental measure ofthe number of video bits per channel use that can be deliveredsubject to the QoS requirement and we refer to it as the sourcespectral efficiency . Next, we solve the alternative problem offairness-based resource allocation whereby the objective is tomaximize the minimum video quality across users and contrastthe solution with the sum quality maximizing policy.
2) Maximal user subset scheduling:
We derive a schedulingpolicy to select a subset of users such that all scheduled userscan meet their statistical delay requirement. We show that theoptimal scheduling policy can be obtained in polynomial timein the number of users and it involves computing the minimumresource allocation required by each user to support their QoSrequirement, using it as a sorting criterion, and scheduling thefirst sorted users such that the sum of their minimum resourcerequirement does not exceed the total available resources.Under the fairness constraint, a similar solution is obtainedwith the major difference that the sorting criterion is the videoquality corresponding to the minimum rate representation ofthe video sequence.
3) Statistical QoS-based user admission:
We extend theproblem to accommodate dynamically changing networkswhereby users can request new sessions with certain targetQoS requirement. We derive the admission policy that ensuresthe admitted user can meet the statistical delay constraintwithout jeopardizing any of the other users’ QoS require-ments. The user admission policy is derived under both sumquality-maximizing resource allocation as well as fairness-based resource allocation and it depends on the minimum raterepresentation available for the video sequences correspondingto each user as well as the QoS requirements of each user.
B. Related Work
The effective capacity link layer model characterizes thecapacity of the wireless channels in the presence of queuesusing QoS exponents that describe the decay rate of thequeue length tail probability and characterize a correspondingstatistical delay bound. For scalable video transmission, effec-tive capacity analysis is applied in [12] to provide statistical delay bounds for scalable video transmission over unicastand multicast links. Considering energy-efficiency as a targetobjective in an ad-hoc network with possible multihop trans-missions, [10] derives energy-efficient transmission schemessuch that the end-to-end delay bounds are satisfied. Powerand rate adaptation with effective capacity-driven quality ofservice provisions is considered in [7]. While addressingstatistical delay bounds for video transmission, no previouswork directly optimizes video quality, but instead minimizesresource utilization [11], [12], maximizes rate or throughput[7]–[9], or minimizes energy consumption [10], [11].Multiuser scheduling and resource allocation for videotransmission has been studied in [13], [16]–[19]. In [13], theknowledge of the variations in the dependence of perceivedvideo quality to the compression rate is utilized for resourceallocation to improve video QoE across multiple users. Ac-counting for users’ QoE variability in this manner allowsadapting resource allocation in a content-aware manner. Anasymptotically optimal online algorithm for optimizing usersQoE is proposed that allows realizing tradeoffs across meanQoE, variance in QoE, and fairness. This work applies forstored video streaming as it assumes reliable transport andlarge buffer-ahead which enables tolerating large delays. In[16], a cross-layer packet scheduling scheme that streams pre-encoded video over wireless downlink packet access networksto multiple users is presented. A scheduling scheme is used inwhich data rates are dynamically adjusted based on channelquality as well as the gradients of a video quality utilityfunction. In addition to video distortion, the design of theutility function takes into account decoder error concealment.In [17], the problem of multiuser resource allocation for uplinkOFDMA is studied taking into account the rate-distortioncharacteristics of different users. A subcarrier assignmentand power allocation algorithm is devised to minimize theaverage video distortion among users in the system based onthe CSI and rate-distortion. In [18], a cross-layer resourceallocation and packet scheduling scheme is developed takinginto account the time-varying nature of the wireless channelsto minimize the expected distortion of the received sequence.Complex error concealment is taken into account in estimatingvideo distortion and the gradients of the distortion are usedto efficiently allocate resources across users. In [19], theproblem of scheduling and resource allocation for multiuservideo streaming over downlink OFDM channels is addressedfor SVC with quality and temporal scalability. A schedulingalgorithm prioritizes the transmissions of different users byconsidering video contents, deadline requirements, and trans-mission history.A significant body of literature is devoted to point-to-point optimizations applied to real-time video transmission.Such adaptive video transmission techniques can be roughlycategorized into joint source-channel coding (JSCC) [20]–[23], unequal error protection (UEP) [23]–[27], prioritizedscheduling [28], and loss visibility-based prioritization [29]–[31]. While these point-to-point optimizations are useful forimproving video quality and error resilience for individualvideo streams, they don’t capture hybrid user dynamics interms of rate-distortion behavior and differentiated QoS re- ....
User 1User 2User K γ K γ γ Rate-DistortionCharacteristics B B B K Rate Adaptation and Resource Allocation
Rate-distortion characteristicsDelay QoS requirementsChannel state information ....
Delay QoS Requirements R K Q K (R K )Q (R ) R Q (R ) R Fig. 1. Proposed system block diagram for quality-driven resource allocation and rate adaptation of delay-constrained video streams. quirements which is the key focus of this paper.Previous work on video transmission in a multiuser setupfocuses on stored video use cases (e.g. [13], [16]–[18]) inwhich content can be buffered ahead opportunistically andlarge delays can be tolerated. In that scenario, the problemreduces to rate-distortion optimizations and buffer manage-ment policies. For real-time video, however, such as livestreaming or video conferencing, whereby the content in notpre-encoded, large buffers are not feasible and stringent delayconstraints need to be guaranteed. For that important use case,no previous work addresses the problem of optimizing a videoquality-based utility function while guaranteeing a statisticaldelay bound per user.
C. Paper Organization
The rest of the paper is organized as follows: We presentthe system model in Section II. The background for providingstatistical delay bounds using the theory of effective capacity ispresented in Section III. In Section IV, we solve the problem ofquality-maximizing resource allocation under statistical delayconstraints as well as fairness-based resource allocation. InSection V, we present the maximal subset scheduling solutionand the user admission criterion. In Section VI, we present thealgorithm for joint scheduling and resource allocation underthe quality-maximizing policy and the fairness-based policy.We present results and analysis of the corresponding gainsin Section VII. Finally, concluding remarks are provided inSection VIII. II. S
YSTEM M ODEL
Consider the downlink of a base station where K videousers orthogonally share a bandwidth B Hz. Each user k isallocated a bandwidth B k Hz such that P Kk =1 B k = B . Theindividual user bandwidths are assumed narrowband so thatthe corresponding wireless channel experiences flat fading. Wenote that in a wideband system, with orthogonal frequencydivision multiple access (OFDMA), this assumption can beensured by allocating adjacent subcarriers to individual users.The channel coherence time is T . Further, the timescale ofvideo rate adaptation is much larger than T . This is typicallythe case in practice because the rate of the video source isadapted at the GoP timescale which is typically on orderof a second. The channel variation is assumed to be in the milliseconds timescale. For slow fading channels whereby thechannel varies at the same timescale as the source, statisticaldelay guarantees are not needed since the service rate is almostdeterministic. Thus, the challenging case of interest is whenthe channel state variation is considerably faster the sourcerate variation.Since resource allocation and rate adaptation are done at thesource variation timescale and the video source experiencesthe ergodic capacity of the channel, the base station (BS) oraccess point (AP) only require channel distribution information(CDI). Let f k ( γ ) represent the SNR distribution for user k .Moreover, { γ k } is modeled as an ergodic and stationary block-fading process uncorrelated among consecutive packets j . Weassume that each user experiences Gaussian noise such that,with capacity-achieving codes, the instantaneous transmissionrate for user k is C k = B k log(1 + γ k ) .The video segment intended for user k is transmitted atrate R k . A queue is inserted for each user stream to ab-sorb the mismatch between the arrival and service rate dueto the channel variations. The video stream for each user k is characterized by a rate distortion model Q k ( R k ) thatdetermines the mapping from the source rate to the videoperceptual quality. The rate-quality mapping function Q k ( . ) is concave and continuous. In practice, rate distortion curvescorresponding to practical video codecs always follow thisconcave behavior because diminishing returns in quality areachieved as the rate increases.The corresponding system block diagram is shown in Figure1. The three major per-user dynamics are: (1) rate-distortioncharacteristics Q k ( R k ) , (2) channel statistics f k ( γ ) , and (3)delay QoS requirements { D k, th , P k, th } . We focus on resourceallocation and rate adaptation for users served by a singlecell. We note, however, that the results in the paper applyto a system with other-cell interference if the interference istreated as noise where f k ( γ ) then represents the distributionof the signal to noise and interference ratio (SINR).III. S TATISTICAL D ELAY B OUNDS
In this section, we describe the procedure for providingdelay guarantees by characterizing link-level QoS metricsaccording to the effective capacity link layer model.
A. Queuing Model for Video Transmission
A separate queue is maintained for each video streamat the base station. Given the SNR distribution f k ( γ ) foreach user, the objective is to adapt the source rates R k andthe bandwidth allocation B k such that the following QoSconstraint is satisfiedPr (cid:8) D k > D th k (cid:9) ≤ P th k ∀ k (1)where D k is the queuing and transmission delay for user k video stream, D th k is the statistical delay-bound, and P th k isthe target delay-bound violation probability.The behavior of the queue-length process in queuing-basedcommunication networks is extensively treated in [32]. For anergodic and stationary arrival and service processes, the queuelength at time t of each queue can be bounded exponentiallyas t → ∞ Pr (cid:8) l k ( t ) > l th k (cid:9) . = e − θ k l th k ≤ k ≤ K (2)where l k is the queue length at queue k and l th k is the queue-length threshold. The parameter θ k , termed the QoS exponent ,determines the decay rate and is used to characterize delay.More stringent QoS requirements are characterized by larger θ k while looser QoS requirements require smaller θ k . B. Effective Capacity for Statistical Delay Bounds
The effective capacity (EC) channel model captures a gen-eralized link-level capacity notion of the fading channel bycharacterizing wireless channels in terms of functions that canbe easily mapped to link-level QoS metrics, such as delay-bound violation probability. Thus, it is a convenient tool fordesigning QoS provisioning mechanisms [1], [12].We denote by C k ( θ k ) and A k ( θ k ) the effective capacityand effective bandwidth functions, respectively, for the k th user. Given an arrival process { A k } , its effective bandwidth,denoted by A k ( θ k ) (bits/GoP), is defined as the minimumconstant service rate required to guarantee a specified QoSexponent θ k . In contrast, for a given service process { C k } , itseffective capacity, denoted by C k ( θ k ) (bits/GoP), is defined asthe maximum constant arrival rate which can be supported by { C k } subject to the specified QoS exponent θ k .The effective capacity can be thought of as the capacity ofthe underlying channel from the perspective of upper protocollayers. The effective capacity theory states that considering thedelay as the performance metric instead of the queue lengthin (2), the QoS guarantee can be written equivalently as afunction of the effective capacity as followsPr (cid:8) D k ( t ) > D th k (cid:9) . = e − θ k C k ( θ k ) D th k ≤ k ≤ K. (3)Moreover, for a stationary and ergodic service process { C k } that is uncorrelated across time frames, the effective capacitycan be expressed as [7] C k ( θ k ) = − θ k ln (cid:0) E γ { e − θ k C k } (cid:1) = − θ k ln (cid:16) E γ { e − θ k B k T log(1+ γ k ) } (cid:17) · (4) To provide the QoS guarantee θ k for user k , the effectivecapacity on the k th link should be equal to the effectivebandwidth [3], [12], [33], i.e., C k ( θ k ) = A k ( θ k ) ∀ k = 1 , · · · , K. The effective bandwidth for the arrival process described aboveis A k ( θ k ) = R k T . Thus, for the QoS constraint in (1) to besatisfied, we need to ensure that C k ( θ k ) ≥ ¯ C k = R k T . TheQoS constraint reduces to R k ≤ − T θ k ln (cid:16) E γ { e − θ k B k T log(1+ γ k ) } (cid:17) . Furthermore, we can find θ k by solving C k ( θ k ) = R k T using(3) as follows θ k = ln(1 /P th k ) T R k D th k . (5)IV. Q UALITY - DRIVEN R ESOURCE A LLOCATION AND R ATE A DAPTATION
In this section, we formulate and solve the problem ofresource allocation and rate adaptation for maximizing thesum video quality across users subject to the statistical delayconstraint.
A. Sum Quality-Maximizing Policy
The problem of optimizing the source rate vector R = { R k } Kk =1 and the bandwidth allocation vector B = { B k } Kk =1 to maximize the sum video quality subject to a statistical delayconstraint per user is formulated as follows max R , B K X k =1 Q k ( R k )s . t . K X k =1 B k = B ; B k ≥ ∀ k (6)Pr (cid:8) D k > D th k (cid:9) ≤ P th k ∀ k (7)Using the theory of effective capacity, Lemma 1 provides themaximal video source rate supported by each user subject tothe statistical delay constraint. Lemma 1.
For a given bandwidth allocation vector B andset of QoS exponents θ , the optimal source rate R k for user k is R ∗ k ( B k , θ k ) = − T θ k ln (cid:16) E γ n e − θ k B k T log(1+ γ k ) o(cid:17) = − T θ k ln (cid:16) E γ n (1 + γ k ) − θkBkT ln(2) o(cid:17) . TABLE IC
OMMONLY USED NOTATION
Video source / QoS metrics (User k ) Wireless Channel / System D th k Delay constraint K Number of video users P th k Delay violation probability B k User k BW allocation θ k QoS exponent B Total bandwidth R k Source rate C k ( θ k ) User k effective capacity R min k Minimum rate representation f k ( γ ) Fading distribution Q k ( R k ) Rate-quality mapping T Channel coherence time d k BW-QoS exponent product R k /B k Source spectral efficiency
Proof:
Since Q k ( R k ) is increasing, increasing the sourcerate for any user improves the objective function. Further-more, the delay bound violation probability Pr (cid:8) D k > D th k (cid:9) is increasing in the source rate. Thus, the optimal solutionmust satisfy Pr (cid:8) D k > D th k (cid:9) = P th k ∀ k . The equality inthe statistical delay bound implies that the effective capacityconstraint C k ( θ k ) ≥ ¯ C k = R k T is satisfied with equality,that is, C k ( θ k ) = R k T . Plugging in C k ( θ k ) from (4), weobtain R ∗ k ( B k , θ k ) = − ln (cid:0) E γ { e − θ k B k T log(1+ γ k ) } /T θ k (cid:1) andthe result follows.Since the QoS exponent in (5) is also a function of thesource rate which in turn is a function of the bandwidth allo-cation, it cannot be computed beforehand. Thus, we compute θ k jointly with B k as part of the optimization problem. Thefollowing Lemma re-writes the delay constraint in terms of theQoS exponent and the bandwidth allocation using the effectivecapacity expression for an uncorrelated channel. Lemma 2.
The statistical delay constraint (7) can be writtenequivalently as E γ n (1 + γ k ) − θkBkT ln(2) o = p k where p k = P th k /D th k . Furthermore, the solution to theproblem requires that θ ∗ k = d k /B ∗ k where the bandwidth-QoS exponent product d k is a constant dependent only onthe channel distribution information and the delay constraint.Proof: From Lemma 1, the statistical delay constraintshould be satisfied with equality. Thus, we can rewrite thestatistical delay constraint as follows e − θ k C k ( θ k ) D th k = P th k θ k T R ∗ k ( B k , θ k ) D th k = − ln( P th k )ln (cid:16) E γ n (1 + γ k ) − θkBkT ln(2) o(cid:17) = ln( P th k ) D th k (8) E γ n (1 + γ k ) − θkBkT ln(2) o = P th k /D th k = p k (9)where (8) follows using Lemma 1. Furthermore, since p k in(9) is a constant independent of the channel statistics and thebandwidth allocation, it follows that θ k B k should be constant.Thus, θ ∗ k = d k /B ∗ k for some constant d k .Applying Lemma 1 and Lemma 2, we rewrite the resourceallocation problem equivalently as follows min B ,θ − K X k =1 Q k (cid:18) − T θ k ln (cid:16) E γ { (1 + γ k ) − θkBkT ln(2) } (cid:17)(cid:19) s . t . K X k =1 B k = B (10) E γ n (1 + γ k ) − θkBkT ln(2) o = p k ∀ k (11) B k ≥ ∀ k. (12)First, we prove the convexity of the problem above both in B and θ in Lemma 3. Lemma 3.
The problem is convex in B and θ . Furthermore,a feasible point always exists if Q k ( R k ) is defined for every R k ≥ . Proof: First, we show convexity of the objective functionin B k . Since − Q k ( R k ) is convex in R k and non-increasing, itsuffices to show that R k = − T θ k ln (cid:16) E γ { (1 + γ k ) − θkBkT ln(2) } (cid:17) is concave in B k . We have αR k ( B k ) + (1 − α ) R k ( B k )= − T θ k (cid:20) α ln (cid:18) E γ (cid:26) (1 + γ k ) − θkB kT ln(2) (cid:27)(cid:19) + (1 − α ) ln (cid:18) E γ (cid:26) (1 + γ k ) − θkB kT ln(2) (cid:27)(cid:19)(cid:21) = − T θ k ln E γ (cid:26) (1 + γ k ) − θkB kT ln(2) (cid:27) α E γ (cid:26) (1 + γ k ) − θkB kT ln(2) (cid:27) − α ! . Now, apply Lyaponuv’s inequality which states that E [ | Y | s ] r × E [ | Y | l ] − r ≥ E [ | Y | m ] where r = ( l − m ) / ( l − s ) and < s < m < l . We assume B k < B k w.l.o.g. andapply the inequality for | Y | = (1 + γ k ) − , r = α , s = θ k T B k / ln(2) , and l = θ k T B k / ln(2) . Thus, α =( θ k T B k / ln(2) − m ) / ( θ k T B k / ln(2) − θ k T B k / ln(2)) . Solv-ing for m , we obtain m = θ k T ln(2) ( αB k + (1 − α ) B k ) . Applyingthe inequality, we obtain αR k ( B k ) + (1 − α ) R k ( B k ) ≤ − T θ k h ln (cid:16) E γ n (1 + γ k ) − θkT ln(2) ( αB k +(1 − α ) B k ) o(cid:17)i = R k ( αB k + (1 − α ) B k ) . Thus, R k ( B k ) is concave in B k and the objective functionis convex in B k . Next, we show that the objective func-tion is also convex in θ k , It is clear that the component ln (cid:16) E γ { (1 + γ k ) − θkBkT ln(2) } (cid:17) is convex in θ k as well by theexact arguments used above. Thus, we have R k ( θ k ) = − T θ k f ( θ k ) where f ( θ k ) is convex and non-increasing in θ k . Also, T θ k is convex and non-increasing. The product oftwo convex and non-increasing functions is convex. Thus, T θ k f ( θ k ) is convex and R k ( θ k ) is concave in θ k . Applyingthe same arguments, we can further show that the constraint(11) is convex in B k and θ k . Thus, the optimization problemis convex and has a unique solution if it feasible.We further show that the problem is always feasible if Q k ( R k ) is defined for every R k ≥ , i.e., the video source ratecan be arbitrarily reduced. Using Lemma 2, we can alwaysfind a positive d k = B k θ k that satisfies the statistical delayconstraint. For any B k , θ k that satisfies B k θ k = d k , we useLemma 1 to find a corresponding source rate R k that can besupported subject to the delay bound. To summarize, givenany delay bound and channel conditions, there exists a smallenough source rate such that the delay bound is met.Theorem 1 provides the optimal resource allocation solutionby showing that the optimal operating point per user is suchthat the rate-distortion slope is the inverse of the supportedvideo source rate per unit bandwidth. The maximum sourcerate per unit bandwidth is a fundamental measure of thenumber of video bits per channel use that can be deliveredsubject to the QoS requirement which we refer to as the sourcespectral efficiency . Theorem 1.
Sum Quality-Maximizing Resource Allocation:
The optimal bandwidth allocation B k and source rate R k foreach user k is such that ∂Q k ( R ∗ k ( B ∗ k , θ ∗ k )) ∂R ∗ k ( B ∗ k , θ ∗ k ) | {z } Rate distortion slope × R ∗ k ( B ∗ k , θ ∗ k ) B ∗ k | {z } Source spectral efficiency = ρ where R ∗ k ( B k , θ k ) = − T θ k ln (cid:16) E γ { (1 + γ k ) − θkBkT ln(2) } (cid:17) and ρ is a constant chosen such that P Kk =1 B k = B . Further, R ∗ k ( B k , θ ∗ k ) /B k is independent of B k and is only a functionof the fading distribution f k ( γ ) , the delay bound D th k , and thedelay bound violation probability P th k .Proof: We write the Lagrangian function for the re-source allocation problem and the Karush-Kuhn-Tucker (KKT)conditions to derive the optimal solution. The Lagrangianfunction L ( B , θ , ρ, λ , µ ) of the resource allocation problemis as follows L ( B , θ , ρ, λ , µ ) = − K X k =1 Q k (cid:18) − T θ k ln (cid:18) E γ { (1 + γ k ) − θkBkT ln(2) } (cid:19)(cid:19) − ρ K X k =1 B k − B ! + K X k =1 λ k (cid:18) E γ (cid:26) (1 + γ k ) − θkBkT ln(2) (cid:27) − p k (cid:19) − K X k =1 µ k B k where ρ , λ = { λ k } Kk =1 , and µ = { µ k } Kk =1 are the Lagrangemultipliers corresponding to constraints (10), (11), and (12)respectively. The KKT conditions are ∂L∂B k (cid:12)(cid:12) B k = B ∗ k = 0 , ∂L∂θ k (cid:12)(cid:12) θ k = θ ∗ k = 0 ∀ kµ ∗ k ≥ , B ∗ k ≥ , µ ∗ k B ∗ k = 0 ∀ k P Kk =1 B ∗ k = B E γ (cid:26) (1 + γ k ) − θ ∗ kB ∗ kT ln(2) (cid:27) = p k ∀ k. Taking the derivative of L ( B , θ , ρ, λ , µ ) with respect to B k ,we obtain ∂L∂B k = ∂Q k ( R ∗ k ( B k , θ k )) ∂R ∗ k ( B k , θ k ) × E γ (cid:26) (1 + γ k ) − θkBkT ln(2) log(1 + γ k ) (cid:27) E γ (cid:26) (1 + γ k ) − θkBkT ln(2) (cid:27) − ρ − λ k θ k T (cid:18) E γ (cid:26) (1 + γ k ) − θkBkT ln(2) log(1 + γ k ) (cid:27)(cid:19) − µ k = θ k T E γ (cid:26) (1 + γ k ) − θkBkT ln(2) log(1 + γ k ) (cid:27)(cid:18) ∂Q k ( R ∗ k ) /∂R ∗ k θ k T p k − λ k (cid:19)| {z } g ( B k ,θ k ,λ k ) − ρ − µ k where we used the last KKT condition and we defined g ( B k , θ k , λ k ) as stated above. Thus, we have ∂L∂B k = θ k g ( B k , θ k , λ k ) − ρ − µ k . Note that ∂Q k ( R ∗ k ) /∂R ∗ k is the slope of the rate distortionfunction at R ∗ k . Now, we take the derivative of the Lagrangianwith respect to θ k as follows ∂L∂θ k = ∂Q k ( R ∗ k ) ∂R ∗ k (cid:20) ln ( p k ) T θ k + B k θ k p k E γ (cid:26) (1 + γ k ) − θkBkT ln(2) log(1 + γ k ) (cid:27)(cid:21) − λ k B k T (cid:18) E γ (cid:26) (1 + γ k ) − θkBkT ln(2) log(1 + γ k ) (cid:27)(cid:19) = B k T E γ (cid:26) (1 + γ k ) − θkBkT ln(2) log(1 + γ k ) (cid:27) (cid:18) ∂Q k ( R ∗ k ) /∂R ∗ k θ k T p k − λ k (cid:19)| {z } g ( B k ,θ k ,λ k ) + ∂Q k ( R ∗ k ) ∂R ∗ k ln ( p k ) T θ k = B k g ( B k , θ k , λ k ) + ∂Q k ( R ∗ k ) ∂R ∗ k ln ( p k ) T θ k . Substituting ∂L∂B k (cid:12)(cid:12) B k = B ∗ k = 0 and ∂L∂θ k (cid:12)(cid:12) θ k = θ ∗ k = 0 , we obtainthe following set of equations ( θ ∗ k g ( B ∗ k , θ ∗ k , λ ∗ k ) − ρ ∗ − µ ∗ k = 0 B ∗ k g ( B ∗ k , θ ∗ k , λ ∗ k ) + ∂Q k ( R ∗ k ( B ∗ k ,θ ∗ k )) ∂R ∗ k ( B ∗ k ,θ ∗ k ) ln( p k ) T θ ∗ k = 0 . Substituting g ( B ∗ k , θ ∗ k , λ ∗ k ) = ( µ ∗ k + ρ ∗ ) /θ ∗ k into the secondequation, we obtain B ∗ k ( µ ∗ k + ρ ∗ ) θ ∗ k + ∂Q k ( R ∗ k ( B ∗ k , θ ∗ k )) ∂R ∗ k ( B ∗ k , θ ∗ k ) 1 T θ ∗ k ln ( p k ) = 0 . Equivalently, ∂Q k ( R ∗ k ( B ∗ k , θ ∗ k )) ∂R ∗ k ( B ∗ k , θ ∗ k ) ln ( p k ) = − T B ∗ k θ ∗ k ( ρ ∗ + µ ∗ k ) . To obtain another form of the expression that hasa useful interpretation, we substitute R ∗ k ( B k , θ k ) = − ln (cid:18) E γ (cid:26) (1 + γ k ) − θ ∗ kB ∗ kT ln(2) (cid:27)(cid:19) / ( T θ k ) . Thus, we have ∂Q k ( R ∗ k ( B ∗ k , θ ∗ k )) ∂R ∗ k ( B ∗ k , θ ∗ k ) = ( ρ ∗ + µ ∗ k ) B ∗ k R ∗ k ( B ∗ k , θ ∗ k ) . First, we analyze µ ∗ k . If user k is not assigned any resources, µ ∗ k > and B ∗ k = 0 . This requires either θ ∗ k = ∞ or p k = 1 tosatisfy constraint (11). Substituting θ ∗ k = ∞ into (3), we obtain P th k = 1 , i.e., the target delay bound violation probability isone. On the other hand, p k = P th k /D th k = 1 is equivalent toeither P th k = 1 or D th k = ∞ . Thus, the optimal bandwidthallocation is always non-zero unless the target delay bound isinfinite or the target delay bound violation probability is one.Since this case is not meaningful and impractical, we have µ ∗ k = 0 and B ∗ k > ∀ k . Therefore, ∂Q k ( R ∗ k ( B ∗ k , θ ∗ k )) ∂R ∗ k ( B ∗ k , θ ∗ k ) = ρ ∗ B ∗ k R ∗ k ( B ∗ k , θ ∗ k ) . Equivalently, we have ∂Q k ( R ∗ k ( B ∗ k , θ ∗ k )) ∂R ∗ k ( B ∗ k , θ ∗ k ) × R ∗ k ( B ∗ k , θ ∗ k ) B ∗ k = ρ ∗ . We refer to the term R ∗ k ( B ∗ k , θ ∗ k ) /B ∗ k as the source spectralefficiency as it represents the number of video bits that canbe delivered per channel use subject to the QoS constraint. The interpretation of the result is that the optimal resourceallocation is such that the slope of the rate distortion curvemultiplied by the source spectral efficiency is the same forall users. Furthermore, that constant determined by ρ ∗ iscomputed such that the sum bandwidth constraint is satisfiedwith equality. Note that as ρ decreases, B ∗ k increases forall k and vice versa, thus reducing the problem into aone-dimensional root-finding problem involving solving for P k B ∗ k ( ρ ) − B = 0 to find ρ .We make the following observations about the result ob-tained in Theorem 1:1) The source spectral efficiency jointly character-izes the channel and the QoS requirement:
Thesource spectral efficiency R ∗ k ( B k ) /B k is independentof the resource allocation B k . It is only a function ofthe channel statistics, the delay bound, and the delaybound violation probability. Specifically, R ∗ k ( B k ) /B k =ln (cid:0) /P th k (cid:1) /D th k T d k .2) The resource allocation policy is source spectralefficiency-optimal:
The optimal solution to the quality-maximizing problem is source spectral efficiency-optimal, since R ∗ k ( B k ) is the maximum rate that couldbe supported for the given delay bound and violationprobability and R ∗ k ( B k ) /B k is independent from B k .Thus, ln (cid:0) /P th k (cid:1) /D th k T d k is a fundamental measure ofthe maximum number of video bits per channel use thatcan be delivered subject to the QoS requirement.The result in Theorem 1 applies to any fading distribution.In practice, however, solving for the optimal resource alloca-tion and source rates requires numerical computation of d k bysolving E γ n (1 + γ k ) − dkT ln(2) o = p k . The following corollarysimplifies the expression for the special case of a Rayleighchannel. Corollary 1.
For a Rayleigh channel with average SNR ¯ γ k such that f k ( γ ) = γ k e − γ/ ¯ γ k , the delay constraint reduces to ln (cid:18) E dkT ln(2) (cid:18) γ k (cid:19)(cid:19) = ln( P th k ) D th k − γ k + ln(¯ γ k ) where E a ( x ) = R ∞ ( e − xt /t a ) dt is the exponential integral.Proof: We reduce the delay constraint from Lemma 2 asfollows p k = E γ n (1 + γ ) − dkT ln(2) o = 1¯ γ k Z ∞ (1 + γ ) − dkT ln(2) e − γ/ ¯ γ k d γ = − γ k e / ¯ γ k (1 + γ ) − dkT ln(2) E dkT ln(2) (cid:18) γ ¯ γ k (cid:19)(cid:21) ∞ = 1¯ γ k e / ¯ γ k E dkT ln(2) (cid:18) γ k (cid:19) where E a ( x ) = R ∞ ( e − xt /t a ) dt is the exponential integral.Substituting p k = P th k /D th k and taking the natural logarithmon both sides, the result follows. B. Fairness-driven Resource Allocation
Next, we consider the alternative objective of maximizingthe minimum video quality across all served users to providea notion of fairness in the resource allocation policy. Underthis objective, the problem can be formulated as follows max R , B min k Q k ( R k )s . t . K X k =1 B k = B Pr (cid:8) D k > D th k (cid:9) ≤ P th k ∀ kB k ≥ ∀ k where R is the source rate vector and B is the bandwidthallocation vector. Corollary 2.
The optimal resource allocation vector B is suchthat B ∗ k = T d k D th k ln(1 /P th k ) Q − k ( q ) where q is selected such that P Kk =1 T d k D th k Q − k ( q ) / ln(1 /P th k ) = B .Proof: Given any resource allocation { B k } Kk =1 , let k ′ =argmin k Q k ( R ∗ k ( B k ) . Note that, from Lemma 3, for anyresource allocation { B k } Kk =1 , a feasible solution ( R ∗ , B ∗ )always exists if Q k ( R k ) is defined for every R k ≥ .Thus, k ′ is well defined. By increasing B k ′ , the objectivefunction min k Q k ( R k ) = Q k ′ ( R k ′ ) keeps improving until argmin k Q k ( R ∗ k ( B k ) = k ′ . Thus, it follows that the optimalsolution requires Q k ( R ∗ k ( B ∗ k ) = Q j ( R ∗ j ( B ∗ j , θ ∗ j )) = q ∀ j, k .Furthermore, given that Q k ( R k ) is defined for every R k ≥ , asmall enough q always exists. Given any target video quality q for each user, using Lemma 1 and Lemma 2, the correspondingresource allocation for user k is B k = d k θ k = T d k D th k ln(1 /P th k ) Q − k ( q ) Thus, the required sum bandwidth for all users to a videoquality q for each user is K X k =1 B k = T K X k =1 d k D th k Q − k ( q )ln(1 /P th k ) and B ∗ k can be found by numerically solving for q such that P Kk =1 B k = B . C. Practical Considerations
In practice, the number of operating points on the rate-distortion curve is finite and determined by the differentrepresentations of the video sequence available at the server.Let ( R min k , Q min k ) correspond to the operating point on user k rate distortion curve that provides the lowest available rate. If R ∗ k ( B ∗ k , θ ∗ k ) < R min k , then the solution corresponding to user k cannot be realized. Thus, practically, user k cannot use any ofthe available video descriptions and still meet the target delayconstraint. V. M
AXIMAL U SER S UBSET S CHEDULING AND U SER A DMISSION
In this section, we solve the problem of selecting thelargest subset of users to serve such that each user in thesubset can meet their delay constraint. We solve the problemunder sum quality-maximizing resource allocation as well asfairness-based resource allocation. Afterwards, we present auser admission policy that determines the criterion for a newuser to be admitted into the system given the delay constraintand the delay constraints for the existing users so that thedelay constraint of each user is not compromised.Theorem 2 summarizes the main result of the section. Itpresents the optimal scheduling policy that selects a maximalsubset of users such that all user meet their statistical delayconstraint. To summarize, the optimal scheduling policy canbe obtained in polynomial time in the number of users and itinvolves computing the minimum bandwidth required by eachuser to support their QoS requirement, using it as a sortingcriterion, and scheduling the first sorted users such that thesum of their minimum bandwidth requirement does not exceedthe total bandwidth. Under the fairness constraint, a similarsolution is obtained with the major difference that the sortingcriterion is the video quality corresponding to the minimumrate representation of the video sequence.
Theorem 2.
Maximal User Subset Scheduling:
Under max-imum sum quality resource allocation, the scheduling policyto maximize the number of users that can meet their QoSrequirement is as follows. Define B min k = D th k d k T ln(1 /P th k ) R min k (13) and let ℓ ( i ) be the sorting operation in increasing order on B min k so that B min ℓ ( i ) is the i th sorted element ∀ i = 1 , · · · , K ,then the maximum number of users supported N ∗ is N ∗ = argmax N N X i =1 B min ℓ ( i ) s . t . N X i =1 B min ℓ ( i ) ≤ B and the corresponding scheduled users are ℓ (1) , · · · , ℓ ( N ∗ ) .Under fairness-based resource allocation, the schedulingpolicy to maximize the number of users that can meet theirQoS requirement is as follows. Define q min k = Q k ( R min k ) andlet m ( i ) be the sorting operation in increasing order on q min k so that q min m ( i ) is the i th sorted element ∀ i = 1 , · · · , K , thenthe maximum number of users supported N ∗ is N ∗ = argmax N N X i =1 B m ( i ) ( q min m ( N ) ) s.t. N X i =1 B m ( i ) ( q min m ( N ) ) ≤ B and the corresponding scheduled users are m (1) , · · · , m ( N ∗ ) .Proof: If user k is scheduled, the minimum bandwidthrequirement is such that the user’s delay constraint can be metwith the minimum source rate. As the source rate increases,a larger bandwidth is required to maintain the same delayconstraint. Thus, the minimum bandwidth B min k required tomeet the statistical delay bound is B min k = B ∗ k ( R min k ) = D th k d k T ln(1 /P th k ) R min k . Given the minimum bandwidth required per user, the problemof finding a maximal user subset reduces to a special case ofthe Knapsack problem [34] where all items (users) have equalvalue v k = 1 and weights (bandwidths) w k = B min k . Since v k = 1 ∀ k , the exact solution corresponds to the N ∗ userswith the smallest w k such that P w k < B . More formally,define ℓ ( i ) to be the sorting operating in increasing order sothat B min ℓ ( i ) is the i th sorted element ∀ i = 1 , · · · , K , then themaximum number of users supported N ∗ is N ∗ = argmax N N X i =1 B min ℓ ( i ) s . t . N X i =1 B min ℓ ( i ) ≤ B and it follows that the corresponding scheduled users are ℓ (1) , · · · , ℓ ( N ∗ ) .Under the fairness-based resource allocation, define m ( i ) be the sorting operating in increasing order so that q min m ( i ) isthe i th sorted element ∀ i = 1 , · · · , K . Corollary 2 states that,under the fairness-based policy, all users should achieve thesame target video quality ¯ q . Defining q min k = Q k ( R min k ) , itfollows that if q min k > ¯ q , user k cannot be served. Thus, user m ( i ) can only be served if m (1) , · · · , m ( i − are also served.This reduces the number of possible solutions to K + 1 . Thefirst possibility is serving no users, the second is serving user m (1) , all the way to option K +1 which requires serving users m (1) , · · · , m ( K ) . Furthermore, if users m (1) , · · · , m ( N ) areserved, the minimum target quality is q min m ( N ) since all users arerequired to maintain the same quality and any ¯ q < q min m ( N ) isnot feasible for at least one user. Thus, the maximum numberof users supported N ∗ can be found as follows N ∗ = argmax N N X i =1 B m ( i ) ( q min m ( N ) ) s . t . N X i =1 B m ( i ) ( q min m ( N ) ) ≤ B and it follows that the corresponding scheduled users are m (1) , · · · , m ( N ∗ ) .Next, we contrast the scheduling solution under maximumsum quality resource allocation with that under fairness-basedresource allocation. First, with maximum sum quality resourceallocation, the system can always support at least the samenumber of users as with the fairness-based policy. This can beseen in the following inequalities B min ℓ ( i ) ≤ B min m ( i ) ≤ B m ( i ) ( q min m ( N ) ) where the first inequality follows since ℓ ( i ) is the i th sortedelement according to the B min k criterion and the secondinequality follows by the definition of B min k . Furthermore,no scheduling policy can serve more users than that undermaximum sum quality resource allocation since it is based oneach served user operating at the minimum bandwidth requiredto maintain the QoS requirement. Finally, the fairness-basedpolicy and the quality-maximizing policy support the samenumber of users if Q k ( R ) = Q ( R ) ∀ k and R min k = R min ∀ k ,that is, all users have the same rate-distortion behavior and minimum rate representations. While this is typically not thecase in practice, it provides the following useful intuition:Achieving fairness is least costly when the rate-distortionbehavior of the users in the network is similar such thatall users achieve similar incremental gains in quality whenincreasing their source rate.We next consider the user admission problem, whereby aset of users are already using the wireless system resourcesfor video streaming, and a new user enters the system. Forthis scenario, we derive the user admission criterion thatensures that (1) the admitted user meets the target QoSrequirement, and (2) the current users’ QoS requirements’are not jeopardized by the admitted user. This user admissionpolicy is presented in Theorem 3. Theorem 3.
User Admission Criteria:
Under the quality-maximizing policy, a new user with QoS requirements { D th K +1 , P th K +1 } can be admitted into the system with K existing users operating at { R k } Kk =1 if R min k R k ≤ − R ∗ K +1 B × d K +1 T D th K +1 ln(1 /P th K +1 ) ∀ k = 1 , · · · , KR min K +1 R ∗ K +1 ≤ . Under the quality fairness policy where each of the K useroperates with quality q init = Q i ( R i ) , user K + 1 with QoSrequirements { D th K +1 , P th K +1 } can be admitted if R min i R i ≤ − Q − K +1 ( q final ) B × d K +1 T D th K +1 ln(1 /P th K +1 ) ∀ i = 1 , · · · , KR min K +1 Q − K +1 ( q final ) ≤ where q final is computed such that P K +1 k =1 B ∗ k = B .Proof: For user K to be admitted, two conditions need tobe satisfied. First, the other users need to be able to maintaintheir own QoS constraint. Second, the admitted user needsto be allocated enough resources to meet the target QoSrequirement. If admitted, user K + 1 gets bandwidth B ∗ K +1 ( R K +1 ) = d K +1 θ ∗ K +1 = R K +1 d K +1 T D th K +1 ln(1 /P th K +1 ) . The new resource allocation for the rest of the users is obtainedby allocating an equivalent total bandwidth of B − B ∗ K +1 .The source rate scales linearly in the bandwidth according toTheorem 1, i.e., R k ( B k ) /B k is independent of B k for eachuser k . Thus, since the old source rate for user k is R k = B × ( R k /B ) , admitting user K +1 reduces user k ’s source rateto R new k = ( B − B ∗ K +1 ) × ( R k /B ) . Thus, the user admissioncondition reduces to R min k ≤ R new k = R k ( B − B ∗ K +1 ) B = R k − R K +1 d K +1 T D th K +1 B ln(1 /P th K +1 ) ! . Thus, the condition can be written equivalently as R min k R k ≤ − R ∗ K +1 B × d K +1 T D th K +1 ln(1 /P th K +1 ) . (14)If the condition in (14) is satisfied for each user k , the QoSconstraint is maintained after user K + 1 is admitted. Inaddition, for user K + 1 to satisfy their own QoS requirement,the maximum source rate required should exceed the minimumrate representation R min K +1 , i.e., R min K +1 ≤ R ∗ K +1 . Under thefairness-based policy, the same analysis follows with theexception that R ∗ K +1 = Q − K +1 ( q final ) such that, after useradmission, all users have equal video quality q final and q final isselected such that P K +1 k =1 B k = B .VI. J OINT S CHEDULING AND R ESOURCE A LLOCATION A LGORITHM
In this section, we present the Algorithm that jointly selectsa maximal user subset from a set of candidate video users withdelay QoS constraints. Among the scheduled users, resourcesare allocated either to maximize the sum video quality orminimum video quality. The Algorithmic description for thesum quality maximizing policy is provided in Algorithms 1.First, we use the delay constraint D th k and the delay con-straint violation probability P th k to compute p k the bandwidth-QoS exponent product d k . Next, we compute the minimumbandwidth requirement B min k for each user to meet their QoSconstraint and the corresponding video quality q min k . Based onthe channel conditions and the QoS requirement, the maximumsource spectral efficiency R ∗ k ( B k ) /B k is computed for eachuser. The next step involves scheduling a maximal user subsetbased on Theorem 2 depending on the selected utility function.After a subset of users is selected, the optimal bandwidthallocation B ∗ k is obtained using Theorem 1. Algorithm 1
Quality-maximizing Joint Scheduling and Re-source Allocation.
Given K users with delay QoS requirements { D th k , P th k } Kk =1 , fadingdistributions f k ( γ ) , and total bandwidth B . Step 1. Compute QoS-related Metrics for k = 1 → K do
1) Find p k = P th k /D th k ∀ k .2) Given f k ( γ ) , find the bandwidth-QoS exponent product d k = B ∗ k θ ∗ k ∀ k by solving E γ (cid:26) (1 + γ k ) − dkT ln(2) (cid:27) = p k .3) Compute minimum bandwidth requirement per user B min k = D th k d k T ln(1 /P th k ) R min k and minimum quality q min k = Q k ( R min k ) .4) Compute the maximum source spectral efficiency R ∗ k ( B k ) /B k =ln (cid:0) /P th k (cid:1) /D th k T d k . end for Step 2. Select Maximal User Subset
1) Sort B min k with operator ℓ ( i ) such that B min ℓ ( i ) is the i th sortedelement ∀ i = 1 , · · · , K
2) Compute maximum number of users N ∗ = argmax N P Ni =1 B min ℓ ( i ) s.t. P Ni =1 B min ℓ ( i ) ≤ B
3) Schedule users ℓ (1) , · · · , ℓ ( N ∗ ) . Step 3. Allocate Resources
1) Set ρ = ρ initial
2) Solve ∂Q k ( R ∗ k ( B ∗ k ,θ ∗ k )) ∂R ∗ k ( B ∗ k ,θ ∗ k ) = Td ∗ k ( ρ ∗ )ln( p k ) for R ∗ k ∀ k = ℓ (1) , · · · , ℓ ( N ∗ )
3) Solve θ ∗ k = − ln( p k ) / ( T R ∗ k ) ∀ k = ℓ (1) , · · · , ℓ ( N ∗ )
4) Solve B ∗ k = d k /θ ∗ k ∀ k = ℓ (1) , · · · , ℓ ( N ∗ )
5) If P B ∗ k < B , decrease ρ and repeat 2,3,4, otherwise increase ρ andrepeat 2,3,4 until | P B ∗ k − B | < ǫ . Delay bound D th (sec) D e l a y b o und v i o l a t i o np r o b a b ili t y P t h . . . . . . . . . −4 −3 −2 −1 Source Spectral Efficiency R ∗ /B ∗ (bps/Hz)for ¯ γ = 0 dB PracticallyInfeasibleRegion (a) Average SNR ¯ γ = 0 dB. Delay bound D th (sec) D e l a y b o und v i o l a t i o np r o b a b ili t y P t h . . . −4 −3 −2 −1 Source Spectral Efficiency R ∗ /B ∗ (bps/Hz)for ¯ γ = 20 dB PracticallyInfeasibleRegion (b) Average SNR ¯ γ = 20 dB.Fig. 2. Contour plot of source spectral efficiency, the maximum source bits per channel use, for a Rayleigh channel vs. delay bound D th and delay boundviolation probability P th . In the lower left regime, the steep decline puts a practical limitation on the feasible QoS operating points. VII. R
ESULTS
In this section, we present results and analysis to demon-strate the performance of the proposed scheduling and resourceallocation algorithms for real-time video transmission. First,we present analysis of the source spectral efficiency underdifferent QoS requirements and channel conditions. Next, weanalyze the resource allocation for the two user case withhybrid QoS requirements. Finally, we present the general caseof multiple users with joint scheduling and resource allocation.
A. Source Spectral Efficiency under Delay Constraints
Figure 2 shows a contour plot of the maximum source spec-tral efficiency supported by a Rayleigh channel as a function ofthe delay bound and delay bound violation probability. Note,from Theorem 1, that the maximum source spectral efficiencycan be expressed as ln (cid:0) /P th k (cid:1) /D th k T d k and is achieved bythe proposed rate adaptation algorithm. The main observationfrom the plot is that there exists a boundary beyond whichthe source spectral efficiency declines very rapidly, makingpractical realization of the QoS constraint impossible since itwould require either using a very large bandwidth or operatingat a very small source rate. For example, to achieve D th k = 0 . sec and P th k = 10 − , we require a bandwidth 1000 timesthe source rate even if the average SNR is ¯ γ = 20 dB. Thisprovides a practical insight into the range of feasible QoSconstraints.Figure 3 shows the achievable source spectral efficiencyover a range of SNRs for D th k = 0 . sec. It can be seen thatthe advantage of good channel conditions is minimal undervery stringent QoS requirements whereas relaxing the targetdelay violation probability enables supporting high sourcerates even at poor channel conditions. We note that theseresults are considered as an extreme case since they considera Rayleigh channel and uncorrelated fading instances. Withan uncorrelated channel, the randomness in the service rate islargest, thus exacerbating the problem of maintaining a certaindelay constraint. −10 0 10 20 30 10 −4 −2 D e l a y b o u n d v i o l a t i o np r o b a b ili t y P t h Average SNR ¯ γ S o u r ce Sp ec t r a l E ffi c i e n c y R ∗ / B ∗ ( bp s / H z ) Fig. 3. Source spectral efficiency for a Rayleigh channel vs. average SNR ¯ γ and target delay bound violation probability P th for a target delay bound D th = 0 . sec. B. Resource Allocation under Hybrid Delay QoS Require-ments
Next, we consider two users with different QoS require-ments sharing a wireless channel. User 1 has a delay constraint D th1 = 2 sec corresponding to a typical live video streamingapplication and user 2 has a delay constraint D th2 = 0 . sec corresponding to a typical interactive video conferencingapplication. The target delay bound violation probability is . for both users. To understand the effect of unequal delayrequirements in isolation, we consider the case where bothusers have the same rate distortion characteristics. Specifi-cally, we use the Foreman video sequence [35] encoded withH.264/AVC. The GoP structure is IBP BP · · · and the GoPduration is 16 frames. The MB size is × and we usethe CIF resolution of × . The different rate-distortionoperating points are obtained by modifying the quantization User 1 Average SNR ¯ γ (dB) U s e r A v e r ag e S N R ¯ γ ( d B ) −20 −15 −10 −5 0 5 1002468101214161820 User 1ServedUser 2ServedNoUsersServed Both UsersServed (a) Service regions.
User 1 Average SNR U s e r A v e r ag e S N R
38 1510 521 −20 −15 −10 −5 0 5 1002468101214161820 Mean qualityper served user 3530252015105 γ γ (dB) ( d B ) (b) Corresponding average quality (STRRED) per user.Fig. 4. Two user case study with unequal QoS requirements, that is, D th1 = 2 sec, D th2 = 0 . sec, and P th1 = P th2 = 0 . . The minimum source rate for auser to be served is R min1 = R min2 = 185 Kbps, and B = 5 MHz. The significant difference in SNR to achieve the same video quality shows that schedulingbased only on channel conditions is highly suboptimal. Note that lower STRRED values corresponds to higher video quality. parameters of the discrete cosine transform (DCT) coefficients.We measure video quality using the spatio-temporal reducedreference entropic differencing (STRRED) index [36]. It isa reduced reference (RR) video quality metric that usescombined spatial and temporal information differences andcorrelates quite well with subjective quality. Further, it requiresminimal exchange of side information from the video server,making it suitable for resource allocation at the networkedge. We specifically use
STRRED M corresponding to thevertically oriented subband which is shown in [36] to performbest. Note that the higher values of STRRED correspond tolower video quality. Thus, our proposed algorithm uses therate-quality function Q k ( R k ) = − STRRED M ( R k ) .The minimum rate representation is R min = 185 Kbps andit provides a corresponding STRRED of . The total systembandwidth is 5 MHz. Figure 4a shows the range of SNRs overwhich: (1) None of the users can be served, (2) only user 1 canbe served, (3) only user 2 can be served, and (4) both userscan be served simultaneously. The main observation is that theSNR range is very different for the two users. User 1 can beserved with as low as - dB while user 2 requires at least8 dB to satisfy their QoS requirement. This shows that MaxSNR-based scheduling is highly suboptimal for video userswith different QoS requirements and QoS-aware schedulingachieves significant gains. For instance, for − ≤ ¯ γ ≤ and ¯ γ < ¯ γ , a Max SNR scheduler always favors user 2,although it is clear that in this entire operating region, user2 cannot meet the QoS requirement while user 1 can. Figure4b shows a contour plot of the corresponding average videoquality per user at each operating point. C. Multiuser Scheduling and Resource Allocation
Next, we consider the general case where there is a largenumber of users to be served. We apply the scheduling solutionderived in Theorem 2 to select a user subset followed bythe corresponding resource allocation policy. We consider a single cell setup where the users are distributed accordingto a poisson point process (PPP) in the cell. Half the videousers have a delay constraint D th1 = 2 sec corresponding tolive video users and the other half have a delay constraint D th2 = 0 . sec corresponding to a typical two-way videoconferencing user. The target delay bound violation probabilityis . for both sets of users and the total system bandwidthis 20 MHz. We use three video sequences [35] encoded withH.264/AVC. The GoP structure is IBP BP · · · and the GoPduration is 16 frames. The MB size is × and we use theCIF resolution of × . The minimum rate representationof the three different video are R min k = 160 , , and Kbps. For accurate channel modeling, we derive the averageSNRs ¯ γ k from the distances d k using the following relation ¯ γ k = 10 log ( P t ) − K dB − δ log ( d k ) −
10 log( N B k ) where K dB is the pathloss constant, δ is the pathloss exponent,and N is the power spectral density of AWGN noise. In oursimulations, K dB = 21 . dB, δ = 3 . , and N = 4 × − W/Hz.Figure 5a shows the number of users supported using theproposed maximal user subset scheduling algorithm alongwith (a) sum quality-maximizing resource allocation, and (b)fairness-based resource allocation. As we proved in Theorem2, the maximal user subset scheduling algorithm with sumquality-maximizing resource allocation outperforms any otherscheduling/resource allocation combination under the samedelay constraints. To distinguish the resource allocation andscheduling gains, we consider two baselines. In both baselines,scheduling is based on QoS-aware Max SNR, that is, themaximum number of the highest SNR users that can besupported such that they can meet their delay constraint isscheduled. In the first baseline, the available bandwidth isdivided equally among the scheduled users. In the secondbaseline, the bandwidth is allocated according to the sumquality-maximizing resource allocation. Thus, the difference User Density (Users/Km ) N u m b e r o f u s e r ss e r v e d Scheduling:
Maximal User Subset
Resource Allocation:
Max Sum Quality
Scheduling:
Maximal User Subset
Resource Allocation:
Max Min Quality
Scheduling:
Max SNR
Resource Allocation:
Max Sum Quality
Scheduling:
Max SNR
Resource Allocation:
Equal BW SchedulingGainFairnessPenaltyBaselinesResourceAllocationGain (a) Average number of users supported
User Density (User/Km ) S t a nd a r dd e v i a t i o n o f D M O S s c o r e s a c r o ss u s e r s Scheduling:
Maximal User Subset
Resource Allocation:
Max Sum Quality
Scheduling:
Maximal User Subset
Resource Allocation:
Max Min Quality
Scheduling:
Max SNR
Resource Allocation:
Equal BWBaselineSum quality-maximizingresource allocationFairness-basedresource allocation (b) Standard deviation of DMOS-mapped video qualityFig. 5. Analysis of proposed scheduling and resource allocation algorithms in comparison to the baselines vs. user density for P t = 30 dBm and B = 20 MHz.The scheduling baseline is QoS-aware Max SNR, that is, the maximum number of the highest SNR users that can meet their delay constraint are scheduled.In contrast to Max sum quality and fairness-based resource allocation, the resource allocation baseline shares the resources equally among scheduled users. between the baselines is the resource allocation gain and thedifference between the second baseline and the proposed algo-rithm is the scheduling gain. We observe that significant gainsare achieved in terms of the total number of users supportedin the system. The resource allocation gain corresponds to1.6x increase in capacity due to the better partitioning of thewireless system resources. The scheduling gain corresponds to2.2x-3.5x increase in capacity. Furthermore, under the fairnessconstraint, the number of users supported drops because forc-ing users with hybrid QoS requirements and different channelconditions to operate at the same video quality increases theresource consumption, particularly for cell edge users. Still, itoutperforms both baselines.Figure 5b shows the standard deviation of the video qualityacross users vs. user density under the sum quality-maximizingpolicy. To obtain a meaningful interpretation of the standarddeviation, we first map the STRRED quality scores to dif-ferential mean opinion score (DMOS) which is linear in userjudgments and ranges from 0 to 100. For that purpose, we usethe fit in [36]. We deduce from Figure 5b that the deviationof quality across users is reasonable. Thus, maximizing sumquality comes at a little cost to cell edge users’ performanceand maintaining complete fairness by maximizing the mini-mum video quality in the cell is too costly to the capacity.To understand how the significant capacity gains areachieved, Figure 6 shows the users covered by the proposedscheduling and resource allocation algorithm in comparisonto the first baseline at P t = 25 W = 44 dBm. The maximaluser subset scheduling algorithm supports a total of 198users corresponding to 184 live streaming sessions and 14video conferencing sessions. In contrast, Max SNR schedulingsupports only 64 users corresponding to 35 live streamingsessions and 29 video conferencing sessions. Thus, in essence,the gains stem from dropping a small fraction of users thatrequire excessively large amount of resources to meet theirQoS requirement such that the total number of users supportedis maximized. Another way to visualize the gains is through the coverage radius. With the baseline, the coverage radius is2.1 Km. Under the proposed algorithm, the coverage radiusfor video conferencing users is reduced to 1.5 Km such thatlive streaming users can be served up to a 5 Km radius.VIII. C ONCLUSION
In this paper, we used the concept of effective capacityto provide a framework for statistical delay provisioningfor multiple users sharing a wireless network. Sum quality-maximizing resource allocation policies as well as fairness-based policies were derived. Furthermore, maximal user subsetscheduling was proposed to maximize the number of sched-uled users that can meet their QoS requirement. Significantgains in capacity, measured in terms of number of userssupported in the system, are achieved due to QoS-awarescheduling and resource allocation.R
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