Delay Budget Partitioning to Maximize Network Resource Usage Efficiency

Transcription

1 Deay Budget Partitioning to Maximize Network Resource Usage Efficiency Kartik Gopaan Tzi-cker Chiueh Yow-Jian Lin Forida State University Stony Brook University Tecordia Technoogies Abstract Provisioning techniques for network fows with endto-end QoS guarantees need to address the inter-path and intra-path oad baancing probems to maximize the resource utiization efficiency. This paper focuses on the intra-path oad baancing probem: How to partition the end-to-end QoS requirement of a network fow aong the inks of a given path such that the deviation in the oads on these inks is as sma as possibe? We propose a new agorithm to sove the end-to-end QoS partitioning probem for unicast and muticast fows that takes into account the oads on the constituent inks of the chosen fow path. This agorithm can simutaneousy partition mutipe end-to-end QoS requirements such as the end-to-end deay and deay vioation probabiity bound. The key concept in our proposa is the notion of sack, which quantifies the extent of fexibiity avaiabe in partitioning the end-to-end deay requirement across the inks of a seected path (or a muticast tree). We show that one can improve network resource usage efficiency by carefuy seecting a sack partition that expicity baances the oads on the underying inks. A detaied simuation study demonstrates that, compared with previous approaches, the proposed deay budget partitioning agorithm can increase the tota number of ong-term fows that can be provisioned aong a network path by up to.2 times for deterministic and 2.8 times for statistica deay guarantees. I. INTRODUCTION Performance-centric network appications such as Voice over IP (VoIP), video conferencing, streaming media and onine trading have stringent Quaity of Service (QoS) requirements in terms of end-to-end deay and throughput. In order to provide QoS guarantees, the network service provider needs to dedicate part of network resources for each customer. Hence an important probem faced by every provider is how to maximize the utiization efficiency of its physica network infrastructure and sti support heterogeneous QoS requirements of different customers. Here utiization efficiency is measured by the amount of customer traffic supported over a fixed network infrastructure. Traffic engineering techniques in Muti-Protoco Labe Switched (MPLS) networks seect expicit routes for Labe Switched Paths (LSP) between a given source and destination. Each LSP coud act as a traffic trunk carrying an aggregate traffic fow that requires QoS guarantees such as bandwidth and deay bounds. In our terminoogy, a fow represents a ong-ived aggregate of network connections (such as a VoIP trunk) rather than short-ived individua streams (such as a singe VoIP conversation). The key approach underying traffic engineering agorithms, such as [], is to seect network paths so as to baance the oads on the network inks and routers. Without oad baancing, it is possibe that resources at one ink might be exhausted much earier than others, thus rendering the entire network paths unusabe. For rea-time network fows that require end-to-end deay guarantees, there is an additiona optimization dimension for baancing oads on the network inks, namey, the partitioning of end-to-end QoS requirements aong the inks of a seected network path. Specificay we are interested in the variabe components of end-to-end deay, such as queuing deay at intermediate inks and smoothing deay before the ingress, rather than the fixed deay components, such as propagation and switching deays. In this paper, we use the term deay to refer to variabe components of end-to-end deay. Consider the path shown in Figure in which each ink is serviced by a packetized rate-based scheduer such as WFQ [2]. Given a request for setting up a rea-time fow F i aong this path with a deterministic end-to-end deay requirement D i, we need to assign deay budgets D i, to F i at each individua ink of the path. Intuitivey, ow deay typicay requires high bandwidth reservation. In other words, since fow F i s packet deay bound at each ink is inversey proportiona to its bandwidth reservation, the amount of deay budget aocated to F i at a ink determines the amount of bandwidth reservation that F i requires at that ink. The question is, how do we partition the end-to-end deay requirement D i into per-ink deay budgets such that (a) the end-to-end deay requirement D i is satisfied and (b) the amount of traffic admitted aong the muti-hop path can be maximized in the ong-term? This is the Deay Budget Partitioning probem for deay constrained network fows. Most categories of rea-time traffic, such as VoIP or video conferencing, can toerate their packets experiencing end-toend deays in excess of D i within a sma deay vioation probabiity P i. Such statistica deay requirements of the form (D i, P i ) can assist in reducing bandwidth reservation for rea-time fows by expoiting their toerance eves to deay vioations. When we consider partition of the end-to-end deay vioation probabiity requirement P i, in addition to the deay requirement D i, the deay budget partitioning probem is generaized to statistica deay requirements for network fows. This paper makes the foowing main contributions. Firsty, we present a unified agorithm tempate, caed Load-based Sack Sharing (LSS) for the deay budget partitioning probem, and describe its appication to unicast and muticast fows with deterministic and statistica deay guarantees. Secondy, our agorithm can hande mutipe simutaneous fow QoS require-

2 End to end deay D i 2 3 Partition D = D min i, + D = D min + D = D min i, S i, i,2 i,2 S i,2 i,3 i,3 + S i,3 Admission Criteria sum (D min i, ) <= D i Deay Sack S i = S i, + S i,2 + S i,3 = D i sum (D min i, ) Fig.. Exampe of partitioning end-to-end deay budget D i over a three-hop path. The sack S i indicates the amount of fexibiity avaiabe in partitioning the deay budget D i. ments such as end-to-end deay bounds and deay vioation probabiity bounds. Earier approaches handed ony a singe end-to-end QoS requirement at a time. Thirdy, we introduce the notion of partitioning sack in end-to-end QoS rather than directy partitioning the entire end-to-end QoS as in earier approaches. Sack quantifies the amount of fexibiity avaiabe in baancing the oads across inks of a muti-hop path and wi be introduced in Section II. Finay, we provide the detaied admission contro agorithms for unicast and muticast fows that can be used in conjunction with any scheme of partitioning end-to-end deay and deay vioation probabiity requirements. We mode our work on the ines of Guaranteed Service architecture [3] where the network inks are serviced by packetized rate-based scheduers [2] [4] [5] [6] and there exists an inverse reationship between the amount of service rate of a fow at a ink and the corresponding deay bound that the fow s packets experience. A rate-based mode with GPS scheduers was aso adopted in [7]. Note that we address the probem of partitioning additive end-to-end deay requirement and the mutipicative deay vioation probabiity requirement. Specificay, the probem we address is not the same as partitioning botteneck QoS parameters. Indeed, for packetized ratebased scheduers such as WFQ, the end-to-end deay bound for a fow contains both additive (per-ink) and botteneck deay components. In Section IV, we show how these two components can be separated and how the QoS partitioning probem is reevant in the context of rate-based scheduers. Considering the bigger picture, any scheme for end-to-end QoS partitioning aong a path is not sufficient by itsef to satisfy traffic engineering constraints. Rather, end-to-end QoS partitioning is one of the components of a arger network resource management system [8] in which network resources need to be provisioned for each fow at three eves. At the first eve, a network path is seected between a given pair of source and destination that satisfies the fow QoS requirements, and at same time baances the oad on the network. At the second eve, the end-to-end QoS requirement of the new fow is partitioned into QoS requirements at each of the inks so as to baance the oad aong the seected path. This is the intra-path QoS partitioning probem that we address in this paper. Finay, at the third eve a resource aocation agorithm determines the mapping between the QoS requirements and the actua ink-eve resource reservation. The rest of the paper is organized as foows. Section II introduces the notion of sack sharing which is centra to our work. Section III paces our work in the context of reated research in deay budget partitioning. Section IV formuates our mode of the network and reviews standard resuts for endto-end deay bounds. In Section V, VI and VII we describe a series of deay budget partitioning agorithms for unicast and muticast network paths having deterministic and statistica end-to-end deay requirements. In Section VIII we anayze the performance of the proposed agorithms and Section IX concudes with a summary of main research resuts. II. NOTION OF SLACK AND SLACK SHARING The extent of fexibiity avaiabe in baancing the oads across inks of a muti-hop path is quantified by the sack in end-to-end deay budget. For each ink of the muti-hop path in Figure, we can compute the minimum oca deay budget Di, min guaranteed to a new fow F i at the ink provided that a residua (unreserved) bandwidth at is assigned for servicing packets from the new fow. The difference between the end-to-end deay requirement D i and the sum of minimum deay requirements Di, min, i.e. D i = D i 3 Dmin i,, represents an excess sack that can be shared among the inks to reduce their respective bandwidth oads. The manner in which sack is shared among inks of a fow path determines the extent of oad baance (or imbaance) across the inks. When the inks on the seected network path carry different oads, one way to partition the sack is to share it based on the current oads on the network inks. For exampe, assume that the three inks in Figure carry a oad of 4%, 8% and 2% respectivey. Given a tota sack in deay budget D i, one coud partition the sack proportionay as 2 7 D i, 4 7 D i, and 7 D i, respectivey, rather than assigning each 3 D i. The reason that the former assignment is better from the viewpoint of oad-baancing is that a more oaded ink shoud be assigned a arger deay budget in order to impose a ower bandwidth demand on it. The atter scheme, on the other hand, woud ead to second ink getting bottenecked much earier than the other two, preventing any new fows from being admitted aong the path. In fact, as we wi show ater, we can do even better than proportiona partitioning described above if we expicity baance the oads across different inks.

3 III. RELATED WORK Equa Aocation (EA) scheme, proposed in [9], divides the end-to-end oss rate requirement equay among constituent inks. The principa concusion in [9] is simiar to ours, that is the key to maximize resource utiization is to reduce the oad imbaance across network inks. The work focused on optimizing the minima (botteneck) ink utiity by equay partitioning the end-to-end oss rate requirements over the inks. The performance of this scheme was shown to be reasonabe for short paths with tight oss rate requirements but deteriorated for onger paths or higher oss rate toerance. Proportiona Aocation (PA) scheme, proposed in [7], considered partition of end-to-end QoS over inks of a muticast tree in proportion to the utiization of each ink. The performance of PA was shown to be better than EA since it accounts for different ink oads. Our work is different from the above two proposas in the foowing aspects. First, we use a heuristic that directy baances the oads on different inks instead of indirecty addressing the objective via equa/proportiona aocation. Secondy, instead of partitioning the sack in QoS, the above two proposas partition the entire end-to-end QoS requirement directy among the inks. If an equa/proportiona partition resuts in tighter deay requirement than the minimum possibe at some ink, then the proposa in [7] assigns the minimum deay at that ink and then performs equa/proportiona aocation over remaining inks. With such an approach, the minimum deay assignment converts the corresponding ink into a botteneck, disabing a the paths that contain this ink. In contrast, we partition the sack in end-to-end deay, instead of the end-to-end deay, which heps prevent the formation of botteneck inks as ong as non-zero sack is avaiabe. Efficient agorithms to partition the end-to-end QoS requirements of a unicast or muticast fow into per-ink QoS requirements have been proposed in [] [] [2]. The optimization criteria is to minimize a goba cost function which is the sum of oca ink costs. The cost functions are assumed to be weaky convex in [] and increase with the severity of QoS requirement at the ink whereas [] addresses genera cost functions. On the other hand, [3] addresses the probem in the context of discrete ink costs in which each ink offers ony a discrete number of QoS guarantees and costs. For the agorithms in [] [] [2] [3] to be effective, one needs to carefuy devise a per-ink cost function that accuratey captures the goba optimization objective in our case that of maximizing number of fows admitted by baancing the oads among mutipe inks of the path. As we wi demonstrate in Section VIII, the best cost function that we coud devise to capture the oad-baancing optimization criteria, when used with agorithms in [], does not yied as high resource usage efficiency as the expicit oad-baancing approach proposed in this paper. Instead of indirecty addressing the oad-baancing objective via a cost function, our LSS agorithm expores ony those QoS partitions that maintain expicit oad-baance among inks. Furthermore, the above agorithms consider ony singe QoS dimension at a time. In contrast, our agorithm can hande mutipe simutaneous QoS dimensions, such as both deay and deay vioation probabiity requirements. The probem of partitioning as we as QoS routing has been addressed in [4] [5]. The goa of [4] [5] is different from ours, namey that of maximizing the probabiity of meeting the QoS requirements of a fow. Whie we do not address the routing probem in this paper and focus on the intra-path QoS partitioning probem, an approach to integrate our proposa in this paper with QoS routing schemes has been outined in [8]. A rea-time channe abstraction with deterministic and statistica deay bounds, based on a modified eariest deadine first (EDF) scheduing poicy, has been proposed in [6]. However, equa aocation scheme is used to assign per-ink deay and packet oss probabiity bounds. An approach to provide end-to-end statistica performance guarantees has been proposed in [7] when the traffic sources are modeed with a famiy of bounding interva-dependent random variabes. Rate controed service discipine is empoyed inside the network. However, the work does not address the issue of how to ocay partition the end-to-end deay requirement. IV. NETWORK MODEL A rea-time fow F i is defined as an aggregate that carries traffic with an average bandwidth of ρ avg i and burst size σ i. We assume that the amount of fow F i traffic arriving into the network in any time interva of ength τ is bounded by (σ i + ρ avg i τ). The (σ i, ρ avg i ) characterization can be achieved by reguating F i s traffic with reativey simpe eaky buckets. We focus this discussion on unicast fows and wi generaize to muticast fows when necessary. We consider the framework of smoothing at the network ingress and bufferess mutipexing in the network interior as advocated in [8] [5]. Specificay, as shown in Figure 2, a reguated fow F i first traverses a traffic smoother foowed by a set of rate-based ink scheduers at each of the intermediate inks aong its muti-hop path. The first component smoothes the burstiness in F i s traffic before the ingress node. Each rate-based scheduer at intermediate ink services the fow at an assigned rate ρ i, ρ avg i. The manner in which rates ρ i, are assigned wi be described ater in Sections V and VI. The smoother reguates fow F i s traffic at a rate ρ min i = min{ρ i, } i.e., at the smaest of per-ink assigned rates. Since fow F i s service rate at each ink scheduer is greater or equa to smoothing rate at the ingress, F i s traffic does not become bursty at any of the intermediate inks. Competey smoothing F i s traffic before the ingress node has the advantage that it aows the interior ink mutipexers to empoy sma buffers for packetized traffic. Additionay, as shown beow, it permits decomposition of fow s end-to-end deay requirement D i into deay requirements at each network component aong the fow path. Muticast fows have a simiar setup except that fow path is a tree in which each node repicates traffic aong outgoing branches to chidren. We now proceed to identify different components of endto-end deay experienced by a fow F i. The first component is the smoothing deay. The worst-case deay experienced at

4 avg ( σ i, ρ i ) Smoother ρ i min S Rate based Link Scheduers ρ i, ρ i,2 ρ i,m S2 Sm avg min ρ i, ρ i ρ i = min { ρ i, } Fig. 2. Network components aong a muti-hop fow path. Fow F i that has (σ i, ρ avg i ) input traffic characterization passes through a smoother foowed by a set of rate-based ink scheduers. F i s service rate at each ink is ρ i, ρ avg i and the bursts are smoothed before the ingress at a rate of ρ min i = min{ρ i, }. the smoother by a packet from fow F i can be shown to be as foows [5]. D i,s = σ i /ρ min i () The maximum input burst size is σ i, the output burst size of the smoother is, and the output rate of the smoother is = min{ρ i, } ρ min i The second component of end-to-end deay is the accumuated queuing deay at intermediate inks. We assume that packets are serviced at each ink by the Weighted Fair Queuing (WFQ) [9] [2] scheduer which is a popuar approximation of the Generaized Processor Sharing (GPS) [2] cass of rate-based scheduers. It can be shown [2] that the worstcase queuing deay D i, experienced at ink by any packet beonging to fow F i under WFQ service discipine is given by the foowing. D i, = δ i, + L max + L max (2) ρ i, ρ i, C δ i, is F i s input burst size at ink, L max is the maximum packet size, ρ i, is the reservation for F i at ink, and C is the tota capacity of ink. The first component of the queuing deay is fuid fair queuing deay, the second component is the packetization deay, and the third component is scheduer s non-preemption deay. Since our network mode empoys bufferess mutipexing at interior inks, the input burst δ i, is at each ink. The deay bound of Equation 2 aso hods in the case of other rate-based scheduers such as Virtua Cock [2]. In genera, for any rate-based scheduer, we assume that a function of the form D (.) exists that correates the bandwidth reservation ρ i, on a ink to its packet deay bound D i,, i.e. D i, = D (ρ i, ). We are interested in rate-based scheduers since, in their case, the reationship between per-ink deay bound and the amount of bandwidth reserved at the ink for a fow can be expicity specified. In contrast, even though non rate-based scheduers (such as Eariest Deadine First (EDF) [2]) can potentiay provide higher ink utiization, in their case the resource-deay reationship for each fow is difficut to determine, which in turn further compicates the admission contro process. In Figure 2, the end-to-end deay bound D i for fow F i over an m-hop path is given by the foowing expression [22] [23] when each ink is served by a WFQ scheduer. D i = σ i ρ min + i ( Lmax ρ i, + L max C ) (3) Here ρ i, ρ avg i and ρ min i = min{ρ i, }. In other words, end-to-end deay is the sum of traffic smoothing deay and per-ink queuing (packetization and non pre-emption) deays. For muticast paths, end-to-end deay is the sum of smoothing deay at ingress and the maximum end-to-end queuing deay among a unicast paths from the source to the eaves. V. UNICAST FLOW WITH DETERMINISTIC DELAY GUARANTEE We now propose a series of agorithms for deay budget partitioning that we ca Load-Based Sack Sharing (LSS) agorithms. The first agorithm, presented in this section, addresses deterministic deay guarantees for unicast fows. The second agorithm addresses statistica deay guarantees for unicast fows and the third agorithm extends LSS for muticast fows; these are presented in subsequent sections. The deterministic and statistica agorithms for unicast fows are named D-LSS and S-LSS respectivey and those for muticast fows are named D-MLSS and S-MLSS. Let us start with the case where a fow F N is requested on a unicast path and requires an end-to-end deterministic deay guarantee of D N i.e, none of the packets carried by F N can exceed the deay bound of D N. Assume that the network path chosen for a fow request F N consists of m inks and that N fows have aready been admitted on the unicast path. The tota capacity and current bandwidth oad on the th ink are represented by C and L respectivey. The goa of deay budget partitioning is to apportion F N s end-to-end deay budget D N into a set of deay budgets D N, on the m network inks and the smoothing deay D N,s at the smoother, such that the foowing partition constraint is satisfied D N,s + D N, D N (4) and the number of fows that can be admitted over the unicast path in the ong-term is maximized. We saw in Section IV that for rate-based scheduers ike WFQ, there exists a function of the form D (ρ i, ) that correates a fow F i s bandwidth reservation ρ i, on a ink to its packet deay bound D i,. The specific form of reation D (ρ i, ) is dependent on the packet scheduing discipine empoyed at the inks of the network. For exampe, if a ink is managed by a WFQ scheduer, then the reation is given by Equation 2.

5 δ =.5; b = δ; whie() { for = to m { ρ N, = max{(c L β C b), ρ avg D N, = D (ρ N, ); /* Deay at ink */ } N } D N,s = σ N / min{ρ N, }; /* Smoother deay */ sack = D N D N, D N,s; if (sack and sack DT hreshod) return D ˆ N ; δ = δ/2; if (sack > ) b = b + δ; ese b = b δ; } Fig. 3. D-LSS: Load-based Sack Sharing agorithm for a unicast fow with deterministic deay guarantee. The agorithm returns the deay vector Dˆ N =< D N,, D N,2,, D N,m >. A. Admission Contro Before computing D N,, one needs to determine whether the fow F N can be admitted into the system in the first pace. Towards this end, first we cacuate the minimum deay budget that can be guaranteed to F N at each ink if a the residua bandwidth on the ink is assigned to F N. Thus the minimum deay budget at ink is given by D (C L ), where C is the tota capacity and L is the currenty reserved capacity. From Equation, the corresponding minimum smoothing deay is σ N / min{c L }. The fow F N can be admitted if the sum of minimum smoothing deay and per-ink minimum deay budgets is smaer than D N ; otherwise F N is rejected. More formay, σ N min {C L } + D (C L ) D N (5) B. Load-based Sack Sharing (D-LSS) Once the fow F N is admitted, next step is to determine its actua deay assignment at each ink aong the unicast path. We define the sack in deay as D N = D N D N,s D N, (6) If the fow F N can be admitted, it means that after assigning minimum deay budgets to the fow at each ink, the sack D N is positive. The purpose of sack sharing agorithm (D-LSS) is to reduce the bandwidth requirement of a new fow F N at each of the m inks in such a manner that the number of fows admitted in future can be maximized. A good heuristic to maximize number of sessions admissibe in future is to apportion the sack in deay budget across mutipe inks traversed by the fow such that the oad across each intermediate ink remains baanced. By minimizing the oad variation, the number of fow requests supported on the network path can be maximized. The D-LSS agorithm for unicast fows with deterministic deay guarantee is given in Figure 3. Let the remaining bandwidth on the th ink after deay budget assignment be the form β C b. The agorithm essentiay tunes the vaue of b in successive iterations unti the resuting sack fas beow a predefined DThreshod. Since β C b represents the remaining capacity of the th ink, β can be set differenty depending on the optimization objective. If the optimization objective is to ensure that a ink s remaining capacity is proportiona to its raw ink capacity, β shoud be set to. If the optimization objective is to ensure that a ink s remaining capacity is proportiona to the current oads on the inks, then β shoud be set to be proportiona to L / m i= L i for a inks. Smaer the vaue of DThreshod, the coser LSS can get to the optimization objective. VI. UNICAST FLOW WITH STATISTICAL DELAY GUARANTEE Now we consider the case where a new fow F N requires statistica deay guarantees (D N, P N ) over a m-hop unicast path, i.e, the end-to-end deay of its packets needs to be smaer than D N with a probabiity greater than P N. Since the deay bound does not need to be stricty enforced at a times, the network resource demand can be presumaby smaer for a fixed D N. The S-LSS agorithm needs to distribute D N and P N to constituent inks of the m-hop path. In other words, it needs to assign vaues D N, and P N,, such that the foowing partition constraints are satisfied D N,s + D N, D N (7) m ( P N, ) ( P N ) (8) and the number of fow requests that can be admitted into the system in the ong-term is maximized. Here we assume there exist correation functions D (ρ i,, P i, ) and P (ρ i,, D i, ) that can correate the bandwidth reservation ρ i, to a statistica deay bound (D i,, P i, ). D i, = D (ρ i,, P i, ) (9) P i, = P (ρ i,, D i, ) () In Section IV, we gave a concrete exampe of such correation functions in the context of deterministic deay guarantees where ink was serviced by a WFQ scheduer. At the end of this section, we wi provide an exampe of how such correation functions can be determined for statistica deay guarantees using measurement based techniques. Note that the above condition on partitioning the endto-end deay vioation probabiity is more conservative than

6 for = to m do { P N, = ; D N, = D (C L, ); } whie ( (D N,s + D N, > D N ) and { m ( ( P N,) ( P N )) ) k = index of ink such that reduction in deay D N,k D k (C k L k, P N,k + δ) is maximum among a inks; P N,k = P N,k + δ; D N,k = D k (C k L k, P N,k ); } m if ( ( P N, ) < ( P N )) Reject fow request F N ; ese Accept fow request F N ; Fig. 4. The admission contro agorithm for unicast fow F N with statistica deay requirements (D N, P N ). necessary. In particuar, we assume that a packet can satisfy its end-to-end deay bound ony if it satisfies its per-hop deay bounds. For instance a packet coud exceed its deay bound at one ink, be serviced eary at another ink aong the path and in the process sti meet its end-to-end deay bound. However, modeing such a genera scenario is difficut and depends on the eve of congestion at different inks and their inter-dependence. Hence we make the most conservative assumption that a packet which misses its oca deay bound at any ink is dropped immediatey and not aowed to reach its destination. This heps us partition the end-to-end deay vioation probabiity into per-hop deay vioation probabiities as mentioned above. A. Admission Contro The admission contro agorithm for the case of statistica deay guarantees is more compicated than the deterministic case because it needs to check whether there exists at east one set of {< D N,, P N, >} that satisfy the partitioning constraints 7 and 8. The detaied admission contro agorithm is shown in Figure 4. It starts with an initia assignment of minimum deay vaue D (C L, ) to D N, assuming that a the remaining capacity of the th ink is dedicated to F N and P N, =. Since the initia assignment might vioate endto-end deay constraint, i.e. D N,s + m D N, > D N, the agorithm attempts to determine if there exists a ooser P N, such that the deay partition constraint can be satisfied. Thus the agorithm iterativey increases the vaue of some P N,k by a certain amount δ and recomputes its associated D N,k, unti either D N,s + m D N, becomes smaer than D N, or m ( P N,) becomes smaer than ( P N ). In the first case, F N is admitted since a constraint satisfying partition exists; in the atter case even assigning a the avaiabe Initiaize D ˆ N and P ˆ N with the fina D N, and P N, vaues computed from the admission contro agorithm; do { D ˆ N = D ˆ N ; P ˆ N = P ˆ N ; Dˆ N =reax deay( P ˆ N ); Pˆ N =reax prob( D ˆ N ); } whie( ( Dˆ N D ˆ N > threshd) or ( PN ˆ ˆ P N > threshp ) ); Fig. 5. S-LSS: Load-based Sack Sharing agorithm for unicast fows with statistica deay guarantee. resources aong the F N s path is insufficient to support the QoS requested and F N is rejected. In each iteration, that ink k is chosen whose deay budget reduces the most when its vioation probabiity bound is increased by a fixed amount, δ, i.e, the one that maximizes D N,k D k (C k L k, P N,k + δ). B. Load-based Sack Sharing (S-LSS) In the context of statistica deay guarantees, the goa of sack sharing agorithm is to apportion both the sack in deay D N and the sack in assigned probabiity P N over m network inks traversed by the fow F N. D N is cacuated using Equation 6 and P N is cacuated as foows. m P N = ( P N,) () ( P N ) Let the deay vector < D N,, D N,2,, D N,m > be represented by D ˆ N. Simiary, et PN ˆ represent the probabiity vector < P N,, P N,2,, P N,m >. The S-LSS agorithm for unicast fows with statistica deay guarantees is given in Figure 5. The agorithm starts with a feasibe assignment of minimum deay vector Dˆ N and probabiity vector PN ˆ obtained during admission contro. In every iteration, the agorithm first reaxes the deay vector Dˆ N assuming fixed probabiity vector PN ˆ, and then reaxes the probabiity vector whie fixing the deay vector. This process repeats itsef ti the distance between the vaues of Dˆ N or between P ˆ N from two consecutive iterations fas beow a predefined threshod. Since the PN ˆ vaues affect the Dˆ N vaue reaxation step and vice-versa, mutipe rounds of aternating reaxation steps are typicay required to arrive at the fina partition. The reax deay() procedure is simiar to the deterministic D- LSS agorithm in Figure 3 except that the resource correation function D (ρ N, ) is repaced by D (ρ N,, P N, ). Figure 6 gives the reax prob() procedure which is simiar to reax deay(), except that the correation function is P (ρ N,, D N, ) and the sack in probabiity is defined as in Equation. In evauations described in Section VIII, we empiricay observe that this two-step iterative reaxation agorithm typicay converges to a soution within 2 to 5 iterations. C. Deay to Resource Correation In this section, we briefy give an exampe of a inkeve mechanism to determine the correation functions D (.) and P (.) for statistica deay guarantees using measurement

7 δ =.5; b = δ; whie() { for = to m do { ρ N, = max{(c L β C b), ρ avg N } P N, = P (ρ N,, D N, ); /* Vioation probabiity at ink */ } m sack = ( P N,) ; ( P N ) if(sack and sack P T hreshod) return P ˆ N ; δ = δ/2; if(sack > ) b = b + δ; ese b = b δ; } Fig. 6. reax prob() routine to reax probabiity assignment vector Pˆ N, given a deay assignment vector Dˆ N. based techniques. The approach, caed Deay Distribution Measurement (DDM) based admission contro, is described in detai in [8]. The DDM approach reduces the resource requirement for each rea-time fow at a ink by expoiting the fact that worst-case deay is rarey experienced by packets traversing a ink. CDF construction: Assume that for each packet k, the system tracks the run-time measurement history of the ratio r k of the actua packet deay experienced Di, k to the worst-case deay Di, wc, i.e., r k = Di, k /Dwc i, where r k ranges between and. The measured sampes of ratio r k can be used to construct a cumuative distribution function (CDF) P rob(r). Figure 7 shows an exampe of a CDF constructed in this manner in one simuation instance [8] using aggregate VoIP fows. We can see from the figure that most of the packets experience ess than /4 th of their worst-case deay. Resource mapping: The distribution P rob(r) gives the probabiity that the ratio between the actua deay encountered by a packet and its worst-case deay is smaer than or equa to r. Conversey, P rob (p) gives the maximum ratio of actua deay to worst-case deay that can be guaranteed with a probabiity of p. The foowing heuristic gives the correation function D (ρ i,, P i, ). D (ρ i,, P i, ) = ( Lmax ρ i, + L ) max P rob ( P i, ) (2) C The term (L max /ρ i, +L max /C ) represents the deterministic (worst-case) deay from Equation 2 with δ i, =. In other words, to obtain a deay bound of D i, with a deay vioation probabiity bound of P i,, we need to reserve a minimum bandwidth of ρ i, which can guarantee a worst-case deay of Di, wc = D (ρ i,, P i, )/P rob ( P i, ). The corresponding inverse function P (ρ i,, D i, ) can be derived from Equation 2 Cumuative Distribution Function Ratio of actua to worst-case deay Fig. 7. Exampe of cumuative distribution function (CDF) of the ratio of actua deay to worst-case deay experienced by packets. X-axis is in og scae. as foows. P (ρ i,, D i, ) = P rob ( L max ρ i, D i, + Lmax C ) (3) An important aspect of DDM approach is that the measured CDF changes as new fows are admitted. Hence, before using the distribution P rob(r) to estimate a new fow s resource requirement, DDM needs to account for the new fow s future impact on P rob(r) itsef. Detais of the impact estimation technique are presented in [8]. VII. PARTITIONING FOR MULTICAST FLOWS It is reativey straightforward to generaize the agorithms in Sections V and VI to muticast fows. We ca these agorithms D-MLSS and S-MLSS for deterministic and statistica versions of LSS agorithm for muticast fows. In this section, we focus on the deterministic version (D-MLSS). The statistica version (S-MLSS) can be derived in a manner simiar to the unicast case (S-LSS) and a detaied description of S-MLSS is provided in [8]. A rea-time muticast fow F N consists of a sender at the root and K receivers at the eaves. We assume that the end-to-end deay requirement is the same vaue D N for each of the K eaves, athough the number of hops from the root to each of the K eaves may be different. A muticast fow with K receivers can be ogicay thought of as K unicast fows, one fow to each eaf. Athough ogicay treated as separate, the K unicast fows share a singe common reservation at each common ink aong their paths. Here we briefy describe the essence of the admission contro and deay partitioning agorithms for muticast fows and the detais are presented in [8]. A. Admission Contro A K-eaf muticast fow with an end-to-end deay requirement can be admitted if and ony if a of the constituent K unicast fows can be admitted. The admission contro agorithm for muticast fow thus consists of appying a variant of the unicast admission contro agorithm in Section V to each of the K unicast paths. The variation accounts for the fact that the K unicast paths are not competey independent. Specificay, since the K unicast paths are part of the same

8 tree, severa of these paths share common inks. Before verifying the admissibiity of j th unicast path, the D N, vaues on some of its inks may have aready been computed whie processing unicast paths to j. Hence, we carry over the previous D N, assignments for inks that are shared with aready processed paths. B. Load-based Sack Sharing (D-MLSS) To compute the fina deay partition for the inks of a K- eaf muticast path, we appy a variant of the D-LSS agorithm in Figure 3 to the K unicast paths one after another. Two variations deserve mention here. First the deay reaxation is appied to unicast paths j = to K in the increasing order of their current sack in deay budget (D N D N,s m j D N,), where D N,s is the smoothing deay defined in Equation, and m j is the ength of j th unicast path.. This processing order ensures that sack partitioning aong one path of the tree does not vioate end-to-end deay aong other paths that may have smaer sack. Secondy, when processing j th unicast path, the deay reaxation ony appies to those inks which are not shared with paths to j, i.e. those inks in the j th path whose D N, vaues have not yet been determined. VIII. PERFORMANCE EVALUATION In this section, we evauate the performance of the LSS and MLSS agorithms for unicast and muticast fows against three other schemes. The first scheme, named Equa Sack Sharing (ESS), is based on the Equa Aocation (EA) scheme proposed in [9]. EA equay partitions the end-to-end deay among the constituent inks in the path of a unicast fow. As discussed in Section III, ESS is an improvement over EA since it partitions the sack in end-to-end QoS (deay and/or deay vioation probabiity) equay among the constituent inks. MESS is a muticast version of ESS in which the variations proposed in Section VII are appied. The second scheme, named Proportiona Sack Sharing (PSS), is based on the Proportiona Aocation (PA) scheme proposed in [7]. PA directy partitions the end-to-end QoS in proportion to oads on constituent inks of unicast/muticast path. As with ESS scheme, PSS is a variant of PA that partitions the sack in endto-end QoS in proportion to the oads on constituent inks and MPSS is a muticast variant of PSS. The third scheme is the Binary-OPQ (or OPQ for short) proposed in [24] for unicast paths, which requires that the goba optimization objective be expressed as the sum of per-ink cost functions. We use squared sum of per-inks oads, i.e. m (L /C ) 2, as the cost to be minimized for goba optimization since it captures overa oads as we as variation in oads across different inks. Thus we defined the per-ink cost in OPQ as (L /C ) 2. Again, we partition the sack in end-to-end QoS rather than the endto-end QoS directy. MOPQ is the muticast version of OPQ proposed in [24]. Since OPQ and MOPQ operate with ony a singe end-toend QoS requirement, we compare them ony against the deterministic D-LSS and D-MLSS versions. On the other hand, it is straightforward to extend ESS, PSS and their muticast versions to hande the two simutaneous QoS requirements of end-to-end deay and deay vioation probabiity. Hence we compare these schemes for both deterministic and statistica cases. As a note on terminoogy, D-ESS and S-ESS refer to deterministic and statistica versions of ESS scheme for unicast fows, D-MESS and S-MESS refer to the same for muticast fows and so on for PSS. The admission contro agorithm that we use for ESS, PSS, OPQ, and their muticast versions is exacty the same as what we propose in this paper for LSS and MLSS. In other words, before sack sharing is performed using any of the schemes, the decision on whether to admit a new fow is made by comparing the accumuated end-to-end minimum QoS against the required end-to-end QoS. Thus the differences shown in performance resut soey from different techniques for sack sharing. A. Evauation Setup We evauate the performance of different sack sharing agorithms using both unicast and muticast paths. The first topoogy for unicast paths (Unicast-) has 45 Mbps capacity at the ast ink and 9 Mbps capacity at a other inks. The second topoogy for unicast paths (Unicast-2) has a mix of ink capacities between 45 Mbps to 2 Mbps. Simiary, the Muticast- topoogy consists of the tree in which destinations are connected to 45 Mbps inks whereas interior inks have 9 Mbps capacity. The Muticast-2 topoogy consists of a mix of ink capacities. Both unicast and muticast agorithms have aso been compared over a grid topoogy and a genera topoogy based on Sprint s North American IP backbone and, due to space considerations, the resuts are presented in [8]. A evauations of LSS agorithms in this paper use β = in the sack sharing phase since it eads to perfect oad baancing with the topoogies considered here. In the context of genera network topoogies, different vaues of β may be chosen depending upon the optimization objective of the routing agorithms being empoyed in the network. For instance, with network-wide oad-baancing as the optimization criteria, resuts in [8] show that β vaues that are set in inverse proportion to expected ink utiization vaues form a good choice for genera network topoogies. Agorithms for deterministic end-to-end deay guarantees (D-* schemes) are evauated using C++ impementations whereas those for statistica deay guarantees (S-* schemes) are evauated using dynamic trace driven simuations with the ns-2 network simuator. Each rea-time fow traffic in trace driven simuations consists of aggregated traffic traces of recorded VoIP conversations used in [25], in which spurt-gap distributions are obtained using G.729 voice activity detector. Each VoIP stream has an average data rate of around 3 kbps, peak data rate of 34 kbps, and packet size of L max = 28 bytes. We temporay intereave different VoIP streams to generate 5 different aggregate traffic traces, each with a data = kbps. The resuts shown in statistica experiments are average vaues over test runs with different random number seed used to seect VoIP traces and initiate rate of ρ avg i

9 Scenario Topoogy ESS PSS OPQ LSS Unicast/Deterministic Unicast D i = 6ms Unicast Unicast/Statistica Unicast- 8 2 N/A 39 D i = 45ms P i = 5 Unicast N/A 292 Muticast/Deterministic Muticast D i = 6ms Muticast TABLE I FLOWS ADMITTED WITH ESS, PSS, OPQ, AND LSS ALGORITHMS. LENGTH=7, ρ avg i = Kbps AND σ i = 5 Kbits Unicast-2 Topoogy D-ESS D-OPQ D-PSS D-LSS Tria Number Muticast-2 Topoogy D-MESS D-MOPQ D-MPSS D-MLSS Tria Number Fig. 8. with Unicast-2 and Muticast-2 topoogies with deterministic deay guarantees. Hops=7, ρ avg i = Kbps, D i = 6ms, σ i = 5 Kbits. new fows. Fow requests arrive with a random inter-arriva time between to 5 seconds. We perform evauations mainy for the static case in which fow reservations that are provisioned once stay in the network forever. The WFQ [2] service discipine is empoyed for packet scheduing at each ink in order to guarantee the bandwidth shares of fows sharing the same ink. In the rest of the section, we present performance resuts for cases of unicast fows with deterministic and statistica deay requirements, and muticast fows with deterministic deay requirements. For muticast fows, the memory requirements in the case of statistica trace driven simuations do not scae in our current system and hence their resuts are not presented. B. Effectiveness of LSS Agorithm We first take a snapshot view of the performance of LSS agorithm in comparison to ESS, PSS and OPQ agorithms and ater examine the impact of different parameters in detai. Tabe I shows the number of fows admitted over a 7-hop paths for unicast and muticast fows with deterministic and statistica deay requirements. Figure 8 pots the number of fows admitted with deterministic deay guarantees under Unicast-2 and Muticast-2 topoogies for different mixes of ink bandwidths. The tabe and figures demonstrate that in a scenarios, LSS consistenty admits more number of fows than a other agorithms. This is because LSS expicity attempts to baance the oads across different inks. In contrast, ESS agorithm does not optimize any specific metric and PSS agorithm does not expicity baance the oads among the inks. Simiary we see that the performance obtained with OPQ agorithm is worse than that with LSS. The main probem ies not within the OPQ agorithm itsef, but in coming up with a cost metric that accuratey captures the oad-baancing criteria. In this case, OPQ agorithm performs its work of coming up with a soution that is cose to optima in minimizing the specific cost metric; however, the best costmetric we can construct turns out to be ony an approximation of the fina oad-baancing optimization objective. Instead of impicity capturing the oad-baancing criteria by means of a cost function, the LSS agorithm approaches the probem in a reverse fashion by exporing ony those sack partitions that maintain expicit oad baance among the inks. C. Capacity Evoution In order to understand why the LSS agorithm admits a higher number of fow requests than other agorithms, we compare their resource usage patterns. Figure 9 pots the evoution of avaiabe ink bandwidth on the constituent inks of the Unicast-2 topoogy when fows require a deterministic end-to-end deay bound of 6ms. Figure pots the same curves when fows require statistica end-to-end deay bound of 45ms and deay vioation probabiity of 5. We used a smaer statistica deay bound of 45ms compared to the deterministic deay bound of 6ms in Figure 9 because, for statistica case, the difference in performance among different agorithms is evident ony at smaer deay bounds. Figure pots the same curves for Muticast-2 topoogy with tree depth of 7 when muticast fows require end-to-end deterministic deay bound of 6ms. At any point in time, LSS is abe to expicity baance the oads on constituent inks. On the other hand, the ink oads are imbaanced in the case of ESS, PSS and OPQ agorithms. Specificay, the bandwidth of the inks with ower capacity is consumed more quicky than that of inks with higher capacity and consequenty fewer number of

10 D-ESS on Unicast-2 Topoogy 68 Mbps (-) 9 Mbps (-2) 58 Mbps (2-3) 2 Mbps (3-4) 35 Mbps (4-5) 3 Mbps (5-6) D-PSS on Unicast-2 Topoogy 68 Mbps (-) 9 Mbps (-2) 58 Mbps (2-3) 2 Mbps (3-4) 35 Mbps (4-5) 3 Mbps (5-6) D-OPQ on Unicast-2 Topoogy 68 Mbps (-) 9 Mbps (-2) 58 Mbps (2-3) 2 Mbps (3-4) 35 Mbps (4-5) 3 Mbps (5-6) D-LSS on Unicast-2 Topoogy 68 Mbps (-) 9 Mbps (-2) 58 Mbps (2-3) 2 Mbps (3-4) 35 Mbps (4-5) 3 Mbps (5-6) Fig. 9. Evoution of avaiabe ink capacity with the number of fows admitted for unicast fows with deterministic deay guarantee. Hops=7, ρ avg i = Kbps, D i = 6ms, σ i = 5 Kbits S-ESS on Unicast-2 Topoogy 68 Mbps (-) 9 Mbps (-2) 58 Mbps (2-3) 2 Mbps (3-4) 35 Mbps (4-5) 3 Mbps (5-6) S-PSS on Unicast-2 Topoogy 68 Mbps (-) 9 Mbps (-2) 58 Mbps (2-3) 2 Mbps (3-4) 35 Mbps (4-5) 3 Mbps (5-6) S-LSS on Unicast-2 Topoogy 68 Mbps (-) 9 Mbps (-2) 58 Mbps (2-3) 2 Mbps (3-4) 35 Mbps (4-5) 3 Mbps (5-6) Fig.. Variation in avaiabe ink bandwidth with the number of fows admitted for unicast fows with statistica deay guarantee. Hops=7, ρ avg i = Kbps, D i = 45ms, P i = 5 and σ i = 5 Kbits. M-ESS on Muticast-2 Topoogy M-PSS on Muticast-2 Topoogy M-OPQ on Muticast-2 Topoogy M-LSS on Muticast-2 Topoogy Mbps (-) 2 Mbps (-2) 94 Mbps (2-3) 68 Mbps (3-4) 78 Mbps (4-5) 9 Mbps (5-6) 5 Mbps (6-7) Mbps (-) 2 Mbps (-2) 94 Mbps (2-3) 68 Mbps (3-4) 78 Mbps (4-5) 9 Mbps (5-6) 5 Mbps (6-7) Mbps (-) 2 Mbps (-2) 94 Mbps (2-3) 68 Mbps (3-4) 78 Mbps (4-5) 9 Mbps (5-6) 5 Mbps (6-7) Mbps (-) 2 Mbps (-2) 94 Mbps (2-3) 68 Mbps (3-4) 78 Mbps (4-5) 9 Mbps (5-6) 5 Mbps (6-7) Fig.. Evoution of avaiabe ink capacity with the number of fows admitted for muticast fows with deterministic deay guarantee. Tree depth=7, ρ avg i = Kbps, D i = 6ms, σ i = 5 Kbits. fows are admitted. Note that a singe ink with insufficient residua capacity is enough to render the entire network path unusabe for newer reservations. Among ESS, PSS and OPQ agorithms, OPQ and PSS have simiar performance foowed by the ESS. The differences in performance arise from the extent to which each agorithm accounts for oad-imbaance between inks. For muticast fows in Figure, the capacity evoution curves are not perfecty baanced in the case of D- MLSS due to the fact that the assigned bandwidth on some of the inks is ower-bounded by the Kbps average rate of fows which is arger than the deay-derived bandwidth required to satisfy the deay budget at those ink. D. Effect of End-to-end Deay Figure 2 pots the variation in number of admitted fows over unicast and muticast topoogies as their end-to-end deterministic deay requirement is varied. With increasing deay, a the four agorithms admit more number of fows, since a ess strict end-to-end deay bound transates to ower resource requirement at intermediate inks. Again, LSS admits more fows than others since it performs oad-baanced partitioning of the sack in end-to-end deay. A maximum of 45 fows with Kbps average data rate can be admitted by any of the agorithms since the smaest ink capacity is 45 Mbps. E. Effect of End-to-end Deay Vioation Probabiity Figure 3 pots the variation in average number of admitted fows over unicast and muticast paths as their deay vioation probabiity bound is varied from 6 to for a 45ms endto-end deay bound. The LSS agorithm is abe to admit far more fows than ESS and PSS agorithms since it can perform oad-baanced sack sharing aong both the dimensions of deay as we as deay vioation probabiity. The performance gain for LSS over ESS and PSS is much arger than in the case of deterministic deay requirements (Figure 2) because even a sma increase in deay vioation probabiity yieds a significant reduction in resources assigned to a fow. F. Effect of Path Length Figure 4 pots the variation in number of admitted fows having deterministic deay requirements of 6ms, as the ength of the unicast path increases. For a the four agorithms, there

11 D-ESS D-OPQ D-PSS D-LSS Unicast-2 Topoogy Deay (ms) D-MESS D-MOPQ D-MPSS D-MLSS Muticast-2 Topoogy Deay (ms) Fig. 2. vs. deterministic end-to-end deay bound for unicast and muticast paths. Hops/Tree depth=7, ρ avg i = Kbps and σ i = 5 Kbits. Average number of fows admitted S-ESS S-PSS S-LSS Unicast-2 Topoogy e-7 e-6 e-5 e-4 e-3 e-2 e- Deay vioation probabiity Fig. 3. Average number of fows admitted vs. end-to-end deay vioation probabiity bound. Hops=7, D i = 45ms, ρ avg i = Kbps and σ i = 5 Kbits. Averages computed over ten simuation runs with different random seeds. is a drop in the number of fows admitted with increasing path ength because the same end-to-end deay now has to be partitioned among more number of intermediate hops. Thus increasing the path ength has the effect of making the endto-end requirement more strict, which in turn transates to higher resource requirement at the intermediate inks. The LSS agorithm sti outperforms the other three agorithms in terms of number of fows it can support since it manages the decreasing sack in deay to counter oad imbaance among inks. G. Effect of Burst Size Figure 5 pots the impact of increase in burst size on the number of fows admitted with deterministic deay requirement of 6ms. For a the four agorithms, the number of fows admitted drops with increasing burst size. Reca that burst size contributes to the smoothing deay σ i /ρ min i before the ingress node. For arger burst size a bigger fraction of endto-end deay is consumed at the smoother in Figure 2 and consequenty a smaer fraction of the deay is eft for partition among inks of the unicast/muticast path. As before, the LSS agorithm sti outperforms the other three agorithms in terms of number of admitted fows. IX. CONCLUSION Resource provisioning techniques for network fows with end-to-end deay guarantees need to address an intra-path oad baancing probem such that none of the constituent inks of a seected path exhausts its capacity ong before others. By avoiding such oad imbaance among the inks of a path, resource fragmentation is ess ikey and more fows can be admitted in the ong term. We have proposed a oad-aware deay budget partitioning agorithm that is abe to sove this intra-path oad baancing probem for both unicast and muticast fows with either deterministic or statistica deay requirements. In particuar, the proposed Load-based Sack Sharing (LSS) agorithm aocates a arger share of the deay sack to more oaded inks than to ess oaded inks, thus reducing the oad deviation among these inks. Through a detaied simuation study, we have shown that the LSS agorithm can indeed admit more number of unicast or muticast fows in comparison with three other agorithms proposed in the iterature. The improvement is up to.2 times for fows with deterministic deay bounds and 2.8 times for statistica deay bounds. In a arger context, the proposed deay partitioning agorithm is just one component of a comprehensive network resource provisioning framework. Whie the proposed agorithms are described in the context of a given unicast path or muticast tree, eventuay these agorithms need to be integrated more cosey with other components such as QoSaware routing or inter-path oad baancing to achieve goba optimization. Quantitativey exporing the inherent interactions between intra-path and inter-path oad baancing schemes is a fruitfu area for future research. ACKNOWLEDGMENT We woud ike to thank Henning Schuzrinne and Wenyu Jiang for providing the VoIP traces used in our simuations. We woud aso ike to thank Pradipta De, Ashish Raniwaa, Srikant Sharma and anonymous reviewers for their insightfu suggestions that heped improve the presentation of this paper. REFERENCES [] M.S. Kodiaam and T.V. Lakshman, Minimum interference routing with appications to MPLS traffic engineering, in Proc. of INFOCOM 2, Te Aviv, Israe, March 2, pp