An Approach to Pricing, Optimal Allocation and Quality of Service Provisioning In High-Speed Packet Networks Jakka Sairamesh
Donald F. Ferguson
Yechiam Yemini
Ctr. for Telecomm. Research Columbia University New York, NY, 10027
IBM Research Division T. J. Watson Research Ctr. Hawthorne, NY, 10532
Computer Science Dept. Columbia University New York, NY, 10027
[email protected]
d
[email protected]
Abstract In this paper, we propose a new methodology based on economic models to provide Quality of Service (QoS) guarantees to competing trac classes (classes of sessions) in packet networks. We consider an economic model of a packet network where resources are priced. Trac classes compete for network resources and they purchase them to satisfy their QoS needs. Our contributions are the following: 1) We provide a new de nition for QoS provisioning based on economic models (Pareto eciency). 2) We obtain the set of optimal resource allocations (Pareto optimal) which provide QoS guarantees to competing traf c classes. 3) We show the impact on equilibrium prices and optimal allocations due to trac load and variability, and QoS requirements. 4) We propose packet scheduling and admission policies to provide QoS guarantees to trac classes based on available QoS and prices in the network.
1 Introduction Rapid advances to ber-optics and VLSI technology have led to the evolution of high-speed broadband networks. These networks can support various real-time applications such as video, voice, image and data which are envisioned to share network bandwidth, bu ers, switching and processing resources. Allocating network resources eciently and satisfying the diverse QoS requirements of various sessions in such shared networks is a challenging problem. Early work in providing QoS in packet networks was done by Lazar et al. [14]. They proposed scheduling mechanisms for multi-class trac in broadband networks under QoS constraints. Recently, several researchers [10, 18] have argued that providing QoS per session is necessary to satisfy the diverse
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requirements of the applications. An excellent survey of QoS provisioning techniques is given by Kurose [10]. He classi ed these techniques into four types, one such type is to provide bounds on QoS parameters such as cell (packet) loss in multiplexers per session in ATM (Asynchronous Transfer Mode) networks [19, 21, 18, 20]. Nagarajan et. al [15] have investigated transient and steady-state models for appropriate QoS parameter de nitions per session in a network. A few researchers have used economic models for QoS provisioning in networks [14, 13, 4, 5, 7, 8, 11, 12]. In such models, users1 compete for resources and try to obtain maximal bene t by purchasing the appropriate amount of resources (under their wealth constraint). In the economy, resources are not only utilized eciently, but also allocated in such a way that users with diverse requirements achieve maximal bene t (individual optimality) [1, 2, 3]. Kadabha and Ja e [13] proposed optimal ow control mechanisms in networks based on utility functions which represented throughput-delay trade-o s of sessions. Kurose et. al. [7, 8] used an exchange economy2 model to design multiple access protocols, and veri ed that competitive equilibrium exists. This work was extended by Don Ferguson et. al. [4, 5, 6] to study ow control in packet networks and resource allocation in distributed systems. Ron et.al. [12] have developed pricing schemes for networks (Internet) to support multiple services. In their model, prices are set based on service classes. Their pricing schemes are static and do not take into consideration trac load (demand), variations in ser1 A user is either a session or an application or a trac class, ie. one who uses the resources. 2 Competing users exchange goods (trans. probabilities) till a point where any more exchanges will reduce the bene t of either one of users.
vice and trac parameters. Service provisioning in ATM Networks using economic models was also considered by Steven [11]. Our work is considerably different. In this paper, we present economic models for packet based networks to provide QoS to trac classes. Our goals are summarized below:
Non−Convex Preferences Convex Preferences Iso−quants (contours)
Iso−quants
Direction of Increasing Preference X(1)
X(1)
Ex: Buffer
Ex: Buffer
Utility functions and demand: De ne and char-
acterize the QoS preferences of trac classes (or sessions). Obtain appropriate utility functions for users, then compute the set of allocations that provide QoS. Price and Competitive equilibrium: Establish competitive equilibrium and obtain equilibrium prices and Pareto allocations under QoS constraints. Develop a rationalized pricing scheme, which is sensitive to the trac load, variations in trac parameters and QoS requirements.
Allocation principles and Control Mechanisms:
Develop practical resource allocation algorithms (scheduling and admission) and control principles for QoS provisioning in packet networks using mathematical economic models.
The paper is organized as follows. In section 2, we discuss the preference model of a user3 . We also discuss the economy and formulate economic based problems for allocating resources. In section 3, we illustrate the problem and model using well known M=G=1=B type trac models. We compute Pareto optimal allocations under QoS constraints and prices under equilibrium conditions. We present control mechanisms based on optimal allocations to provide QoS guarantees. We also show the impact of trac variability on prices using G=D=1=B (ATM) trac models. In section 4, we describe our future work and conclusions.
Direction of Increasing Preference
X(2) Ex: Link Capacity
(a)
Ex: Link Capacity
X(2)
(b)
Figure 1: Convex Preferences in Figure (a), NonConvex preferences in Fig. (b) the convex curves are called iso-quants or indi erence curves, they are contours of a function which have a constant value. In Figure 1 (a), the direction of preference is shown to be moving away from the origin. This implies that a user prefers more amounts of x1 (eg. bu er space) and x2 (eg. link capacity) for higher QoS. The user prefers the upper contour set4 of the resource space [2, 3]. Another example is shown in Figure 1(b) where the preferences are non-convex. The user prefers lower amounts of resource x1 and more amounts of resource x2 for higher QoS.
2.1 Utility Functions
A wide variety of applications with various preferences are envisioned to use the high bandwidth networks of the future, thus a exible service architecture is needed. For example, video applications have a wide variety of QoS requirements [22, 23]. The preferences of the applications (users) can be mapped to the resources they consume. In Figure 1 (a), an example is shown where the preferences of a user are convex in two resources (eg. bu er and link capacity). Each of
Preferences of users are represented by utility functions, these are functions of the resources the user consumes. They could be mappings from a resource vector space to a real number. If U (:) is a utility function of a user, then it prefers x y, when U (x) > U (y), where x and y are resource allocations. Several types of users (eg. video, image, voice and data sessions) share resources at each node in the network. The QoS parameters of each of these users can be represented as functions of these resources. Utility function is a function of these QoS functions in such a way the preferences of the user are re ected. Let x = fx1; x2; ::::xM g be the allocation vector (M is the number of resources), and q = fq1; q2; q3; ::::; qng be the set of QoS parameters per user, where qi is a function: c : ?1+( c )1+b
(3)
The above function is continuous and di erentiable for all c 2 [0; C ], and for all b 2 [0; B ]. We assume that b 2 < for continuity purposes of the utility function.
Theorem 1 The utility function (packet loss probability) for an M=M=1=B system is decreasing convex in c for c 2 [0; C ], and decreasing convex in b; 8b 2 [0; B ]. proof given in [9].
be interpreted as the e ective number in an equivalent M=M=1 system, where the system utilization = (1 ? U ). Two Classes: We consider two competing trac classes. Using the equilibrium conditions, the equilibrium price ratio is given by, pc = N1 ? b1 = N2 ? b2 (5) pb c1 log 1 c2 log 2 By using the resource constraints c1 + c2 = C and b1 + b2 = B , the equilibrium conditions become a function of just two variables. The Pareto surface is the set of allocations that satisfy equation 5. The functions Ni and Ui (for all i 2 f1; 2g) have several interesting properties for di erent values of i . We study the properties of these functions for various regions of 1 and 2 , where 1 and 2 are utilizations of TC1 and TC2 respectively. 1 < 1; 2 < 1: As bu er is varied to in nity, the utility function (loss probability) becomes 0, and the e ective average number (N1 ; N2) become the average number in an M=M=1 queue. The limb1 !1 N1 = 1?11 ; limb1 !1 U1 = 0. The quantity b1 ? N1 0 for b1 2 [0; 1). 1 > 1, 2 > 1: The allocated capacity is less than the mean rates of TC1 and TC2. We consider the case where the bu er tends to in nity. limb1 !1 N1 = 1; limb1!1 U1 = 0. The quantity b1 ? N1 < 0 for b1 2 [0; 1). 1 ! 1, 2 ! 1: The quantity N1 = b1, N2 = b2. The equilibrium condition for o ered loads equal b2 (b2 +1) to 1 is b1 2(b1+1) 1 = 22 . Several other cases such as 1 > 1; 2 = 1 are omitted in this paper, but are essential in determining the Pareto surface. K Classes: For K trac classes of M=M=1=B type competing for resources (link and bu er), the following theorem is stated: Theorem 2 Let f (ck ; bk ; k) be smooth convex Utility function the resource constraints: PK c =forC class Pk.K Given and b = B for all i; k 2 [1; K ], i=1 i i=1 i
The problem is to minimize the cell loss probability under the budget constraint. Each trac class computes a demand set. The Lagrangian is given as follows: min[f (c; b; ) ? L (pc c + pb ? w)], where L is the Lagrange multiplier. The utility function is smooth, convex and compact, therefore the demand set is just one element [1, 2]. @U=@c = pc =pb. The price The price ratio is given by @U=@b ratio is a function of the resource allocations and the trac parameter . Using the utility function, the price ratio is given by, pc = N ? b ; N = (1 ? U ) (4) pb c log 1 ? (1 ? U ) where function N is the ratio of the e ective queue utilization ((1 ? U )) to the e ective queue emptiness (1 ? (1 ? U )), where = =c. This can also
Pareto optimal allocations exist. Price equilibrium exists. Proof is given in [9].
The following gives the equilibrium condition, from which the Pareto allocations and the corresponding equilibrium price ratios can be computed.
50
4
1
(1) (2) (3)
45
(1) (2)
(1) L(1)=L(2)=10 (2) L(1)) 0, then, pc = pc ? (+) c If (Db ? B ) < (>) 0, then, pb = pb ? (+) b Go back to step (2). 5 Else if Dc = C at pc, and Db = B at pb , then the equilibrium is attained and prices are at equilibrium. The algorithm computes iteratively the equilibrium prices in a competitive economy using the utility functions. Given the wealth in the economy, the prices converge to a point on the Pareto surface, which can be computed using the rst-order conditions. There is a minimum price p that each trac class has to pay, if the equilibrium prices are lower than p. work-conserving here means that the link scheduler (or controller) is never idle when there are packets to be served in the bu er. 9 In this process, prices are adjusted to make demand equal to the supply. The prices that converge to this are the equilibrium prices. 8
3.2.4 Admission Control Algorithm
Wealth based Admission Algorithm:
Each session that arrives is mapped to a corre-
sponding service class based on its QoS requirements, and adds its wealth to the trac class. At each node (or switch), the new equilibrium point is computed based on the change in wealth. If the QoS guarantees can be provided, then the session is admitted, otherwise it is rejected. If the session is admitted, the scheduling algorithm adapts to the new allocations. The prices of the bu er and link capacity change with the admission of the new session (and completion of a session). In order to avoid computation of prices for every new session that arrives, the prices can be updated after every y% change in average arrival rate or load.
3.2.5 Max and Average Delay Constraints
Max Delay Constraint: Max delay constraint imposes
a constraint on the bu er allocation if the networks are of ATM type, where the service time at each switch for each packet is xed. First the QoS surface for loss probability constraints will be computed, and from that the set of allocations that meet the bu er constraint will be extracted. This new set will provide loss and max delay guarantees. A trac class will select the appropriate set of allocations that meet the QoS requirements under the wealth constraint. An illustration is given in Figure 4 (b). Average Delay Constraint: The average delay for an M=M=1=B system is convex and decreasing in c 8 c 2 (; 1) and concave and increasing in b 8 b 2 (0; 1) [9]. Therefore, the preferences for average delay are non-convex. Due to this, if the utility is just average delay, then there does not exist price equilibrium or Pareto optimality. However, if loss probability and average delay are the QoS parameters, then the QoS surface under the loss constraint can be chosen such that they meet the average delay constraint as well.
3.3 Service and Trac Variability In this section, we study the impact of service and trac variability on the Pareto surface and equilibrium prices for two classes of trac. Several trac classes of the M=H2 =1=B type compete for resources. The hyper-exponential service distribution has a squared co-ecient of variation Cv2
100
50 Cv=1 Cv=3 Cv=5
45
T=10 T=5 6.5 Equilibrium Price Ratio p(c)/p(b)
80
40
70 Buffer Capacity
35 Buffer Capacity
7 Cv=1 Cv=10 Cv=20
90
30 25 20
60 50 40 30
15
20
10
6
5.5
5
4.5
10 4
5
20
0 0
10
20
30 40 Link Capacity
50
60
25
30 35 Link Capacity
(a)
40
0
2
4
6 8 10 12 14 Squared Co-efficient of Variation
16
18
20
(b)
Figure 8: Pareto Surfaces for Cv2 = f1; 3; 5g using the M=H2=1=B type trac model where C = 60; B = 50.
Figure 9: Pareto Surface for Cv2 = f1; 10; 20g (a), Equilibrium Prices: T = 5 and T = 10 (b).
1. When Cv2 = 1, it emulates the M=M=1=B system. We vary Cv2 from 1 to R(> 1) to study the impact on the Pareto surface. The utility function (loss probability) is given in [17]. In Figure 8 (b), the Pareto surfaces versus allocation to class (1) for various Cv2s is shown. For Cv2 = 1, the surface is the same as the M=M=1=B case. As Cv2 increases, the Pareto surface shows that the link capacity becomes a valuable commodity for small bu er allocations and then bu er becomes a valuable commodity as more link capacity is allocated to class (1). We use ON-OFF trac models to study the impact of trac variability on the Pareto surface at a link in ATM type networks. The tail distribution of the queue when sources are multiplexed into an in nite bu er queue is given in [16]. This is a reasonable approximation to the cell loss probability which is given as follows:
Using the constraints c1 + c2 = 60 and b1 + b2 = 100, the Pareto surface is obtained. In Figure 9(a), Pareto surfaces for di erent Cv2 are shown ( xing the other parameters). As Cv2 increases from 1 to 20, the Pareto surface tends to show that bu er space and link capacity are becoming more and more valuable. The equilibrium price ratios versus Cv2 are shown in Figures 9 (b). The price ratios increase as Cv2 increases. A higher Cv2 implies a higher cell loss probability and therefore more resources are required, therefore higher price ratio (link capacity is more valuable compared to bu er). In the Figure 9 (b), the allocations are c = 30; b = 50 for TC1 and TC2.
?b U = S c rp 1 + S r (12(c??)S2 (Cr2p )+ 1) T (7) p v
models. A new de nition for QoS provisioning based on Pareto ecient allocations is given. These allocations are not only ecient (from a Pareto sense), but also satisfy the QoS constraints of competing trac classes (or users). We have provided a methodology for the network service provider to price services based on the demands placed by the users, and techniques to allocate bu er and link resources optimally to each of the trac classes at a single link. These allocations can be used for bu er dimensioning, packet scheduling and QoS control in the packet switches. We proposed simple weighted round-robin type work-conserving scheduling policies based on QoS surface, which will guarantee QoS to the trac classes. From the QoS surface, the range of possible prices can be obtained, therefore giving a choice to the network service provider to price users and allocate re-
A TC consists of S identical (homogeneous) ONOFF sources, which are multiplexed to a bu er. Each source has the following trac parameters: fT; rp ; ; Cv2g, where T is the average ON period, rp is the peak rate of the source, Cv2 is the squared coecient of variation of the ON period, and is the mean rate. The conditions for a queue to form are as follows: S rp > c (peak rate of the TC is greater than the link capacity), and S rp < c (mean rate less than link capacity). In the numerical examples, we use two trac classes (with the same values). There are S1 = S2 = 10 sessions in each trac class, and T = 5; rp = 1; = 0:5.
4 Conclusion and Future Work In this paper, we have developed a decentralized framework for QoS provisioning based on economic
sources. The impact on prices due to trac load and variations in trac processes, and QoS constraints is shown. Therefore, users must be charged based on their trac characteristics such as burstiness (correlated and peaky trac). Prices are excellent indicators of available QoS at each node and link in the network. We are currently investigating market based mechanisms to admit and route sessions using resource price information in large networks, where each packet switch is supplier. Economic models can provide new insights into resource sharing and QoS provisioning in future networks which will connect millions of users, and provide a large number of services. Such networks require decentralized mechanisms to control access to services. Pricing and competition can provide solutions to reduce the complexity of service provisioning and eciently utilize the resources.
References
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