Twenty years ago, the Internet was dedicated only to a small community of researchers. Now, it is an ... services, such as e-mail, short messages, file transfers, voice-over-IP, video-conference, e- commerce ..... (Internet access, Web hosting). IP ...
Bandwidth Sharing in Circuit-Switching Networks Helio Waldman, Marcio S. Savasini, Gustavo S. Quiterio Optical Networking Laboratory - OptiNet State University of Campinas - UNICAMP R. Albert Einstein, 400, C.P. 6101, 13083-852 Campinas, SP - Brasil {waldman, savasini, gustavo}@decom.fee.unicamp.br http://www.optinet.fee.unicamp.br
1
Introduction
Twenty years ago, the Internet was dedicated only to a small community of researchers. Now, it is an institution of paramount importance in the life of a large part of the world population. Today, the Internet is the infrastructure for a variety of vital communication services, such as e-mail, short messages, file transfers, voice-over-IP, video-conference, ecommerce, e-gov, and so on. Low production costs for electronic components have enabled the massive use of computers, cell-phones, and communication equipment. These devices are connected to the Internet by several access media: the traditional public switched telephone network (PSTN); broadband networks such as LAN’s, cable TV networks, and fiber; or wireless networks such as cellular, Bluetooth, Wi-Fi, or infra-red. Incoming traffic from several access networks is aggregated into channels of ever larger capacity and needs to be transmitted with specified quality to each user. Optical networks are seen as the major candidates for emerging access, as well as the only currently feasible option for the backbone, due to the remarkable growth of transmission capacity of optical fibers when using WDM. These networks will stand as integrated transport platforms, with smart common resources able to provide simultaneous differentiated services to different users. The need to accommodate different services in the same infrastructure will generate the formulation of policies for service differentiation, since different applications may require different levels of quality-of-service (QoS). Quality-of service may differ in traffic throughput, in connection request blocking probability, in packet loss, or in delays incurred by the data traffic. Hence the need to guarantee service integrity through a negotiation between service provider and user aiming at the definition of quality parameters specified for each connection request, thus shaping a contract known as a Service Level Agreement (SLA). In this contract, the user must state the transport service specifications that will guarantee satisfactory service quality to his application, thus allowing the network operator to allocate its resources efficiently under well-defined constraints. In this way, the objectives of both parties may be reached [1]. MPLS networks set up end-to-end connections (Label Switched Path - LSP) by assigning labels. The data are forwarded by each router according with the label they carry. In this way, an LSP may aggregate traffic from different sources offering the same level of QoS, assigning them the same sequence of labels along the same route. GMPLS enables MPLS capabilities to be extended to optical networks, thus allowing for the provision of differentiated services. 1
At connection request, the quality-of-service specified by each user will generate corresponding labels. According with their origin, destination and QoS, each data flow may be embedded in LSP’s with higher capacity in the network. Load balance and flow control within each LSP will try to guarantee the quality required by the user from origin to destination. The price each user is willing to pay is closely related with the quality-of-service offered by the provider, as well as with the guarantee of compliance with the contract [2]. In addition to this price, users may differ in other attributes, such as the required minimum capacity to be purchased, or yet they may have some maximal capacity above which no marginal utility gain is perceived. The Telecom operating companies that provide Internet access to their customers and interconnect their networks will normally charge for this kind of service (voice or data) according with the resources that are used, geographical distance between origin and destination, and connection time. A better quality of service will require more resources at each point of the network. The access to each service can be made from a fixed or mobile terminal. The paying terminal may be in the connection origin or destination. Distance between traffic ingress and egress points depends on the location of the users and the connection time is associated with the type of service. The traditional operating model of the Telecommunications sector is organized as shown on Table 1. Layer I is operated by the transmission and switching equipment, including the fixed and mobile telephone sets. Layer II is responsible for the traffic transport through circuit-switching. Layer III includes the services offered by the operating companies. The dominant service in this model is voice, with a differentiated service (0800) and data transfer through fax devices, through voice channels. The voice service is charged through a variety of rates that take into account mainly distance (local, national or international), and time of day (business, normal, or late). Data service is charged per use (quantity of data) or by a flat rate. New applications are now under development, based on the ever higher availability of the Internet and its TCP/IP interface in society. A new category of service emerges: the content services. The utility of content, which can be delivered electronically and may become more interactive in the future, is highly dependent on user expectation and previous exposure. In this way, the same service, but with different contents, may have a higher or lower value, depending on the utility of the pair service-content to each user. In this scenario a new agent is gaining importance in the negotiation of the price of service-content: the content/application provider. These agents control and distribute software-based services and solutions to their customers from a data warehouse. A negotiation between the user and the content/application provider is necessary at service request time or even continually. The emergence of these new agents means that the Telecommunications sector operating model must be modified so that their action may be taken into consideration (Table 2) leading to the rise of the Infocommunications sector [3]. Layers I and II continue in the hands of the operating companies, including equipment, transmission systems and network layer. Layer III provides connectivity to the network for the access equipment of users, service providers and content providers. Layer IV provides the softwares and functionalities that are necessary to use both services and contents. Content applications are available in Layer V and users are in Layer VI. 2
Primary traffic is generated by high level services, that have a direct interface with the user. Low level services are invisible to the users, but are indirectly used. They may be data exchange services for communication between origin and destination terminals (clients and servers) of high level traffic. The quality of high level services depends on the quality of low level transport service [4]. The next section discusses how the behavior of users can be modelled taking into account their willingness to pay, as well as minimal (necessary) and maximal (sufficient) bandwidth requirements. Next we present a model for service differentiation in the network, leading to maximization of the bandwidth that can be shared among differentiated users. Some simulations are presented at the end, together with the conclusions.
2
Economic Context
For each user a utility function U (θ) is defined, where θ is the amount of resources used by the user. Loosely speaking, the utility function may be understood as the price the user is willing to pay for the amount θ of resources. The most QoS-demanding users need to use more resources for the same traffic intensity. The utility function U (θ) must then be described as a function of the bandwidth that must be allocated for a certain QoS class. Elastic services, such as data transfer, have decreasing marginal gains with increasing bandwidth, since they tolerate lower QoS levels. A typical non-decreasing, concave utility curve for these services is shown on Fig. 1. Current Internet access is featured as an elastic service. The utility of a higher transmission capacity will never decrease, but it will increase by smaller and smaller amounts with increasing bandwidth. If the user cannot use more than a certain capacity, then the utility will remain constant above this maximal usable level. For services with strong real time requirements, such as voice, the utility curve presents a step at the value of bandwidth required by the application, so it is not uniformly concave, as in Fig. 2. This happens because there is a certain minimum bandwidth required to make the application performance satisfactory. When there is no sufficient bandwidth, the service has no utility to the user. This study is based on the utility curve proposed by [5], which is given by: ( U (θ) =
³ mθ, ³ ´´ 0 ≤ θ ≤ γ mγ 1 + ln γθ , γ < θ ≤ π ³ ³ ´´ mγ 1 + ln πγ , π < θ,
(1)
where θ is the capacity to be requested by the user, m is the maximum price per bandwidth unit the user is willing to pay, γ is the minimal capacity required by the service, and π is the maximal capacity the user is able to use. Notice the U (θ) is non-decreasing and concave. We assume that the user will request an amount θ of resources that will maximize his or her well being, defined as the positive difference between the utility and the price paid for θ resources. If M ≤ m is the price per unit bandwidth in the network and θ∗ (M ) is the resulting requested amount of resources, we have:
3
max ∴
[U (θ) − M θ] θ∗ (M )
=
(2)
(U 0 )(−1) (M ),
In (1), the utility is linear up to capacity γ with slope (unit price) m. The solution of (2) in this region yields any value of capacity between 0 and γ for unit price M = m, as shown in Fig. 3, that plots the requested capacity as a function of unit price. We assume that, in this circumstance, the user will request γ resources. As M is taken to lower values than m, θ∗ (M ) will be given by: mγ =M θ mγ θ∗ (M ) = M
U 0 (θ) = ∴
(3)
Therefore, the requested resources will increase for decreasing price M , keeping revenue constant, equal to mγ, until the requested capacity reaches π, corresponding to unit price mγ/π. At this point, the utility function becomes linear again, but now with slope (corresponding to marginal utility) zero. Further price reductions from this point will not generate further increase the requested resources, which will then saturate at π, as shown in Fig. 3. If γ = 0 and π = ∞ , utility function (1) would be strictly concave and would represent a strictly elastic user [6]. The utility function in (1) represents a user with elastic behavior in a range of prices and requested resources, thus approaching a description of current Internet utility. Classes of users may be defined according to their utility functions and required QoS. In the next Section, we consider users that may require different levels of maximum blocking probability for their connection requests, and may also have different utility functions. However, the network has only a limited amount of resources that must be shared among these differentiated users. We then compare network differentiated access configurations concerning their ability to accommodate a maximum number of differentiated users.
3
Bandwidth Sharing in the Network
Let us consider a set of K circuits between a source and a destination nodes: in the optical framework, for example, it could be a WDM link with K wavelengths. The circuits may be requested by a small number N of users that do not cooperate, inasmuch as each user only knows how many circuits he is using, with no knowledge about how many circuits are being used by the remaining users, not even in the aggregate. The users do not compete with one another either, in the sense of trying to maximize their participation in the total capacity usage: on the basis of some economic calculus (e.g. the maximization of individual well-being), user 1 just calculates the number of circuit θ1 he wishes to keep on the average. In order to reach this number, he will raise his rate of requests gradually from zero until this desired average capacity is assigned to him with an acceptable blocking probability, previously agreed with the provider in a contract. 4
This context of bandwidth sharing by non-cooperative users has been studied extensively [5] [7]. The most common assumptions are the following: all users produce requests under a Poissonian arrival regime; call holding periods are exponentially distributed; and total undifferentiated sharing over all K available circuits, thus precluding service differentiation on the basis of acceptable blocking probability. Nevertheless, total bandwidth sharing may still result in unequal sharing due to asymmetries between users in their utility functions. In general, a unique equilibrium point is reached as long as the sum of all demands is below total capacity K: N X
θn ≤ K
(4)
n=1
However, as total demand approaches network capacity, one or more users may have its QoS guarantees violated. In this paper, we consider relaxation of the hypothesis of total bandwidth sharing mentioned above. Instead of providing equal access opportunity to unequal users, the network is aware of the following differentiated user features: 1. each user or class of users may contract with the network a maximum acceptable blocking probability; 2. bandwidth sharing may be differentiated, or partial. In the latter case, only a priority user (or class) will have the right to occupy the only remaining free circuit when K − 1 circuits are busy. To each of the remaining users the network will assign a threshold Ti < K so that a request of user i will be blocked when Ti circuits are already busy. Service differentiation creates the necessary prerequisite for price differentiation, which may change economic calculations of the users, leading them to modify the amount θi of demanded circuits, thus generating a new economic and operational equilibrium point. In this Section, though, we are not concerned with this global equilibrium. Our aim is to investigate how partial bandwidth sharing may contribute to enhance the range of vectors θ = {θ1 , θ2 , ..., θN } that may be supported by the network under equilibrium without violating the maximum blocking probability guarantees contracted with each user.
3.1
Total Resource Sharing
Without user differentiation, all classes share the resources under equal foot. The blocking probability will be the same for all classes, given by the Erlang-B formula: νK Pb (ν) = KK! X νi i! i=0
where ν =
λ µ
is the sum of the intensities of traffic from all classes.
5
(5)
The total number of circuits occupied by total traffic ν will be given by: θn (ν) = νn (1 − Pbn (ν))
(6)
The traffic accommodated within each class will be proportional to its demand for resources. If a class needs more capacity to maximize its well-being, it will generate more requests and will receive proportionately more resources (6). This may happen when users are m-asymmetric or γ-asymmetric due to differences in the price sensitivity of their demand curves. π-asymmetric users behave differently only in the region where they demand their maximal desirable capacity πi , with identical utilities in the remaining regions [5]. With no service differentiation, the constraint on the blocking probability must be the same to all classes, so they must all receive the strictest guarantee among all. In this way, compliance with all contracts is guaranteed. However, those less demanding users in terms of blocking probability will receive a free better quality service than required. For this reason, it seems interesting to investigate the performance of differentiation schemes in the utilization of available resources, yielding priority to those users that demand better quality.
3.2
Resource Segregation
One way of differentiating between classes is to segregate resources in order to dedicate each segregated part of the network to each class [8] [9]. This differentiation policy may improve the accommodation of traffic from a priority class. At least a logical - and sometimes a physical - separation of resources is needed to accommodate requests that demand differentiated QoS levels. The higher the demanded QoS, more resources must be reserved exclusively to the priority class. In this way, each segregated set of resources is dedicated to only one class of service. Each service blocking will be caused exclusively by homologous (of its own class) traffic and the amount of resources allocated to that class. Blocking probability will follow the Erlang-B formula for each pair class-resource set. Resources are allocated to each class according to its average demand for circuits and its required upper limit for the blocking probability. This solution, however, eliminates sharing that might be useful to global improvement in the utilization of the network resources. Some resources dedicated to the priority class will often remain idle while they could be used to accommodate requests from lesser priority classes. The waste will be enhanced when resource partition is static [10], so that a fixed group of resources is dedicated to each class.
3.3
Partial Resource Sharing
In order to illustrate the operation gains that may be obtained through partial bandwidth sharing, let us consider a simple case with only two users: user 1 is prioritized, and user 2 is not. Accordingly, requests from user 1 will be blocked only when all K circuits are busy, while those from user 2 will be blocked whenever at least T2 = K − P circuits are busy, where P < K is a positive integer. In this scenario a fixed amount of resources, but not necessarily a segregated group of resources, is reserved for the prioritized class. 6
Such dynamic grouping of resources is able to improve the accommodation of prioritized traffic, with a lesser impact on the non-prioritized class when compared with the resource segregation scheme. Let λ1 and λ2 be the rates of requests from users 1 and 2 at equilibrium (if feasible), respectively; and let µ1 and µ2 be the release rates for connections of users 1 and 2 respectively. Let k1 (t) and k2 (t) be the numbers of circuits held by users 1 and 2 at time t, respectively; and let k(t) = k1 (t) + k2 (t) be the total number of busy circuits in the network at time t. The demand combination (θ1 , θ2 ), the average demands of users 1 and 2, is given by: θ1 = E[k1 (t)]
(7)
θ2 = E[k2 (t)] We shall make the usual assumption of Poissonian request arrivals and exponentially distributed connection-holding times. Then, k1 (t) and k2 (t) are jump processes that can be stochastically described by a continuous-time Markov chain [11] where each state is given by one possible value of (k1 , k2 ). Fig. 4 shows a Markov chain for K = 5 and P = 2. States are labeled with the corresponding pair (k1 , k2 ). The arrows denote the allowed transitions and are labeled by the corresponding transition rates. In general, up to four transitions are possible from any state (k1 , k2 ): 1. to (k1 + 1, k2 ), with rate λ1 , as long as k1 + k2 < K; 2. to (k1 , k2 + 1), with rate λ2 , as long as k1 + k2 < K − P ; 3. to (k1 − 1, k2), with rate k1 µ1 ; and 4. to (k1 , k2 − 1), with rate k2 µ2 . (K + 1)(K + 2) P (P + 1) There are S = − states. All states with k1 + k2 ≥ K − P 2 2 will cause blocking of requests from user 2. Similarly, all states with k1 + k2 = K will block requests from user 1. Therefore, blocking probabilities Pb1 and Pb2 , seen by users 1 and 2 respectively, can be calculated from the steady-state probabilities of the states of the Markov chain. For this purpose, all states must be ordered in some arbitrary order from 1 to S. Let qij be the steady-state probability of state i, and let π = {π1 , π2 , ..., πS } be the stationary probability vector. We define a S × S matrix Q as follows. For i 6= j, element qij is the rate of transitions from state i to state j, i.e. qij is the label shown in Fig. 4 on the arrow that goes from state i to state j, and is zero when there is no such arrow. Finally, the diagonal elements of Q are given by:
qii = −
S X
qij
i = 1, 2, ..., S.
j=1 j6=i
7
(8)
The steady-state probabilities qij may then be obtained from the following global balance equation: πQ = 0
(9)
under the condition: S X
πi = 1
(10)
i=1
Once the steady-state probabilities are calculated, the blocking probabilities are obtained from the following summations: Pb1 =
K X
πk1 ,K−k1
(11)
k1 =P
Pb2 =
K−p P X X
πk1 ,K−p−k1
(12)
p=0 k1 =P −p
3.3.1
A condition for reversibility
When µ1 = µ2 = µ , both Pb1 and Pb2 can be obtained from simple formulas, thus avoiding the need to perform calculations with matrices. The simplification is based on a reduced-state Markov chain where states are denoted only by the total number of circuits k = k1 + k2 , so only K + 1 states are sufficient to describe the process [12]. The chain is shown in Fig. 5, where λ = λ1 + λ2 , and its reversibility may be established by the following properties [11]: 1. if qij is positive for any i 6= j, then qji is also positive; and 2. removal of any bidirectional edge ij in which both qij and qji are positive splits the chain into two separate chains, characterizing the graph as a tree. The reversibility property allows the stationary probabilities πi of the chain states to be obtained from local balance equations, given by: πi qij = πj qji
(13)
Applying this equation to the chain shown on Fig. 5 yields: λ πi−1 , i = 1, 2, ..., K − P iµ λ1 πi = πi−1 , i = K − P + 1, ..., K iµ πi =
8
(14)
Using these recurrence equations to express all stationary probabilities as a function of π0 , we have for ν = µλ e ν1 = λµ1 : πi = π0
νi , i!
i = 1, 2, ..., K − P
(15)
(i−K+P )
πi = π0 Finally, since
K X
ν (K−P ) ν1 i!
, i = K − P + 1, ..., K
πi = 1 , we have:
i=0 max(0,i−K+P )
ν min(i,K−P ) ν1 πi = "K−P X ν j ³ ν ´(K−P ) + j! ν1 j=0
K X j=K−P +1
(16)
# ν1j i! j!
The blocking probabilities may then be expressed as: ν (K−P ) ν1P Pb1 = πK = "K−P K X ν j ³ ν ´(K−P ) X + j! ν1 j=0
Pb2 =
K X i=K−P
j=K−P +1
ν (K−P ) πi =
# ν1j K! j!
K (i−K+P ) X νi i!
i=K−P K−P X j=0
νj j!
+
³ ν ´(K−P ) ν1
(17)
K X j=K−P +1
ν1j j!
(18)
Finally, notice that making P = 0 in (17) yields the well-known Erlang-B formula for undifferentiated service.
4
Results
A link with K = 16 circuits accessed by two non-cooperative users is used to verify, evaluate and compare different network configurations, concerning the accommodation of several demand combinations. User 1 has priority in the partial resource sharing policy. The maximum acceptable blocking probability for users 1 and 2 are 1% and 10%, respectively. In each of the following network configurations, the viable satisfactory equilibrium region (in agreement to the SLA conditions) was obtained on the (θ1 , θ2 ) plane: 1. Total resource sharing, with maximum acceptable blocking probability limited to 1% for both users, since it is not possible to differentiate them in this configuration; 2. Partial resource sharing, with sharing threshold K − P = 1, 2, ..., K − 1 and different µ1 . values for r = µ2 9
The comparison between the traffic servicing performances of the two network configurations is presented in Fig. 6. Each curve delimitates the feasible capacity region for each configuration. Total resource sharing restricts more the acceptance of non-priority traffic, since it guarantees the lower requested blocking probability for both traffic classes. With no service differentiation the network can only serve a maximum traffic intensity νmax , under its blocking restriction (5). The sum of the demanded capacities (θ1 + θ2 ) is limited by θmax (6). For the partial resource sharing network configuration, the viable region is obtained with the maximum traffic supported by the network in every one of the K − 1 possibilities for P . For K = 16, P > 2 results in regions contained in the P = 1 or P = 2 regions, so these two regions are sufficient to describe the range of traffics that may be accommodated by the network complying with QoS contracts. In Fig. 6 are presented the curves for partial resource sharing with r = 1, r = 0.1 and r = 0.01, and total resource sharing. For every partial resource sharing case, the viable regions contain the total resource sharing viable region. Therefore, there is always more demand combinations available in this case. As r decreases, the gain obtained using partial sharing increases, expanding the viable capacity region. This means that the number of demand combinations is greater if connections with longer holding times are prioritized. Since these circuits will remain longer in the network, they shall be better accommodated facing a lower blocking probability. The more dynamic traffic will encounter a higher blocking probability, but the utilization of partial bandwidth sharing can serve a greater number of these connections than the case of no service differentiation. Users 1 and 2 can differ in their willingnesses to pay for the services, or they might have different minimal required bandwidths or maximal usable bandwidths. For each of these asymmetries the users will request different (θ1 , θ2 ) pairs. As already seen, service differentiation can accommodate higher values of θ2 , when θ1 is fixed. For a constant total revenue, prices for user 2 can be lower with service differentiation. For symmetric users, the minimal prices for both users are shown in Fig. 7. Both users have the same characteristics, m = 100, γ = 1 and π = 16. In Fig. 8 the minimal prices for m-asymmetric users are shown. For this case, m2 = 50 as the other parameters remain the same. Minimal prices for γ-asymmetric and π-asymmetric users are show in Fig. 9 and in Fig. 10, respectively. The asymmetries are γ1 = 4 and π2 = 8 for each case. When operating on points on the curves, both users will be facing the maximum agreed blocking probability. Higher prices will make the users require less capacity, releasing link resources. The m-asymmetry limits even more the price user 2 is able to pay per unit. The γ-asymmetry rises the minimum price the network operator will charge user 1. This user needs a higher minimum bandwidth for his QoS requirements, limiting the amount of bandwidth he is able to require to the network. π-asymmetry reduces the price charged for user 2, since his maximum usable bandwidth is less than the total capacity. For all the cases studied, as r decreases and the feasible capacity region expands, the lower will be the price for the non-priority service. This indicates that service differentiation with partial resource sharing can make users pay less and use more services.
10
5
Conclusions
The Internet has become the infrastructure for communications services and content distribution around the globe, modifying the operating model for the Telecom sector. It can be accessed from a variety of electronic devices, via cables or wireless. As the demand for bandwidth grows with uprising applications, optical networks are seen the major candidates for the cable access, as well as the only feasible solution for the backbone. Different applications and different users may require different levels of QoS in the network. Service differentiation is crucial to satisfy network clients. Resource segregation dedicates a group of resources to each user. This can make some resources to remain idle, even if other users demand its use. Partial resource sharing avoids this idleness reserving an amount of resources, not a specific group of resources, to the priority class. Partial resource sharing expands the feasible capacity region when compared to the total resource sharing (no service differentiation). The performance of this type of differentiation is enhanced when connections with longer duration are prioritized. More traffic from the non-prioritized class can be accommodated in the network, still respecting the conditions agreed between client and network operator (SLA). User asymmetries in willingness to pay for the service, minimal required bandwidths and maximum usable bandwidth can change the price to be charged from users by the service/content providers. The prioritization of longer connections can lead to lower prices for the more dynamic requests.
6
Acknowledgment
This work has been supported by CPqD (GIGA/FINEP Project), Pronex/Fapesp Program (National Center of Excellence in Optical Networking) and CNPq.
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[7] B. Ninan and M. Devetsikiotis, “Pricing mediated bandwidth allocation for the next generation internet,” Proc. Globecom 2003 , pp. 3030–3034, 2003. [8] X. Q. et al., “Supporting integrated voice and data traffic over EGPRS,” Proc. IEEE Intl. Conf. on Commun. ICC’01 6, pp. 1748–1753, 2001. [9] N. Andriolli, T. Jakab, L. Valcarenghi, and P. Castoldi, “Separate wavelength pools for multiple-class optical channel provisioning,” Proc. Networks , pp. 1748–1753, 2004. [10] Q. Zhang, V. M. Vokkarane, J. P. Jue, and B. Chen, “Absolute QoS differentiation in optical burst-switched networks,” IEEE J. Select. Areas Commun. 22, pp. 1781–1795, Nov. 2004. [11] A. Kumar, D. Manjunath, and J. Kuri, Communication Networking: an Analytical Approach, Elsevier, 2004. [12] R. Wolff, Stochastic Modelling and the Theory of Queues, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.
Figures and Tables
12
Table 1: Traditional Operating Model: Telecom Sector Services Layer ( voice, fax, 0800 services) II Network Layer (circuit-switched network) I Equipment Layer (switches, transmission systems, customer premises equipment)
III
VI V IV
III
II
I
Table 2: Operating Model: Infocommunications Sector Customers Applications/Content Layer (Web design, on-line information services, broadcasting services) Navigation & Middleware Layer (browsers, portals, search engines, directory assistance, security, electronic payment) Connectivity Layer (Internet access, Web hosting) IP Interface Network Layer (optical fiber network, DSL local network, radio access network, Ethernet, frame relay, ISDN, ATM) Equipment & Software Layer (switches, transmission equipment, routers, servers, CPE, billing software)
Figure 1: Elastic Utility Function
13
Utility
Utility Function
0
γ
Capacity
Figure 2: Real-Time Service Utility Function
Figure 3: Elastic Demand
14
Figure 4: Markov Chain for K = 5, P = 2 - States are labeled (k1 , k2 )
15
Figure 5: Markov Chain - reversibility
12 Total Sharing Partial Sharing: r=1 Partial Sharing: r=0.1 Partial Sharing: r=0.01
Non−Priority Traffic [Erl.]
10
8
6
4
2
0 0
1
2
3
4
5
6
7
Priority Traffic [Erl.]
Figure 6: Viable Capacity Regions 16
8
9
symmetric users Total Sharing Partial Sharing: r=1 Partial Sharing: r=0.1 Partial Sharing: r=0.01
100 m1 = m2 = 100
γ1 = γ2 = 1 π1 = π2 = 16
90 80
Price 2
70 60 50 40 30 20 10 0 10
20
30
40
50
60
70
80
90
100
Price 1
Figure 7: Symmetric Users m−asymmetric users 55 50
m1 = 100; m2 = 50
45
γ1 = γ2 = 1 π1 = π2 = 16
Total Sharing Partial Sharing: r=1 Partial Sharing: r=0.1 Partial Sharing: r=0.01
40
Price 2
35 30 25 20 15 10 5 0 10
20
30
40
50
60
70
80
Price 1
Figure 8: m-asymmetric Users 17
90
100
gamma−asymmetric users
Total Sharing Partial Sharing: r=1 Partial Sharing: r=0.1 Partial Sharing: r=0.01
100 m1 = m2 = 100
γ1 = 4; γ2 = 1 π1 = π2 = 16
90 80
Price 2
70 60 50 40 30 20 10 0 40
50
60
70
80
90
100
Price 1
Figure 9: gamma-asymmetric Users pi−asymmetric users Total Sharing Partial Sharing: r=1 Partial Sharing: r=0.1 Partial Sharing: r=0.01
100 m1 = m2 = 100
γ1 = γ2 = 1 π1 = 16; π2 = 8
90 80
Price 2
70 60 50 40 30 20 10 0 10
20
30
40
50
60
70
80
Price 1
Figure 10: pi-asymmetric Users 18
90
100
