iBusiness, 2013, 5, 95-106
http://dx.doi.org/10.4236/ib.2013.53012 Published Online September 2013 (http://www.scirp.org/journal/ib)
Quality of Service on Queueing Networks for the Internet*
Hans W. Gottinger
STRATEC, Munich, Germany.
Email: hg528@bingo-ev.de
Received February 28th, 2013; revised March 30th, 2013; accepted April 11th, 2013
Copyright © 2013 Hans W. Gottinger. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Most studies of resource allocation mechanisms in Internet traffic have used a performance model of the resource pro-
vided, where the very concept of the resource is defined in terms of measurable qualities of the service such as utiliza-
tion, throughput, response time (delay), security level among others. Optimization of resource allocation is defined in
terms of these measurable qualities. One novelty introduced by an economic mechanism design approach is to craft a
demand-driven system which takes into account the diverse QoS requirements of users, and therefore, uses multiobjec-
tive (utility) optimization techniques to characterize and compute optimum allocations. Economic modelling of com-
puter and communication resource sharing uses a uniform paradigm described by two level modelling: QoS require-
ments as inputs into a performance model that is subject to economic optimization.
Keywords: Internet Economics; Network Economy; Queueing Systems; Mechanism Design; Performance
1. Introduction
In the context of congested networks for Internet traffic,
the phenomenon of packet loss is due to two reasons: the
first, packets arrive at a switch and find that the buffer is
full (no space left), and therefore are dropped. The sec-
ond is that packets arrive at a switch and are buffered,
but they do not get transmitted (or scheduled) in time,
then they are dropped. A formal way of saying this: for
real-time applications, packets, if delayed considerably in
the network, do not have value once they reach the des-
tination, and a job that misses the deadline may have no
value at all. The sort and variety of delay can severely
impact the operability and efficiency of a network and
therefore is of eminent interest for economic analysis
(Radner 1, van Zandt 2, Mount and Reiter 3).
A way to look at the network economy is to invoke
mechanism design principles supporting market mecha-
nisms (Deng et al. 4; Neumann et al. 5).
The present paper builds on previous work (Gottinger
6) and expands on Quality of Service principles and
management procedures toward creating demand for
service in the Internet economy.
1.1. Quality of Service (QoS)
With the Internet we observe a single quality of service
(QoS): “best effort packet service.” Packets are trans-
ported first come, first-served with no guarantee of suc-
cess. Some packets may experience severe delays, while
others may be dropped and never arrive. Different kinds
of data place different demands on network services
(Shenker, 7). Email and file transfer requires 100 per-
cent accuracy, but can easily tolerate delay. Real-time
voice broadcasts require much higher bandwidth than file
transfers, and can tolerate minor delays but they cannot
tolerate significant distortion. Real-time video broadcasts
have very low tolerance for delay and distortion.
Through the widespread usage of VOIP the Internet has
become more and more sensitive data. Because of these
different requirements, network allocation algorithms
should be designed to treat different types of traffic dif-
ferently but the user must truthfully indicate which type
of traffic he/she is preferring, and this would only happen
through incentive compatible pricing schemes which is at
the very core of economic mechanism design.
*In the real world the invisible hand of free markets seems to yiel
surprisingly good results for complex optimization problems.This
occurs despite the many underlying difficulties: decentralized control,
uncertainties, information gaps, computational power, etc.One is
tempted to apply similar market based ideas in computational scenar-
ios with similar complications, in the hope of achieving similarly goo
Noam Nisan, “Algorithms for Selfish Agents: Mechanism Design for
Distributed Computation”, in STACS 99 Trier, Springer: Berlin 1999.
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet
An early pathbreaking analytical framework for in-
voking economic design principles into the present day
Internet has been proposed by Ferguson et al. 8. As
measurable ingredients for QoS they identified perform-
ance criteria such as average response time, maximum
response time, throughput, application failure probability
and packet loss. QoS can be affected by various factors,
both quantitative (i.e., latency, CPU performance, storage
capacity etc.) and qualitative that proliferate through
reputation systems hinging on trust and belief and as the
Internet matures in promptness, reliability, availability
and foremost security for a certain quality service level
targeted, thus embracing a “Generalized Quality of Ser-
vice” level. If you relate performance to utilize as a
multiplicative factor, performance varies over the range
of utility in 0,1. This would result in service level ar-
rangements (SLAs) comprising service reliability and
user satisfaction (Macias et al. 9).
From the work of Ferguson et al. 8 network pricing
could be looked at as a mechanical design problem. The
user can indicate the “type” of transmission and the
workstation in turn reports this type of the network. To
ensure truthful revelation of preferences, the reporting
and billing mechanism must be incentive compatible.
1.2. A Simple Mechanism Design
There are k agents in an Internet economy that collec-
tively generate demand competing for resources from a
supplier. The supplier herself announces prices entering a
bulletin board accessible to all agents (as a sort of trans-
parent market institution). In a simple form of a trading
process we could exhibit a “tatonnement process” on a
graph where the agents set up a demand to the supplier
who advertise at the prices on a Bulletin Board which are
converted to new prices in interaction with the agents.
The tatonnement process in economics is a simple
form of an algorithmic mechanism design (AMD), Nisan
and Ronen 10, that in modern computer science
emerged as an offspring to algorithmic game theory (Ni-
san et al. 1).
The approach to mechanical design would enable users
of applications to present their QoS demands via utility
functions defining the system performance requirements.
The resource allocation process involves economic
actors to perform economic optimization given schedul-
ing policies, load balancing and service provisioning.
Along this line such approaches have been the basis of
business models for grid and cloud service computing
(Mohammed et al. 2).
Distributed algorithmic mechanism design for internet
resource allocation in distributed systems is akin to an
equilibrium converging market based on economy where
selfish agents maximize utility and firms seek to maxi-
mize profits and the state keeps an economic order pro-
viding basic public goods and public safety (Feigenbaum
et al. 3).
In fact, it’s a sort of Hayek type mechanism limited to
a special case of a diversified Internet economy (Myer-
son [14]).
1.3. Pricing Congestion
The social cost of congestion is a result of the existence
of network externalities. Charging for incremental capac-
ity requires usage information. We need a measure of the
user’s demand during the expected peak period of usage
over some period, to determine the share of the incre-
mental capacity requirement. In principle, it might seem
that a reasonable approach would be to charge a premium
price for usage during the pre-determined peak periods (a
positive price if the base usage price is zero), as is
rountinely done for electricity pricing (Wilson 5).
However, in terms of internet usage, peak demand peri-
ods are much less predictable than for other utility ser-
vices. Since the use of computers would allow to sched-
ule some activities during off-peak hours, in addition to
different time zones around the globe, we face the prob-
lem of shifting peaks. By identifying social costs of net-
work externalities, the suggestion by MacKie-Mason and
Varian 6 is directed toward a scheme for internalizing
this cost as to impose a congestion price that is deter-
mined by a real-time Vickrey auction. The scheme re-
quires that packets should be prioritized based on the
value that the user puts on getting the packet through
quickly. To do this, each user assigns his/her packets a
bid measuring his/her willingness-to-pay (indicating ef-
fective demand) for immediate servicing. At congested
routers, packets are prioritized based on bids. In line with
the design of a Vickrey auction, in order to make the
scheme incentive compatible, users are not charged the
price they bid, but rather are charged the bid of the low-
est priority packet that is admitted to the network. It is
well-known that this mechanism provides the right in-
centives for truthful revelation. Such a scheme has a
number of desirable characteristics. In particular, not
only do those users with the highest cost of delay get
served first, but the prices also send the right signals for
capacity expansion in a competitive market for network
services. If all of the congestion revenues are reinvested
in new capacity, then capacity will be expanded to the
point where its marginal value is equal to its marginal
2. Quality of Service Parameters
2.1. Internet Communication Technologies
The Internet and Asynchronous Transfer Mode (ATM)
have strongly positioned themselves for defining the fu-
ture information infrastructure. The Internet is success-
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Quality of Service on Queueing Networks for the Internet 97
fully operating one of the popular information systems,
the World Wide Web (WWW), which suggests that the
information highway is forming on the Internet. However,
such a highway is limited in the provision of advanced
multimedia services such as those with guaranteed qual-
ity of service (QoS). Guaranteed services are easier to
support the ATM technology. Its capability far exceeds
that of the current Internet, and it is expected to be used
as the backbone technology for the future information
infrastructure. ATM proposes a new communications
paradigm. ATM allows integration of different types of
services such as digital voice, video and data in a single
network consisting of high speed links and switches. It
supports a Broadband Integrated Services Digital Net-
work (B-ISDN), so that ATM and B-ISDN are some-
times used interchangeably, where ATM is referred to as
the technology and B-ISDN as the underlying technical
standard. ATM allows efficient utilization of network
resources, and simplifies the network switching facilities
compared to other proposed techniques in that it will
only require one type of switching fabric (packet switch).
This simplifies the network management process. The
basic operation of ATM, and generally of packet-
switched networks, is based on statistical multiplexing.
In order to provide QoS, the packets need to be served by
certain scheduling (service) disciplines. Resource alloca-
tion algorithms depend heavily on the scheduling
mechanism deployed. The scheduling is to be done at the
entrance of the network as well as the switching points.
The term “cell” designates the fixed-size packet in ATM
networks. ATM allows variable bit rate sources to be
statistically multiplexed. Statistical multiplexing pro-
duces more efficient usage of the channel at the cost of
possible congestion at the buffers of an ATM switch.
When the congestion persists, buffer overflow occurs,
and cells are discarded (or packets are dropped). There-
fore, resources (i.e. bandwidth and buffer space) need to
be carefully allocated to meet the cell loss and the delay
requirements of the user (Gottinger [17]).
The delay and the cell loss probability that the user
wishes the network to guarantee are referred to as the
QoS parameters. Overall, QoS is usually defined in terms
of cell loss probability, delay bounds and other delay and
drop-off parameters. How can one provide such QoS
with guarantees? The general approach is to have an ad-
mission or performance algorithm that takes into account
the traffic characteristics of the source and assigns suit-
able amounts of resources to the new connection during
channel establishment. The admission algorithm is re-
sponsible for calculating the bandwidth and buffer space
necessary to meet the QoS requirements specified by the
user. The algorithm depends on how the traffic is char-
acterized and the service disciplines supported by the
Although the Internet is capable of transporting all
types of digital information, it is difficult to modify the
existing Internet to support some features that are vital
for real time communications. One important feature to
be supported is the provision of performance guarantees.
The Internet uses the Internet Protocol (IP), in which
each packet is forwarded independently of the others.
The Internet is a connectionless network where any
source can send packets any time at speeds that are nei-
ther monitored nor negotiated. Congestion is bound to
happen in this type of network. If congestion is to be
avoided and real-time services are to be supported, then a
negotiation (through pricing or rationing) between the
user and the network is necessary. ATM (Asynchronous
Transfer Mode) is a connection-oriented network that
supports this feature. A virtual channel is established,
and resources are reserved to provide QoS prior to data
transfer. This is referred to as channel establishment.
2.2. Traffic in B-ISDN
In a B-ISDN environment high-bandwidth applications
such as video, voice and data are likely to take advantage
of compression. Different applications have different
performance requirements, and the mechanisms to con-
trol congestion should be different for each class of traf-
fic. Classification of these traffic types is essential in
providing efficient services. There are two fundamental
classes of traffic in B-ISDN: real-time and non real-time
(best effort) traffic. The majority of applications on the
Internet currently are non-real-time ones based on TCP/
IP. TCP/IP is being preserved with an ATM technology.
The Internet can support data traffic well but not real-
time traffic due to the limitations in the functionality of
the protocols. B-ISDN needs to support both non-real-
time and real-time traffic with QoS guarantees. Most data
traffic requires low cell loss, but is insensitive to delays
and other QoS parameters. Standard applications such as
Telnet require a real-time response and should therefore
be considered real-time applications. Video is delay-
sensitive and, unlike Telnet, requires high bandwidth.
High throughput and low delay are required of the ATM
switches for the network to support video services to the
clients. This puts a constraint on the ATM switch design
in that switching should be done in hardware and the
buffer sizes should be kept reasonably small to prevent
long delays. On the other hand, best effort traffic tends to
be bursty, and its traffic characteristics are hard to predict.
This puts another, opposite constraint on an ATM switch,
which requires large buffers at the switching point, fur-
ther complicating its design.
2.3. Congestion Control
Statistical multiplexing can offer the best use of re-
sources, however, this is done at the price of possible
congestion. Congestion in an ATM network can be han-
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Quality of Service on Queueing Networks for the Internet
dled basically in two ways: reactive control and preven-
tive control. Reactive control mechanisms are commonly
used in the Internet, where control is triggered to allevi-
ate congestion after congestion has been detected. Typi-
cal examples of reactive control are 1) explicit conges-
tion notification (ECN), 2) node to node flow control and
3) selective cell discarding. In the more advanced pre-
ventive control approach, congestion is avoided by allo-
cating the proper amount of resources and controlling the
rate of data transfers by properly scheduling cell depar-
tures. Some examples of preventive control mechanisms
are 1) admission and priority control, 2) usage parameter
control and 3) traffic shaping. Appropriate mathematical
tools rooted in AMD are available for congestion control
(Srikant [18]).
Reactive and preventive control can be used concur-
rently, but most reactive controls are unsuitable for high-
bandwidth real-time applications in an ATM network,
since reactive control simply is not fast enough to handle
congestion in time. Therefore, preventive control is more
appropriate for high speed networks.
2.4. Service Discipline
Traffic control occurs at various places in the network.
First, the traffic entering the network is controlled at the
input, second, the traffic is controlled at the switching
nodes. In either case, traffic is controlled by scheduling
the cell departures. There are various ways how to
schedule departure times, and these mechanisms are part
of service disciplines. The service discipline must trans-
fer traffic at a given bandwidth by scheduling the cells
and make sure that it does not exceed the buffer space
reserved (or the delay bound assigned) for each channel.
These functions are usually built into the hardware of the
ATM switch and into the switch controller. When im-
plementing a service discipline in an ATM network, it is
important to choose it simple enough that it can be easily
integrated into an ATM switch. However, the discipline
must support the provision of quality of service guaran-
tees. This also means that the service discipline is re-
sponsible for protecting “well-behaved” traffic from the
“ill-behaved” traffic and must be able to provide certain
levels of QoS guarantees. The service discipline also
needs to be flexible enough to satisfy the diverse re-
quirements of a variety of traffic types, and to be effi-
cient, that is, to permit a high utilization of the network.
Various service disciplines have been proposed, and
many of them have been investigated thoroughly and
compared. An important class is that of disciplines used
in rate-allocating servers.
2.5. Bandwidth-Buffer Tradeoff
A simple example on the representation of QoS parame-
ters is the bandwidth-buffer tradeoff. Bandwidth can be
traded for buffer space and vice versa to provide the
same QoS. If a bandwidth is scarce, then a resource pair
that uses less bandwidth and more buffer space should be
used. Resource pricing is targeted to exploit this tradeoff
to achieve efficient utilization of the available resources.
The pricing concept for a scarce resource is well-known
in economics, but in the context of exploiting the band-
width-buffer tradeoff, Low and Varaiya [19] use non-
linear optimization theory to determine centralized opti-
mal shadow prices in large networks, it shows wide-
spread applicability of mechanism design theory (Vohra
[20]). With respect to large scale application, however,
the complex optimization process limits the frequency of
pricing updates, which causes inaccurate information
about available resources. In order to make pricing in the
context of a buffer-bandwidth tradeoff more adjustable
and flexible it should be based on decentralized pricing
procedures according to competitive bidding in large
markets where prices will be optimal prices if the mar-
kets are efficient. This would also allow flexible pricing
which results in accurate representation of available re-
sources in that prices are updated as the instance connect
request arrives. The subsequent procedure is based on
distributed pricing as a more feasible alternative to opti-
mal pricing.
3. Network Economy
The economic mechanism design model consists of the
following players: Agents and Network Suppliers. Con-
sumers or user classes: Consumers (or user classes) re-
quest for QoS. Each user class has several sessions (or
user sessions). Users within a class have common pref-
erences. User classes have QoS preferences such as pref-
erences over packet-loss probability, max/average delay
and throughput. Users within an class share resources.
Agents and Network Suppliers: Each user class is rep-
resented by an agent. Each agent negotiates and buys
services (resource units) from one or more suppliers.
Agents demand for resources in order to meet the QoS
needs of the user classes. Network providers have tech-
nology to partition and allocate resources (bandwidth and
buffer) to the competing agents. In this competitive set-
ting, network providers (suppliers) compete for profit
Multiple Agent-Network Supplier Interaction: Agents
present demands to the network suppliers. The demands
are based on their wealth and QoS preferences of their
class. The demand by each agent is computed via utility
functions which represent QoS needs of the user classes.
Agents negotiate with suppliers to determine the prices.
The negotiation process is iterative, where prices are ad-
justed to clear the market; supply equals the demand.
Price negotiation could be done periodically or denpend-
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet 99
ing on changes in demand.
Each agent in the network is allocated a certain
amount of buffer space and link capacity. The buffer is
used by the agent for queueing packets sent by the users
of the class. A simple FIFO queueing model is used for
each class. The users within a class share buffer and link
resources. This makes sense to create and maintain net-
work stablity in large networks (Weinard [21]).
Agent and supplier optimality: Agents compete for
resources by presenting demand to the supplier. The
agents, given the current market price, compute the af-
fordable allocations of resources (assume agents have
limited wealth or budget). The demand from each agent
is presented to the supplier. The supplier adjusts the
market prices to ensure demand equals supply.
The main issues from the economic model are:
Characterization of class QoS preferences and traffic
parameters via utility functions, and computation of
demand sets given the agent wealth and the utility
Existence and computation of Pareto optimal alloca-
tions for QoS provisioning, given the agent utility
Computation of equilibrium price by the supplier
based on agent demands, and conditions under which
price equilibrium exists. Price negotiation mecha-
nisms between the agents and suppliers.
This is in adjustment to computing classical economic
equilibrium models such as in Scarf [22].
3.1. Problem Formulation
Network model: The network is composed of nodes
(packet switches) and links. Each node has several output
links. Each output link is associated with an output buffer.
The link controller, at the output link, schedules packets
from the buffers and transmits them to other nodes in the
network. The switch has a buffer controller that can par-
tition the buffer among the traffic classes at each output
link. We assume that a processor on the switch aids in
control of resources.
We have confined ourselves to problems for a single
link (output link) at a node, but they can be applied to the
network as well. Let B denote the output buffer of a link
and C be the corresponding link capacity. Let {ck, bk}be
the link capacity and buffer allocation to class k on a link,
where k
[1, K].
be the price per unit link capacity and unit buffer at a link,
and wk be the wealth (budget) of traffic class k. The
utility function for TCk is
,, .
The traffic of a class is represented by a vector of traffic
parameters (Trk) and a vector of QoS requirements (such
as packet loss probabilities, average packet delay and so
Agent (TC: traffic class) buys resources from the net-
work at the given prices using its wealth. The wealth
constraint of agent TCk is:
bk ckk
pb pc w
A budget set is the set of allocations that are feasible
under the wealth constraint (budget constraint). The
budget set is defined as follows:
:, k
BpxxXpx w
Computation of demands sets: The demand set for
each agent is given by the following:
As set up by Ferguson et al. [8] the goal of TCk is to
compute the allocations that provide maximal preference
under wk and p. Each TCk performs the following to ob-
tain the demand set (defined above): solve {ck,bk} such
max, ,
and budget constraints
bk ckkk
pbpcwcC bB
3.2. Utility Parameters
In the previous section, we showed a general utility func-
tion which is a function of the switch resources; buffer (b)
and bandwidth (c). The utility function could be a func-
tion of the following:
Packet loss expected utility
Average packet delay
Packet tail utility
Max packet delay
Security level
Pr |Um
where m
is a
compromised message accessible by other agents ,
and |mm|0
is true message preserving.
The variables b and c in the utility functions refer to
buffer space allocation and link bandwidth allocation. In
the utility functions Ub and Uc; the parameters bT and cT
are constants. For example, the utility function
for max packet delay is simply a constant as b increases,
but drops to 0 when T
and remains zero for any
further increase in b.
We look at utility functions which capture packet loss
probability of QoS requirements by traffic classes, and
we consider loss, max-delay and throuput requirements.
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet
After this we proceed to utility functions that capture
average delay requirements, followed by utility functions
that capture packet tail utility requirements. We also
could give examples of utility functions for agents with
multiple objectives; agents have preferences over several
QoS parameters.
3.3. Packet Loss
The phenomenon of packet loss is due to two reasons:
the first, packets arrive at a switch and find that the
buffer is full (no space left), therefore, are dropped. The
second is that packets arrive at a switch and are buffered,
but they do not get transmitted (or scheduled) in time,
then they are dropped. A formal way of saying this: for
real-time applications, packets, if delayed considerably in
the network, do not have value once they reach the des-
A proper way to deal with it is through queueing sys-
We consider K agents, representing traffic classes of
M/M/1/B type, competing for resources from the net-
work provider. The utility function is packet loss utility
(U1) for the user classes. We choose the M/M/1/B model
or traffic and queueing for the following reasons:
The model is tractable, where steady state packet loss
utility is in closed-form, and differentiable. This helps
in demonstrating the economic models and the con-
There is interest in M/M/1/B or M/D/1/B models for
multiplexed traffic (such as video), where simple his-
togram based traffic models capture the performance
of queueing in networks (Kleinrock [23]).
In this case we look at the simple queueing system, for
example with Poisson inputs, and exponential interarrival
and general service time distribution B at a single server
(Kleinrock and Gail [24] p. 7,21).
For more complex traffic and queueing models (ex-
ample of video traffic) we can use tail utility functions to
represent QoS of the user class instead of loss utility.
In the competitive economic model, each agent prefers
less packet loss, as the more packet loss, the worse the
quality of the video at the receiving end. Let each agent
TCk have wealth wk, which it uses to purchase resources
from network provider.
Let each TC transmit packets at a rate λ (Poisson arri-
vals), and let the processing time of the packets be expo-
nentially distributed with unit mean. Let c, b be alloca-
tions to a TC. The utility function U for each TC is then a
function of the allocations and the Poisson arrival rate.
3.4. Loss Probability Requirement: Utility
In view of queueing discipline, we consider K agents,
representing traffic classes of M/M/1/B type, competing
for resources from the network provider. The utility
function is packet loss probability (Ut) for the user
classes. We choose the M/M/1/B model of traffic and
queueing for the following reasons. The model is tracta-
ble, where steady state packet loss probability is in
closed form, and differentiable. This helps in demon-
strating the economic models and concepts. Models such
as M/M/1/B or M/D/1/B for multiplexed traffic (such as
video) are appropriate where simple histogram based
traffic models capture the performance of queueing in
For more complex traffic and queueing models (say,
example of video traffic) we can use tail probability
functions to represent QoS of the user class instead of
loss probability. In the competitive economic model,
each agent prefers less packet loss, the more packet loss
the worse the quality of the video at the receiving end.
Let each agent TCk have wealth wk which it uses to pur-
chase resources from network providers.
Let each TC transmit packets at a rate λ (Poisson arri-
vals), and let the processing time of the packets be expo-
nentially distributed with unit mean. Let c, b be alloca-
tions to a TC. The utility function U for each TC is given
as follows:
 
cc cc
cccc c
 
 
 
 
The above function is continuous and differentiable for
0, ,cC and for all
0, .bB We assume b
for continuity purposes of the utility function.
3.5. Loss Probability Constraints
The loss probability constraint is defined as follows: it is
the set of (bandwidth, buffer) allocations
:, c
xXUx L
Ux is the utility function (loss probability
function where lower loss is better) and Lc is the loss
constraint. The preferences for loss probability are con-
vex with respect to buffer and link capacity.
Computation of the QoS Surface by Supplier: assume
that the supplier knows the utility functions of the agents,
which represent the QoS needs of the traffic classes, then
the supplier can compute the Pareto surface, and find out
the set of Pareto allocations that satisfy the QoS con-
straints of the two agents.
This set could be a null set, depending on the con-
straints and available resources.
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet 101
The QoS surface can be computed by computing the
points A and B with a bandwidth-buffer space pair on the
burstiness curve used for resource allocation. The bursti-
ness curve represents the buffer size necessary to avoid
cell losses at each service rate level.
Point A is computed by keeping the utility of (say)
class 1 constant at its loss constraint and computing the
Pareto-optimal allocation by maximizing the preference
of (say) class 2. Point B can be computed in the same
way. The QoS surface is the set of allocations that lies in
A, B. The same technique can be used to compute the
QoS surface when multiple classes of traffic compete for
resources. There are situations where the loss constraints
of both the traffic classes cannot be met. In such cases,
either the demand of the traffic classes must go down or
the QoS constraints must be relaxed. This issue is treated
as an admission control problem, where new sessions are
not admitted if the loss constraints of either class is vio-
3.6. Max and Average Delay Requirements
A max delay constraint simply imposes a constraint on
the buffer allocation, depending on the packet sizes. If
the service time at each switch for each packet is fixed,
the max delay is simply the buffer size or a linear func-
tion of buffer size. Once the QoS surface for loss prob-
ability constraints are computed, then the set of alloca-
tions that meet the buffer constraint will be computed.
This new set will provide loss and max delay guarantees.
A traffic class will select the appropriate set of alloca-
tions that meet the QoS requirements under the wealth
A class of interesting applications would require aver-
age delay constraints on an end-to-end basis. Some of
these applications include file transfers, image transfers,
and lately Web based retrieval of multimedia objects.
Consider a traffic model such as M/M/1/B for each traf-
fic class, and consider that several traffic classes (repre-
sented by agents) compete for link bandwidth and buffer
resources at a link with QoS demands being average de-
lay demands.
Let us now transform the average delay function into a
normalized average delay function for the following rea-
sons: average delay in a finite buffer is always less than
the buffer size. If a user class has packet loss probability
and average delay requirements, then buffer becomes an
important resource, as the two QoS parameters are con-
flicting with respect to buffer. In addition, the switch
buffer needs to be partitioned among the traffic classes.
Another way to look at this: a user class can minimize
the normalized average delay to a value that will be less
than the average delay constraint. This normalized aver-
age delay function for an M/M/1/B performance model,
for an agent, is shown below:
 
ccbc cb
bb c
 
 
This function is simply the average delay divided by
the size of the finite buffer. This function has convexity
properties. Therefore, an agent that prefers to minimize
the normalized average delay, would prefer more buffers
and bandwidth from the packet switch supplier.
THEOREM. 1: The utility function (*) (normalized
average delay) for an M/M/1/B system is decreasing con-
vex in c for
0, ,cC and decreasing convex in b for
Proof. Using standard techniques of differentiation one
can show very easily that U’ is positive.
 
log 1
Ucc cc
 
lim1 2.
The second derivative is also positive:
 
bb b
Ucc ccc
 
  
lim1 .
Consider that agents use such utility functions to ob-
tain the required bandwidth and buffers for average delay
requirements. Then competition exists among agents to
buy resources Due to convexity properties, the following
theorem is stated:
THEOREM 2: Consider K agents competing for re-
sources at a switch with finite buffer and finite band-
width (link capacity) C. If the K agents have a utility
function as shown in (*), then both Pareto optimal allo-
cation s and equilibrium prices exist.
Proof. An intuitive proof can be based on the fact that
the traffic classes have, by assumption, smooth convex
prefereences in
cc C and
0, ,
bb B
and that the utility functions are decreasing convex in the
allocation variables. The prices can be normalized such
that cb
By normalizing the prices, the budget
set B(p) does not change, therefore the demand function
of the traffic classes (utility under the demand set Φ(p) is
homogeneeous of degree zero in the prices. It is also well
known that if the user (traffic class) has strictly convex
preferences, then their demand functions will be well
defined and continuous. Therefore, the aggregate demand
function will be continuous, and under the resource con-
straints, the excess demand functions (which is simply
the sum of the demands by the K traffic classes at each
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet
link minus the resource constraints at each link) will also
be continuous.
The equilibrium point is defined as the point where the
excess demand function is zero. Then using fixed point
theorems (Brouwer’s fixed point theorem), the existence
of the equilibrium price for a given demand can be
shown. Different sets of wealth inputs of the traffic
classes will have different Pareto allocations and price
If the user preferences are convex and smooth, then
under the resource constraints, a Pareto surface exists.
This can also be shown using fixed-point theorems in an
exchange economy type model, where each user (traffic
class) is given an initial amount of resources. Each user
then trades resources in the direction of increasing pref-
erence (or increasing benefit) until a point where no more
exchanges can occur and the allocation is Pareto optimal.
The proof is the same when using the unit price simplex
An agent can use a utility function which is a combi-
nation of the packet loss probability and normalized av-
erage delay function.
3.7. Tail Probability Requirements: Utility
Here we assume that agents representing traffic classes
have tail probability requirements. This is similar to loss
probability. Some applications prefer to drop packets if
they spend too much time in the network buffers. More
formally, if a packet exceeds its deadline in a certain
buffer, then it is dropped. Another way to formalize this
is: if the number of packets in a buffer exceed a certain
threshold, then the new incoming packets are dropped.
The main goal of the network supplier is to minimize the
probability that the number of packets in a buffer cross a
threshold. In queueing terminology, if the packet tail
probability exceeds a certain threshold, then packets are
dropped. The problem for the agent is to minimize packet
tail probability. The agents compete for resources in or-
der to reduce the tail probability. First we discuss tail
probability for the M/M/1 model, and then we consider
agents which represent traffic classes with on-off models.
Which are of particular relevance to ATM networks. We
assume all the traffic classes have the same requirement
of minimizing tail probability which implies competing
for resources from the supplier.
3.7.1. Tail Probability with M/M/1 Model
Consider agents representing traffic classes with tail
probability requirements, and consider an infinite buffer
M/M/1 model, where the main goal is to minimize the
tail probability of the queueing model beyond a certain
threshold. Formally,
Tail.Prob. b
PX bc
 (3)
The system assumes that λ < c. From the above equa-
tion, the tail probability is decreasing convex with re-
spect to c as long as λ < c, and is decreasing convex with
respect to b as long as λ < b.
Consider agents using such a utility function for ob-
taining buffer and bandwidth resources, then using the
convexity property and the regions of convexity being (λ
c). Using the equilibrium condition, as derived in Got-
tinger [25], we obtain for Pareto optimal allocation and
price equilibrium:
 
 
pp bcc
bc c
bc c
We assume K agents competing for buffer and band-
width resources, with tail probability requirements as
shown in (3). For the case of two agents in competition,
the equilibrium condition is as follows:
log log11
 
 
For equilibrium in network economies we can interpret
(5) as the ratio of the logs of the utilizations of the class-
es is proportional to the ratio of the time spent in clearing
the buffer contents.
3.7.2. Tail Probability with On-Off Models
In standard performance models the utility functions are
derived using simple traffic models such as Poisson, with
the mean arrival rate as the main parameter. Here we use
on-off (bursty) traffic models in relation to the competi-
tive economic model. The traffic parameters are mean
and variance in arrival rate. We show how the traffic
variability has an impact on the resource allocation, and
in general the Pareto surface at a link. We assume an
ATM type network where packet sizes are of fixed size
(53 bytes).
On-off models are commonly used as traffic models in
ATM networks (Kleinrock [23]). These traffic sources
transmit ATM cells at a constant rate when active and
nothing when inactive. The traffic parameters are aver-
age burst length, average rate, peak rate, and variances in
burst length. The traffic models for such sources are
on-off Markov sources (Ross [26]). A source in a time
slot (assuming a discrete time model) is either “off” or
“on”. In the on state it transmits one cell and in the off
state it does not transmit any cell. When several such
(homogeneous or heterogeneous) sources feed into an
infinite buffer queue, the tail distribution of the queue is
given by the following formula:
 
Pr, ,,, ,,b
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet 103
where and
,, , v
,, , v
cb C
are functions
of traffic parameters and link capacity c.
Such functions are strictly convex functions in c and b.
These functions are currently good approximations to
packet loss probability in finite buffer systems, where
packet sizes are of fixed size. These approximations be-
come very close to the actual cell (packet) loss for very
large buffers. The utility function is as follows: A TC
consists of S identical (homogeneous) on-off sources
which are multiplexed to a buffer. Each source has the
following traffic parameters:
Tr C
where T is
the average on period, rp is the peak rate of the source,
is the squared coefficient of variation of the on pe-
riod, and ρ is the mean rate. The conditions for a queue
to form are: S rp > c (peak rate of the TC is greater than
the link capacity) and S rp ρ c (mean rate less than link
The packet tail distribution of the queue when sources
are multiplexed into an infinite buffer queue then has the
121 1
USr ccSrSrCT
 
Using a numerical example, we use two traffic classes
(with the same values). There are sessions
in each traffic class,
1,0.5.5, p
Using the
constraints 12 and 12 the Pareto
surface is obtained. As increases from 1 to 20, the
Pareto surface tends to show that buffer space and link
capacity are becoming more and more valuable. The
equilibrium price ratios
vs. v
pc pbC
increase as increases. A higher implies a higher
cell loss probability and therefore more resources are
required, therefore a higher price ratio (link capacity is
more valuable compared to buffer).
4. Specific Cases
Now we consider some specific cases of agents with dif-
ferent QoS requirements.
4.1. Loss and Average Delay
Consider the following case, where two agents have dif-
ferent QoS requirements, one of them (agent 1) has a
packet loss probability requirement and the other (agent
2) has an average delay requirement. We assume that the
network supplier has finite resources, C for link band-
width and B for buffer. Using the properties of loss
probability and average delay with respect to bandwidth
and buffer, the Pareto optimal solution is simply: all
buffer to agent 1, as agent 2 does not compete for link
buffer. The competition is for link bandwidth between
agent 2 and agent 1. Let w1 be the wealth of agent 1,
annd w2 for agent 2, then the equilibrium prices of buffer
and bandwidth are the following:
bb c
ppp wwC
Since there is no competition for buffer space, the cost
of the buffer is simply the fixed cost
p. The Pareto
allocations are
,BCww w
for agent 1 and
0,Cww w
for agent 2.
4.2. Loss and Normalized Average Delay
Consider the following case, where agent 1 and agent 2
have preferences on loss probability and normalized av-
erage delay requirements (transforming average delay
requirements into normalized average delay require-
ments). In this case, the two agents have different utility
functions, however, their preferences are such that more
buffer and more bandwidth is required and this causes
the agents to compete for both resources.
The utility function for agent 1 is as follows:
11loss 1delay1
 
 0,1
The utility function for agent 2 is as follows:
22 loss2delay2
1,where 0UU U
 
 ,1
For example, agent 1 might prefer more weight on loss
probability than normalized average delay compared to
agent 2 who weighs normalized average delay more than
loss probability. Let agent 1 choose 10.9,
and agent
2 choose 20.1.
Due to the convexity properties of
the loss probability function and the normalized average
delay function, the resultant multi-objective utility func-
tion is decreasing convex with respect to bandwidth and
buffer, respectively. Under the equilibrium condition, the
equilibrium prices for the resources have the familiar
prroperty that the ratio of prices is equal to the ratio of
marginal utilities with respect to the resources, for each
Using the resource constraints
cc C
12 ,bb B
we can obtain the Pareto surface. To compute a specific
Pareto allocation one uses the following parameters:
agent 1 and agent 2 have the same traffic arrival rate
The performance model is the M/M/1/B
model for both agents. Using the atonement process,
where agents negotiate with the link supplier to buy
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet
bandwidth and buffer resources, the process converges to
a price equilibrium. The Pareto optimal allocation is split
evenly with respect to buffer and bandwidth among the
agents. The price of link bandwidth is higher than the
price of buffer.
5. Service Economy: Architecture for
Consider a large scale distributed information system
with many consumers and suppliers. Suppliers are con-
tent providers such as web servers, digital library servers,
multimedia database and transaction servers. Consumers
request for and access information objects from the vari-
ous suppliers and pay a certain fee or no fee at all for the
services rendered.
Consider that third party suppliers provide information
about suppliers to consumers in order to let consumers
find and choose the right set of suppliers.
Access and dissemination: consumers query third-
party providers for information about the suppliers, such
as services offered and the cost (price). Likewise, suppli-
ers advertise their services and the costs via the third
party providers in order to attract consumers. Consumers
prefer an easy and simple way to query for supplier in-
formation, and suppliers prefer to advertise information
securely and quickly across many regions or domains.
For example, consider a user who wishes to view a mul-
timedia object (such as a video movie). The user would
like to know about the suppliers of this object, and the
cost of retrieval of this object from each supplier.
Performance requirements: users wish to have good
response time for their search results once the queries are
submitted. However, there is a tradeoff. For more infor-
mation about services offered, advanced searching
mechanisms are needed, but at the cost of increased re-
sponse time. In other words, users could have prefer-
ences over quality of search information and response
time. For example, users might want to know the service
costs in order to view a specific information object. In
large networks, there could be many suppliers of this
object, and users may not want to wait forever to know
about all the suppliers and their prices. Instead, they
would prefer to get as much information as possible
within a certain period of time (response time).
From the above example, in order to let many con-
sumers find suppliers, a scalable decentralized architec-
ture is needed for information storage, access and up-
Naming of services and service attributes of suppliers
becomes a challenging issue when hundreds of suppliers
spread across the globe. A simple naming scheme to
connect consumers, across the internet, with information
about suppliers is essential. The naming scheme must be
extensible for new suppliers who come into existence. A
name registration mechanism for new suppliers and a de-
registration mechanism (automatic) to remove non-exis-
tent suppliers is required. In addition, naming must be
hierarchical, domain based (physical or spatial domains)
for scalability and uniqueness. Inter-operability with re-
spect to naming across domains is an additional chal-
lenging issue not covered in this paper.
The format of information storage must be simple
enough to handle many consumer requests quickly within
and across physical domains. For better functionality and
more information, a complex format of information
storage is necessary, but at the cost of reduced perform-
ance. For example, a consumer, in addition to current
service cost, might want to know more information such
as the cost of the same service during peak and off-peak
hours, the history of a supplier, its services, and its repu-
tation, in order to make a decision. This information has
to be gathered when requested. In addition, the storage
formats must be inter-operable across domains.
Performance: a good response time is important to
make sure consumers get the information they demand
about suppliers within a reasonable time period, so that
decision-making by consumers is done in a timely fash-
ion. In addition, the design of the right architectures for
information storage and dissemination is necessary for a
large scale market economy to function efficiently. Using
the previous example, consumers and suppliers would
prefer an efficient architecture to query for and post in-
formation. Consumers would prefer good response time
in obtaining the information, and suppliers prefer a se-
cure and fast update mechanism to provide up-to-date
information about their services.
Security in transferring information and updating in-
formation at the bulletin boards (name servers) is crucial
for efficient market operation and smooth interaction
between consumers and suppliers. For this the third party
suppliers (naming services) have to provide authentica-
tion and authorization services to make sure honest sup-
pliers are the ones updating information about their ser-
6. Conclusions
We show some applications of mathematical economics
and operations research to resource management pro-
blems in distributed systems and computer networks.
These concepts are used to develop effective market
based on control mechanisms, and to show that the allo-
cation of resources is Pareto optimal.
We propose novel methodologies of decentralized
control of resources, and pricing of resources based on
varying, increasingly complex QoS demands of users.
We bring together economic models and performance
Copyright © 2013 SciRes. IB
Quality of Service on Queueing Networks for the Internet 105
models of computer systems into one framework to solve
problems of resource allocation and efficient QoS provi-
sioning matching large-scale e-commerce applications.
The work can be applied to pricing services in ATM,
networks and (wireless) Integrated Services Internet of
the future. We address some of the drawbacks to this
form of modelling where several agents have to use
market mechanisms to decide where to obtain service
(which supplier?). If the demand for a resource varies
substantially over short periods of time, then the actual
prices of the resources will also vary, causing several
side effects such as indefinite migration of consumers be-
tween suppliers. This might potentially result in degrada-
tion of system performance where the resources are un-
derutilized due to the bad decisions (caused by poor
market mechanisms) made by the users in choosing the
suppliers. As in real economies, the resources in a com-
puter system may not easily be substitutable. The future
work is to design robust market mechanisms and ration-
alized pricing schemes which can handle surges in de-
mand and variability, and can give price guarantees to
consumers over longer periods of time. Another draw-
back is that resources in a computer system are indivisi-
ble resulting in non-smooth utility functions which may
yield sub-optimal allocations, and potential computa-
tional overhead.
In addition to models for QoS and pricing in computer
networks, we are also working towards designing and
building distributed systems using market based on me-
chanisms to provide QoS charge users either in a com-
mercial environment or in a private controlled environ-
ment by allocating quotas via fictitious money (charging
and accounting) by central administrators.
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