Generalized quality-of-service routing with resource allocation

Abstract

We present a general framework for the problem of quality-of-service (QoS) routing with resource allocation for data networks. The framework represents the QoS parameters as functions rather than static metrics. The formulation incorporates the hardware/software implementation and its relation to the allocated resources into a single framework. The proposed formulation allows intelligent adaptation of QoS parameters and allocated resources during a path search, rather than decoupling the path search process from resource allocation. We present a dynamic programming algorithm that, under certain conditions, finds an optimal path between a source and destination node and computes the amount of resources needed at each node so that the end-to-end QoS requirements are satisfied. We present jitter and data droppage analyzes of various rate-based service disciplines and use the dynamic programming algorithm to solve the problem of QoS routing with resource allocation for networks that employ these service disciplines.

Full text

Page 1

450IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
Generalized Quality-of-Service Routing
With Resource Allocation
Ahmed R. Bashandy, Member, IEEE, Edwin K. P. Chong, Fellow, IEEE, and Arif Ghafoor, Fellow, IEEE
Abstract—We present a general framework for the problem
of quality-of-service (QoS) routing with resource allocation for
data networks. The framework represents the QoS parameters
as functions rather than static metrics. The formulation incorpo-
rates the hardware/software implementation and its relation to
the allocated resources into a single framework. The proposed
formulation allows intelligent adaptation of QoS parameters and
allocated resources during a path search, rather than decoupling
the path search process from resource allocation. We present a
dynamic programming algorithm that, under certain conditions,
finds an optimal path between a source and destination node and
computes the amount of resources needed at each node so that the
end-to-end QoS requirements are satisfied. We present jitter and
data droppage analyzes of various rate-based service disciplines
and use the dynamic programming algorithm to solve the problem
of QoS routing with resource allocation for networks that employ
these service disciplines.
IndexTerms—Quality-of-service(QoS),rate-basedqueueing,re-
source allocation, routing.
I. INTRODUCTION
N
remote servers and databases. Transmission of multimedia data
over a broadband network requires special support from the
underlying routers and switches. The objective of the network
support is to guarantee the quality-of-presentation (QoP) re-
quired by the multimedia client(s) at the destination(s) [27]. To
achieve this objective, the network must provide guaranteed
quality-of-service (QoS) under diverse network conditions
and resource constraints. Several researchers have tackled
this problem from a variety of aspects, such as end-to-end
strategies [1], individual node resource management strategies
[2], QoS-routing strategies without resource consideration
[16], [22], [25], [31], [32], and QoS routing with resource
consideration [7], [19]–[21], [28].
In a data network, the output interface of a node employs
a specific packet scheduling algorithm, commonly known as a
EXT-GENERATION Internet applications involve ac-
cessing large volume of multimedia information from
Manuscript received November 1, 2003; revised May 15, 2004. This research
was supported in part by the National Science Foundation (NSF) under Grant
CISE/EIA-9972883, Grant ECS-0098089, Grant ANI-0099137, and Grant
ANI-0207892.
A. R. Bashandy is with Cisco Systems, San Jose, CA 95134 USA (e-mail:
bashandy@cisco.com).
E. K. P. Chong is with the Department of Electrical and Computer Engi-
neering, Colorado State University, Fort Collins, CO 80521 USA (e-mail:
echong@goku.engr.colostate.edu).
A. Ghafoor is with the School of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN 47907 USA (e-mail: ghafoor@
ecn.purdue.edu).
Digital Object Identifier 10.1109/JSAC.2004.839423
service discipline. The service discipline determines the rela-
tion among the QoS parameters provided by the node and the
allocatedresources.Theexistenceoftheserelationschangesthe
perspective about the problem of QoS routing with resource al-
location. In this paper, we present a general formulation of this
problem that employs the knowledge of the service discipline
to combine QoS routing and resource allocation into a single
framework. The formulation exhibits three important features.
First, it takes into consideration the relationship between the
QoSmetricsandtheallocated resources.Second,it allowsinde-
pendent allocation of different resources coupled with routing.
Third, it captures the interrelationship among various QoS met-
rics. These features are particularly important if the QoS re-
quirements and the resource constraints are specified in such
a way that one stringent requirement can be satisfied at the ex-
pense of another one that has some degree of tolerance. Based
on the proposed formulation, we present a dynamic program-
ming algorithm to find a route between the source and desti-
nation nodes and determine the amount of resources along the
intermediate nodes to satisfy the QoS requirements.
The QoS requirements for a path can be specified by four
parameters: the maximum variation in end-to-end jitter delay
, the minimum reliability or percentage of data droppage
, the minimum long term average bandwidth
and the maximum end-to-end delay
requirements involves two aspects: routing and resource al-
location. Traditionally, a network is modeled by a directed
graph and the QoS parameters are captured by a fixed metric(s)
assigned to the edges [7], [16], [19]–[22], [25], [26], [28], [32].
The process of route establishment is usually decoupled from
the problem of resource reservation and the ability to vary
resources and quality metrics. In reality, such metrics are not
fixed and can be changed during the process of path hunting. In
this paper, we focus on two key network resources, namely; the
bandwidth and the buffer space used in the queues of service
disciplines.
This paper is organized as follows. In the next section, we
provide an overview of the recent results in the area of QoS
routing. In Section III, we present a network model that al-
lows capturing of relation among the node implementation, al-
located resources, and the QoS parameters, and specify condi-
tionswhenaroutesatisfiestheQoSrequirements.InSectionIV,
we present a general mathematical formulation for the problem
of QoS routing with resource allocation, analyze the formula-
tion, and present a dynamic programming algorithm to solve
this problem. In Section V, we illustrate the applicability of the
framework presented in Sections III and IV by examining some
of the well-known service disciplines, and derive the relation
,
. Satisfying these
0733-8716/$20.00 © 2005 IEEE

Page 2

BASHANDY et al.: GENERALIZED QoS ROUTING WITH RESOURCE ALLOCATION451
between resources and the level of QoS provided by these dis-
ciplines. Section VI summarizes the distinction between the re-
search results presented in this paper and earlier work in this
area. In Section VII, we conclude the paper and propose some
future work.
II. BACKGROUND
The problem of QoS routing has been the center of attention
inbothacademicandindustrialcommunitiesforsometime.The
research in this area can be broadly divided into two areas: QoS
routing without resource consideration and QoS routing with
resource consideration. In this section, we provide an overview
of some of the relevant results in both areas.
In the area of QoS routing without resource consideration
[16], [18], [22], [25], [26], [31], [32], [34], the problem is typ-
ically formulated as follows. The network is modeled as a di-
rected graph
. QoS parameters are captured as func-
tionsthatmapthesetofedges
usually referred to as weights or metrics. These metrics repre-
sent the level of QoS provided by each edge. In determining
these metrics, interaction among resources is ignored. Instead,
each metric is assumed to be an independent numerical value
that can be aggregated using ordinary addition. The solution to
the QoS routing problem is then a simple path hunting between
the source and destination nodes such that the path satisfies all
the QoS requirements. The QoS routing problem formulated in
thismannerisequivalenttoaconstrainedshortestpathproblem,
which is NP-complete [14]. Some researchers have suggested
approximatealgorithmswithtimecomplexitythatgrowsnonex-
ponentially as the solution becomes more accurate. Other have
provided pseudopolynomial algorithms with time complexity
dependent on the numerical values of the arguments. In some
cases, exact algorithms with exponential time complexity have
been proposed. In [32], Wang and Crowcroft divided metrics
into three categories: additive, multiplicative, and concave (bot-
tleneck). They showed that a combination of one additive or
multiplicative metric and one concave metric is not NP-com-
plete,whileacombinationofmorethatoneadditiveand/ormul-
tiplicative metric is NP-complete. Usually, researchers attempt
to solve the optimization version of the problem, which is to
find a path that minimizes one of the metrics, while satisfying
the constraints on the others.
For the second problem that deals with QoS routing with
resource consideration [7], [19], [21], [28], the network is mod-
eledasadirectedgraphwithQoSparameterscapturedasmetrics
assigned to each edge. Resources are also defined as numer-
ical values associated with each edge. Based on the available
resources, the traffic description, and, in some cases, the em-
ployed service discipline, the metrics associated with each edge
are computed. In some cases, only a single resource is consid-
ered, which is bandwidth [7], [20], [28]. In some studies, the
problem of reliability (data droppage) is not addressed, even
thoughaservicedisciplineisassumedtobeemployed[19],[21].
Mostoftheexistingworkinthisareaappliestospecialcasesand
does not address the general formulation of the problem of QoS
routingwithresourceallocation.Inaddition,intheexistingwork
both the allocated resources and the metrics are assumed to be
staticandcannotbechangedduringtheprocessofpathhunting.
toasetofnonnegativeintegers,
The discussion above highlights the observation that earlier
researchintheareaofQoSroutingattemptstosolvetheproblem
using one of the following two approaches: the first approach
models the problem as a constrained shortest path without con-
sidering any resources. In the second approach, fixed metrics
are assigned to each link based on the available resources and
the traffic description prior to applying a path search algorithm.
However, the results of these approaches do not incorporate the
effect of multiple routers, connected in tandem, on the QoS pa-
rameters of a path, rather addition is used to aggregate the met-
rics. Also, these results do not include the possibility of tuning
the allocated resources during the path search in order to estab-
lish a route that satisfies all the QoS requirements collectively.
For example, suppose that the jitter requirement is very tight
whilethedatadroppagerequirementhassomeflexibility.Inthis
case, it may be essential to reduce the queue size to achieve re-
ducedjittercausedbythequeueingdelay,eventhoughabundant
memory may be available. In other words, assigning the met-
rics prior to the path searching algorithm may preclude finding
a paththatsatisfiestheQoSrequirements, eventhoughone such
path may exist. Varying QoS metrics, if required, is usually car-
ried out prior to a path search. In summary, existing results do
not provide a single framework that allows routers and switches
to utilize the combined knowledge of hardware/software imple-
mentation and the available resources to take intelligent routing
decisions.
III. NETWORK MODEL
As mentioned in Section II, the values of the QoS parameters
are sometimes computed based on the available resources, the
trafficshape,andservicedisciplines.Thesevaluesarecomputed
prior to searching a path and are not allowed to change during
the path search. The main drawback of this approach is that it
does not fully incorporate the knowledge of the network ele-
ments implementation and the traffic shape inside the network
in relation to the allocated resources. In this section, we extend
the graph theoretic model to incorporate this knowledge. The
extension to the graph theoretic model is based on the concept
of jitter graph proposed in [4]. A jitter graph consists of jitter
nodes and jitter links. We proceed to discuss these two entities
and the way they abstract the physical characteristics of a data
network. Based on the jitter graph model, we identify the rela-
tionships between the QoS parameters and different resources
inside a network.
A. Concept of Jitter Graph
A jitter graph consists of a set of jitter nodes V and a set of
jitter links E. The set of nodes
the cross-connections residing in switches, routers, computers,
etc. The set of links
represents the service disciplines em-
ployed at the output interface of the switches and the physical
communication lines to which they are connected. Traffic tra-
verses such networks through established channels. A channel
canbeviewedasapaththroughajittergraph,whichissequence
ofjitternodesconnectedviajitterlinks.Thecorrespondencebe-
tween the physical network and proposed model is depicted in
Fig. 1.
represents the processors and

Page 3

452IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
Fig. 1. Network model.
Ajitterlink
terized by five basic attributes: 1) maximum propagation delay
; (2) minimum propagation delay
mission capacity
; 4) reliability
The maximum propagation delay,
time difference between transmitting a packet by node
receiving this packet at the node
the end-to-end delay of point-to-point connections that employ
service disciplines with bounded queueing delay, such as PGPS
(WFQ) [23],
[6], and VC [37]. The parameter
also represent a statistical bound for networks with arbitrarily
large delays. The attribute
between transmitting a packet by node
input of node . This attribute captures the physical propaga-
tion delay and the service time, which is usually the size of a
packet in bits divided by the physical capacity of the link in
bits per second. The third attribute
transmission capacity of a link in bits per second. The fourth
attribute
is the minimum reliability of the jitter link
Formally, let
be the total number of packets transmitted
by the jitter node
toward the jitter node
so far. Letbe the total number of packets received
at the jitter node . The jitter link
reliability value of
if
Several researchers [12], [30], [36] define jitter as the differ-
encebetweenthemaximumand minimumpossiblepropagation
delaysacrossthelink
,thatis
this is a valid and widely accepted definition—and we use it in
Section V—for the sake of generality, we provide a less restric-
tive definition of jitter. We define jitter for a jitter link
the variation in the delay between submitting a packet for trans-
mission at the node
till it is completely received at the input
of node . For a certain class of applications, jitter can be taken
to be the maximum difference. For some class of applications,
statistical measures, such as the variance of the delay along the
link, may be more appropriate.
We make further assumptions to facilitate the analysis.
• The reliability value
only by data droppage occurring at the output service dis-
cipline as a result of buffer overflow. This assumption
is justified by the fact that packet loss due to corruption
during signal propagation along optical fibers or coaxial
connectingthetwonodesand ischarac-
; 3) maximum trans-
; and 5) jitter
, is the maximum
.
and
. This parameter captures
can
is the minimum time difference
and receiving it at the
represents the physical
.
along the jitter link
is said to provide a
.
.Although
as
of a jitter linkis affected
cables tends to be negligible when compared with packet
droppage in nodes.
• We assume that a switch is nonblocking [17], that is,
the switch fabric has enough processing power to relay
packets from the input to the output interface. Hence,
we can assume that jitter and delay result only from the
queueing, service time, and/or propagation delay.
B. Effect of Resources on the QoS Parameters
Based on the jitter graph model outlined in Section III-A, we
identify two types of resources:
•
: the buffer allocated for a given data stream in the
service discipline residing in the node
data to the node ;
•
: the bandwidth allocated for a given data stream
along the communication link between the nodes
represented by the jitter link
The above resources are restricted by the maximum values im-
posed by the hardware and/or software limitations of a node.
These restrictions are specified as follows:
•
,thetotalbufferspaceavailableforthedatastream
under consideration to the service discipline transmitting
packets from the node
to the node ;
•
, the physical link capacity;
•
, the available bandwidth or an administra-
tive limit on the maximum amount of bandwidth available
for allocationto the data stream under consideration along
the jitter link
.
Due to the existence of the service discipline, the jitter, reli-
ability, and end-to-end delay are functions of the resources and
the traffic shape. Hence, we have
and transmitting
and
.
where the functions
traffic shape and the service discipline employed along the jitter
link
. In Section V, we provide a derivation of functions
,, andfor various service disciplines.
The next step is to specify QoS parameters of a path
jitter graph. Traditionally, the QoS parameters of a path are as-
sumed to be the sum or the product [9], [32] of the QoS parame-
ters on each individual link. However, in general, a simple sum-
mation or product may not provide accurate tight bounds on the
QoSparametersofapath.Foragivenpath
jitter
, the reliability
and the bandwidth
should be some functions, with argu-
ments corresponding to the amount of resources allocated along
the path. That is
,, andare specified by the
in a
,the
, the maximum delay,
(1)

Page 4

BASHANDY et al.: GENERALIZED QoS ROUTING WITH RESOURCE ALLOCATION453
where the functions
teraction between the service discipline in a path and the traffic
shape. As mentioned in Section I, data streams can specify
their QoS requirements in terms of four parameters: the max-
imum jitter
, the minimum reliability
end-to-end delay
, and the minimum longterm bandwidth
. A pathis said to satisfy the QoS requirements of the
stream if
,,, and are specified by the in-
, the maximum
and
(2)
Earlier studies [6], [13], [15], [22]–[24], [32], [35] have consid-
ered the bandwidth
as a concave or bottleneck function of
the path. In this paper, we adopt the same convention. Hence
(3)
For convenience, we define the parameter
by
for a jitter link
(4)
which represents the log-reliability of the link. Similarly, we
define the path log-reliability parameter
for a pathby
(5)
Hence, a path
satisfies the reliability requirement
, where
if
(6)
This parameter is adopted from the reliability linearization pro-
cedure used in [32].
In the rest of the paper, we pay a special attention to in-
formation retrieval and broadcast types of applications such as
Internet-based radio and TV broadcast, retrieval of preorches-
tratedmultimediadocuments,andmultimediaWebservices.For
this class of applications, jitter,
ical role in terms of end-to-end QoS management, rather than
end-to-end delay,
, and
this paper, we focus on the problem of finding a path in a jitter
graphforaspecificdatastreamsuchthattherequirements(
, and) are satisfied.
Next, we discuss the problem of QoS routing with resource
allocation and formulate it as a nonlinear program.
, andplay a more crit-
. Consequently, for the rest of
,
IV. QOS ROUTING WITH RESOURCE ALLOCATION
In this section, we start by defining the problem of QoS
routing with resource allocation in light of the network model
discussed in Section III. Then, we present the path feasibility,
the subpath addition, and nonincreasing reliability conditions
that lead to a polynomial time algorithm for solving the routing
problem. At the end of this section, we present a dynamic
programming algorithm that solves the problem in polynomial
time.
A. Problem Definition
Consider a jitter graph
representing the source and destination nodes, respectively.
For each jitter link
, we have the following resources: the
buffer size
, where
width
, where
Definition IV.1: For a given path path
problem is defined as follows:
and two nodes ,
, and the band-
.
, the path-resource
(7)
Note that the constraint
the convention that the bandwidth of a path
function of the path as specified in (3).
A solution to the path-resource problem is represented by nu-
merical values assigned to the decision variables
for all. Specifying these values provides a
methodology of allocating resources along a path
data stream so as to satisfy its QoS requirements. Based on (7),
we define a feasible path as follows.
Definition IV.2: A path
is said to be feasible for the QoS
requirements specified by (
path-resource problem defined in (7) for the path
Definition IV.3: Let
the nodes
and . The problem of QoS routing with resource
allocation is defined as follows:
results from adopting
is a bottleneck
and
for a given
, ) if the feasible set in the
is nonempty.
be the set of all paths between
(8)
This definition of the QoS routing problem is slightly nontradi-
tionalastheproblemofQoSrouting,asdefinedintheliterature,
is to find a feasible solution rather than finding an optimized
QoS metric subject to a set of constraints specified on the other
metrics. However, the optimization problem does provide a so-
lution for the feasibility problem.
The QoS routing problem in (8) has certain properties that
makes it distinct from the routing problems addressed in the
literature. The key distinct features of our routing problem are
listed as follows.
• Our definition does not assume that metrics are combined
through addition. Rather, the QoS attributes of a path are
functions that depend on the QoS resulted from the allo-
cated resources and the individual links.
• Our definition incorporates the relation between metrics
and resources in addition to the interaction between
routers. A solution to the problem of QoS routing with
resource allocation defined by (8), if it exists, depends on
the local functions
and
and the global functions
multiple jitter links connected in tandem.
Routing problem in such a general form is intractable. To
solve this problem, we need to impose certain restrictions on
the parameters
,,,
for each jitter link
, and
,
,constructed by
, and. In the next subsection,

Page 5

454IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
we present the path feasibility, the subpath addition, and the
nonincreasing reliability conditions, which result in a polyno-
mial time algorithm for solving this problem. In Section V, we
analyze various service disciplines to derive instances for the
problem of QoS routing with resource allocation based on the
properties of these service disciplines.
B. Optimality Conditions
In this subsection, we present sufficient conditions for the al-
gorithm presented in Section IV-C to solve the problem of QoS
routingwithresourceallocationdefinedin(8).Theseoptimality
conditionscapturesomepropertiesofthehardwareandsoftware
implementation of the routers and switches.
Definition IV.4: Let
the two paths
andsuch that the last node in
first node in
are identical. The binary operator
as follows:
returns the sequence of nodes and edges
constructed by concatenating the path
path
.
Note that the operator
than a path or a trail [33], where a walk is a sequence
such that
path is walk with distinct vertices.
The correctness of the proposed algorithm is based on the
path feasibility, the subpath addition, and the nonincreasing re-
liability conditions defined below.
Definition IV.5: Let
be a jitter graph with the parameters
,, anddefined for every edge
, , anddefined for every path. Let the QoS
requirements for the bandwidth and jitter be defined by be the
positive pair (
,). The graph
feasibility condition if, for each path
the requirement
small positive values of
.
The path feasibility condition may appear to be unsatisfiable.
However,theintuitionbehinditisasfollows.Formostqueueing
systems, the queueing delay, which is the main source of jitter,
can be reduced by increasing the bandwidth or decreasing the
queue size. The bandwidth cannot be increased arbitrarily due
tothephysicalandtheadministrativelimitsofanode.However,
in most of the cases, the queue length can be reduced to an arbi-
trarilysmallsizetosatisfyanarbitrarilysmalljitterrequirement.
Definition IV.6: Let
be a jitter graph with the parame-
ters
,, and defined for every edge
and, , and defined for every path
sider two paths
and
, where is the value of the optimum cost
function of the path-resource problem defined in (7). Consider
. Let
. Let both paths
sible. The jitter graph
is said to satisfy the subpath addition
condition if
be a jitter graph. Consider
and the
is defined
to the end of the
produces a walk rather
, while a
and
is said to satisfy the path
such that
can be satisfied for arbitrarily
,
. Con-
such that
and
and be fea-
The intuition behind the subpath addition condition is that, in
some cases, concatenating the same edge to two paths results in
the same reduction in reliability.
Definition IV.7: Let
,, and
,
be a path from
The jitter graph
condition if
be a jitter graph with the parameters
defined for every edge
defined for every path . Let
. Letbe a path from
is said to satisfy the nonincreasing reliability
and
, and
toto .
(9)
The nonincreasing reliability condition defined by the relation
(9) is satisfied for almost all practical networks. This is because
the probability of data droppage in the additional path
increases the probability of data droppage for the whole path,
unlesssuch a pathemploys an error correctiontechnique,which
rarely occurs for high volume multimedia traffic.
C. QoS Routing Algorithm
Inthissection,wepresenta dynamicprogrammingalgorithm
to solve the problem of QoS routing with resource allocation
defined in (8). The algorithm defined by the following recursive
relation:
(10)
where
is the path computed at the
. The derivation of (10) can be found
in Appendix A-1.
Based on (10), the algorithm QoS-Routing specified in
Algorithm IV.1 finds a solution to the problem of QoS routing
with resource allocation.
iteration and
Algorithm IV.1: QoS-Routing( ,
,)
1 Pre-Process
2Initialization Loop
3 For each
4 For each
5 For each
6 Main Loop
7 for
8
for each node
9
10
11
for each node
12
13
14
15
if
16
17
else
18
,,
-
-
Before we analyze this algorithm, we point out an important
issue. The algorithm is givenin a general form, and relies on the
subalgorithms Pre-Process and Path-Opt. Prior to executing
the main loop, the subalgorithm Pre-Process performs some
preliminary processing such as deleting edges with insufficient

Page 6

BASHANDY et al.: GENERALIZED QoS ROUTING WITH RESOURCE ALLOCATION455
resources. The subalgorithmPath-Opt solvesthepath-resource
problem defined in (7). Implementations of these subalgorithms
depend on the characteristics of the network elements (routers
and switches) employed by the network and the interaction be-
tween these elements when they are connected in tandem. The
properties and interaction are completely specified by the func-
tions
,,,, and . In Section V, we examine some
of the rate-based service disciplines to derive the functional
forms for
,,,, and
crete implementations for the subalgorithms Pre-Process and
Path-Opt.
The analysis of the proposed QoS-Routing algorithm is as
follows.
and, hence, produce con-
1) Time Complexity: Assume that the time complexity of
Path-Opt is
and that of Pre-Process is .
Intheinitializationloop,alltheedgesarescannedonce.
In the main loop, all the edges are scanned
Hence, therunningtime ofthealgorithm is
.
2) Correctness of the QoS Routing Algorithm: To prove the
correctness of the algorithm, we need to show that the ap-
plication of this algorithm on a jitter graph yields a solu-
tion to the problem of QoS routing with resource alloca-
tion as defined by (8), given that the path feasibility, the
subpath addition, and the nonincreasing reliability con-
ditions are satisfied. Such a proof involves two aspects.
First, we need to prove that the subgraph
with
we need to prove that, at the termination of the algorithm,
for all nodes
reachable from .
times.
is acyclic. Second,
that are
The next theorem provides a formal statement of the correct-
ness of Algorithm IV.1.
Theorem IV.1: Given the jitter graph
rameters
,,,,
sume that the path feasibility, the subpath addition, and the non-
increasing reliability conditions given in Definitions IV.5, IV.6,
and IV.7, respectively, are satisfied. Assume that the algorithm
Path-Opt solves the path-resource problem defined in (7) if a
path exists or returns
otherwise. Then, at the termination of
Algorithm IV.1, the following statements are true:
with pa-
as defined in (1) and (3). As-
1)
for every node such that there
is no path from the source node
to ;
2)
for every node such that there
is no path from the source node
to ;
3)
for every node such that
there exists a path from the source node
4) the graph
to ;
such that
contains the
path
from the source node
for every nodeif there exists a path
to .
Proof: Due to space limitations, we only provide an out-
line of the proof. Refer to [3, Sec. 5.7] for details of the proof.
The proof is carried out by induction on the variable
AlgorithmIV.1.Intheinductionbasis,thetheoremisprovedfor
. That is, the theorem is proved for one edge only. In the
used in
induction step, we prove the theorem for
is correct for .
In this section, we have presented an algorithm that solves
the problem of QoS routing with resource allocation based on
the path feasibility, the subpath addition, and the nonincreasing
reliability conditions, which, in effect, impose certain relation-
shipsbetweenthedifferentQoSparametersandtheavailablere-
sources. In the next section, we examine some of the rate-based
service disciplines to derive relationships between the QoS pa-
rameters and the available resources. We illustrate the condi-
tions under which the path feasibility, the subpath addition, and
the nonincreasing reliability conditions are satisfied.
assuming that it
V. APPLICATIONS TO RATE-BASED SERVICE DISCIPLINES
In the previous sections, we have presented a general frame-
work that captures the effect of service disciplines, network re-
sources, and the interaction of nodes (connected in tandem), on
the level of QoS provided by a network. The proposed frame-
workprovidesageneralformulationoftheQoSroutingproblem
withresourceallocation.Thekeyfeatureoftheframeworkisthe
representation of the QoS parameters as functions of resources
rather than fixed value metrics. In this section, we demonstrate
the applicability of the framework by examining some of the
rate-based service disciplines and deriving functional forms for
,,,, and . The end-to-end delay
can be easily derived from the jitter
By studying several service disciplines, such as the ones dis-
cussed in [6], [13], [15], [23], [24], and [37], it can be noticed
that the jitter and data losses in a network depend on three fac-
tors: 1) the traffic shape; 2) the interaction between service dis-
ciplines connected in tandem; and 3) the allocated resources. To
elaborate,letthearrivalfunctionattheoutputqueueofthenode
be denoted by. Then, for the session , the jitter
link
has well defined functions
on the function
, the resources allocated at the node
, and the properties of the communication link between the
nodes
and . Leaky bucket constrained sources are commonly
used to model deterministically bounded data sources [6], [8],
[15], [23], [29]. The analysis and results obtained in the liter-
ature are generally based on the assumption that data sources
are leaky bucket constrained. In the usual leaky bucket notation,
with
andrepresenting the bucket size, and average traffic
rate, respectively, an arrival function
form to
, or
and
and.
and that depend
is said to con-
, if, for any interval
. For the rest of this
,
section, we use leaky bucket constrained data sources to model
the traffic entering a network.
We examine four service disciplines in this section: gener-
alized processor sharing (GPS) [10], [23], packet by packet
generalized processor sharing (PGPS) [23], [24], worst case
fair weighted fair queueing
fair queueing (SCFQ) [15]. It is shown in [13], [35] that the
resource requirements and QoS behavior of virtual clock (VC)
[37] is identical to PGPS. Hence, we can apply the results for
networks employing the PGPS discipline to the ones employing
the VC technique. The GPS, PGPS,
disciplines have some common characteristics. For example,
[6], and self-clocked
, and SCFQ service

Page 7

456IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
they all provide an upper bound on the queueing delay, which
is equivalent to the link jitter
obtained for these service disciplines assume the leaky bucket
constrained data sources. Note, the aforementioned service
disciplines are analyzed to illustrate the generality and applica-
bility of the definitions and algorithm defined in Section IV.
Let
•
be the maximum number of backlogged bits
from the session
in the output service discipline queue
of the node
during the life time of the session ;
•
be the maximum number of bits backlogged in-
side the path
•
be the number of sessions traversing the link
•
be the size of the buffer in bits allocated for the
session at the output service discipline of the node ,
which transmits packets to the subsequent node ;
•
be the bandwidth (in bits per second) allocated
for the
session along the communication link
•
be the capacity (in bits per seconds) of the link
connecting the nodesand ;
•
be the maximum packet size in bits.
The values of
and
allocatedforthe
sessionattheoutputservicedisciplineofthe
node
and along the jitter link
theparameters
andofthejitterlink
the service discipline, are functions of the resources
and . Also, for a path , the parameters
are functions of the resources
each link
.
For the GPS, PGPS,
, and SCFQ disciplines, we per-
form the following tasks.
• Analyze the service discipline to determine
, , and , as functions of the resources.
• Specialize the path-resource problem defined in (7) to re-
flect the properties of the service discipline. The resulting
path-resource and QoS routing problems are considered
to be instances of the general problems defined in (7) and
(8), respectively.
• Solve the instance of the path-resource problem.
• Derive a solution for the instance of the path resource
problem for the path
given that we have a solution for the subpath
. The objective is to reduce the time com-
plexity of the subalgorithm Path-Opt.
• Discuss the optimality conditions and describe how to use
AlgorithmIV.1tosolvetheinstanceoftheproblemofQoS
routing with resource allocation.
• Computethetimecomplexityneededtosolvetheproblem
of QoS routing with resource allocation corresponding to
the service discipline under consideration.
In the following sections, we analyze the above mentioned
service disciplines.
. Also, the analysis and results
;
;
;
represent the resources
. Hence, for each session ,
,representing
, , and
forand
,,
A. Generalized Processor Sharing (GPS)
Generalized processor sharing (GPS) was first proposed in
[10] under the name weighted fair queueing (WFQ). It is also
known as the fluid fair queueing (FFQ) [15], [35]. GPS has also
been analyzed in [23] and [24], where the authors provide guar-
anteed bandwidth and worst case queueing delay on a session
basis. In GPS, packets are assumed to be infinitesimally divis-
ible and the server can serve multiple packets simultaneously.
1) The Instance of the Path-Resource Problem: The deriva-
tion of the QoS parameters for the th session along the path
consisting of GPS servers connected in tandem is given in
Appendix A-2. Given
,
path-resourceproblemfor apathconsistingofGPSserverscon-
nected in tandem by the following optimization problem:
, and , we can define the
subject to
(11)
where
lowed for the
variables of this problem are
and
.Thebandwidthistheminimumbandwidthalong
the path. If we choose the real number
the fraction of the link bandwidth assigned to session
Appendix A-2), such that
represents the maximum buffer size al-
session on the link . The decision
,,
representing
(see
, then
(12)
It can easily be shown that if the value of the function
is optimum, then the value of
as given in (12). After finding the optimum
value of the decision variable
decision variables
2) Solving the Instance of the Path-Resource Problem
Given the Solution for
: Define the variable
is at least
, we are left with the
.,
. The solution to the optimiza-
tion problem in (11) has three cases.
Case 1)
Case 2)
and
.
Case 3)
and
.
In Case 1), the number of bits accumulated in the queues
inside the path cannot possibly increase the jitter beyond
the maximum required value
solution can be obtained by having
for all
. Hence, an optimum
and
.

Page 8

BASHANDY et al.: GENERALIZED QoS ROUTING WITH RESOURCE ALLOCATION457
For Cases 2) and 3), the maximum number of queued
bits can result in increasing the jitter along the path be-
yond the maximum allowed value
of buffer space
,
duced to avoid violation of the jitter requirement. Therefore,
we need to maximize the value of
. Hence, the amount
, must be re-
such that
. The optimum solution
for this case is given by
(13)
(14)
To show that the solution provided in (13) and (14) in-
deed yields the optimum value of
and 3), we need to consider two cases. The first case is
when
for Cases 2)
. Let
.
In this case, nothing can be done because increasing any
other value
,
of
, which is dictated by
second case is when
all
. In this case, if we increase the value of
for some
the value of
for another
the total number of queued bits may result in violating the jitter
constraint. This results in having
, will not affect the value
. The
for
, we must decrease
, otherwise
(15)
which leads to a smaller value of
3) Solving the Path-Resource Problem for
the Solution for
: Suppose we solve the path-resource
problem for the path
we concatenate a jitter link (
to produce. We can use the solution of the pathre-
source problem for
to solve the path-resource problem
for
. This technique is valid due to the following
two observations. First, if the path-resource problem for
belongs to either Case 2) or 3), then the path resource problem
for
must also belong to Case 2) or 3) because adding
a jitter link representing a GPS server can never reduce the
overall jitter. In other words, the variable
larger or stay unchanged. Second, in solving the path-resource
problem for
, there is no need to recompute the values
of the QoS parameters of the required resources for each link in
. We only need to recompute the global parameters which
include
,,
and
to determine whether the path-resource problem for
belongs to Case 1), 2), or 3). We can update these global param-
eters every time an edge is concatenated to a path. The values
of
for each individual link can be computed after the
optimum path between the source node
in the graph is computed. Based on these observations, we can
.
Given
. In addition, suppose
) to the end of,
can only get
,
. In addition, we need
and each node
solve the problem for
applying the following steps.
1) Set
2) If
belongs to Case 2) or 3), then
a) If
belongs to Case 2) and
using the solution ofby
.
, then
belongs to Case 2). Hence,
.
b) If
,
then
3).
belongstoCase
Hence,
.
3) If the path-resource problem for
then
belongs to Case 1), then
belongs to Case 1),
may belong to Case 1), 2), or 3). If
. If
belongs to Case 2) or 3), then apply steps 2a) or 2b),
respectively.
4) Optimality Conditions: A GPS network satisfies the path
feasibilityconditiondefinedinDefinitionIV.5becausethesolu-
tions to the instance of the path-resource problem presented in
(27),(13),and(14)arevalidforanypositivevalueof
it can be noticed that concatenating a jitter link to path does not
increase the overall reliability, which means that GPS networks
satisfy the nonincreasing reliability condition.
The subpath addition condition defined in Definition IV.6 is
satisfied if
, for a path
concatenating the link
. However, if
creases because the new link has a smaller available bandwidth,
then the subpath addition condition may not be satisfied. To
solvetheproblemofQoSroutingwithresourceallocationunder
this condition, we can apply the technique used in [19] and [21]
in conjunction with the algorithm QoS-Routing specified in
Algorithm IV.1. In this technique, the set of links
tioned into
disjoint subsets
such that the links belonging to the same subset
samevalueof
.Denotethevalueof
by. The subsets are sorted such that
Thealgorithm QoS-Routingis applied
th time all the links having
for all the remaining edges.
AnimplementationofthesubalgorithmPath-Optcanusethe
steps in Section V-A3 to solve the path-resource problem for
a path consisting of GPS servers. We call this implementation
GPS-Path-Opt.
An implementation for the subalgorithm Pre-Process used at
the beginning of the algorithm QoS-Routing specified in Algo-
rithm IV.1 scans the set of links
ifor if
imum required bandwidth for the session under consideration.
The time complexity of such an implementation is
5) Time Complexity: It can be noticed from Algorithm IV.1
that while solving the path-resource problem for the path
from the solution the path
insteps. This value is used by the al-
gorithm for updating the attribute
timum path between the source node
.Also,
, remains constant after
de-
is parti-
, where,
have the
forthesubset
.
timessuch thatatthe
are deleted and
and deletes any link
, whereis the min-
.
, we can find the value of
while searching for an op-
and any node .

Page 9

458IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
After finding the optimum path to every node, we compute the
value of the resources for the path from the source node
the destination node
using the GPS-Path-Opt algorithm. The
time complexity of the GPS-Path-Opt algorithm is
because the longest path in a graph has at most
Hence, the time complexity of the QoS-Routing algorithm is
. As we apply the QoS-Routing algorithm
times, where , the time complexity for solving the
problem of QoS routing with resource allocation for GPS net-
works is
.
to
edges.
B. PGPS and
The PGPS [23], [24] and
based on the GPS discipline. The GPS service discipline is not
a realistic service discipline because it assumes that packets are
infinitesimally divisible and that the server can serve multiple
packetssimultaneously.ForthePGPSservicediscipline,theau-
thors proposed a server that serves packets from the backlogged
sessions in the same order as their finish service time under
the GPS discipline. For the
not consider all the backlogged packets; rather it considers only
those packets that have already started service, and possibly fin-
ished, under GPS. Details on how to choose the next packet to
serve can be found in [6], [23], and [24].
1) The Instance of the Path Resource Problem: The deriva-
tion of the QoS parameters for the th session along the path
consisting of PGPS or
is given in Appendix A-3. Given
specialize the path-resource problem defined in (7) for a path
consisting of PGPS or
servers as follows:
[6] service disciplines are
discipline, the server does
servers connected in tandem
, , and, we can
subject to
(16)
2) SolvingtheInstanceofthePath-ResourceProblem: Asin
the case of GPS,
is given the same value as in (12).
Thus, the decision variables for this problem are
.
Let the variable
. Just like the path-resource problem for GPS de-
fined in (11), the solution to the optimization problem in (16)
has three cases.
Case 1)
,
Cases 2 and 3)
In Case 1), the number of bits accumulated in the queues in-
side the path
cannot possibly increase the jitter beyond
the maximum required value
. Hence, an optimum solu-
tion can be achieved by having
for all, and
.
In Cases 2) and 3), the accumulation of bits inside the
queues can cause the jitter constraint to be violated. Hence, we
need to reduce the maximum number of queued bits
by reducing the maximum queue size
jitter link
. Note that we cannot decrease
jitter by increasing the bandwidth
value
is already as large as possible. Thus, the
problem reduces to maximizing
. The solution to this problem is
specified by assigning values to the variables
.
Optimum values of
by
for each
because the
subject to having
for
, are given
(17)
and the corresponding optimum value
is given by
(18)
The derivation of the solution given in (17) and (18) is omitted
due to space limitations. Detailed derivation can be found in
[3, Sec. 5.5.2].
3) Solving the Path-Resource Problem for
the Solution for
: Similar to the case of GPS, we need to
show that if we have the solution of the path-resource problem
for the path
, we do not need to resolve the problem for
the path
constructed by concatenating the jitter link
to the end of. This can be proved by using the
same arguments used for the case of GPS discipline. Thus, we
can solve the problem for
by applying the following steps.
1) Set
2) If
belongs to Case 2) or 3), then
a) If
belongs to Case 2) and
Given
using the solution of
.
, thenbe-
longs to Case 2). Hence,
.
b) If
, then
is belongs to Case 3). Hence,
.
3) If the path-resource problem for
case (1), then
belongs to Case 1), then
belongs to
belong Case 1), 2), or 3). If
.
If
and 2b), respectively.
belongs to Case 2) or 3), then apply steps 2a)

Page 10

BASHANDY et al.: GENERALIZED QoS ROUTING WITH RESOURCE ALLOCATION459
An implementationof thesubalgorithm Path-Optcan follow
the steps presented in this subsection to solve the instance of the
path-resource problem for PGPS and
such an implementation as PGPS-
4) Optimality Conditions and Time Complexity: Just like
GPS, jitter graphs representing PGPS and
plines satisfy the path feasibility and nonincreasing reliability
conditions defined in Definition IV.5 and IV.7, respectively. The
subpath addition condition is satisfied only if the bandwidth
of a path does not decrease when a link is concatenated to
its end. Thus, we can apply the technique used in solving the
problem of QoS-routing with resource allocation for GPS net-
works to solve the same problem for PGPS or
An implementation for the algorithm Pre-Process for PGPS
or
performs the same tasks as those for GPS networks.
The time complexity is also the same as that of GPS networks.
. We can refer to
-Path-Opt.
service disci-
networks.
C. Self-Clocked Fair Queueing (SCFQ)
SCFQ service discipline is presented as an example of a ser-
vice discipline that can be incorporated into the framework pre-
sented in Sections III and IV but applying Algorithm IV.1 to a
network that employs such a service discipline may not yield
an optimum solution for the problem of QoS routing with re-
source allocation. Instead, we provide a suboptimal solution for
this service discipline.
In the PGPS and
, computing the tag for selecting the
nextpackettobetransmittedmaybetoocomplexforhighspeed
networks. For SCFQ, an approximate method for computing
suchataghasbeenproposedinearlierliterature.Duetosuchap-
proximation, packets under SCFQ service discipline can suffer
more queueing delay than PGPS and
SCFQ in [35] and provided bounds on the buffer space require-
ments and the queueing delay. As mentioned in Sections V-A
and V-B, we assume that the arrival function
.
1) The Instance of the Path-Resource Problem: The deriva-
tion of the QoS parameters for the
consisting of SCFQ servers connected in tandem is provided
in Appendix A-4. Given the functional forms of
, andfor a path
problem for SCFQ can be defined as follows:
. Zhang analyzed
session along the path
,
, the path-resource
subject to
(19)
2) Solving the Instance of the Path-Resource Problem: As-
suming that
define the variable
to be the same as that used for PGPS
and
. By replacingwith
, we can
such that
(20)
we can apply the procedure used in solving the path-resource
problem for PGPS and
problem for SCFQ.
3) Optimality Conditions: The subpath addition is satisfied
if the bandwidth
of a path does not change when a link is
concatenated to its end. Also, the nonincreasing reliability con-
dition is satisfied. However, due to the existence of the term
to solve the path-resource
in (40) (Appendix A-4), the
pathjitterconstraintmaynotalwaysbesatisfiedforsmallvalues
of
. Hence, the path feasibility condition is not satisfied for
SCFQ networks. An implementation of the subalgorithm Pre-
Process for SCFQ networks performs the same tasks as those
performed for PGPS and
to perform an extra task because the path feasibility condition is
not satisfied. This extra task is to run a shortest path algorithm
[5], [11] on the jitter graph with
metric for each link
. If thesum of themetrics of theedges
of the shortest path from the source node s to the destination
node
is not less than, then there are no feasible paths in
the current network and the algorithm QoS-Routing terminates
without any success. Otherwise, to solve the instance of the
problem of QoS routing with resource allocation for SCFQ net-
works, we apply the same procedure used for PGPS or
networks, but use req defined in (20) instead of
fact that the path feasibility condition is not satisfied can lead
to a suboptimal solution to the instance of the problem of QoS
routing with resource allocation for SCFQ networks.
networks. In addition, it needs
assigned as a
. The
VI. COMPARISON WITH PREVIOUS RESEARCH
In Section II, we gave an overview of the previous research
in the area of QoS routing. In this section, we emphasize some
of the aspects that make this paper distinct from the previous
research. Some of these aspects are: the network model, the
solution approach, and combining queueing analysis with
routing.
As elaborated in Section III, the proposed model combines
QoS routing and resource allocation into a single framework.
It captures the relationship between the QoS metrics and the
allocated resources as well as the relationship between the met-
rics themselves. We model the QoS metrics of each link in the
network as functions of the allocated resources rather than fixed
valued weights. The functional forms are dependent on the
traffic description and the hardware/software implementation
of the network element. For a given path , a QoS parameter is
also modeled as a function of the resources allocated along the
path. Unlike previous models, the functional form depends on
the interaction between the network elements when connected
in tandem, rather assuming simple aggregation functions, such
as addition or multiplication. In short, this model is more real-
istic, makes less assumption about the network, and captures
the intelligence derived from the knowledge about network
elements implementation.
Based on the proposed model, we have presented a novel for-
mulation of the problem of QoS routing with resource alloca-
tion in (7) and (8). Unlike previous research, the solution to the
problem yields both a feasible path and the amount of resources
needed to provide the level of QoS along this path. We provide

Page 11

460IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
an algorithm that yields an exact solution under certain condi-
tion. We show, in Section V, that these conditions are satisfiable
for some packet scheduling algorithms.
It is quite common to consider queueing analysis and QoS
routingasseparatefieldsofresearch.InSectionV,weshowthat,
by combining these two apparently different research areas, one
can provide more insight into both the problems and use this in-
sight to solve the problem of QoS routing. The use of queueing
analysis of various service disciplines is the first step toward
employing the knowledge of the hardware and software imple-
mentation to solve the problem of QoS routing with resource
allocation.
VII. CONCLUSION
In this paper, we presented a generalized formulation for the
problemofQoSroutingcombinedwithresourceallocation.The
key feature in the new formulation is that theQoS parameters of
the links are taken to be functions, rather than fixed value met-
rics. Another important aspect is that the formulation captures
the fact that the different metrics and resources are interrelated
and modifying one of the resources leads to the change of more
thanonemetric.Thenewformulationhastheadvantageofusing
the knowledge of the hardware and software of the nodes to in-
telligently satisfy the multiple conflicting QoS requirements of
the end users.
Based on the new formulation, we presented a dynamic pro-
gramming algorithm that solves the problem of QoS routing
with resource allocation under certain conditions. The correct-
ness of the algorithm relies on the path feasibility and the sub-
path addition conditions. To show the universality of the algo-
rithm, we analyze some of the well-known service disciplines
to derive instances of the general formulation that reflects the
properties of these service disciplines.
The contribution in this paper can be summarized as follows.
• A novel generalized formulation for the QoS routing with
resource allocation problem.
• Analysis of the problem and the dynamic programming
algorithm for solving this problem.
• Applying the formulation and the dynamic programming
algorithm to existing service disciplines, which shows the
generality of the proposed formulation and QoS routing
algorithm.
• The jitter and data droppage analysis of some of the
rate-based service discipline under insufficient resources
condition.
The proposed formulation can be extended in several direc-
tions. One direction is to analyze the interaction between dif-
ferent service disciplines connected in tandem to derive func-
tional forms for jitter, reliability, and bandwidth along a path.
Another direction is to develop and analyze a parallel imple-
mentations to Algorithm IV.1, which can lead to an effective
QoS routing with resource allocation protocol. A third direction
is to derive general functional forms for jitter, reliability, and
bandwidth so as to accommodate a large number of the com-
monly used service disciplines.
We believe that the results presented in this paper provide a
new perspective to the problems of QoS routing and resource
allocation. It also presents a foundation for further research in
the quest of designing intelligent networks.
APPENDIX
A. Derivation of the QoS Routing Algorithm
In this appendix, we derive the dynamic programming algo-
rithm to solve the problem of QoS routing with resource alloca-
tion defined in (8). The development of the algorithm is similar
to that of the Bellman–Ford shortest-path algorithm [5].
Let
be the set of all nodes
Let
be a path that solves the QoS routing with resource
allocation problem defined in (8) for a node
problem can be solved using the following recursive equation:
such that.
. The QoS
(21)
For an acyclic jitter graph, an algorithm can be derived di-
rectly from the recursive relation given in (21). However, (21)
is not given in a sequential form and, hence, we cannot use it di-
rectly to derive an algorithm for an arbitrary jitter graph. Equa-
tion (21) needs to be modified as described below.
Let
be the set of simple paths with at most
between the node
and the node . Define
edges
by
(22)
If there is no path from
to be
simple path between any two nodes has at most
Hence,
(22).
towith at mostedges, we define
. Note thatsince a
edges.
. Equation (10) follows from
B. QoS Parameters for GPS
Each session
nodes
sponding to the fraction of the link bandwidth assigned to
that session. Suppose that the session
instance . Then, the rate of serving the session at the instance
is given by
is the set of sessions that are backlogged at the instance .
Define
by
traversing the communication link between
is assigned a real numberand corre-
is backlogged at some
b/s, where
(23)
representstheminimumbandwidthguaranteedforthe
session at any time instance where data from the th sessions
is backlogged in the queue of the output service discipline of
the node . Thus, the value of
source
, which the bandwidth allocated for the th ses-
sion. If the real numbers
for all
then
and the maximum queueing delay is
is equivalent to the re-
are chosen such that
and,

Page 12

BASHANDY et al.: GENERALIZED QoS ROUTING WITH RESOURCE ALLOCATION 461
bounded by
service discipline of a node is represented by the jitter
If
dropped. Hence, the minimum reliability along the jitter link
is less than 1. Thus, the parameters of the jitter link
representing GPS are given by
. Recall, that queueing delay at the
.
, then part of the data may be
(24)
(25)
(26)
where
link
is the bandwidth allocated for the th session in bits per second,
is defined in (23), and
allocated for the output queue for the th session.
The next step is to compute the parameters
and
for a path
servers in tandem. Referring to [24], we can see that the
maximum number of bits
, which represents the GPS servers at
the output of the nodes
bandwidth
of the path
is the reliability for the th session along the jitter
is the jitter along the jitter link,,
is the buffer space
,,
consisting of GPS
queued in the jitter links
, is given by
is taken as
. Also, the
(27)
Thus,themaximumqueueingdelayalongthepath
by
. By closely examining the GPS service dis-
cipline, we can see that under the worst case conditions, the
maximum number of bits
can be buffered at the output queue of the
node
represented by the jitter link
buffer available at this node is less than
of the bits may be dropped. Hence, the minimum reliability
for the pathis given by
isgiven
backlogged in the subpath
. If the total
, then some
(28)
As for the queueing delay, although the sizes of the queues
in individual nodes may be less than
simultaneously buffered inside multiple nodes. As long as no
node has zero buffer space, the total number of bits backlogged
inside the path
can be as large as
jitter
along the path
, there may be bits
. Thus, the maximum
is given by
(29)
C. QoS Parameters for PGPS
Let
be the number of bitsfed into the output queue
service discipline during the interval
for the session . Assume that
of the PGPS or
with and , where
is defined in (23) and
sion traversing the jitter link
, if the real numbers
chosen proportional to the long term average
is the number of ses-
. For both PGPS and
, , are
, we have
and the maximum queueing delay is
. Recall that the queueing delay
transmitting data to node
. If
bounded by
at the service discipline of a node
is represented by the jitter
some packets may be dropped. Hence, the minimum reliability
along the jitter link
is less than 1. Thus, the parameters of
the jitter link
representing PGPS or
, then
are given by
(30)
(31)
(32)
where
link
sion in bits per second,
for the output queue for the th session,
(23), and
is the maximum packet size in bits.
The next step is to compute the bandwidth
, and the reliability
consisting ofhops of PGPS or
tandem. For
, we can see from [6], [24], and [35] that we
can use (27) used for GPS.
Forthereliability,denoteby
bits that can be queued in the subpath
sumethat,foralljitterlinks
allocated for each session traversing the link
tional to its session long term averagebandwidth
to[6],[23],[24],and[35],wecanseethatthemaximumnumber
of bits queued in the subpath
is the reliability for the
is the bandwidth allocated for the th ses-
is the buffer space allocated
session along the jitter
,
is defined in
, the jitter
for a path
servers connected in
themaximumnumberof
. As-
,thebandwidth
is propor-
. Referring
is given by
(33)
where
ditions, the maximum number of queued bits
the subpath
can be queued at the output service discipline
of the last node
represented by the jitter link
Hence, the reliability
is the maximum packet size. Under the worst case con-
inside
.
of the pathis given by
(34)
The number of bits queued inside a subpath
the number of hops from
along the path
do not have enough buffer space to hold the
maximum possible queue size, then the maximum number of
bits queued inside the subpath
depends of
to. If some of the jitter links
is given by
(35)
and the initial condition is given by
(36)

Page 13

462IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 23, NO. 2, FEBRUARY 2005
where
stream at the output of the node
Even though some or all of the jitter links may not have
enough buffer space to hold the maximum possible number
of queued bits
,
can occur if
jitter links of the path
. Hence,
is the buffer allocated for the th data
.
, the maximum jitter
bits are distributed among the
is given by
(37)
D. QoS Parameters of SCFQ
The parameters
senting SCFQ can be computed as follows:
andfor the jitter link repre-
(38)
(39)
where
size of the output queue allocated for the th session,
is defined in (32) and
the link
.
For the path parameters
for a path
presented in [35]. First,
for GPS, PGPS, and
is the same as thatof PGPS and
The maximum possible number of bits queued in a subpath
is given by, which is the same as that of PGPS
and
. Thus, if some of the jitter links along the path
do not have enough buffer space, the maximum number of bits
queued inside a subpath
is given
and (36). The jitter
is given by
is the maximum packet size in bits,is the
is the number of session traversing
, , and
, we use the analysis of SCFQ
for SCFQ is the same as that
given in (27). Also, the reliability
givenin (34).
defined in (35)
(40)
REFERENCES
[1] S. Baqai, M. F. Khan, M. Woo, S. Shinkai, A. Khokhar, and A. Ghafoor,
“Quality-based evaluation of multimedia synchronization protocols
for distributed multimedia information systems,” IEEE J. Sel. Areas
Commun., vol. 14, no. 7, pp. 1388–1403, Sep. 1996.
[2] R. Barden, L. Zhang, S. Berson, S. Herzog, and S. Jamin. (1997,
Sep.) Resource reservation protocol (RSVP) version 1 functional
specifications. Request For Comments 2205. [Online]. Available:
http://www.cis.ohiostate. edu/bin/rfc/rfc2205.html
[3] A. R. Bashandy, “Quality of service routing and traffic regulation for
distributed multimedia systems,” Ph.D. dissertation, Purdue Univ.,
West Lafayette, IN, Aug. 1999. [Online]. Available: http://multi-
media.ecn.purdue.edu/~bashandy/papers/thesis.ps.
[4] A. R. Bashandy, E. K. P. Chong, and A. Ghafoor, “Network modeling
and jitter control for multimedia communication over broadband net-
work,” in Proc. IEEE Int. Conf. Comput. Commun. INFOCOM, vol. 2,
New York, Mar. 1999, pp. 559–566.
[5] R. Bellman, “On a routing problem,” Quart. Appl. Math., vol. 16, no. 1,
pp. 87–90, 1958.
[6] J. C. R. Bennett and H. Zhang, “?? ?: Worst case fair weighted fair
queueing,” in Proc. IEEE Int. Conf. Comput. Commun. INFOCOM, vol.
1, 1996, pp. 120–128.
[7] I. Cidon, R. Rom, and Y. Shavitt, “Multipath routing combined with re-
source reservation,” in Proc. IEEE INFOCOM, Kobe, Japan, Apr. 1997,
pp. 92–100.
[8] R. L. Cruz, “A calculus for network delay, Part I: Network elements in
isolation,” IEEE Trans. Inf. Theory, vol. 37, no. 1, pp. 114–121, Jan.
1991.
[9]
, “A calculus for network delay, Part II: Network analysis,” IEEE
Trans. Inf. Theory, vol. 37, no. 1, pp. 114–121, Jan. 1991.
[10] A. Demers, S. Keshav, and S. Shenker, “Analysis and simulation of a
fair queueing algorithm,” Internetworking, Res. Experience, vol. 1, no.
1, pp. 3–26, Sep. 1990.
[11] E. W. Dijkstra, “A note on two problems in connection with graphs,”
Numerische Mathematik, vol. 1, pp. 269–271, 1959.
[12] D. Ferrari, “Distributed delay jitter control in packet-switching Internet-
works,” Internetworking: Res. Experience, vol. 4, no. 2, pp. 1–20, Mar.
1993.
[13] N. R. Figueira and J. Pasqual, “An upper bound on the delay for the
virtual clock service discipline,” IEEE/ACM Trans. Netw., vol. 3, no. 4,
pp. 399–408, Aug. 1995.
[14] M.R.GareyandD.S.Johnson,ComputersandIntractability,AGuideto
the Theory of NP-Completeness.
[15] S. J. Golestani, “A self clocked fair queueing scheme for broadband ap-
plications,” in Proc. IEEE INFOCOM, Toronto, ON, Canada, Jun. 1994,
pp. 636–646.
[16] J. M. Jaffe, “Algorithms for finding paths with multiple constraints,”
Networks, vol. 14, pp. 95–116, 1984.
[17] M. Karol, M. Hluchyj, and S. Morgan, “Input versus output queueing
on a space-division packet switch,” IEEE Trans. Commun., vol.
COMM–35, pp. 1347–1356, Dec. 1987.
[18] T. Korkmaz and M. Kruz, “Multiconstrained optimal path selection,” in
Proc.IEEEInt.Conf.Comput.Commun.INFOCOM,vol.2,Anchorage,
AK, Mar. 2001, pp. 834–843.
[19] Q. Ma and P. Steenkiste, “Quality of service routing for traffic with per-
formance guarantees,” in Proc. Int. Workshop on Quality of Service’97
(IWQOS’97). New York, 1997.
[20]
, “Supporting dynamic interclass resource sharing: A multiclass
QoS routing algorithm,” in Proc. IEEE Int. Conf. Comput. Commun. IN-
FOCOM, vol. 1, New York, Mar. 1999, pp. 649–660.
[21] A. Orda, “Routing with end to end QoS guarantees in broadband net-
works,” in IEEE Proc. Int. Conf. Comput. Commun. INFOCOM, vol. 1,
San Francisco, CA, Apr. 1998, pp. 27–34.
[22] A.OrdaandA.Sprinnston,“Per-computationschemesforQoSrouting,”
IEEE/ACM Trans. Netw., vol. 11, no. 4, pp. 578–591, Aug. 2003.
[23] A. K. Parekh and R. G. Gallager, “A generalized processor sharing ap-
proach to flow control in integrated services networks: The single-node
case,” IEEE/ACM Trans. Netw., vol. 1, no. 3, pp. 344–357, Jun. 1993.
[24]
, “A generalized processor sharing approach to flow control in in-
tegrated services networks: The multilenode case,” IEEE/ACM Trans.
Netw., vol. 2, no. 2, pp. 137–150, Apr. 1994.
[25] H. F. Salama, D. R. Reeves, and Y. Viniotis, “A distributed algorithm
for delay-constrained unicastrouting,” in Proc.IEEEINFOCOM, Kobe,
Japan, Apr. 1997.
[26] J. L. Sobrinho, “Algebra and algorithms for QoS path computation and
hop-by-hop routing in the Internet,” IEEE/ACM Trans. Netw., vol. 10,
no. 4, pp. 541–592, Aug. 2002.
[27] R. Steinmetz, “Human perception of jitter and media synchronization,”
IEEE J. Sel. Areas Commun., vol. 14, no. 1, pp. 61–72, Jan. 1996.
[28] The ATM Forum Technical Committee. (1996, Mar.) Private
network-networkinterfacespecification
af-pnni-0055.000. [Online]. Available: ftp://ftp.atmforum.com/pub/ap-
proved-specs/af-pnni-0055.000
[29] J. Turner, “New directions in communication, or which way to the in-
formation age,” IEEE Commun. Mag., vol. 24, pp. 8–15, 1986.
[30] D. Verma, H. Zhang, and D. Ferrari, “Delay jitter control for real-time
communication in a packet switching network,” in Proc. TriComm’91,
1991.
[31] R. Vogel, R. G. Herrtwich, W. Kalfa, H. Wittig, and L. C. Wolf, “QOS-
based routing of multimedia streams in computer networks,” IEEE J.
Sel. Areas Commun., vol. 14, no. 7, pp. 1235–1244, Sep. 1996.
[32] Z. Wang and J. Crowcroft, “Quality-of-service routing for supporting
multimedia applications,” IEEE J. Sel. Areas Commun., vol. 14, no. 7,
pp. 1228–1234, Sep. 1996.
[33] D. B. West, Introduction to Graph Theory.
Prentice-Hall, 1996.
San Francisco, CA: Freeman, 1979.
version1.0(PNNI).
Upper Saddle River, NJ:

Page 14

BASHANDY et al.: GENERALIZED QoS ROUTING WITH RESOURCE ALLOCATION 463
[34] X. Yuan, “Heuristic algorithms for multiconstrained quality-of-service
routing,” IEEE/ACM Trans. Netw., vol. 10, no. 2, pp. 244–256, 2002.
[35] H. Zhang, “Service disciplines for guaranteed performance service in
packetswitchingnetworks,”Proc.IEEE,vol.83,no.10,pp.1373–1396,
Oct. 1995.
[36] H. Zhang and D. Ferrari, “Rate controlled static priority,” presented at
the INFOCOM, vol. 1, San Francisco, CA, Apr. 1993.
[37] L. Zhang, “Virtual clock: a new traffic control algorithm for packet
switching networks,” in Proc. ACM SIGCOM’90, Philadelphia, PA,
Sep. 1990, pp. 19–29.
AhmedR.Bashandy(S’93–M’95)receivedtheB.S.
degree in electronics engineering from Cairo Univer-
sity, Cairo, Egypt, in 1990, the M.S. degree in com-
puter science and the Ph.D. degree in computer engi-
neering from Purdue University, West Lafayette, IN,
in 1995 and 1999 respectively.
He joined Cisco Systems Inc., San Jose, CA, in
1999,wherehehasworkedonvariousprojectsinuni-
cast and multicast routing. His current interests are
network security and routing.
Edwin K. P. Chong (S’87–M’91–SM’96–F’04)
received the B.E. (honors) degree with First-Class
Honors from the University of Adelaide, South
Australia, in 1987, and the M.A. and Ph.D. degrees
both from Princeton University, Princeton, NJ, in
1989 and 1991, respectively, where he held an IBM
Fellowship.
He joined the School of Electrical and Computer
Engineering, Purdue University, West Lafayette, IN,
in 1991, where he was named a University Faculty
Scholar in 1999, and was promoted to Professor in
2001. Since August 2001, he has been a Professor of Electrical and Computer
Engineering and a Professor of Mathematics at Colorado State University, Fort
Collins. He coauthored the recent best-selling book An Introduction to Opti-
mization (New York, NY: Wiley, 2001) 2nd edition. His current interests are in
communication networks and optimization methods.
Dr. Chong received the NSF CAREER Award in 1995 and the ASEE
Frederick Emmons Terman Award in 1998. He was a corecipient of the 2004
Best Paper Award for a paper in the Computer Networks Journal. He was on
the Editorial Board of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and
is currently an Editor for Computer Networks. He served as an IEEE Control
Systems Society Distinguished Lecturer.
Arif Ghafoor (S’83–M’83–SM’86–F’99) received
the B.Sc. degree in electrical engineering from the
Engineering University of Science and Technology,
Lahore, Pakistan, in 1976, the M.Sc., M.Phil., and
the Ph.D. degrees from Columbia University, New
York, in 1977, 1980, and 1984, respectively.
He is currently a Professor in the School of
Electrical and Computer Engineering, Purdue Uni-
versity, West Lafayette, IN, and is the Director of
Distributed Multimedia Systems Laboratory. He has
been actively engaged in research areas related to
multimedia systems, information security, and parallel and distributed com-
puting. He has coedited Multimedia Document Systems in Perspectives and has
coauthoredSemanticModelsforMultimediaDatabaseSearchingandBrowsing
(Norwell, MA: Kluwer, 2000). He has published over 150 technical papers
in leading journals and conferences. He has served on the Editorial Boards
of various journals including ACM/Springer Multimedia Systems Journal, the
Journal of Parallel and Distributed Databases, and the International Journal
on Computer Networks. He has served as a Guest/Co-Guest Editor for various
special issues of numerous journals including ACM/Springer Multimedia
Systems Journal, the Journal of Parallel and Distributed Computing, and the
International Journal on Multimedia Tools and Applications.
Dr. Ghafoor has received the IEEE Computer Society 2000 Technical
Achievement Award for his research contributions in the area of multimedia
systems. He has served as a Guest/Co-Guest Editor for the IEEE JOURNAL
ON SELECTED AREAS IN COMMUNICATIONS and the IEEE TRANSACTIONS ON
KNOWLEDGE AND DATA ENGINEERING.