IOP Generic Communication
(IOP GenCom)
Quality of Service for Personal Networks at Home
Deliverable
D 2.4 Refinement of new QoS protocols and mechanisms
Abstract
This document presents a refinement of the state of the art QoS protocols and
mechanisms for Personal networks as presented in the previous deliverable.
In the first part of the document efficient searching for resources and services in
Personal Networks is addressed. There are two common methods employed for
searching: Flooding and Random Walks (RW), where for Personal Networks we
propose the use of multiple RWs. The performance of searching with multiple RWs
on two graph topologies, ER random graphs and power law graphs generated using
preferential attachment are studied. An efficient way of searching graphs using RWs
with no repetition of steps (i.e., memory) is shown, and in addition we show that an
optimal value of the memory Μ depends on the topology of the network.
The second part of the document deals with QoS differentiation in Personal Networks.
By giving different priorities to users, their performance can be increased or decreased.
We study the impact of the priority settings on the average waiting time of a packet for
each customer class. A mathematical model is developed to approximate the 1-limited
polling system that is used to model the network.
The third part of the document analyses the throughput of Personal Networks, taking
into account the impact of interference due to the wireless transmissions. A new theorem
is stated and proven that provides an upper bound on the throughput a network can
achieve.
The results presented in this document can be used to refine existing protocols and
serve as a basis for developing new ones. They provide insights in the different
aspects affecting the QoS a Personal Network can obtain.
IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
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July 1, 2007
1.0
Tom Coenen, Santpal Dhillon
Tom Coenen
Network Architecture and Services (NAS), TUDelft
Wireless Mobile Communication (WMC), TUDelft
Design and Analysis of Communication Systems (DACS), UTwente
Stochastic Operations Research (SOR), UTwente
WP2 QoS control and modeling in both traffic and routing
Public
38
Revision History
Version Date
1.0
1 July 2007
Editor
Tom Coenen
Summary
IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
Table of Content
1. INTRODUCTION....................................................................................................2
1.1 PURPOSE ...............................................................................................................3
1.2 SCOPE ...................................................................................................................3
1.3 ASSUMPTIONS .......................................................................................................3
1.4 OUTLINE OF THIS DOCUMENT ................................................................................4
2. SEARCHING IN PERSONAL NETWORKS.......................................................5
2.1 RELATED WORK ....................................................................................................5
2.2 SEARCH STRATEGIES .............................................................................................6
2.2.1 Random Walk........................................................................................................... 6
2.2.2 Random Walk with memory M................................................................................. 6
2.3 SEARCHING WITH MULTIPLE RW QUERIES ............................................................7
3. QOS DIFFERENTIATION IN PERSONAL NETWORKS..............................15
3.1 MODELING DESCRIPTION .....................................................................................15
3.2 POLLING MODEL ..................................................................................................16
3.3 ANALYSIS ...........................................................................................................16
3.3.1 General case .......................................................................................................... 16
3.3.2 Special cases .......................................................................................................... 19
3.4 VALIDATION .......................................................................................................20
3.4.1 General case .......................................................................................................... 20
3.4.2 Special cases .......................................................................................................... 21
3.5 CONCLUSION .......................................................................................................22
4 AN UPPER BOUND ON NETWORK THROUGHPUT....................................24
4.1 AD HOC INTERFERENCE MODEL ...........................................................................25
4.2 THE EXTENDED MULTICOMMODITY FLOW PROBLEM ...........................................27
4.3 EXAMPLES...........................................................................................................29
4.4 CONCLUSION .......................................................................................................31
5 CONCLUSION .......................................................................................................32
6 REFERENCES........................................................................................................33
APPENDIX.................................................................................................................37
A. ACRONYMS ......................................................................................................37
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
1. Introduction
This document presents a refinement of the state of the art QoS protocols and
mechanisms for Personal networks as presented in the previous deliverable. Personal
Network (PN) [1] is a user-centric concept that is proposed to realize the users’ request to
access information and services with their mobile devices at hand anywhere and anytime
(Fig. 1). Recent advances in wireless communications, mobile computing and hand-held
electronic devices have made the concept possible for development. Niemegeers and
Heemstra de Groot had an in-depth analysis in [1] of the concept and the research issues
that are crucial in developing personal networks. In brief, personal networks extend the
user’s personal operating space of a PAN, which is normally less than 10 meters around
the user, to a global coverage of his home network, corporate network, other personal
resources and resources belonging to others regardless of their geographical locations.
Thus a Personal Network is essentially a combination of various sub networks which may
be ad hoc, cellular, wired or wireless networks etc.
Figure 1 Personal Networks
Novel QoS routing approaches are required to efficiently manage and support Personal
Network features such as distributed multimedia services, mobile users and networks,
heterogeneous inter-networking, service guarantees, point-to-multipoint communications,
real time applications, which require hard time constraints etc. The problem of routing in
wired, cellular and ad hoc networks has been discussed extensively in literature and
current Internet and cellular networks are examples of routing protocols being used
successfully. In single hop wireless technologies, such as WLAN (IEEE 802.11), no
routing protocols are needed, but these networks are being extended to multihop
networks. However, QoS support is missing in these routing protocols. Since most of the
networking in PN is ad hoc based and delivering end-to-end QoS in ad hoc networks is
intrinsically linked to the performance of routing protocols used, the main focus of this
document is on routing protocols and models for ad hoc networks.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
Routing in ad hoc networks is a complex issue due to inherent non uniform propagation
characteristics of wireless transmissions and the possibility that nodes may move at any
instant. If only two hosts, located closely together, are involved in the ad hoc network, no
real routing protocol or routing decisions are necessary. In contrast, since most of the ad
hoc networks have a large number of hosts, packets make multiple hops to travel from
any source to a destination node. Thus routing protocols are necessary in multihop
wireless ad hoc networks.
1.1 Purpose
The purpose of this document is to refine various routing protocols for ad hoc networks
that will be utilized in the proposed personal network architecture and solutions to ensure
that the QoS requirements are being met. However, the design of a new QoS routing
protocol or use of any previous proposed routing protocol for personal networks should
not be limited by this document. An analysis of searching strategies in Personal Networks
and the possible throughput of a network, as well as the modeling of QoS differentiation
give insight in how to refine existing protocols, or give a basis for the design of new
ones.
1.2 Scope
In this document, we will provide insight in how to refine various routing protocols that
have been proposed for ad hoc networks. A search strategy using multiple random walk
queries is analyzed. Delivering end to end quality of service in Personal networks is
linked to the performance of the routing protocol used. We give a mathematical model to
determine the impact of QoS settings on the waiting time of packets in a network.
Furthermore, for a general type of network, an upper bound is derived for the throughput
that can be achieved. All these results provide valuable insight needed when developing
QoS aware networks and their routing protocols.
1.3 Assumptions
There are some assumptions when discussing the requirements of personal networks.
• Devices: the devices we are discussing are all digital communicating devices that
are equipped with at least one communicating interface either wireless interfaces
such as UMTS, 802.11 and 802.15, or wired links such as Ethernet, xDSL. Some
devices may be multimode with more than one interface type.
• Ad hoc networks: the ad hoc networks we are referring to in this document may
consist of heterogeneous wireless technologies such as 802.11, 802.15. Therefore,
what we are discussing are multihop hybrid ad hoc networks with partially mesh
topology. Partial mesh topology refers to a topology of an ad hoc network where
nodes may connect to only some, not all of the other nodes. Partial mesh topology
is partly the result of the heterogeneous wireless technologies that may be
involved in the ad hoc networks.
• WLAN (IEEE 802.11): the WLAN technology we discuss here is in mixed mode,
that is, a WLAN node can dynamically switch between ad hoc mode and
infrastructure mode during communication.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
1.4 Outline of this document
The outline of this document is as follows. First of all, several search strategies for
wireless networks are listed. Then we elaborate on what we find to be the best strategy
for Personal Networks and evaluate this type of strategy, using multiple random walk
queries.
The second part of the document models the QoS aspects in wireless networks. The
impact of the differentiation between users is captured to see impact on the waiting time
of packets for different classes of customers.
The third part states a theorem that has been developed to provide insight in the
throughput that can be achieved in a general type of network. Using the theorem, an
upper bound can be derived, or for given demands between users a factor can be
computed that shows the percentage of these demands that can be met.
The final part concludes this document by summarizing the obtained results.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
2. Searching in Personal Networks
Efficient searching for resources and services is an important issue in Personal Networks.
There are two common methods employed for searching - flooding and random walks
(RW). In wireless ad-hoc networks, reactive protocols such as AODV and DSR use
flooding to locate the destination [23]. In peer-to-peer networks, both flooding and RW
have been employed to locate services and resources [3], [4], [14]. RWs have been shown
to induce lower overhead than constrained flooding for searching in peer-to-peer
networks [4], [14]. Multiple RWs have been proposed for searching on unstructured peerto-peer networks by Lv et al. [4]. In web graphs, search engines use breadth first search
to perform a complete search of the web. However, to reduce the overhead of searching,
agents or spiders based on RW, where the next hop is chosen uniformly among the
neighbors of the node, or variations of the RW such as the RW strategy where the next
hop is chosen as the node with maximum degree are widely used [21].
We propose the use of multiple RWs for searching in Personal Networks.
We study the performance of searching with multiple RWs on two graph topologies - ER
random graphs and power law graphs generated using preferential attachment. Both these
graph topologies are important since ad-hoc wireless network, a constituent of the PN,
can be modeled as ER random graphs [18] while the overlay or peer-to-peer topology
constructed for PN can be modeled as power law random graphs [14]. We do not
consider dynamic topologies in this paper. In case of multiple RW queries, we study the
optimal number of queries and the Τ Τ Λ of queries for ER random graphs and
preferential attachment power law graphs. We also show an efficient way of searching
graphs using RWs with no repetition of steps (i.e., memory). In addition, an optimal
value of the memory Μ depends on the topology of the network.
Adaptive techniques based on RWs have been proposed for searching by Bisnik and
Abhouzeid [22]. In the searching technique proposed in [22], the number of RW queries
used for searching is varied depending on the previous performance of searching. Our
work differs from previous approaches since we study the optimization between the
number of queries and the time-to-live (Τ Τ Λ) of queries for different graph topologies.
2.1 Related work
Unstructured overlay networks such as Gia proposed by Chawathe et al. [3] and Gnutella
build a random graph and use flooding or RWs to discover data stored at different nodes.
RWs have been shown to induce lower overhead than constrained flooding used by the
current versions of Gnutella [4], [14]. In the original Gia [3], the RWs were biased to
prefer nodes with higher capacity but Castro et al. [2] have shown that preferring nodes
with higher degree leads to a higher success rate and a lower delay. Thus, further
improvements have been proposed to Gia in which RWs are biased towards the higher
degree nodes [2]. Also, variations of RWs have been proposed in which there are no
loops [2]. Different search algorithms for scale-free and power law graphs have been
analyzed in [1], [9], [10]. The term local search algorithm or path finding strategies is
also used for different variations of RWs [1], [10]. In [1] and [10], RW strategies where
the next hop is chosen as the node with the highest degree and without retracing of steps
have been analyzed in terms of expected hopcount and weight. We show that strategies
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
such as RW using highest degree and RW without retracing of steps can lead to
deadlocks. Therefore, the comparison of different RW strategies should also include an
analysis of deadlocks, in addition, to the expected hopcount comparison.
In mobile agent based routing, the mobile agents perform a RW or a variant of the RW
while searching for the destination. In Ant-Net, loop-erased RWs are used by the mobile
agents [27]. Mobile agents using RW have been proposed for providing membership
services for ad-hoc networks by Dolev et al. [8]. As a sampling technique, RWs have
been used for providing membership services in ad hoc networks [5], [19] that provide
the nodes in the network with a view of the other nodes and that are used by various
applications such as location services, peer sampling services and random overlay
constructions [19]. Bar-Yossef et al. [19] develop a membership service for ad hoc
networks based on RW using highest degree. They show that the performance of such
membership service is superior to other existing membership services based on gossiping
or flooding [19].
The analysis of RWs has been an active topic of research [7], [8], [16]. Lovász [8] and
Chung [11] presents a detailed survey of the RWs. An exact analysis of RWs on graph
topologies such as the lattice and the torus has been studied in [7].
2.2 Search strategies
2.2.1 Random Walk
In searching with RW, the next hop is chosen uniformly among the neighbors of the
nodes. Thus, no topology information is required to search with RWs.
2.2.2 Random Walk with memory M
A first-in first-out (FIFO) list, called the memory list M, is maintained. The memory list
M contains the node identifiers νϕ of the last Μ nodes visited during the RW, i.e. M =
{ν1, ν2,…, νΜ}, where Μ = |M | represents the number of elements in the memory list M.
The next hop is chosen uniformly among the neighbours of the node that are not in the
memory list M. In our implementation of the RW strategy with memory Μ (RWM), the
node identifier of the current node is not stored in the memory list M and no self-loops are
allowed (The next hop cannot be chosen as the node itself.) The idea of using RWM on a
graph is similar to the RW with unbounded memory in a continuous 3-dimensional space
which has been referred to as self-avoiding RW [26]. The RWM strategy is equivalent to
the search queries proposed for overlay networks where the structure is used to ensure
that nodes are visited only once during a query and to control the number of nodes that
are visited accurately [4].
From the routing point of view, the major shortcoming of RWs on graphs is the existence
of loops in the path while traveling from the source to the destination node. To prevent
loops, the simplest method is to introduce memory in the RWs. The one hop loops can be
prevented by using Μ = 1, both the two hop and one hop loops can be prevented by using
Μ = 2 and so on. Thus, a complete memory Μ = Ν − 1 totally eliminates loops in the
RWs. But the introduction of memory (Μ ≥ 1) in RWs can lead to a deadlock. Figure 1
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
explains the concept of deadlock in a RW with memory. Let us assume that the RW has a
memory Μ = 3 represented as M= {ν1, ν2, ν3} where ν3 represents the last node visited.
A3
A
A4
A1
A2
C
B
1
4
2
3
Figure 1: Explanation of deadlock in (a) RW with memory: The path 1→2→3→4 is shown by
arrows. At node 4, the dashed lines indicate the links that cannot be used with memory M≥3. (b)
RW using highest degree: Node A1 chooses node 1 as next hop and node 1 chooses node A1 as
next hop leading to an infinite loop.
Consider the situation where the RW enters the cluster B at node 1. Suppose that the RW
leads to a path 1 → 2 → 3 → 4 in the cluster B. At node 2, the memory list M contains and
at node 3 the memory list M contains {1, 2}. At node 4, the memory list M contains
{1, 2, 3} and since all the neighbors of node 4 are in the memory list M and no self loops
are allowed, the RW cannot move forward nor backward. Therefore, a deadlock prevents
the RW from ever reaching the destination. Keeping the memory Μ equal to the degree
of nodes is not sufficient to prevent loops. Consider the same situation as above but with
memory equal to degree of node 4, i.e. Μ = 2. At node 4, the memory list M contains
{2, 3} and the next hop can be chosen as node 1.
Thus, introducing memory may remove the loops in the RW but can induce deadlocks. In
the implementation of Gia, Castro et al. [4] have used a query in RWs which consists of
all the previously visited hops. This is similar to using complete memory in our analysis.
The above analysis shows that there are two distinct regimes possible for RWs. Without
memory, i.e. Μ = 0, the RWs can have loops but no deadlocks. For complete memory, the
RWs can only have deadlocks and no loops. When the value of the memory Μ is such
that 0 < Μ < Ν − 1, the RWs can have loops and deadlocks.
2.3 Searching with multiple RW queries
We show that using multiple queries reduces the overhead for searching as compared to
flooding with sequence numbers. Searching (packet) overhead is defined as product of
number of hops for each query (packet) and number of queries (packets) i.e., the total
number of packets exchanged. A single RW query has an overhead and time to discover
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
of Ο(Ν log Ν ) for ER random graphs and BA power law graphs [6], [24]. Since the time
to discover is large in RW, we split a single RW query into Θ multiple queries each with
a given Τ Τ Λ. Each RW query stops when the Τ Τ Λ is reached or the destination is
located (In RWM, the query also stops if there is a deadlock). Since each RW query
makes at most Τ Τ Λ hops, the worst-case searching overhead is Θ*Τ Τ Λ. The expected
number of hops is less since any query stops once the destination is found. However,
there is a probability that a destination is never located by the multiple queries. Thus,
with multiple queries we define the probability of success as the probability that the
destination is located by at least one query out of the Θ queries generated. In flooding
with sequence numbers, the probability of success is defined as the probability that a
destination is located with a given Τ Τ Λ. We want to minimize Τ Τ Λ and Θ and
maximize Pr [success].
To analyze the performance of multiple ΡΩ queries, we define the efficiency as the
inverse of expected packet overhead needed to discover the destination node. The
efficiency is normalized by multiplying by Ν. The gain of searching in one scheme over
another is defined as the ratio of efficiency for the corresponding schemes.
η=
Number of iterations(Pr[success]
×N
Total number of packets
⎛ RW M ⎞ η (RWM )
g⎜
⎟=
⎝ RW ⎠ η (RW )
(1)
(2)
Consider a single RW and RWM on a complete graph ΚΝ . The searching overhead can be
approximated by the product of expected hopcount to discover destination and the
number of iterations. Since in RW and RWM, the expected hopcount is Ν − 1 and (Ν −
1)/2 respectively, the gain for a complete graph ΚΝ. In flooding with Τ Τ Λ = 1, × is
Ν/(Ν − 1) and with Τ Τ Λ = 2, × is Ν/ (Ν − 1)2. Thus, the efficiency of flooding depends
on the value of Τ Τ Λ and the efficiency decreases with Τ Τ Λ.
Table II and III show the efficiency for a single RW, RWM query and flooding with
sequence numbers for different values of Ν and π. The results for flooding in Tables II
and III are for optimized values of Τ Τ Λ such that Pr [success] is close to 1. The gain
obtained by using RW over flooding is significant, particularly, when the link density π is
large. In addition, RWM=N-1 is a more efficient way of searching than RW since the
expected number of hops required to find destination or deadlock is less. However, the
probability of deadlocks is high when the link density π is small. As the link density π is
decreased, efficiency for both RW and RWM decreases. This is in contrast to flooding
where the efficiency increases as the link density is decreased.
Table I
Efficiency of searching by using flooding, and a single RW or RWM query for a dense ER
random graph (πΝ=80) for different values of Ν.
pN=80
N
g
g (RWM
× (RW)
× (RWM) ×
(flooding) (RW/flooding) /RW)
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
100
200
400
800
1.17
1.08
1.05
1.01
1.82
2.16
1.99
2.0
0.015
0.03
0.06
0.12
78
36
17.5
8.5
1.55
2
1.9
1.98
Table II
Efficiency of searching by using flooding, and a single RW or RWM query for a dense ER
random graph (πΝ=10) for different values of Ν.
N
× (RW)
100
200
400
800
0.84
0.84
0.82
0.8
RWM -×
(flooding)
1.6 (0.73)
1.54 (0.66)
1.47 (0.59)
1.4 (0.52)
pN=10
g
g (RWM
×
(flooding) (RW/flooding) /RW)
0.14
6
1.89
0.21
4
1.83
0.11
7.5
1.79
0.14
5.7
1.76
A single RW leads to lower overhead to locate a destination than flooding with sequence
numbers. However, the time to search for destination is much larger than in flooding.
Since we want to maximize Pr[success] and minimize Τ Τ Λ, we split the single RW or
RWM into multiple queries with fixed Τ Τ Λ such that Θ * Τ Τ Λ = Ν log Ν . Table III and
IV show the results for multiple RW and RWM queries for ER random graph with Ν = 400
and link density π = 0.015 and 0.2 respectively. In Table IV, since the link density π is
large, Pr [success] ' 1 and is not shown.
The probability of success is very low in RWM with only a single query when the link
density π is small. However, when split into multiple queries, Pr [success] increases. As
the Τ Τ Λ is decreased and Θ is increased, the efficiency decreases for both RW and
RWM. The decrease in efficiency occurs with small Τ Τ Λ since most of the queries
search only the neighboring nodes which have been visited already by other queries.
There is also a decrease in efficiency because the number of queries and the Τ Τ Λ is
fixed. Thus, multiple queries might locate the destination. The terminating conditions can
be included which improve the efficiency but increases the complexity of searching
algorithms. For example, a scheme is proposed in [4], where the query checks with the
source node whether the destination is located. As shown by tables IV and V, the
efficiency decreases by a factor of 3 as the Τ Τ Λ is decreased from 2400 to 120.
Moreover, when the Τ Τ Λ is small, the gain obtained by using RWM over RW is small.
Therefore, only for large values of link density π and Τ Τ Λ, RWM is a more efficient way
of searching than RW. If we use Θ > 1 with Τ Τ Λ = 2400, the efficiency decreases.
Ν=400.
pN=6
N=400
Table III
Efficiency of searching by multiple RW or RWM queries for π=0.015 and
×
RW
Pr[success]
×
RWM
Pr[success]
g
(RWM
/RW)
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
Q=1, TTL =
2400
Q=20, TTL =
120
Q = 120, TTL =
20
Q = 400, TTL =
6
Ν=400.
0.7
0.996
1.23
0.28
1.8
0.27
0.96
0.33
0.97
1.2
0.24
0.95
0.26
0.97
1.1
0.15
0.73
0.18
0.82
1.2
Table IV
Efficiency of searching by multiple RW or RWM queries for π=0.2 and
N=400
Q=1, TTL =
2400
Q=20, TTL =
120
Q = 120, TTL =
20
Q = 400, TTL =
6
× (RW)
0.7
× (RWM)
1.997
g (RWM /RW)
2
0.27
0.298
1
0.24
0.285
1
0.15
0.261
1
Figure 2 shows Pr [success] versus the searching overhead (number of packets exchanged)
and the worst time to discover the destination. Since the size of network is not known a
priori, we also show simulations for query split into multiple queries with a different
Τ Τ Λ. We use a linearly increasing Τ Τ Λ and the maximum time to discover is given by
maximum Τ Τ Λmax.
Τ Τ Λ(λ) = Τ Τ Λmin + (λ − 1) ∗ αδδ_Τ Τ Λ
(3)
Using (3), the αδδ_Τ Τ Λ parameter can be expressed as Τ Τ Λmax−Τ Τ Λmin/Θ−1. Figure
2 shows that efficiency of searching by RW and RWM decreases with Τ Τ Λ. This is in
contrast to the assumptions made in the analysis in [22], where searching with multiple
queries and independent uniform sampling are assumed to be equivalent. Also, the linear
query performs as good as sending multiple queries with a large Τ Τ Λ(100). Thus, the
simulations show that searching by using a single RW query with Τ Τ Λ is more efficient
than sending Θ RW queries with Τ Τ Λ’ (Τ Τ Λ = Θ*Τ Τ Λ’) for ER random graphs.
Figure 3 shows the results for searching with multiple queries in BA power law graph. In
these graphs, RWM=2 gives the best performance in terms of reducing search overhead.
RWM=N-1 performs worse than RWM=2 since many of the queries end in a deadlock. This
reduces the efficiency of the RWM=N-1 strategy and queries with larger Τ Τ Λ need to be
sent to achieve the same probability of success as RWM=2. Moreover, the improvement in
performance of different RW strategies compared to flooding is limited. Even with a
large Τ Τ Λ = 1000, the RW does not perform better than flooding.
Figure 4 compares the performance of searching with RW and RWM in ER random graph
and BA power law graph using the same Τ Τ Λ = 400. The results are for Ν = 10000 and
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
for ER random graph the link density π = 0.001. Figure 4 also shows the results for
searching for a high-degree destination node in BA power law graph (the average degree
of the destination is 72). Searching in ER random graphs performs better than in BA
power law graph. This can be attributed to the fact that most of the nodes in ER random
graph have a larger degree than the degree of nodes in BA power law graph (section
III.E). Thus, in BA power law graph, the RW makes large number of hops among the low
degree nodes while searching for the destination node. Moreover, in BA power law
graph, since the degree of uniformly chosen destination and source nodes is small,
performance of searching for a uniformly chosen node is much worse than searching for a
high degree node.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
1.0
0.8
Pr[success]
0.6
Flooding with sequence num bers
RW (TTL = 6)
RW (TTL = 100)
RW M = N-1 ( TTL = 100 )
RW ( Q = 20 )
RW M = N-1 ( Q = 20 )
linear query
RW ( TTL m in = 20, add_TTL = 2 )
RW M = N-1 ( TTL m in = 20, add_TTL = 2 )
0.4
0.2
N = 400
p = 0.015
0.0
500
1000
Number of messages
1500
1.0
Pr[success]
0.8
0.6
Flooding with sequence num bers
Q = 20, RW M = N-1
Q = 20, RW
Linear Query
RW (TTLm in = 20, add_T T L = 2)
RW M = N-1 (T T L m in = 20, add_T T L = 2)
0.4
N = 400
p = 0.015
0.2
0.0
0
50
100
150
200
Worst case time to discover
Figure 2: Performance of searching using RW and RWM with multiple queries in ER random
graph for N = 400 and p = 0.015.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
1.0
0.8
N = 10000
Pr[success]
0.6
Flooding with sequence numbers
RW
TTL min = 100, add_TTL = 20
TTL = 10
TTL = 1000
0.4
RWM
M = N-1, TTL min = 100, add_TTL = 20
M= N-1, TTL = 100
M = 2, TTL = 100
M = 2, TTL min = 100, add_TTL = 20
0.2
0.0
0
10
20
30
40x10
3
Number of messages
1.0
0.8
Pr[success]
0.6
N = 10000
0.4
flooding with sequence numbers
RW
RWM
0.2
TTLmin = 100, add_TTL = 20
M = 2, TTLmin = 100, add_TTL = 20
M = N-1, TTLmin = 100, add_TTL = 20
0.0
0
1000
2000
3000
4000
Worst case time to discover
Figure 3: Performance of searching using RW and RWM with multiple queries for BA power law
graph (N = 10000).
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
1.0
0.8
N = 10000
p = 0.001 (ER graph)
Pr[success]
0.6
ER random graph
Flooding
RW, TTL = 400
RWM = N-1, TTL = 400
BA model
Flooding
RW, TTL = 400
High degree destination
RW, TTL = 400
0.4
0.2
0.0
0
10
20
30
40
Number of messages
50
60x10
3
Figure 4: Comparison of searching by multiple RW and RWM queries on ER random graph and
BA power law graph. (N = 10000 and for ER random graph p = 0.0001)
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
3. QoS differentiation in Personal Networks
In this part we present a novel queueing analysis approach to determine the waiting
time of packets at a node in a network with QoS differentiation. In the network we
consider, the differentiation between the users is done by giving them different
probabilities of being allowed to transmit a packet. We model the network by a polling
model: we model the nodes in the network as queues and the shared transmission
capacity as a single server moving between the queues. Such an approach has also been
used for the analysis of for instance Bluetooth Piconets [30],[31], where a cyclic polling
model mimics the protocol in use. Opposed to previous work, we do not assume a cyclic
polling model, but capture the random nature of the CSMA/CA based MAC protocol, as
is used in e.g. IEEE 802.11, by setting probabilities for the nodes to be polled next, thus
arriving at a random polling model. What we consider to be the main contribution of our
work is an analysis of non-saturated nodes, with Poisson packet arrivals and thus taking
into account the dynamic nature of the network. For various settings, we determine the
waiting time at the different nodes to obtain insight into the effect of service
differentiation in a communication network.
By viewing the nodes in the network as isolated queues with a server that goes on
vacations, of which the length depends on the state of the other nodes, we analyze the
behaviour of such a queue. In case of saturated nodes, this analysis is exact. Another
exact analysis is possible for a network with identical nodes, where a polling model with
switchover times can be used to analyze the waiting time of packets at a node.
Previous work on QoS differentiation has focused on the impact on the throughput and
level of differentiation due to IEEE 802.11 parameters, such as the backoff values, AIFS
and TXOP limits, which determine the time until a transmission attempt, the waiting time
after a transmission and the maximum number of packets to send during a transmission.
For the basic IEEE 802.11 protocol, Bianchi [28] used a Markov chain analysis to
determine the transmission and collision probabilities to compute the throughput of the
network under saturation conditions. Many extensions have been made to this model to
include the frame retransmission limit [37], capture [36] and flow level dynamics [29] as
well as non-saturation conditions [38]. For the IEEE 802.11e protocol, analytical results
also often build on the work of Bianchi, for example [39]. The approach presented in the
present paper differs substantially from these models, as we differentiate between users
by the probability of being allowed to transmit a packet. We propose a novel way of
approximating the waiting time of a packet for a communication network with QoS
differentiation, but the background of our paper with the listed previous work coincides.
3.1 Modeling description
We consider a network consisting of two types of users (high and low priority) that
want to transmit packets over a wireless channel. The network is a so called "single cell"
where the transmission capacity is fully shared as all nodes are within each others
interference range. Hence only one transmission can take place at any point in time.
Quality of Service (QoS) differentiation is made between the nodes by for instance
setting different backoff window values, interframe spaces and TXOP limits as in the
IEEE 802.11e protocol. These parameters influence the time it takes for a node to attempt
a transmission, the time a node waits after the channel becomes idle and the amount of
packets the node is allowed send at once on its turn to transmit. We assume that the
TXOP limit is the same for all nodes, being one single packet, and that the other settings
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
only influence the probability of a node being next in turn to transmit a packet. We also
take into account the random access scheme in IEEE 802.11, due to the CSMA/CA based
MAC protocol. We also assume that the size of a packet is the same for all nodes. We
analyze the impact of the distinction between the two types of nodes by considering the
expected waiting time of a packet for each type of node.
3.2 Polling model
To model the system, we make the analogy between the nodes competing for the
channel to transmit a packet and one server representing the shared transmission capacity.
A node being allowed to transmit coincides with the service of a packet at the queue by
the server. As the channel is used by the different nodes in a random order (due to the
random access scheme of the MAC protocol, which depends on the parameters), this can
be modelled as the server polling the different nodes. The distinction between the two
types of nodes is made by the probability that the server will move to one of the nodes.
We will take the probability of moving to a high priority (HP) node to be a times as high
as moving to a low priority (LP) node. Each node has an infinite queue and packets arrive
according to a Poisson process with rates l(HP) and l(LP) at the two different types of
queues. The service time of a packet is considered to be deterministic with value as the
packet sizes in the system are equal for all nodes and the channel speed is considered to
be constant at all times. As a node is only allowed to transmit one packet when being
polled, the service discipline is 1-limited. We assume that there are no switchover times
between the queues, so that consecutive transmissions of packets can take place.
3.3 Analysis
In the following, we will derive expressions for the average waiting time of a packet
for any type of queue. We start with considering one high priority node, surrounded by n
low priority nodes. At both queues, packets arrive according to a Poisson process. The
server will move to the HP queue with probability a/(n+a) and to a certain LP queue with
probability 1/(n+a). We present an algorithm to approximate the waiting time of a packet
for both types of queue. For specific situations, being that either the HP or LP nodes are
saturated, meaning they always have packets ready to be transmitted, exact results are
presented. Exact results are also given for the situation where all nodes are equal.
3.3.1 General case
Consider the situation where customers arrive at the HP and LP node according to a
Poisson process with rates l(HP) and l(LP) respectively, such that the nodes are not
saturated. To determine the average waiting time of a packet in the queue, we consider
the queues separately, as if they are isolated. From the point of view of the HP node, the
server is either present and serving a customer, or away while serving an LP node. We
thus can consider the system as an M/D/1 queue with vacations (c.f. [34],[40]), where the
absence of the server while serving other nodes are the vacations. The length of these
vacations, which depend on the state of the other queues, influences the waiting time of
the packets in the queue. For illustratory reasons, we first give the analysis for the
situation where there are two nodes, one high priority and one low priority node, which,
as we show later, can be extended to any number of nodes.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
Two nodes
In the two node situation, each node can be considered separately as a queue with a
server that goes on vacation as described before. The duration of a vacation now depends
on the state of the other queue. We approximate the distribution of the length of the
vacation V(x), given the number of customers N(y) at the other queue (HP or LP) using
the following recursion:
∞
P (V ( x) = kτ | N ( y ) = i ) = q( y ) ∑ P(V ( x) = (k − 1)τ | N ( y ) = j ) P ( A( y ) = j − i + 1)
j = i −1
⎧1, i = 0
P (V ( x) = 0 | N ( y ) = i ) = ⎨
⎩1 − q( y ), i = 1..∞
where V(x) is the length of the vacation seen by the node x, whereas N(y), q(y) and A(y)
are the number of customers at node yP, the probability of the server polling node yP and
the number of arriving customers during a service time at node yP respectively. The
variable x can be HP or LP and y then automatically is the other queue. As the number of
customers in a queue will not tend to infinity in a stable system, we will truncate the
queues as if there is a maximum of M customers in the queue for numerical purposes.
The vacation length distribution is then determined using
P(V ( x) = kτ ) = ∑ P(V ( x) = kτ | N ( y ) = i ) P ( N ( y ) = i ) ,
i
where we use the steady state distribution of a node y for the values of P(N(y)=i). As this
steady state distribution is not known, we start with a random distribution assuming the
node to be empty with high probability to prevent analyzing an instable system. Using
this distribution, the vacation distribution for the other queue is obtained.
Having obtained the vacation time distribution, we derive the probability generating
function (pgf) φ(z) of the number of customers left behind by a departing customer of the
HP node. This also gives us the steady state distribution of the number of customers in
the queue (c.f. [43]), so that by using Little's law we acquire the expected waiting time of
a packet. The pgf is given by (c.f. [OMM91])
φ(z)=[V∗(z)-V(z)]S(z)π(0)/(z-V(z)S(z)),
(1)
where V(z) is the pgf of the number of customers arriving during a vacation and V*(z) is
the pgf of the number of customers arriving during a vacation, given that at least one
customer arrived. S(z) is the pgf of the number of customers arriving during a service
time and π(0) denotes the steady state probability of the queue being empty. For the
derivation of this formula we refer to the appendix. Note that for the standard M/D/1
queue (without vacations) we would have V*(z)=z and V(z)=1. In our situation however,
when the server leaves the queue empty this will lead the server to serve a customer in
the LP node. A customer arriving at the HP node during this service time has to wait for
the server to become available, which differs from the immediate service it would receive
in the M/D/1 queue, even if the HP node gets an infinite higher priority.
We now have obtained the steady state distribution of the number of packets at the
considered queue. This steady state can in turn be used to determine the vacation time
distribution of the other queue as done in the previous step. This leads to the following
iteration:
1. Set the initial steady state queue length distribution
2. Compute the vacation time distribution
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
3. Determine the steady state queue length distribution of the queue
4. Loop steps 2 and 3 until convergence occurs
This approach mimics the interaction between the two queues and thus provides insight
in the proportion of customers at each queue. The approach estimates the number of
customers found at the other queue to determine the vacation time for the considered
queue. When this vacation time is underestimated, the server might switch to the other
queue earlier than in the real situation and starts servicing a packet at the considered
queue (when available), thus leaving the server busy. When however the vacation time is
overestimated, the approach leaves the server at the other queue, whereas this queue
actually would be empty, thus leaving the server idle while it could process jobs in the
considered (non-empty) queue. The presented approach hence underestimates the
capacity of the server, but equally for both queues. The average waiting time of all
customers in the total system, which can be seen as an M/D/1 queue as it is work
conserving, is known and given by
EW(total)= τ(2-ρ)/2(1-ρ),
where ρ=(λ(LP)+λ(HP))τ, the load of the total system. The results obtained by the
iteration give a higher average waiting time due to underestimation of the server capacity.
The waiting time of each type of customer should hence be scaled down, so that the
average waiting time of all customers in the system is correct. This leads to in improved
estimation of the average waiting type of a customer per type of queue.
Multiple queues
Note that this approach can easily be extended to multiple queues of any priority class.
The vacation length of a considered queue then depends on the state of all the other
queues, but can be computed in a similar way. However, the number of iterations needed
and the numerical efforts involved will grow exponentially with the number of queues in
the system. Take for example the situation with three queues, one HP queue and two LP
queues. The vacation length distribution of a queue is then given by
P(V ( x) = kτ | N (1) = i, N (2) = j ) =
∞
∞
q (1) ∑∑ P (V ( x) = (k − 1)τ | N (1) = a, N (2) = b) P( A(1) = a − i + 1) P( A(2) = b − j )
a =i −1b = j
∞
+ q(2)∑
∞
∑ P(V ( x) = (k − 1)τ | N (1) = a, N (2) = b) P( A(1) = a − i) P( A(2) = b − j + 1)
a = i b = j −1
⎧1, i, j = 0
⎪q( x) /(1 − q(2)), i = 1..∞, j = 0
⎪
P(V ( x) = 0 | N (1) = i, N (2) = j ) = ⎨
⎪q( x) /(1 − q(1)), i = 0, j = 1..∞
⎪⎩q( x), i, j = 1..∞
The vacation length distribution is found using
∑∑ P(V ( x) = 0 | N (1) = i, N (2) =
i
j ) P ( N (1) = i, N ( 2) = j ) ,
j
where again the steady state distribution of the other nodes is needed. Starting again with
a random distribution (with the queues almost always empty) we find the vacation time
and hence the corresponding steady state distribution of the queue under consideration.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
Now the next queue can be analyzed, using the found distribution of the previous queue.
This is repeated until the steady state distributions converge and the average waiting time
follows using Little's law.
3.3.2 Special cases
Saturated LP nodes
Consider the situation where one high priority node with Poisson(λ(HP)) packet
arrivals is surrounded by n saturated low priority nodes. This situation gives insight in the
impact on the waiting time of a packet when giving priority to a user in a busy network.
We take the probability q of visiting the high priority queue to be
q=(a)/(n+a)
where a denotes the factor of importance given to the high priority node, meaning the
probability of visiting the HP queue compared to the LP queue is times as high. The
vacation length distribution is then given by the geometric distribution
P (V ( x) = kτ ) = (1 − q ) k q ,
as any time the server does not jump to the HP node, it will service exactly one packet at
an LP node. As the average time between arrivals of the server is τ/q and the server only
serves one customer at each visit, the HP queue is stable when q>λτ. With the exact
distribution of the vacation length known, we can use the pgf of the number of customers
in the queue given by (1) to find the average number of customers in the HP queue. The
waiting time then easily follows from Little's law.
Empty HP node
We now consider the situation where the low priority nodes are no longer saturated,
but each have an arrival process of rate λ(LP) and a deterministic service time of value .
Let node n+1 be the HP queue, the conservation law (cf [41]) then states that
n +1
n +1
∑ ρ (i) EW (i, q) = ρ
i =1
∑ λ (i)β (i)
( 2)
i =1
2(1 − ρ )
where EW(i,q) denotes the average waiting time in the queue (not including service) and
ρ(i)=λ(LP)τ for i=1..n and ρ(n+1)= λ(HP) τ, so that and ρ =nρ(LP) + ρ(HP). As the
service time distribution is deterministic for any queue, we have that β (i ) ( 2) =τ² and the
total waiting time of a customer is EW(i)=EW(i,q)+τ.
Consider the situation where there are only n LP nodes, so the arrival rate at the HP
queue is set equal to zero. The stability condition is that ρ = nλ(LP)τ < 1 and it
immediately follows that
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
n
n
∑ ρ ( LP) EW ( LP, q) = ρ
∑ λ ( LP)τ
i =1
2(1 − ρ )
i =1
EW ( LP, q) =
2
ρτ
2(1 − ρ )
(2 − ρ )τ
EW ( LP) =
2(1 − ρ )
Saturated HP node
Now consider the situation where the HP queue is saturated, then we have n identical
queues, and looking from the point of view of the LP nodes a server with vacations. The
server vacations can be modelled as a switchover time between the queues, which can be
zero (the HP queue is not necessarily visited before visiting another LP queue). Let p_{i}
denote the probability of jumping to queue i, s_{i} the average time it takes to switch to
queue i and the average mean switchover time, then we have (cf [35])
n
n
∑ ρ (i)[1 −
i =1
λ (i ) σ
p(i ) 1 − ρ
]EW (i ) = ρ
∑ λ (i)β (i)
i =1
2(1 − ρ )
( 2)
+
σ
1− ρ
n
ρ (i )
∑ p(i)
i =1
n
−∑ ρ (i ) s(i ) +
i =1
ρ
2σ
n
∑ p(i)s(i)
i =1
where for our model we have that λ(i)= λ(LP)= λ, ρ(i)=λτ, ρ=nλτ, β (i ) ( 2) = τ ², p(i)=1/n,
s(i)=qτ /(1-q), s (i ) ( 2) =(q(q+1)τ/(1-q)² and σ =s(i) as all switchover times are equal. Here
q denotes the probability of the server polling the HP node. The expression simplifies to
[1 −
nλqτ
nλτ 2
qτn
q + 1 − 2qτ
]EW ( LP) =
+
+
(1 − q)(1 − nλτ )
2(1 − nλτ ) (1 − q)(1 − nλτ )
2(1 − q)
and applying Little's law the average total number of customers in the queue is found
again. The stability condition for this system is that nλ σ /(1- ρ)<1, as this is the number
of arriving customers during the average cycle time of a queue. This can be rewritten into
n<
1 − aλτ
λτ
where a denotes the factor of the probability of visiting the HP node next, that is P(jump
to a certain LP node)=P(jump to HP node), which in this case is the probability of going
on a vacation. Note that this approach can easily be extended to the situation with
multiple high priority nodes, as only the probability of the server being on vacation
changes, so only the values of s(i) and s (i ) ( 2) need to be adjusted.
3.4 Validation
3.4.1 General case
Two nodes
The following table shows the results of the algorithm compared with simulation
results for different values of the arrival rates at the two queues (one HP and one LP
20
( 2)
IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
queue). The probability of moving to the HP queue is twice as high as for the LP queue
for all situations. We have that q=a/(n+a) is the probability of jumping to the HP node.
Rates
λ(LP) λ(HP)
0.2
0.5
0.5
0.2
0.1
0.01
0.01 0.1
0.4
0.1
0.1
0.4
0.1
0.3
0.3
0.1
Simulation
LP
HP
0.472
1.046
1.169
0.347
0.106
0.010
0.010
0.106
0.612
0.138
0.155
0.593
0.137
0.396
0.408
0.128
Algorithm
LP
HP
0.506
1.009
1.205
0.311
0.106
0.010
0.011
0.105
0.617
0.132
0.171
0.578
0.147
0.385
0.409
0.124
LP
7.278
3.078
0.041
3.974
0.903
10.14
7.391
0.308
Error
HP
3.538
10.412
0.517
0.490
4.343
2.460
2.726
3.534
3.4.2 Special cases
Saturated low priority nodes
Figure 1 shows the average waiting time of a packet in the HP queue, for different
values of n, the number of saturated low priority queues in the system. The arrival rate at
the HP queue is set to λ(HP)=0.01. Three lines represent the results of the model for
a=2...4, where a=2 is the highest line and a=4 the lowest and the dots are the
corresponding simulation results. It clearly follows from the figure that where for a sparse
network (low number of LP nodes) the differentiation has hardly any effect, whereas for
a dense network (high number of LP nodes) a higher priority has a bigger impact.
Figure 1: Saturated LP nodes
Empty and saturated high prioirity node
Figure 2 shows the average waiting time of a packet in an LP queue, for different
values of n. The arrival rate λ(LP) is set to 0.01. In this scenario the HP node is absent (or
empty) and the whole network behaves as a standard M/D/1 queue where each separate
queue has the same average behaviour.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
Figure 2: Empty HP node
Figure 3 shows the situation with a saturated HP node, again for the values of a=2...4,
with a=2 the lowest and a=4 the highest line, still using λ(LP)=0.01. For higher values of
a, the server will more often be processing HP packets, leaving less capacity for the LP
nodes. This shows from the figure as the saturation sets in for lower values of n. When
the network is sparse, we see there is a high impact of the differentiation on the waiting
time of the low priority packets, which diminishes slightly as the number of nodes grows.
Nearing the saturation point, the impact increases again.
Figure 3: Saturated HP node
3.5 Conclusion
This part has studied the impact of service differentiation on the waiting time of
packets of the different classes. The differentiation was made by giving different
probabilities to the nodes for being allowed to transmit a packet next. The network under
consideration was mapped onto a polling system, which was analyzed using polling
models and queueing models with server vacations. For both saturated and non-saturated
queues we have derived either expressions for the waiting time, or given an algorithm to
approximate the waiting time of a packet. Simulation results have shown that for the
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
saturated case the results are exact, whereas for the non-saturated case the algorithm
performs well, with results mostly within a 10% error bound. The impact of the priority
setting has been analyzed, which shows that by doubling the probability of visiting the
HP queue compared to the LP queue deteriorates the waiting time for the LP queue and
can easily lead to instability. Future work may include different TXOP limits, one other
setting of the IEEE 802.11e protocol that differentiates between users. A similar analysis
can be done, using for example the approach presented in [42] to determine the steady
state distribution when the vacation time distribution has been determined.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
4 An upper bound on network throughput
Interference is an important aspect of wireless networks that seriously affects the
capacity of the network. This is especially so in a wireless multi-hop network, where a
transmission on one link interferes with transmissions on links in the vicinity. On a multihop path self-interference may result in substantial degradation of end-to-end network
performance. In this respect, due to interaction among hops, multi-hop wireless networks
considerably differ from wired networks thus calling for new modelling and analysis
techniques that take into account interference constraints.
In the absence of interference, but including capacity constraints on the transmission
rate of nodes, feasibility of a set of traffic demands between pairs of nodes can be
determined by considering the flow allocation in the network as a multicommodity flow
problem. The network is modelled as a graph, with the vertices representing the nodes
where traffic is originated, terminated or forwarded. There is an edge between vertices if
the corresponding nodes are within each others transmission range. Each edge has a
capacity, representing the throughput that is possible over that edge. The multicommodity
flow problem then addresses the question whether there exists a set of paths and real
numbers (fractions) so that: (1) for each traffic demand, there is a set of paths from the
traffic source to the destination; (2) fractions of the traffic demand can be allocated to
each path so that for each source-destination pair the traffic demand is realized and (3)
the capacity constraints are taken into account.
This part considers a generic wireless network configuration and traffic load specified
via parameters such as nodes, transmission and interference ranges, as well as the traffic
matrix indicating the demands between source nodes and sink nodes. We make no
assumptions about the homogeneity of nodes with regard to transmission range or
interference range, nor the capacity of the links. This is in contrast to previous work [46]
that has focused on asymptotic bounds under assumptions such as node homogeneity and
random communication patterns. In [49] a conflict graph is used to address the problem
of finding a feasible flow allocation to realize demands between pairs of nodes. While the
conflict graph provides a more comprehensive modelling of the scheduling problem, it is
also more complicated to deal with. A detailed discussion on the relation of our work
with [49] is presented in Section 4, showing that we obtain a tight upper bound for an
example provided in [49]. In [50] the question of a routing algorithm to find paths
satisfying the traffic demands in a distributed setting is addressed. For an LP relaxation of
the interference problem, [52] presents necessary conditions for link flow feasibility. This
yields an upper bound similar to that of [49]. In addition [52] introduces an edge
colouring problem in which each colour at an edge represents a time slot for
transmission. This problem is solved using a FPTAS, yielding a lower bound for the link
flow allocation. In [53] this work is extended to multi-radio and multi-channel networks.
Our work does not solve any LP's, but provides a good characterization for the feasibility
of the fractional multiflow problem with interference constraints which gives a fast way
of finding upper bounds for a slightly different interference constraint setting.
In this part we introduce a new approach to model interference in a carrier sensing multihop wireless network. To this end, we transform the sustainable load problem into a
multicommodity flow problem that we extend with interference constraints. The main
theorem of this part states a condition for the feasibility of the multicommodity flow
problem with interference constraints, given the demands between source and destination
nodes. Using this theorem, we compute the maximal throughput between a single source
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
and a single destination. We consider the following elements to be the key contribution of
our work:
· The use of polyhedral combinatorics (Farkas' Lemma) to obtain a structural
expression for feasibility of the multicommodity flow problem with interference
constraints, in terms of a `generalized cut condition' analogous to the `max-flow min-cut'
theorem of Ford and Fulkerson [44].
· The generality of our framework which incorporates the following realistic effects: the
transmission range is not necessarily equal to the interference range, the network may
consist of mixed wired and wireless connections and wireless links have different
capacities depending on distance, obstacles or transmission power.
4.1 Ad hoc interference model
Ad-hoc networks use transmissions over a wireless channel to communicate between
users. However, if multiple transmissions take place at the same time over the same
wireless channel, transmissions may collide and the data will be lost. This interference
limits the throughput of ad-hoc networks. Adopting the model of [48], we will model the
interference constraints. To do so, we define the transmission range and the interference
range of a node. When nodes in a wireless network want to communicate, they need to be
close enough to receive each others signals. The transmission range of a node is the
maximum distance from that node to where its received signal strength is insufficient for
maintaining communications. Even though a signal may be too weak to be received
correctly outside the transmission range, the signal can still cause interference preventing
nodes from receiving other signals correctly. The interference range is the maximum
distance from a node to where it prevents other nodes to maintain communications. Note
that in general and in our model the transmission and interference range are not equal. In
the following we adopt a graph representation (see e.g. [47],[49]) in which these ranges
will be represented by arcs. Let V denote the set of nodes, A the set of arcs and let δ + (u )
denote the arcs leaving node u.
In a carrier sensing network, nodes within each others interference range will avoid
transmitting at the same time. We will model this as follows: Let R(v) denote the set of
nodes within the interference range of node v (which includes v itself), that is: if one of
the nodes in R(v)\{v} is transmitting, then v cannot receive other transmissions, nor can v
transmit. Let ρ(u) denote the fraction of time that a node u is transmitting, then
∑ ρ (u ) ≤
f (v )
(1)
u∈R ( v )
where f(v) denotes the interference capacity of the node. The interference capacity
denotes the amount of interference a node can handle and still transmit data itself. For a
wired network one could set f(v)=∞, whereas f(v)=1 ensures that no two nodes within
each others interference range transmit at the same time. Taking f(v)<1 can model a loss
due to for instance non-ideal CSMA/CA effects.
Consider the set I(v) of all arcs leaving v, entering v and leaving the nodes that are in
v's interference range:
I (v) = {a | a ∈ δ + (u ), u ∈ R(v)} .
(2)
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
It follows that for all the arcs that are within I(v) the interference capacity f(v) may not be
exceeded. In particular, if f(v)=1, simultaneous transmissions cannot take place over arcs
a₁,a₂ ∈ I(v). For later use, we also introduce here a dual notion of I(v), viz. J(a), the set of
vertices that experience interference by a transmission over arc a:
J(a)={v ∈ V|a ∈ I(v)}.
(3)
Consider node v ∈ V. The interference arc set I(v) is denoted using bold arcs in Figure
1a and the set of nodes J(a) affected by arc a by the grey nodes in Figure 1b.
Figure 1: I(v) and J(a)
To each arc a a capacity b(a)>0 is assigned. In actual networks, due to e.g. unequal
distances among nodes or external disturbances such as noise, the link capacities may be
different.
Consider a set of source and destination pairs (r(1),s(1)),...,(r(k),s(k)). When the net
amount of flow between a source node r(i) and a destination node s(i) is d(i), we say that
the value of the (r(i),s(i)) flow is d(i). For an allocation, let x{i}(a) denote the amount of
traffic for source destination pair (r(i),s(i)) over link a. We will call this the flow of
commodity i over arc a. To take the link capacities into account, we use the following
interference constraints:
x{i}(a )
≤ f (v ) .
i =1 a∈I ( v ) b( a )
k
∑∑
Note that the interference constraints indicate whether a node can receive correctly and
that we assume that collisions are fatal, which is a worst case scenario as in practice
caption may be possible. However, as we impose condition (3) on all nodes, for a
transmission over link (u,v) to be successful, we have that (3) must hold for both u and v,
so both sender and receiver must be free of interference. This closely resembles the
behaviour of IEEE 802.11 under RTS-CTS, where both sender and receiver must be free
of interference, see e.g. [54]. For a successful communication, the sender must be able to
hear the link layer acknowledgement transmitted by the receiver.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
4.2 The extended multicommodity flow problem
The multicommodity flow problem (MCFP) describes the problem of finding an
allocation of flows over links such that all flows are transferred from their source to their
destination, without exceeding the capacity of the links. The multicommodity flow
problem with interference constraints is as follows.
Given a graph G(V,A), with link capacities b:A→ℝ⁺, interference capacities f:V→ℝ⁺
and source and destination pairs (r(1),s(1)),...,(r(k),s(k)) with demands d(1),...,d(k) ∈ ℝ⁺,
find for each i=1,...,k an (r(i),s(i)) flow x{i} ∈ ℝ₊{|A|} of value d(i), where x{i}(a) is the
amount of traffic of commodity i sent via arc a, and so that for each arc a∈ A and vertex
v ∈ V the capacity and interference constraints are met. Let δ + (U ) ={a=(u,v) ∈ A|u ∈ U,v
∉U} and δ − (U ) ={a=(u,v) ∈ A|u∉U,v ∈ U} so that δ + (v) and δ − (v) denote the arcs
leaving and entering node v respectively. Our multicommodity flow problem with
interference constraints has the following feasibility constraints:
k
∑ x{i}(a) ≤ b(a), ∀a ∈ A
(4)
i =1
∑
x{i}( a) =
a∈δ + ( v )
(5)
a∈δ − ( v )
∑
x{i}( a) −
∑
x{i}(a ) −
a∈δ + ( r ( i ))
a∈δ + ( s ( i ))
∑ x{i}(a), ∀v ∈ V , v ≠ r (i), s(i)
∑ x{i}(a) = d (i), ∀i
(6)
∑ x{i}(a) = −d (i), ∀i
(7)
a∈δ − ( r ( i ))
a∈δ − ( s ( i ))
x{i}(a )
≤ f (v )
i =1 a∈I ( v ) b( a )
k
∑∑
(8)
Equation (4) shows the capacity constraints on the arcs, equation (5) assures flow
conservation, i.e. for each node the flow in must equal the flow out, and (6) and (7)
define that the demands leaving the source and entering the destination. Note that (7) is
redundant as it follows from (5) and (6), but is included here for completeness. Equation
(8) is our interference constraint. Equations (4)-(7) define the multicommodity flow
problem in its standard form, that is included in our formulation by setting the
interference capacities of all nodes to infinity, that is f(v)=∞ for all v ∈ V.
We now formulate a generalized cut condition for the multicommodity flow problem
with interference constraints, so including (8). To this end, define length functions
l:A→ℝ⁺ on all arcs and interference functions w:V→ℝ⁺ on all nodes.
For a given length function l:A→ℝ⁺ and interference function w:V→ℝ⁺, let
dist{l,w}(r(i),s(i)) denote the distance function that incorporates both the length and
interference, where the distance of a path is built up of the distance q{l,w}(a) of the arcs
in the path as follows
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
q{l.w}( a ) = l (a ) +
(9)
w(t )
t ∈J ( a ) b ( a )
∑
q{l , w}( P ) = ∑ q{l , w}( a )
(10)
dist{l , w}( r , s ) = min q{l , w}( P)
(11)
a∈P
P∈P{ r , s}
with J(a) as in (2) and P{r,s} the set of all paths from r to s.
Theorem: The multicommodity flow problem with interference constraints is feasible, if
and only if for all length functions l:A→ℝ⁺ and node interference functions w:V→ℝ⁺ it
holds that
k
∑ d (i)dist{l , w}(r (i), s(i)) ≤ ∑ l (a)b(a) + ∑ w(v) f (v)
i =1
a∈ A
(12)
v∈V
The proof of Theorem 1 is given is omitted here. The interference function w(v) is a
dual variable that can be interpreted as the price paid for a unit of the interference
capacity of a node v, which gives a weighted cut of the nodes by chosing the values of
w(v) to be zero or not (likewise the length function l(a) can be interpreted as the price
paid for a unit of capacity of an arc a, which also gives a weighted cut of the arcs). The
distance of an arc is then the price paid for the capacity of that arc, together with the price
paid for the interference capacity used by that arc for a flow unit. Note that our theorem is
an extension of the cut condition for the multicommodity flow problem (without
interference). This can be seen as follows. Setting all interference capacities to a very
large value the condition of Theorem 1 reduces to
k
∑ d (i)dist{l}(r (i), s(i)) ≤ ∑ l (a)b(a)
i =1
a∈ A
as the inequality only makes sense for w=0. The cut condition by setting a length of 1 to
all arcs in a cut and zero otherwise for the multicommodity flow problem states that
∑
b( a ) ≥
a∈δ + (U )
∑ d (i) .
r ( i )∈U
s ( i )∉U
This cut condition is necessary for the existence of a solution, but not sufficient. The
max-flow min-cut theorem states that if k=1, for every network, there exists a flow (maxflow) for which the amount is equal to the total capacity of the smallest cut in the
network (min-cut), see Ford and Fulkerson [44].
A direct consequence of Theorem 1 is that when there is only one commodity, a bound
on the throughput d of the network can be determined by
d=
∑ l (a)b(a) + ∑ w(v) f (v)
a∈ A
v∈V
dist{l , w}( r , s )
.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
When there are multiple commodities with demands d(1),...,d(k), Theorem 1 can
determine the maximal value 0 ≤ λ ≤ 1 such that for all commodities a throughput of d(i)
can be achieved using
λ=
∑ l (a)b(a) + ∑ w(v) f (v)
a∈A
k
v∈V
∑ d (i)dist{l , w}(r (i), s(i))
.
i =1
4.3 Examples
We can consider the network as depicted in Figure 2 as nodes in a network, connected
by links using 802.11b with a maximum transmission rate of 11 Mbit/s, but with link 3
having a bad connection, due to distance or a disturbance, only reaching the minimal
transmission rate of 5.5 Mbit/s.
Figure 2: Series of nodes with capacity constraints
We want to transmit data from node 1 to node 5. If we solve for the best solution without
considering interference, it is clear that we can send at a speed of 5.5 Mbit/s, as this is
limited by the slowest link. The interference constraints for the network with identical
link capacities would imply that all links can be used one third of the time, leading to an
overall throughput of 1.83 Mbit/s when considering the constraints separately.
For the example of Figure 2 we have the interference constraints
x (1) x (2) x (3)
+
+
≤1
11
11
5 .5
x(2) x (3) x(4)
+
+
≤1
11
5 .5
11
From a direct solution of (4)-(8) it follows that x(a)=2.75 is a feasible solution, higher
than the earlier claimed 1.83 Mbit/s. Using Theorem 1, we find that
2w(1) + 4w(2) + 4w(3) + 3w(4) + w(5)
≤
11
11l (1) + 11l (2) + 5.5l (3) + 11l (4) + w(1) + w(2) + w(3) + w(4) + w(5)
d (1)(l (1) + l (2) + l (3) + l (4) +
which gives by taking the cut w(3)=1 and all other values (including l(i)) equal to zero
4/(11)d(1) ≤ 1.
So x(a)=2.75 is also the optimal solution. The value now found for x(a) can be interpreted
as the fraction of time a link is in use multiplied by the transmission rate of the link. This
shows that links one, two and four get a quarter of the time, whereas link three gets half
of the time. So when considering four slots, link one, two and four each get one slot
(where link one and four use the same slot!) and link three the other two. This way we
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
have an accurate representation of the network incorporating both the capacity and
interference constraints, together with the flow conservation laws.
We now consider the more sophisticated network used in [49] as depicted in Figure 3,
where all arcs have capacity 1.
Figure 3: 3x3 grid
The upper bound on the throughput from node 0 to node 8 for this network obtained in
[49] is 0.667, opposed to the optimal 0.5, even though their algorithm has discovered all
possible cliques in the conflict graph. Using Theorem 1 and taking the cut w(1)=w(3)=1
or w(4)=1 gives the lowest upper bound that can be achieved, resulting in d(1) ≤ 0.5,
which is tight. The reason we obtain a different result than in [49] is that we use
constraints for all nodes, so that considering for example node 0 we have that
x(3)+x(9) ≤ 1, as both signals reach node 0. In the approach of Jain et al., arcs 3 and 9 are
not connected in the conflict graph, as a simultaneous transmission over both arcs is
possible.
There is an interesting relation between the approach presented in this paper and the
results of [49]. Jain et al. use a conflict graph to determine lower and upper bounds for
the throughput of the network. The conflict graph C has vertices corresponding to the
arcs in the transmission graph, where there is an edge between two arcs if and only if the
arcs are not allowed to transmit simultaneously. In our approach, C has as vertex set A,
where there is an edge between a(i) and a(j) (for some 1 ≤ i,j ≤ |A|) if and only if ∃v ∈ V
s.t. a(i),a(j) ∈ I(v). Note that I(v) defines a clique in C for each v in V. In fact, our
interference model adopted here resembles the protocol model of [49], but it is `stronger'
in the sense that for the same network, we have more interference constraints (edges in
the conflict graph) than [49].
In [49] it is shown that a vector x(i):A→ℝ⁺ (corresponding to a flow i), can be
scheduled without interference conflicts if and only if x(i) lies in the stable set polytope
of C. (The stable set polytope is the convex hull of the incidence vectors of the stable sets
in the graph). It is well-known that the stable set polytope is contained in the fractional
stable set polytope. (The fractional stable set polytope is defined by all constraints
indicating that the total flow in a maximal clique in the conflict graph is at most 1.)
In this paper, instead of first defining the conflict graph C and then discovering its
cliques, we directly formulate inequalities corresponding to the cliques I(v) for all v ∈ V.
The polytope defined by these inequalities will therefore contain the fractional stable set
polytope. As a result, the upper bound obtained here cannot always be achieved using a
flow allocation without interference conflicts.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
4.4 Conclusion
In this part we considered multi-hop wireless networks, for which we have stated a
theorem giving a necessary and sufficient condition for the existence of a solution for the
multicommodity flow problem with interference constraints, given a required throughput
between nodes of the network. By extending the multicommodity flow problem with
interference constraints, while still considering the capacity of the links, we have
constructed an LP formulation for assigning flows to links to achieve the required
throughput between certain nodes in the network. Using Farkas' Lemma, we have
developed the corresponding general cut condition which gives an upper bound on the
throughput that can be achieved in the network, given the source-destination pairs. The
upper bounds are found by defining a cut, where the best cut will give a tight bound on
the throughput, analogous to the max-flow min-cut theorem for the multicommodity flow
problem. Several examples illustrate the use of the theorem in finding the optimal
throughput for some simpler networks.
The applicability of this work for wireless multi-hop networks can for example be seen
when designing a network that must be able to support a certain throughput for each user.
Another possible application is for admission control, as our necessary and sufficient
condition determines when adding a commodity will no longer result in a feasible
solution using the same cutvector. If for any cutvector the condition does not hold, there
exists no feasible solution for adding the commodity. If the cutvector however does not
show infeasibility, this is no guarantee for feasibility as a different cutvector may give a
tighter result for the situation with the added commodity. Due to the general nature of the
theorem, any type of network with heterogeneous users and protocols can be modelled.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
5 Conclusion
In this document various topics are discussed for the refinement of QoS protocols and
mechanisms for Personal Networks. These topics build on the various routing protocols,
and QoS mechanisms for ad hoc networks, as presented in the previous deliverable.
Advantages and disadvantages of searching strategies are discussed. In compliance with
the notion of QoS as presented in an earlier deliverable, an analysis has been made of the
impact of QoS differentiation on the waiting time of packets in a network with different
classes of customers. Finally, a theorem is stated providing insight in the ability o a
network to accommodate a certain demand by the users. This theorem can also be used to
derive an upper bound on the total throughput a network can achieve. All these topics
provide insight in how to refine QoS protocols and mechanisms, taking into consideration
the advantages and limitations of the currently available ones.
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
Appendix
A. ACRONYMS
ABR
Associativity Based Routing
AMRoute
Ad Hoc Multicast Routing Protocol
AODV
Ad Hoc On Demand Distance Vector Routing
BB
Black Burst
BF
Bellman-Ford Algorithm
BPS
Batch-arrival Processor Sharing
CGSR
Cluster Gateway Switch Routing
CSMA/CA Carrier Sense Multiple Access with Collision Avoidance
DCF
Distributed Control Function
DPS
Discriminatory Processor Sharing
DSDV
Destination Sequence Distance Vector Routing
DSR
Dynamic Source Routing
FGMP
Forwarding Group Multicast Protocol
GPSR
Greedy Perimeter State Routing
IEEE
Institute of Electrical and Electronics Engineers
IP
Internet Protocol
LAPAR
Location Aided Power Aware Routing
LAR
Location Aided Routing
MAC
Medium Access Protocol
MACA/PR Multiple Access with Collision Avoidance with Piggyback Reservation
MACAW
Media Access Protocol for Wireless LAN's
MCEDAR Multicast Core-Extraction Distributed Ad Hoc Routing Protocol
ODMRP
On-Demand Multicast Routing Protocol
OSPF
Open Shortest Path First
PAMAS
Power Aware Multi Access Protocol with Signaling
PAN
Personal Area Network
PARO
Power-Aware Routing Optimization Protocol
PN
Personal Network
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IOP GenCom QoS for PNs at Home – D2.2 Assessment of proposed QoS protocols and mechanisms
QoS
Quality of Service
RTS/CTS
Request / Clear To Send
SAMCRA
Self Adaptive Multiple Constraints Routing Algorithm
SSR
Signal Stability Routing Protocol
TORA
Temporally Ordered Routing Algorithm
WLAN
Wireless Local Area Network
WRP
Wireless Routing Protocol
ZRP
Zone Routing Protocol
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