Chapter 2 Analysis of multiuser interference
2.3 Summary
In this chapter, we have provided an analytical model to analyze the power consumption and capacity limits of different medium access control schemes in mobile WBAN. Results show that neither the Beacon-based nor detection-based access scheme can effectively support both the low power consumption and high capacity (in terms of WSN nodes) simultaneously. A hybrid mode
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proposed in this chapter can provide higher capacity while the power consumption is still well managed. This implies a mixed method may be a better solution for WBAN MAC.
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Chapter 3
Distributed multiuser resource scheduling
However, unlike a sensor network focused on static or low mobility scenarios [3], a WBAN has a higher moving speed and more frequent topology changes due to user movement [2]. The moving topology of multiple WBANs is similar to that of mobile ad hoc networks (MANETs) [4], but with group-based rather than node-based movement. The “high mobility” and “group-based movement”
make the WBAN neither equivalent to a sensor network nor to a MANET. A WBAN thus has a high chance of encountering other WBANs, which creates new issues of inter-WBAN scheduling (IWS).
Corresponding discussions [16, 18-21] have just been opened and comprehensive studies are still required.
Distributed interference-avoidance scheduling of wireless networks can be modeled by the notion of distributed graph coloring, which is commonly adopted in sensor networks or MANET [9, 22, 23].
Network topology is modeled as a graph G( , )V E . The vertices V of G represent the wireless nodes; the edges E of G represent radio resource conflicts between mutually-interfering node pairs; the color set C in a coloring of G represents the set of distinct resource units (can be time slots, frequency bands, or code sequences). A complete vertex k-coloring of a graph G is a mapping V G( )C, where C k, such that adjacent vertices receive distinct colors. Thus, interference between wireless nodes can be avoided by mapping different colors (resource units) to adjacent vertices (mutually-interfering nodes). A standard message-passing model is adopted in this study. To negotiate the color mapping between vertices, they can have a two-way message exchange
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with its adjacent vertices. We assume the system is fully synchronous. In other words, all vertices start their coloring algorithms at the same time and execute each step of the algorithm simultaneously.
Other terminologies in graph theory refer to [24]. The chromatic number ( )G is defined as the minimum k used to completely color graph G; N v( ) is defined as the set of adjacent vertices of vertex v ; the maximum degree ( )G is defined as the maximum d v( ) and v V G ( ), where vertex degree d v( ) N v( ).
Two basic requirements of IWS are: (i) fast convergence and (ii) high channel utilization. In the case of a WBAN user walking on a sidewalk, network topology changes frequently when the user keeps encountering other WBAN users. Therefore, a quick IWS that rapidly detects and responds to every topology change is expected, which could adopt the quick 1 ( )G coloring for MANET [25].
Also, IEEE 802.15 TG6, the standard task group of WBAN, requires that the WBAN protocol should support at least the sensor density: 60 sensors in a 6 m space [13]. Such dense WBANs create a 3 3 high probability of mutual interference. It can significantly decrease the number of coexisting WBAN users due to poor channel utilization [19]. Thus, high channel-utilization IWS is also required, which could adopt an optimal spatial-reuse coloring for dense sensor networks [22]. Optimal spatial-reuse implies the maximal number of sensors that access the wireless channel at the same time.
However, references [26-28] show that quick (low time-complexity) coloring and optimal spatial-reuse coloring are trade-offs which cannot be simultaneously achieved with conventional complete coloring. Optimal spatial-reuse coloring uses a minimum number of colors, the chromatic number ( )G , to color a graph. The fewer colors used, the more there is color-reuse among vertices.
It implies more wireless nodes can simultaneously transmit packets using the same resource unit, that is, the system has higher spatial-reuse and channel-utilization. However, completely color a graph by
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using ( )G colors is known to be NP-complete. The fastest ( )G -coloring so far still needs time-complexity O(2nnO(1)) [26]. Nevertheless, study [27] indicates that time-complexity can be significantly reduced to O(log )n if colors are increased to 1 ( )G (sacrificing spatial reuse) and distributed coloring techniques are adopted. 1 ( )G ( )G is known as Brooks’ theorem, which is a loose upper bound for ( )G . A recent work [29] adopts a similar 1 ( )G coloring method. It guarantees not only O(log )n time-complexity but also O(log )n bit complexity, which is the amount of information exchanged between vertices during coloring. Low-bit-complexity of a coloring algorithm makes it applicable to low-computing-power applications, such as sensor networks. Although [28] further decreases the time complexity from O(log )n to2 O( log )n by applying vertex priority, 1 ( )G colors are still required. As a result, neither ( )G [26] nor 1 ( )G -coloring [27, 28] may be directly applied to IWS due to their high time-complexity and low spatial-reuse, respectively.
This chapter proposes Random Incomplete Coloring (RIC) to realize quick and high spatial-reuse IWS. RIC consists of 1) a proposed random-value coloring method and 2) an incomplete-coloring approach. The random-value coloring method is a technique which realizes prioritized vertex coloring [28] (so-called oriented coloring) and will be proven to have a low time-complexity3
(2ln ) 2
W n
O e , which is even lower than [28] . Besides, for high spatial-reuse coloring, conventional
complete coloring using ( )G colors is known as the solution for optimal spatial-reuse.
Surprisingly, for designs of wireless resource scheduling, this study found that higher spatial-reuse (on average) than that of ( )G coloring is possible if partial vertices are allowed to be uncolored
2 Soft-O: O g n( ( )) is short for O g n( ( )logkg n( )).
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and only k( )G colors are used. In wireless networks, we observe that uncolored vertices imply wireless nodes with no transmission, which do not interfere each other. Hence, following the nature of wireless communication, there is no conflict between adjacent uncolored vertices (or with the special color: no color). Based on this fact, we can color a subgraph of graph G. This subgraph is constructed using vertices, excluding uncolored ones. Clearly, it is possible to use less colors for the subgraph. For convience, we name this kind of coloring “incomplete” coloring. The proposed Random Incomplete Coloring (RIC) will be implemented as an inter WBAN scheduling protocol with a TDMA framing structure, a common structure used in sensor or body area networks [2], to test its convergence speed and spatial reuse in various mobile WBAN scenarios.
The rest of this chapter is organized as follows: section 3.1 introduces the details of related works and the problem formulation. Section 3.2 reveals the proposed RIC algorithm. The corresponding analytical models of RIC are provided in section 3.3. Section 3.4 presents the simulation settings and results. Section 3.5 concludes this chapter.
3.1 Related Works and the WBAN System Model
3.1.A Oriented and Non-oriented Coloring
Oriented vertex coloring [28] is a distributed coloring technique that utilizes predefined edge orientation (vertex priority) to improve the coloring speed of (non-oriented) random vertex coloring [27] from O(log )n to O( log )n . Here we refer to [27] as non-oriented vertex coloring to distinguish [28] from [27]. The major difference between oriented and non-oriented coloring is the
3 W x( ) is the Lambert W function [18]. W x( ) is solved by inverting the equation,
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manner of solving color conflicts, as shown in Fig. 3-2 Step 3. For non-oriented coloring, when color conflict cucv happens to vertices pair ( , )u v , both vertices give up that color and re-choose new colors for the color contention in the next coloring round. In contrast, oriented coloring tries to force the vertex that has a higher priority (having an outward edge orientation) to win the color. Although an oriented conflict-path may exist (u v w, cu cv cw) and only the vertex u at the start of the path can win the color (no vertex has a higher priority than u has), oriented coloring generates at least one winner for each path in each round (an exception, deadlock circle, will be mentioned later). This is the reason why oriented coloring speeds up coloring.
Given G( , )V E ; ,u v V G ( ); C ru( ) is the set of
Fig. 3-2 Oriented and non-oriented colorings (Pri-arts)
( ) exp[ ( )]
zW z W z , for any complex number z.
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However, to realize the oriented coloring for IWS, two particular issues need to be considered: (i) fairness and (ii) oriented conflict-circle (deadlock circle). First, oriented coloring assumes the priorities of vertices are pre-defined, which is not practical for the dynamic topology of a WBAN.
How to dynamically and fairly decide the priority of a WBAN should be further addressed. The second issue is the oriented conflict-circle (deadlock circle) problem. An oriented conflict-circle is defined as a circle graph with one-way orientation and vertices in the circle contending for the same color (e.g. u v w u c, u cv cw). There is no vertex in that circle that can be colored because there is always another adjacent vertex with a higher priority. Thus, the coloring cannot be completed unless vertices try to contend using different colors in subsequent rounds. To solve these two problems, a method that implements oriented coloring, random value coloring, is proposed.
Moreover, we will show that this method can further decrease the time complexity from O( log )n [28] to O e W(2ln ) 2n
.
Aside from the above implementation issues, conventional complete colorings [26-28], including oriented coloring [28], have an optimum spatial reuse bounded by ( )G . However, for designs of wireless resource scheduling, spatial reuse can be further improved if the coloring rule is relaxed.
The relaxed coloring approach, referred to as incomplete coloring, will be introduced in the next section.
3.1.B CPN-based IWS
A CPN-based IWS will be adopted in this study due to the imbalanced CPN/WSN architecture of a simple star (Fig. 3-1). In a single WBAN, the CPN plays the role of the master and the WSNs are the slaves. The CPN manages the join, leave, and functional-control of the WSNs. In contrast, WSNs
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are passive devices, which are designed to retrieve specific vital signals such as blood pressure or electrocardiograms (ECG) and forwarding them to the CPN. Besides, a CPN is most likely to be embedded in personal devices such as cellular phones or PDAs with larger and rechargeable batteries.
In contrast, a WSN is expected to be light weight (small battery) and even non-rechargeable for certain implantable applications. These two unbalanced features suggest shifting power-consuming network controls from WSNs to the CPN. Thus, a CPN-based two-step IWS, which is illustrated in Fig. 3-3, is assumed in this study. The CPN first negotiates the WBAN resources with other CPNs that are in the mutually-interfering-range. The CPN then assigns reserved resources to its WSNs. As a result, the WSN wakes up only when (1) receiving beacon messages carrying the pre-regularized transmission schedule from the associated CPN and (2) transmitting vital signals following the schedule to that CPN.
Vital signal transmission from WSN to CPN Nego.
Fig. 3-3 CPN-based inter-WBAN scheduling (IWS)
The CPN-based IWS can now be modeled as a unit-disk graph coloring problem. A 2-dimensional randomly constructed graph (in short, random graph) G( , )V E is generated to model the WBAN network. V G( ) represents the set of CPNs; E G( ) represents the set of conflict links between
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CPNs. The first step in generating the graph is to randomly deploy nV G( ) vertices in a field to simulate the random positions4 of WBAN users. Consequently, edges are added to connect vertices if the distance between CPNs is equal to or less than the mutually-interfering-range between WBANs (the radius of the unit-disk). In this sense, the graph of CPN-based IWS is similar to that of MANET scheduling. However, they have different resource scheduling strategies. In MANET, each vertex represents a wireless node. MANET focuses on efficient inter-node communication and routing.
Hence, “edge” coloring, which models the scheduling of “node-to-node communications”, can be adopted. In contrast, in CPN-based IWS, each vertex represents a sensor group (WBAN). CPN-based IWS tries to resolve the interference between sensor groups belonging to different users. Therefore,
“vertex” coloring, which models the scheduling of active “CPN-based sensor groups”, is appropriate.
As a result, for CPN-based IWS, a k-coloring of the random graph is labeled V G( )C, where C k, such that adjacent vertices receive distinct colors. The labels are colors, which are mapped to different resource units for associated data transmissions (WSNs to the CPN). Only k resource mappings can be decided when a k-coloring algorithm is excuted once. We assume that every WSN always has data to be transmitted to the CPN. Hence, the coloring is periodically performed to map wireless resources to all these data transmissions. The target of this study is to devise a coloring method that simultaneously satisfies two IWS requirements: (i) low time-complexity and (ii) high spatial reuse.
4 In some cases, waiting lines for example, user positions follow certain rules instead of random deployments, which yield special graphs such as lines or grids. Although these graphs might reflect more real scenarios in our life than the random graph does, they need more complicated analysis skills due to their restrictions of graph formation. Therefore, to make the discussion of the proposed RIC more intuitive, this study focuses on the performance analysis of the random graph.
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3.2 Random Incomplete Coloring
The proposed random incomplete coloring (RIC) has two major components: (i) a proposed random-value coloring method and (ii) a proposed incomplete coloring approach. The random value coloring is a low time-complexity coloring method, which is designed for quick IWS. On the other hand, incomplete coloring is a high spatial-reuse coloring approach, which modifies the conventional coloring rule to explore the potential high spatial reuse when k( )G .
3.2.A Random Value Coloring
Random value coloring is a method that realizes oriented coloring. It overcomes two major problems of oriented coloring [28] : fairness and oriented conflict-circle (deadlock circle), as outlined in section 3.1.A. The method of random value coloring is to adopt a random value comparison to generate instant priority differences between all adjacent vertices, as illustrated in Fig. 3-4 (except step 6 for the incomplete coloring).
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Random Incomplete Coloring:
Given G( , )V E , u V ; C ru( ) is the set of available colors of u in coloring round r . The initial size of the available color set is Cu(0) k. For each coloring cycle:
While u is uncolored,
1. u chooses a color cu from C ru( ) with a random value,
u.
2. u broadcasts its RVC message including cu and u to ( )
N u .
3. If u receives RVC messages from vN u( ) with
v u
and cv cu, u remains uncolored. Otherwise, u is colored by cu. (Random value coloring)
4. if u wins the color, it broadcasts the color taken notification.
5. u removes the colors taken by N u( ) from C ru( 1). 6. If C ru( 1) 0 , u becomes uncolored. (Incomplete
coloring)
Fig. 3-4 Random incomplete coloring
In Fig. 3-4, the fairness of random value coloring can be supported if the random values from different vertices are generated with an identical uniform distribution5. Due to the symmetry between vertices, it is obvious that the probability that a vertex generates the maximum random value (wins the color) among (n1) competitors is 1n , where “competitors” means adjacent vertices contending for the same color. Thus, each vertex has equal radio resource sharing with its adjacent vertices. Besides, the way that random value coloring avoids an oriented conflict-circle can easily be observed, since it is not possible to have the ordering of the random values of vertices in a circle with
5 f x( ) 1
b a a
, x b, otherwise f x( )0.28
a one way orientation v1 v2 vn v1. The fairness and deadlock-circle-free properties of the proposed method makes oriented coloring possible to realize for quick IWS.
3.2.B Incomplete Coloring
Incomplete coloring improves spatial reuse by greedily coloring a graph with k( )G . Normally it is not possible to completely color a graph using k( )G colors because color conflict is unavoidable and cannot be resolved6.Thus, incomplete coloring allows uncolored vertices to avoid conflict when vertices run out of k colors, as illustrated at step 6 of Fig. 3-4. Note that when applying incomplete coloring to IWS, the uncolored node means a CPN reserves no resource in that coloring. The WBAN of this CPN becomes temporarily inactive and generates no interference with its neighbor WBANs. The example in Fig. 3-5 demonstrates how incomplete coloring improves spatial reuse (on average).
6 Assume k( )G and G can be completely colored by k colors (adjacent vertices receive distinct colors). However, ( )G is defined as the minimum colors to completely color G and thus
( )
k G contradicts k( )G . As a result, there must be adjacent vertices that receive the same color.
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Fig. 3-5 Random incomplete coloring
Definition 3.1 Vertices-per-color (Vpc), which is defined as the average number of vertices colored by each color, is used to evaluate the average spatial reuse (color reuse) of periodical coloring. Higher Vpc implies more wireless nodes can simultaneously transmit packets using the same resource unit (color), that is, the system has higher spatial-reuse on average.
Definition 3.2 A coloring round is defined as the execution of all steps in the while loop of the coloring algorithm (e.g. steps 1 to 6 in Fig. 3-4). A coloring cycle is defined as the execution of all the coloring rounds required to leave the while loop, that is, the end of the algorithm.
Graph G with 4 vertices in Fig. 5 can be completely colored by ( ) 3G colors (case (3a)), written 3. Thus, Vpc
3 43. On the other hand, if two colors are used for incomplete coloring (k2), written '2, there are four possible results at the end of the coloring cycle: 2a to 2d.Because 2a to 2d all have 3 vertices colored by 2 colors, 2a to 2d have identical 3
Vpc 2. The probability that each case happens depends on the color contention scheme. Assume a random value
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coloring method is adopted. Because cases 2a to 2d are symmetric and vertices 1 to 4 have equal priority in the contention, each case has an identical 1
4 probability of showing up. As a result, when the incomplete coloring with k2, '2, is iteratively performed,
'2 3E Vpc 2
for
each coloring cycle. There is an increase of 12.5% over
3 4Vpc 3 on average.
Of course, the 3-color complete coloring schedules one more resource than the 2-color incomplete coloring in each coloring cycle. In periodical coloring, to schedule the same number of resources, a k-color incomplete coloring (k( )G ) needs ( )G k times more coloring cycles than that of a( )G -color complete coloring, which increases the exchange of coloring messages required for periodical coloring. This could affect the collision probability of coloring messages in a practical IWS implementation. The tradeoffs between spatial reuse and affordable coloring-message exchanges will be closely analyzed in section 3.4.
3.3 Analytical Model of Random Value and Incomplete Coloring 3.3.A Upper Bound of the time and bit complexity of RIC
With the proposed random value coloring method, the time complexity of RIC can be further decreased from O( log )n [28] to O e W(2ln ) 2n
, where W x( ) is the Lambert W function [30]. The time-complexity (upper bound) of RIC to color a graph with a constant-degree is calculated. This could be used to observe the time complexity of the 2-D topology model. Because vertices are randomly deployed in the 2-D model, every vertex should have a similar vertex degree. The accuracy of this constant-degree assumption will be verified in the section 3.4.A.
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Definition 4.1 Oriented conflict-path (OCP) is defined as a path with all edges having a one-way orientation and all vertices in this path contending for the same color.
Lemma 4.2 A graph having the longest OCP with l-length (l-vertices) can be colored in at most l rounds.
This lemma is proved by [28], which shows that, in each coloring round, at least one vertex can be colored in the OCP. Thus, a graph with an l-length OCP can be colored within l rounds.
Lemma 4.3 The probability that an OCP has length larger than or equal to i in round rof the
RIC algorithm is less than or equal to
1 1 1
For the RIC algorithm, an OCP is only generated when the random values of vertices in the path are in descending order, v1 v2 v3 (Definition 4.1). Therefore, the probability that an OCP stay the same during the coloring. Hence, to make sure the length of an OCP is larger than or equal to
32 graph may contain multiple OCPs during the coloring. At the end of Phase I, we prove that the length of any OCP of the graph will be confined by the length leW(2ln ) 2n with a high probability.
According to lemma 4.2, this graph can be further colored with at most eW(2ln ) 2n coloring rounds in Phase II. Thus, the total coloring rounds can be proved to be
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For simplicity in the following calculation, the approximation of the factorial term, ! i i rounds that the RIC algorithm requires is the summation of the maximum rounds of Phases I and II,
10 W(2ln ) 2n
W(2ln ) 2n
W(2ln ) 2n
O e O e O e , Q.E.D.
7 Because each OCP could start from any of l eW(2ln ) 2n vertices. Also, each vertex has adjacent vertices in the constant degree graph, so there are i possible combinations to generate an OCP with
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Definition 4.5 As is explained in [29], the bit complexity of a distributed algorithm (per channel) is defined as the total number of bits exchanged (per channel) during its execution.
Theorem 4.6 The bit complexity of the RIC algorithm is O e
W(2ln ) 2n logn
.In each coloring round of the RIC algorithm, each vertex exchanges one random number and one
In each coloring round of the RIC algorithm, each vertex exchanges one random number and one