Chapter 3 Distributed multiuser resource scheduling
3.3 Analytical Model of Random Value and Incomplete Coloring
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 color identification with each neighbor on each channel. We assume the random number has finite resolution and hence can be expressed by bits with a constant bit length C. Also, the necessary colors for a n-vertices-graph coloring can always be identified by O
logn
bits. As a result, for the RIC, the number of bits exchanged in a coloring cycle (O e
W(2ln ) 2n
rounds) is O e
W(2ln ) 2n logn
.3.3.B Spatial Reuse of Incomplete Coloring
Similar to the time complexity analysis, the Vpc of RIC for a constant degree graph is calculated. Pc is defined as the probability that a vertex can be colored by RIC. In the constant degree graph, the Pcs of all vertices are identical due to the symmetric graph structure. Each k coloring decides a k-color vertex-to-color mapping. Thus, Pc
Vpc n
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Proof: The idea of the Pc calculation is based on the probability complementary between each vertex and its adjacent vertices. In RIC, the probability that a vertex can be colored is
1 i
P represents the probability that the vertex is colored by color c . If a vertex can be colored, i its adjacent vertices must have the color combinations without using c or 1 c or or 2 ck ,
Due to the symmetry of each color in the equation,
1 2 k
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3.4 Computer Simulation
A two-stage performance evaluation of RIC is provided in this section. The first stage evaluates the time-complexity and spatial reuse of the RIC. At the second stage, RIC is applied in an IWS with a TDMA framing structure, a common structure used in sensor or body area networks [2]. It tests the convergence speed and channel utilization of IWS in mobile WBAN scenarios. Packet collisions of data packets and coloring messages will be considered in the second stage.
3.4.A Performance Evaluation of RIC
2-D random graphs with various vertex densities are used to evaluate the performance of the RIC.
Following the system model in Section 3.1, n vertices are randomly deployed in a 10 10 m 2 square. The mutually-interfering-range is set as 2m. n can be 12, 25, 50, 100 to simulate low, middle, high, and extremely high densities. The performance metrics used to evaluate time complexity and spatial reuse of coloring are rounds-per-cycle (Rpc) and vertices-per-color (Vpc) respectively. Rpc is defined as the total coloring rounds required to finish a coloring cycle.
Fig. 6 compares the Rpc of the proposed RIC with INC-only. INC-only is an incomplete coloring combined with non-oriented coloring [27]. Due to the low time-complexity of the RIC,
(2ln ) 2
W n
O e
, Fig. 6 shows that RIC can be finished within five rounds in any number of colors (1~15 colors) and vertex densities (12~100 vertices). In contrast, the coloring rounds of INC-only dramatically increase, especially when the vertex density is high or few colors are used. This huge difference comes from the different strategies of RIC and INC-only in dealing with color conflict.
RIC forces each color conflict to generate a winner. On the other hand, for INC-only, vertices in the
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conflict pair both give up the conflict color and re-select new colors in the next coloring round, which prolongs the coloring time. The time-complexity of INC-only grows significantly in the few
k (reduced color choices) or the high density (increasing competitors) scenarios.
0 5 10 15
100 101 102 103
Num of Colors (k)
Rounds per Coloring (Rpc)
RIC 12 Vertices RIC 25 Vertices RIC 50 Vertices RIC 100 Vertices INC-only 12 Vertices INC-only 25 Vertices INC-only 50 Vertices INC-only 100 Vertices
= 2 = 11
Rounds = 5
= 7
= 4
Fig. 3-6 Rounds per coloring cycle (Rpc) of RIC and INC-only
For the spatial reuse analysis of RIC, Fig. 7 compares the analytical model (theorem 4.5 based on the constant-degree graph assumption) with simulation results for RIC in a 2-D random graph. Fig. 7 shows that the analytical model can correctly predict the Vpc of RIC, which monotonously increases while the number of colors k decreases.
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Fig. 3-7 Vertices per color (Vpc) of analytical model and simulations of RIC
Fig. 8 compares the Vpc of RIC with complete coloring [26-28]. Unlike RIC that keeps improving Vpcfollowing the decrease of colors, the improvement of complete coloring stops at
( )G
colors, which are the minimum colors it can use. The Vpc of RIC has an increase of 90%
over the complete coloring using ( )G colors when the WBAN density is larger than the middle density (25 Vertices) and the coloring number is one. Another comparison with RIC in Fig. 8 is the optimum Vpc in one coloring, which is a dual problem of finding the maximum independent set (MIS)8 in the given graph. Fig. 8 shows that RIC has near optimum Vpc at low and middle densities (12 and 25 vertices). Even at high and extremely high densities (50 and 100 vertices), RIC has a performance of around 90% compared with the MIS values.
8 Maximum Independent Set (MIS) is defined as the maximum set of pair-wise non-adjacent vertices in a graph.
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3.4.B Performance Evaluation of CPN-based IWS
Consequently, RIC is applied in a CPN-based IWS to observe how its high coloring-speed and high spatial-reuse improve IWS performance in mobile and dense WBAN scenarios.
Simulation Settings of Coloring-based IWS
A proposed CPN-based IWS adopts a time division multiple access (TDMA) with two distinct communication channels: inter and intra-WBAN channels. It helps to realize and compare different kinds of coloring-based IWSs. A CPN uses both channels for inter-WBAN resource contention and intra WBAN data (vital signals) collection respectively; a WSN only uses an intra-WBAN channel for data (vital signals) transmission. Following the steps of a CPN-based IWS in section 3.1, a CPN first uses the inter-WBAN channel to contend time slots through a coloring algorithm. The CPN then sends a beacon through an intra-WBAN channel to allocate obtained time slots to its WSN. Finally,
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the WSN transmits a data packet to its associated CPN through the intra WBAN-channel at the time slot that was pre-scheduled by the CPN.
The framing structure is illustrated in Fig. 3-9. A coloring cycle is performed for each superframe.
The intra-WBAN channel consists of one beacon slot and p data slots. Each CPN can reserve at most one slot after each coloring cycle. The number of data slots to be scheduled in each coloring cycle is set as k(equals to the number of colors used in the coloring). The duration of each data slot is 1ms. On the other hand, the inter WBAN channel consists of r slot-groups for r coloring rounds in a cycle. Each coloring round has q slots and is subdivided into (q k ) coloring slots and k winner notification slots, where k is the number of colors used for coloring. Coloring slots are used for exchanging coloring messages between CPNs (e.g. Fig. 3-4 step 2). The CPN chooses a coloring slot in each coloring round to transmit its coloring message. The coloring slot is randomly chosen to reduce potential collisions between coloring messages. Once a CPN wins the slot (color), it broadcasts the winner message at the associated winner notification slot (e.g. Fig. 3-4 step 4).
Because the coloring message exchanged across the inter-WBAN channel contains only color and random value information, the duration of coloring and winner notification slots are set as 25μs, which is 1
40 of the data slot at the intra-WBAN channel. In this study, we skip the steps of superframe synchronization between CPNs, which is beyond the scope of this study. Related work on superframe synchronization can be found in [32].
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Fig. 3-9 Superframe for coloring-based IWS
The settings of p(number of data slots in a superframe) and q (number of coloring slots in a coloring round) are two important control variables that dominate two kinds of data collision: (1) out-of-date scheduling and (2) ill-scheduling. Out-of-date scheduling happens when the frequency of the IWS cannot catch up to the frequency of topology changes due to user mobility. The data transmissions of WSNs scheduled by the out-of-date scheduling might collide with other transmissions from un-negotiated WBANs. Because IWS is performed for each superframe, such collisions are dominated by the duration of the superframe p. The second kind of collision is caused by ill-scheduling, which results from the collision of coloring messages (in short, coloring collision) while CPNs broadcast their coloring messages in the same coloring slot. Without correctly receiving coloring messages from adjacent CPNs, a CPN could make a mistake on treating itself as the slot (color) winner. As a result, data transmissions of WSNs belonging to different WBANs could be scheduled to the same data slot and then collide with each other. Since coloring slots are randomly chosen by each CPN, the larger the number of coloring slots, the lower the collision rate. The number
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of coloring slots is decided by the value q. To closely study both kinds of collisions, two types of settings are used: fixed-p and fixed-q. For the fixed-p model, p is set as the number of colors k
(coloring numbers). Thus, each coloring round at the inter-WBAN channel has
1 40
ms25 k k
q s r r , where r is the number of coloring rounds. As for the fixed-q model, we
choose ( , ) (1,5)k r to set the baseline size of q as 8 slots, which introduces the most serious coloring-message collision9 of RV INC . Thus, 8
5 r40 r
p . Because each k coloring provides k slots scheduling, it is more convenient to make p an integral multiple of qINC only . As a result,
RV INC
q is modified as q40kr and the coloring result repeats every INC only slots.
The generation of mobile WBAN topology is similar to the 2-D random graph. The initial positions of n WBANs are randomly located in a 10 10 m 2 square. n can be 12, 25, 50, 100 to simulate low, middle, high, and extremely high WBAN densities. The mutually-interfering-range of WBANs is set as 2 m. Also, to simulate WBAN mobility, the location change of WBANs follows the Gauss-Markov mobility model [33]. We use [33] to simulate the smooth movement path of a human, while avoiding the sudden stops and sharp turns that happen in the random walk mobility model [34].
The Gauss-Markov mobility model has a tuning factor to control the randomness of WBAN movement. is set as 0.3 in this study (0and 1 correspond to Brownian motion and linear motion respectively).
System throughput is the performance index used to evaluate IWS. Without loss of generality, system throughput is defined as effective transmissions per slot (Tps ), which counts data
9 Fig. 6, RIC can be finished within 5 rounds (r5). While r5,k1 is the setting that yields a minimum q from the equation q40kr, which introduces the most serious coloring-message
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transmissions of all WSNs that are actually received by CPNs in the system. Tps is similar to the vertex per color (Vpc) but further considers performance degradation caused by data collision.
System Throughput of IWS
Coloring speed is found to be the key factor that affects the performance of IWS. Fig. 3-10 compares the system throughput of IWS using the RIC and INC-only algorithms. INC-only can perform sub-( )G coloring but has a much higher time-complexity than RIC (see Fig. 3-6). First, in Fig. 3-10, ill-scheduling is temporarily ignored to make the observation of out-of-date scheduling much more clear. Because the collisions of coloring messages are ignored, the performances of RIC and INC-only in the fixed-p model are identical. Also, rRIC5 leads to pRICk, which makes the performance of RIC in fixed-p and fixed-q models equivalent. Fig. 3-10 shows that the fixed-p model overcomes the mobility much better than the fixed-q model. In the fixed-q model,
INC only 5
p r k
k
is prolonged by the high time-complexity (high number of r) of INC-only and
thus fails to respond in a timely fashion to topology changes. On the other hand, in the fixed-p model, p k is independent of coloring rounds r. Throughput is only slightly degraded while mobility is increased from 3m/s to 9m/s. Furthermore, in the fixed-p model, throughput is improved while the coloring number k is decreased. This improved throughput ought to lead to more out-of-date scheduling. Fortunately, the frequency of IWS is inversely proportional to the duration of the superframe (k1). Decreasing k increases IWS frequency to compensate for the collisions caused by throughput improvement.
collision.
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3m/s 12 WBANs Fixed-p (RIC / INC-only) Fixed-q (RIC) r=5 3m/s 50 WBANs Fixed-p (RIC / INC-only) Fixed-q (RIC) r=5 9m/s 12 WBANs Fixed-p (RIC / INC-only) Fixed-q (RIC) r=5 9m/s 50 WBANs Fixed-p (RIC / INC-only) Fixed-q (RIC) r=5 3m/s 12 WBANs Fixed-q (INC-only) r=r(k) 3m/s 50 WBANs Fixed-q (INC-only) r=r(k) 9m/s 12 WBANs Fixed-q (INC-only) r=r(k) 9m/s 50 WBANs Fixed-q (INC-only) r=r(k)
Note:
Fig. 3-10 Transmission per Slot (Tps) of IWS (ignoring ill-scheduling)
Now we focus on the fixed-p model. Ill-scheduling (coloring collision) is found to be the major factor that seriously degrades the performance of IWS. Fig. 3-11 illustrates the system throughput of IWS in the fixed-p model after considering both out-of-date and ill-scheduling collisions. It shows that RIC has a much higher Tps than INC-only. The reason comes from the substantially fewer coloring rounds of RIC than that of INC-only. rINC only might be larger than thousands of slots due to the inefficient re-coloring of INC-only, which makes qINC only much lower than qRIC (q40kr)
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and introduces serious collisions of coloring messages. For INC-only, 40 k ( )
q r k can be increased
by increasing k and decreasing ( )r k (coloring rounds ( )r k decrease while color choices k increase, as shown in Fig. 3-6). The coloring collision of INC-only is relieved when k is larger than 4 and 5 for the scenarios of 12 and 25 WBANs, respectively. However, increasing k also decreases Tps (referring to the decreasing Vpc in Fig. 3-7). For RIC, the tradeoff between coloring collision and Tps reduction yields optimum throughputs when k is 2 and 5 (instead of 1) for the 50 and
Fig. 3-11 System throughput of IWS with the Fixed-p Model
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3.5 Summary
In this work, random incomplete coloring (RIC) is proposed to realize a fast and high spatial-reuse inter-WBAN scheduling (IWS). Unlike conventional complete coloring schemes, RIC is not limited by the tradeoff between coloring speed and spatial reuse. RIC can always provide fast convergence with time complexity O e W(2ln ) 2n in any spatial reuse requirement. Furthermore, RIC can support an increase of up to 90% of spatial reuse over the conventional complete coloring using chromatic
( )G
-colors, which is known to be the optimal coloring of complete coloring. In the simulation, RIC is applied in a CPN-based IWS protocol with TDMA framing structure. Simulation results show that RIC does overcome inter-WBAN collisions and thus provides high system throughput for mobile wireless body area networks.
This study focuses on the scenario of random-user position, which is modeled as a 2-D random graph. In the future, we would like to analyze the performance of RIC in other special scenarios. For example, users in a waiting line, a movie theater, or a coffe bar. These scenarios can be modeled as a line, a grid, and a clustered graph, respectively.
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Chapter 4
Distributed Multiuser QoS Designs
Quality of Service (QoS) for medical applications is an emerging issue for wireless body area networks (WBAN) [1, 35]. To reliably transmit data streams of medical applications (e.g. vital signals or diagnosis audio / video), WBAN QoS is asked to meet more harsh requirements than those of other wireless networks in terms of transmission latency, packet error rate (PER), and energy consumption, as mentioned in [36-42]. Furthermore, WBAN QoS is featured by considering different critical levels of vital signals. For instance, electrocardiograms (ECG) are deemed to have more important information than body temperature to indicate the health status of a person, hence ECG signals are supposed to have higher priority than that of body temperature. Many centralized scheduling technologies of medium access control (MAC) layer have been proposed to support QoS for a single WBAN (single user) [36-42]. In these works, a central processing node (CPN) of a WBAN centrally schedules radio resources of wireless sensor nodes (WSNs) illustrated in Fig. 4-1.
These centralized controls can effectively meet various QoS requirements of vital signals. They also save energy consumptions of WSNs due to their light control loading of WSN in the CPN-centralized controls [42]. Nevertheless, some WBAN scenarios involve co-existence of multiple users, e.g. a hospital waiting room or a crowded subway station. Co-channel and co-location interference happens when WBANs move close to each other. It causes packet collisions and energy waste, which hence impact WBAN QoS. Besides, multiuser scenarios might need extra definitions of critical levels of medical data. The critical levels of vital signals might vary according to not only signal properties (like the ECGs v.s. Body temperature example in a single WBAN QoS) but also
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user status. For instance, vital signals of an injured person might need higher priority than that of a healthy one. Thus, inter-WBAN priority scheme would be necessary. As a result, a new challenge of multiuser QoS that considers above inter-WBAN issues is introduced. To the best of our knowledge, there is no existing works addressing on solutions of multiuser WBAN QoS so far. Comprehensive studies are still required.
Vital Signals CPN
WSN
WBAN WBAN
WBAN
Fig. 4-1 Wireless Body Area Network
QoS designs for overlapped wireless local area networks (WLANs) or Bluetooth piconets might be the closest problems to multiuser QoS. Jiang and Howitt [43] analyze load-balancing between co-channel and co-location (overlapped) WLANs. Access points (APs) properly share bandwidth according to an optimized load-balancing through backhaul (wire-line) communications. On the other hand, for overlapped piconets, the inter-piconet interference is overcome by interconnecting discrete piconets into a scatternet [44-49]. A scatternet (cross piconet) scheduling is thus applied to provide collision free transmissions among overlapped piconets. However, these approaches might not be suitable multiuser QoS solutions for several reasons. First, these approaches are originally designed for non-medical transmissions, which have less strict QoS requirements and lack priority schemes for medical data. Furthermore, the WLAN approach focuses on static or low mobility scenarios. Its backhaul optimization is only suitable for fixed wireless nodes, which is not possible to