Minimum-Cost QoS-Constrained Deployment and Routing Policies for Wireless Relay Networks

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in NLOS to provide up to 75Mb/s data rate with average coverage radius of 4-5 miles.   [7]. By cooperative relaying, exploiting two basic features of wireless propagation: its broadcast nature and its ability to achieve diversity through different channels, the spatial diversity benefits would be obtained both in eliminating multipath fading and aggregating gains. Thus, the reliability of communications in terms of, for example, bit error probability or outage probability will be improved in a given transmission rate [8], [9]. From the aspect of network architecture, the routing path of each OD pair (BS to SS or contrary) is no longer a conventional single path but evolves into a multipath structure in the wireless network composed by one BS, several relay stations (RSs), and many mobile clusters (MCs), which is a brand new issue and both the RSs deployment decision policies and routing algorithms would be very complex and interesting. We use MC to represent a group of mobile users gathering at the same location.

Motivation
The strongest motivation of this paper which is different from conventional network design problems is: we allow multiple source nodes jointly transmit one single 4 information if the signal strength is not robust enough in the link between one source node to the destination. The routing policy is no longer a single path but more complex and interesting multipath algorithms.
When building wireless network architecture with IEEE 802.16j standard in a metropolis, for example, Taipei city, there are many considerations and decisions may affect the result of design, such as QoS requirements, throughput requirements, and total cost. The objective of network designers will always be "to minimize the total building costs under the guarantees to users' requirements of QoS and throughput". Nonetheless, this objective is obviously a tradeoff because of the total building costs will increase if the QoS and throughput requirements get higher. Base on this conventional tradeoff, we further take the multipath routing algorithm into consideration, which is the most critical problem in IEEE 802.16j environment.

Figure 1 -1 Left: visualization of a beamformer [10]; right: MIMO
There are more researchers put emphases on relays and cooperative relaying techniques, the total capacity and reliability can be enhanced through different diversity skills. Antenna technologies, as well, keep improving for the purpose of spatial diversity and multipath fading diminutions, such as beamforming and multiple-input and multiple-output (MIMO), are the physical implements of diversity gain (Figure 1-1).
Most of the researches are focus on one stage cooperative relaying, to maximize the throughput [11] or to minimize the outage probability [9], [12]. In our work, the feature "a wireless node can broadcast to several destinations and receive from several sources" is exploited. We get into the network development problem with multistage scenarios ( Figure 1-2), from which multipath routing problems are derived. We try to plan a wireless network architecture with cooperative transmissions to minimize the total development costs while ensure the QoS and throughput requirements are satisfied.

.1 Relay
Multihop wireless networking have been widely studied and implemented in the ad hoc networks and mesh networks exploit the user diversity concept to improve performances. The early concept of general relaying problems is in [13], [14] and were inspired by the development of the ALOHA system at the University of Hawaii. The relay channel model is comprised of three terminals: a source that transmits information, a destination that receives information, and a relay that both receives and transmits information in order to enhance communication between the source and destination. [15] There are topological differences between the concept of relay network and mesh network. In relay network, it is designed to be a tree based topology and one end of the path is the BS; in mesh network, there are multiple connections and forms a mesh topology. Figure 1-3 represents the differences between relay and mesh networks.  7 Relays are designed to improve the coverage of a BS and overcome the shadows caused by obstacles. The scenario of relays deployed in a city with one BS is illustrated in Figure 1-4, where four relays covers the area around the corner that shadowed from the BS in the cross of the streets (area within oblique lines).
R R R R

Figure 1 -4 City scenario of relays deployment with one BS
Relays have cost efficiency because of the following features: 1. The diameter of a relay is much less than a BS which means that the transmit power is significantly reduced compare to a BS.
2. The location of a relay is not necessary as high as BS which reduce the deployment and maintenance cost.
3. Relays do not have a wired connection to the backhaul. Instead, they receive signal from BS and retransmit to destination users both wirelessly, and vice versa. The leases of wired broadband backhauls can be saved.
There are three types of relay protocols according to their forwarding strategy [16]: 1. Amplify-and-Forward: Relays act as analog repeaters by retransmitting an amplified version of their received signals. The noise is amplified as well.

Decode-and-Forward:
Relays attempt to decode, regenerate and retransmit the same information from the original source, the propagating decoding errors may occur. In the meantime, the compatibility for multihop (>2) environment which is the fundamental scenario of this paper are proposed to be an optional function and cooperatively relaying techniques where the tree-based routing algorithms in our work originate from are considered into the standard as well [17], [18].

Diversity Techniques and MRC (soft handoff)
There are many different types of diversity that have been employed in various systems. Among these are [19] Frequency diversity, transmitting or receiving the signal at different frequencies; In this paper, a signal is received from more than one source at different locations which utilize the space diversity practically. Since the signal is transmitted to the destinations by the relays COOPERATIVELY, the spatial diversity in this paper can be specified as cooperative diversity.
Cooperative diversity [20] is a relatively new class of spatial diversity techniques that is enabled by relaying [13], [14] and cooperative communications [21], [22] more generally. Cooperative communications are distributed radios jointly transmit information in wireless environments. The main motivation of cooperative diversity is to improve the reliability of communications in terms of, for example, outage probability, or symbol-or bit-error probability, for a given transmission rate [15].
Models with multiple relays have been examined in [20].
There are many researches focus on the performance measurement of different relaying algorithms like [9], [16]; in [23], the author derived an algorithm to find out the "best" end-to-end path between source and destination, among several possible relays to achieve approximately the same diversity gains as conventional cooperative diversity.
Besides the reliability, using space-time coding techniques can further improve the spectral efficiency of transmissions [24].
In a broad sense, there are three major diversity signal-processing techniques, and they are known as selection diversity (SD), equal gain combining (EGC) and maximal ratio combining (MRC) [19]. Regarding to our work, we exploit the spatial diversity gains through MRC techniques to achieve the optimal aggregate SNRs values.
MRC, first discussed by Brennan [25], is the optimal form of diversity combining because it yields the maximal SNR achievable, and is widely used in soft handoff in CDMA cellular system and wireless broadband access network, illustrated in Figure  1-6.

Figure 1 -6 Maximal ratio combining in softhandoff
MRC is an optimal, parallel receiver combining technique that aligns the phase of the carriers in two receiver chains (beam steering) and provides gain in proportion to the individual receiver's signal amplitude and in inverse proportion to the individual receiver's noise power [19], [25], [26], [27]. To make MRC work, designers need two (or more) receivers, phase shifters, and variable gain amplifiers demonstrated in Figure   1-7.

Figure 1-7 MRC architecture
For receivers having statistically independent noise, MRC provides the best composite signal-to-noise ratio (SNR) at the output. In MRC, composite SNR can be computed using: combined I SNR SNR

= ∑
Where I SNR represents the individual receiver's output signal to noise ratios.

Proposed Approaches
We model the wireless relay deployment problem as a mixed integer and linear programming (MILP) problem. In the optimization problem, we minimize the total development cost for network designer subject to relay selection constraints, nodal and link capacity constraints, cooperatively relaying constraints, and routing constraints for both UL and DL transmissions. We propose the Lagrangean relaxation method, in conjunction with the optimization-based heuristics to solve the problem.

Thesis Organization
The reminder of this thesis is organized as fellows. A mathematical formulation for the wireless relay networks design problem is first shaped in Chapter 2. In Chapter 3, the Lagrangean relaxation of the problem and the solution approaches for the Lagrangean sub-problems are presented. In Chapter 4, heuristics are developed for calculating good primal feasible solution of each problem. Chapter 5 is the computational results for the problems. In Chapter 6, we present our conclusions and indicate the direction of the future works.

Problem Description
The sequence of the wireless relay network design is described as fellow: First of all, there must have estimations of where the location of each BS is and how many BSs can fully cover the area to be served. Secondly, the set of BSs roughly divide the whole network into several subnetworks, and each of which is rooted by one BS to the wired network. Meanwhile, there are many candidate locations where are suit to deploy relays.
The decision whether to deploy a relay on a specific location is depending upon the users around the location; and once a location is selected, the relay must associate with one of the BSs mentioned above. Where to develop relays and how many relays should be developed will directly lead to the total costs of the whole network.
Another important factor of the wireless relay network design problem is rooting algorithm. As mentioned in chapter 1, for the broadcast nature and the feature to receive from different sources, there will be multipath routings. In our work, we introduce a multicast-tree-based routing algorithm (MTBR) to apply the multipath concept, which is represented below.

Figure 2 -3 One OD pair routing multicast tree in UL transmission
In this paper, we try to find a near optimal RSs development policy to minimize the total development costs; meanwhile, maintaining both DL and UL spanning trees and using MTBR to ensure the BER and data rate requirements of each MC must be satisfied. Whether or not a location should be selected to build a RS

Problem Formulation
The cooperative RSs of each MC The routing paths of an OD pair (a BS to an MC or contrary), which form a multicast tree from the BS to the cooperative RSs selected by each MC The data rate required to be transmitted of MC n in direction dir in (packets/sec) The fix cost of building an RS on location r The configured cost of building RS r , which is a function of configuration k M An arbitrarily large number The maximum SNR can be received by The maximum SNR can be received by The BER requirement for the transmission received by a destination in direction dir where the destination in DL is

MC and in UL is BS
The nodal capacity of RS r in (packets/sec), which is a function of configuration k ( ) uv C SNR The capacity of link uv in (packets/sec), which is a function of the receiving SNR of node v , where , The maximum spatial diversity of a MC in direction dir The summation of SNR received by MC n in DL ns ε The summation of SNR received by node s in UL oriented

Subject to:
Relay selection constraints Nodal capacity constraints Cooperative relay constraints

Explanation of objective function:
The objective function (IP) is to minimize the total cost of RSs deployment: (1) The fixed cost of RS building such as land acquisition and hardware purchases.
(2) The configuration cost of each RS.
Explanation of constraint:

1) Relay assignment constraints:
Constraint (2.1) requires that each location selected to install a RS with exactly only one configuration or none.
Constraint (2.2) requires that each RS can associate with one BS or none in direction dir .
Constraint (2.3) indicates that once an RS r associates with a BS, r must be built.

2) Nodal capacity constraints:
Constraint (2.4) requires that each RS's total amount of traffic in DL and UL cannot be greater than its nodal capacity.
Constraint (2.5) requires that each BS's total amount of traffic in DL and UL cannot be greater than its nodal capacity.

3) Cooperative relay constraints:
Each MC will select an RS r in direction dir only if r is installed in (2.6).
A MC must select either one BS or RS(s) in direction dir in (2.7). Constraint (2.28) requires that if link uv is on the path p adopted by the MC n to reach RS r in direction dir , then dir nuv y must be 1.

5) Link Capacity Constraints:
The aggregate flow of link uv in direction dir is restricted in (2.29).

Introduction to the Lagrangean Relaxation Method
Lagrangean relaxztion method is a general solution approach for solving mathematical optimization problems, and is used to decompose such problems to exploit their special structure. The decomposition can efficiently reduce the complexities and difficulties of the primal problem so that it becomes the one of the most popular methods for solving optimization problems. It can be utilized in many mathematical problems like integer programming, linear programming, and non-linear programming problems. Especially in large-scale mathematical programming applications, Lagrangean relaxation has significant performance and effectiveness.
We can release the constraints and place them in the objective function with associated Lagrangean multipliers instead. The optimal value of the relaxed problem is

Subproblem 2 (related to decision variables
Subject to: Subject to: The algorithm to optimally solve (Sub 3.3.2) is illustrated below: For DL: Step 1. Use SNR function to calculate the SNR value 1 bn π from every BS to MC n .
Step 2. Find the BS b which can result in the smallest coefficient ( ) 3  That is, we at least had the smallest coefficient for further steps, whether it is negative or not.
Step 4. If the coefficient of 1 nb κ in Step 2 is smaller than the summation of coefficient
Subject to: The algorithm to optimally solve (Sub 3.5.2) is illustrated below: For DL: Step  Step 2. For each MC n , if the coefficient ( ) We can calculate the value of (Sub 3.6.1) by examining every n ω exhaustively. Set n ω while it can result in the smallest value of (Sub 3.6.1). Here we applied the interval Δ to be 0.01. Subject to:

Subproblem 7 (related to decision variable
The same situation likes Subproblem 7, we introduced constraint ( Similar to (Sub 3.6.1), (Sub 3.7.1) can be solved by exhaustively examining nv ε to find out the smallest value of this problem, then set nv ε . Here we applied the interval Δ to be 0.01.

The Dual Problem and the Subgradient Method
According to the algorithms proposed above, we could effectively solve the Lagrangean relaxation problem optimally. Based on the weak Lagrangean duality theorem, the objective value of  15  16  17  18  19  20  21  22 , , , , , , , ) μ μ μ μ μ μ μ μ is a lower bound of IP Z . We construct the following dual problem to calculate the tightest lower bound and solve the dual problem by using the subgradient method.

Lagrangean Relaxation Results
By applying Lagrangean Relaxation method and the subgradient method to solve the complex problem, we can get a theoretical lower bound of the primal problem and some hints to get a feasible solution to the primal problem. Because some difficult constraints of the primal problem are relaxed by using Lagrangean Relaxation method, we cannot guarantee that the consolidated result of the Lagrangean Relaxation problem is feasible to the primal problem. We have to ensure that it is a feasible solution, which is satisfied with all constraints of the primal problem, if not, we have to make some modifications.

Getting Primal Feasible Heuristics
To obtain primal feasible solutions for (IP 1), solutions to the Lagrangean
Step Step 1. Among all built RSs, find the amount of dir SD RSs sorted in Sorting_Coef(), then run following procedures:

Build_RS_Algorithm
Step 1. Build one unbuilt RS r with the priority in Sorting_Coef() Step 2. If SNR of link rn with the highest conf. of r is less than min N P , return false, else return true

Min_BER_Algorithm
Step 1. Find a nearest RS r to n and build with highest conf.
Step 2. Find a minimum BER shortest path from r to b among all RSs with highest conf.
Step 3. Run CheckCapacityofNode(); CheckCapacityofLink(); CheckLinkAmount(); if violate these three constraints, eliminate r or uv from the candidate nodes and links, then repeat Step 1 to Step 3 to find a feasible path.
Step 4. Step 1. Find a minimum BER shortest path from r to b .
Step 2. Check the BER value of each link uv on the path, if larger than dir BER , upgrade the conf. of the source node. If the conf. is full, utilize the links selected by dir nuv y to find a minimum BER shortest path between u and v . If still can't meet the BER requirement, find a minimum BER shortest path between u and v among all RSs.
Step 3. For each link and node selected in Step 1 and Step 2, run CheckCapacityofNode(); CheckCapacityofLink(); CheckLinkAmount(); if violate these three constraints, eliminate r or uv from the candidate nodes and links, then repeat Step 1 to Step 3 to find a feasible path.
Step 4. Build the unbuilt RSs selected in Step 2.

CheckCapacityofNode Algorithm
Check the node capacity if the traffic of n is taken into consideration.
If the capacity of b is full, then switch n to the second nearest BS.

CheckCapacityofLink Algorithm
Check the link capacity if the traffic of n is taken into consideration.

CheckLinkAmount Algorithm
Check whether the amount of links on the path is equal to others.
In the following, we show the complete algorithm for solving (IP 1).

begin
Initialize the Lagrangean multiplier vector

Chapter 5 Computational Experiments
In this chapter, we conduct several computational experiments to examine how good of the quality of our solution approach. In the meantime, for the purpose of evaluating the solution quality, we implement two simple algorithms for comparison.

Minimum BER Algorithm (MBA)
The major concept of MBA is, for each MC n , to find the best paths that can generate the smallest BER value n receives in DL and BS b receives in UL. This algorithm will provide every transmission the best QoS (minimum BER).
Step 3. Repeat Step 2 dir SD times to find cooperative RSs and paths.
Step 4. Run MRC() to find the minimum BER multicast trees.
Step 5. Repeat Step 2 to Step 4 to satisfy all MCs transmissions (DL for the first time).
Step 6. Repeat Step 2 to Step 5 to satisfy UL transmissions.

Homing_BS_Algorithm
Step 1. Home each RS to the nearest BS.
Step 2. Home each MC to the nearest BS.

Minimum_BER_Path_Algorithm
Step 1. Find the nearest RS r .
Step 2. Among all RSs, find the minimum cost shortest path to the nearest BS b .
Step 3. Build all RSs selected in Step 1 and Step 2 in highest configuration.
Step 4. For each RS r or link uv selected in Step 1 and Step 2, run CheckCapacityofNode(); CheckCapacityofLink(); CheckLinkAmount(); if violate these three constraints, eliminate r or uv from the candidate nodes and links, then repeat Step 1 to Step 3 to find a feasible path.

MRC_Algorithm
Step 1. For each cooperative RS r , calculate the total BER including BER on the path from BS to r and BER with summation SNRs from cooperative RSs.
Step 2. If total BER is reduced by introducing r , then include r in the cooperative RS set.
Step 3. Repeat Step 1 and Step 2 dir SD times to find a minimum BER value.

CheckCapacityofNode Function
Check the node capacity if the traffic of n is taken into consideration.
If the capacity of b is full, then switch n to the second nearest BS.

CheckCapacityofLink Function
Check the link capacity if the traffic of n is taken into consideration.

CheckLinkAmount Function
Check whether the amount of links on the path is equal to others.

Density Based Algorithm (DBA)
The major concept of DBA is to build a RS with the priority of the unhoming MC density under its coverage. The coverage is the longest distance that RS can reach a MC with highest configuration.
Step 2. For each BS, sort all RSs by the density of unhoming MCs under its coverage.
Step 3. Build one RS r from the sorted list with highest configuration.
Step 5. Repeat Step 4 dir SD times to find cooperative RSs and paths.
Step 7. Repeat Step 2 to Step 6 to satisfy all DL and UL transmissions.

Feasible_BER_Path_Algorithm
Step 1. For each MC under the coverage of r , find a minimum BER shortest path among all built RSs through the nearest RS.
Step 2. Run CheckCapacityofNode(), CheckCapacityofLink(), CheckLinkAmount() in 5.1.1; if violate these three constraints, eliminate r or uv from the candidate nodes and links, then repeat Step 1 to Step 2 to find a feasible path.

MRC_Algorithm
Step 1. If the total BER of the first cooperative path n receives satisfies the BER constraint, then break; else take the second cooperative path into consideration.
Step 2. Repeat Step 1 dir SD times, if total BER still cannot satisfy the BER constraint, remain n to be unhoming.

Experiment Environments
The computational experiments programs are written in C++ and executing on a Pentium IV 3.0GHz, 3144MB ram, Windows 2000 Service Pack 4 environment. Table   5-1 shows the general parameters and test platform of the experiments. Modulation and Coding (AMC) applied in 802.16j is illustrated specifically in Table 5-3 with the same reference to "Mobile WiMAX".

Experiment Designs
For the specialties of this network deployment problem, the given circumstance including BS and MC locations, and RS candidate locations, has no built RS at the beginning. The word "topology" we introduced below represents the geographic distribution (the position) of the locations where we are able to build an RS on.
We propose two topologies, grid and random, with different numbers of RS and MC in one BS environment to analyze the impacts on deployment cost. Then we apply 70 different numbers of RS and MC with two BSs in a random topology to analyze the deployment in multiple BSs environment. Table 5-5     In grid topology, for a given networkscale, the farthest locations from BS to receive signals under BER threshold should be included in the candidate RS locations to reach the minimum cost objective. We can observe this phenomenon in Figure 5-4 and    In random topology, it is difficult to generate a feasible network to satisfy every MC's transmission when RS number is small (ex. RS= 8). In general, the RSs locations won't distribute uniformly enough to fully cover all MCs. The RS locations in Table 5-7 are one successful trial to make the network feasible.   The same persecution with random topology in BS= 1, it is difficult to get a feasible network when RS number is small (ex. RS= 16 here). We still put one successful set in Table 5-8 for examinations. Figure 5-10 illustrates the total deployment costs in random topologies with BS= 1 and BS= 2. Since the RS locations are different in both conditions, it is meaningless to compare the costs between them. But is it still obvious that the gaps are all larger in every scenarios in BS= 2 than BS= 1 for the network complexity.

Future Works
802.16j is a new technology with new specifications, therefore, there are many aspects regarding 802.16j worth to be studied. Different kinds of diversity techniques, like time diversity or frequency diversity, may be implemented on it with the supports of hard wares.
In this research, only path loss attenuation is considered into the model. For more precisely describe the physical network environment, different fading model shall be involved to complete the transmissions of every links.
Finally, different performance matrices, like delay or throughput, can be considered in the model. Based on cooperative relaying techniques, space-time coding [31], [32] can be utilized to enlarge the transmitting bandwidth. Recently, similar coding schemes are developed continuously for different purposes, thus cooperative relaying has more possibilities to enhance the communications in many aspects.