Wireless Network Design: Optimization Models and Solution ...

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8 The Design of Partially Survivable Networks 189 existing link between a cell and the hub (ti j) is generated from a uniform distribution U(100,500). The random number generator used is from Knuth [7]. Ten problem instances were generated in each of the (8 × 4 × 3) = 96 groups. Table 8.1 provides the results from applying the solution procedure, for the high, medium and low capacity levels. |N| is the number of cells and T is the number of time periods. The columns under each of the capacity categories represent the average number of variables generated as a fraction of the total (in %) and the average gaps (in %), where the gaps are obtained as ( UB−LB LB ) × 100, with UB being the objective function value corresponding to the best integer feasible solution obtained after spending a maximum of two minutes on branch and bound, and LB being the linear programming relaxation value of (M2). The variables and gaps reported are averages over the 10 data sets of each type. The code was implemented using the LINDO Callable Library [12] on an Intel Pentium II running at 450 MHz. IP branch and bound was terminated after a maximum of two minutes of execution. The gaps between the column generation LP and the integer solution obtained by switching on the integrality requirements are low in general and decrease with problem size. The low gaps suggest that the linear programming relaxation of formulation (M2) is reasonably tight. In most of the realistic cases (|N| > 20), the gaps are extremely low (well below 1%). This suggests that a branch and price approach is not necessary to obtain near-optimal solutions. Even in cases where the LP-IP gaps were high (essentially when |N| = 10), it was noted that this was a result of the LP bound being relatively weak for the small problems. In general, as the problem size got larger, the performance of the column generation approach improved. This could be partly due to the fact that more time is spent in linear programming column generation, which increases the number of good quality columns generated. The fraction of variables generated decreased with problem size, as would be expected. Conditioning on the number of cells, the gaps increased marginally with the number of time periods. 8.4 The Single Period Problem with Capacity Restrictions When capacities are introduced at the hubs, the problem becomes more difficult. Kubat et al. [9] present a problem of designing interconnect networks for a cellular system that incorporate reliability requirements in the face of capacity constraints, where the backbone has a tree topology with the MTSO at the root. The links in the tree have known capacities, and it is possible for two hubs with distinct logical channels to the MTSO to share the same physical link. In order to provide increased reliability, it is also desirable to have a cell connect not just to multiple hubs, but to multiple subtrees that do not have any physical links in common between the hubs and the MTSO. This is particularly relevant when the backbone topology is a tree, as the failure of even one high capacity line could affect the entire network. As before, the number of nodes (hubs + MTSO) on distinct subtrees to which a cell is required to be connected is its diversity requirement. The objective is to find a least-cost cell-
190 Syam Menon |N| T High Capacity Medium Capacity Low Capacity Var (%) Gap (%) Var (%) Gap (%) Var (%) Gap (%) 10 3 43.33 3.919 45.00 3.374 45.28 3.031 10 5 17.26 5.493 17.67 3.506 19.31 3.144 10 10 1.05 5.247 1.15 3.749 1.21 2.753 10 15 0.05 7.314 0.06 4.732 0.06 3.640 20 3 43.40 1.342 43.89 1.260 43.82 1.118 20 5 15.57 1.235 15.10 0.996 15.26 0.882 20 10 0.93 1.781 0.97 1.356 1.00 1.102 20 15 0.04 2.552 0.04 2.048 0.05 1.012 50 3 39.22 0.107 42.06 0.266 40.69 0.308 50 5 14.44 0.183 13.38 0.187 13.78 0.304 50 10 0.76 0.285 0.75 0.347 0.79 0.265 50 15 0.03 0.524 0.03 0.381 0.04 0.260 100 3 40.07 0.039 39.49 0.047 40.50 0.073 100 5 13.57 0.103 13.44 0.076 13.24 0.045 100 10 0.66 0.354 0.69 0.384 0.67 0.252 100 15 0.03 0.634 0.03 0.553 0.02 0.583 150 3 39.70 0.024 39.11 0.020 37.42 0.026 150 5 13.27 0.044 12.43 0.034 12.35 0.022 150 10 0.63 0.257 0.62 0.230 0.63 0.157 150 15 0.02 0.510 0.03 0.545 0.03 0.314 200 3 36.62 0.012 35.44 0.009 34.69 0.014 200 5 12.89 0.022 12.65 0.011 12.04 0.019 200 10 0.62 0.188 0.63 0.164 0.59 0.063 200 15 0.02 0.404 0.02 0.304 0.02 0.251 250 3 36.97 0.007 36.81 0.004 36.15 0.008 250 5 12.19 0.020 12.03 0.011 11.43 0.015 250 10 0.59 0.176 0.57 0.097 0.57 0.051 250 15 0.02 0.367 0.02 0.370 0.02 0.287 300 3 32.69 0.005 32.00 0.006 31.63 0.006 300 5 12.20 0.008 11.94 0.010 10.98 0.006 300 10 0.56 0.188 0.56 0.161 0.56 0.033 300 15 0.02 0.304 0.02 0.216 0.02 0.168 Table 8.1 Multi-period Column Generation Results to-hub interconnection network that satisfies the diversity requirements and capacity restrictions. Figure 8.3 shows an example network. Hubs a – c are connected by a high capacity line to the MTSO while v – z are the cells which need to be connected. Hub b is on the path connecting hub c to the MTSO and as a consequence, no cell can
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International Series in Operations
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Editors Jeff Kennington Eli Olinick
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vi Preface assistance. The work of
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viii Contents 3.4.1 Experiments to
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x Contents 8.3.2 The Solution Proce
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xii Contents References . . . . . .
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List of Contributors Khaldoun Al Ag
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List of Contributors xvii Nitin Sal
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Chapter 1 Introduction to Optimizat
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1 Introduction to Optimization in W
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1 Introduction to Optimization in W
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Part I Background
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10 Dinesh Rajan The goal of this ch
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12 Dinesh Rajan The source encoder
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14 Dinesh Rajan ping (FH) CDMA, and
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16 Dinesh Rajan of SNR is given in
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18 Dinesh Rajan 2.3 Information The
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20 Dinesh Rajan Perror ≤ e −nE
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22 Dinesh Rajan For instance, for t
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24 Dinesh Rajan 2.4.1 Block Codes I
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26 Dinesh Rajan B(z) = 1 � (1 + z
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28 Dinesh Rajan layers. Two of the
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30 Dinesh Rajan nel can support a p
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32 Dinesh Rajan 2.5.2 Physical laye
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34 Dinesh Rajan matched filter for
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36 Dinesh Rajan codes) leads to bet
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38 Dinesh Rajan a length N Inverse
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40 Dinesh Rajan the transmission of
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42 Dinesh Rajan setting, a node may
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44 Dinesh Rajan weighting of the si
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46 Dinesh Rajan 29. Oestges, C., Cl
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48 K. V. S. Hari terrain of operati
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50 K. V. S. Hari fading is calculat
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52 K. V. S. Hari terms have been pr
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54 K. V. S. Hari Fig. 3.1 Normalize
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56 K. V. S. Hari Table 3.3 Impulse
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58 K. V. S. Hari where m antennas a
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60 K. V. S. Hari 3.5.2.4 Dominant R
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62 K. V. S. Hari March, 1984-13 Sep
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64 K. V. S. Hari 47. Van Rees, J.:
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66 J. Cole Smith and Sibel B. Sonuc
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Part II Optimization Problems for N
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102 Eli Olinick carrying capacity.
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104 Eli Olinick 93 85 160 16 38 21
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106 Eli Olinick Amaldi et al. [7] p
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108 Eli Olinick levels (possibly le
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110 Eli Olinick gmℓ ∑ ∑ m∈M
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112 Eli Olinick Using bk to denote
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114 Eli Olinick 5.3.5 Additional De
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116 Eli Olinick capacity, provide n
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118 Eli Olinick ficult optimization
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120 Eli Olinick and-bound so that t
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122 Eli Olinick solution times. Cai
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124 Eli Olinick 27. Glover, F.: Tab
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Chapter 6 Optimization Based WLAN M
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6 Optimization Based WLAN Modeling
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10 Integer Programming Models for P
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Chapter 11 Improving Network Connec
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11 Improving Network Connectivity i
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Chapter 12 Simulation-Based Methods
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12 Simulation-Based Methods for Net
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296 Hang Jin and Li Guo formance in
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298 Hang Jin and Li Guo (ISI) as lo
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300 Hang Jin and Li Guo ARQ (HARQ)
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302 Hang Jin and Li Guo 13.3 Radio
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304 Hang Jin and Li Guo Table 13.2
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306 Hang Jin and Li Guo is transmit
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308 Hang Jin and Li Guo uplink powe
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310 Hang Jin and Li Guo The link ad
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312 Hang Jin and Li Guo where ρ mi
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314 Hang Jin and Li Guo 13.4.4.3 Ex
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316 Hang Jin and Li Guo Fig. 13.9 P
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318 Hang Jin and Li Guo 1. Guarante
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Chapter 14 Cross Layer Scheduling i
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14 Cross Layer Scheduling in Wirele
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Chapter 15 Major Trends in Cellular
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15 Major Trends in Cellular Network
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