Adv. Wireless Comm. Systems - Cellular Networks - Objectives Understand basic mathematical techniques to evaluate radio resource allocation problems Understand the principles of Fixed and Dynamic Channel allocations Outline Fundamental mathematical tools Fixed Channel Allocation for non-uniform traffic Dynamic Channel Allocation for non-uniform Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Traffic Model & Channel Utilization Generally, offered calls are not uniform across cells Thus, the capacity is no longer a good measure A measure for non-uniform traffic is Channel Utilization, U (4.1) - no. of busy channels at time t The channel utilization in cell i, , is a function of The offered calls process to the cell, and their duration there The no. of channels allocated to that cell Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The M/M/c/c Loss System - No. of channels - Call arrival rate Call arrivals - Call completion rate - Offered load (in Erlang units) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The M/M/c/c Loss System (cont.) Markov chain transition rates: c The stationary (equilibrium) Probabilities are obtained by solving the local balance equations: c j In our case: (4.2) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Erlang–B Formula For a Loss System with a Poisson arrival process and general i.i.d holding times, a call is blocked with probability c Truncated Poisson Distribution (4.3) (4.4) - System utilization (in Erlangs) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Utilization and Blocking Prob. vs. Offered Load c We need to tradeoff between utilization and blocking prob. - e - utilization (in Erlangs) 1 - blocking prob. offered load (in Erlangs) Rule 4.1: Given c, take the largest s.t. Rule 4.2: Given take the smallest c s.t. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Computation With a Large Number of Channels Definition: and are asymptotic equivalent ( denoted by ), if Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Computation With a Large Number of Channels Computing for large c & is numerically unstable We can use its asymptotic equivalent expression: (4.5) - the density function of a standard Normal distribution - the cdf of a standard Normal distribution Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Large Scale Benefit Increasing c while keeping fixed, yields: if (4.6) if Thus, scaling up the number of channels improves the utilization w/o exceeding the blocking prob. Threshold Conclusion: Sharing channels among cells (consolidation) is better than splitting them (separation) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Fixed Channel Allocation (FCA) For uniform traffic we know that channels per cell is best Question: How do we allocate channels using an FCA algorithm when traffic is non-uniform? Step 1: Given - the offered load to cell i - the maximum blocking prob. in cell i Calculate , the no. of channels required by cell i, , by using rule 4.2. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Fixed Channel Allocation (FCA) Note: For any practical system there are always some fixed amount of channels that are solely reserved for users in cell i. That is – they are not shared by other cells. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Fixed Channel Allocation (FCA) Step 2: Determine where the same channels can be reused How do we do that for a general cell layout ? We construct the Reuse Constraints Graph Given a cell layout, define an undirected graph, where each cell is represent by vertex of a graph each vertex pair is connected iff the same channel can’t be used both cells at the same time Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Reuse Constraints Examples Example 4.1: Cells in tandem Cell: 1 2 3 4 5 The graph for reuse distance = 1 Vertices: 1 2 3 4 5 The graph for reuse distance = 2 Vertices: 1 2 3 4 5 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Reuse Constraints Examples (cont.) 1 Example 4.2: Hexagonal cells in a single cluster 6 2 5 3 4 1 6 The graph for reuse distance = 1 5 2 4 3 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Reuse Constraints Examples (cont.) Example 4.3: Hexagonal cells in a 7 cluster layout – with reuse distance = 2 reuse distance = 1 Part of the edges … reuse distance = 2 Node degree = 6 + 12 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Finding The Minimum Bandwidth FCA The Optimal FCA Problem: Given a layout of n cells and their required no. of channels , What is the minimum total no. of channels, S, needed with Fixed Channel Assignment (FCA) ? The 1st algorithm we present is based on maximal cliques In many cases it finds the optimum – but not always! Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Maximal Cliques Definitions: A graph is complete if every pair of vertices is connected. A clique of a graph is a complete subgraph. A clique is maximal if it has the largest number vertices. For an algorithm see, e.g: D.R. Wood, “An algorithm for finding a maximal clique in a graph”, Operation Research Letters, Vol. 21, pp. 211-217, 1997. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Clique Examples This graph This graph 1 2 3 4 5 1 2 3 4 5 has 4 maximal cliques has 3 maximal cliques 1 2 2 3 1 2 3 3 4 4 5 4 4 5 2 3 4 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Clique Examples (cont.) 1 6 This graph 5 2 has 6 maximal cliques 4 3 1 3 5 2 4 6 2 4 6 3 5 1 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Clique Examples (cont.) Hexagonal cells in a 7 cluster layout with reuse distance = 2 x x x x x x x Each collection of 7 cells marked With x form a maximal click x x x x x x x Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Finding Small S for FCA using Maximal Cliques For every maximal clique K, S must satisfy: (4.7) Reuse Constraints Example 4.4: Hexagonal cells with one cluster and reuse distance 1 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Finding Small S for FCA using Max Cliques (cont.) Algorithm 4.1: Clique-Based, Cell-Independent, Traffic-Dependent FCA (CB-CI-TD FCA) Take Step 1 above to calculate the required , Take Step 2 above to form the Reuse Constraints Set S to (NP-Hard) For cells 1 to n, subsequently allocate channels by reusing channels that don’t violate the Reuse Constraints 4.1 If for some cell, the S channels are not sufficient, increase S by the missing amount Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Finding Small S for FCA using Max Cliques (cont.) Q: Why do we need step 4.1 of the CB-CI-TD FCA algorithm? A: There are for which ,S - the total req. no. A counter-example (1) (1) (1) But we need 1+1+1 = 3 channels (1) (1) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring For the case of , the (proper) coloring of the Reuse Constraint Graph yields the minimum S, namely, S = C x (Chromatic_number) For C=1, it is clear. Assume it holds for C-1 For C, each vertex cannot use neither its previous local colors (channels), nor its neighbors colors Thus, new Chromatic_number of channels are needed Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring (cont.) Example 4.5: 3-tier hexagonal cells with reuse distance = 1 Reuse Constraint Graph Cell layout Chromatic no. = 3 For C=1 we use 3 channels None of these can be reused for C=2 Given a new color in we must have 2 additional colors 3xC is the no. of colors for any n-tier hexagonal cells with reuse dist. = 1 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring (cont.) Example 4.6: n-tier hexagonal cells with reuse distance = 2 Reuse Constraint Graph Cell layout Chromatic no. = 7 For this reuse constraint graph Chromatic no. = Max Clique size Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring (cont.) How do we handle the case with non-uniform ? Does do the job ? For this case it does ! Indeed, 7 and 10 are a MUST 6 cannot be taken from the 7 and 10 (6) (10) (7) (10) (4) S= 10+7+6 = 23 (10) (2) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring (cont.) For this case however, it over shoots ! Indeed, 3 and 5 are a MUST For node y, 2 can be reused from node x that is far enough Giving a total of 3+5+7=15 (7) (3) 5+3+9 = 17 (3) (5) 7+2 = (9) Node x Node y (3) (3) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring (cont.) The answer is by using the weighted coloring algorithm Each vertex of the Reuse Constraints Graph is assigned the weight The weighted coloring problem (also called set coloring) is to color with minimum colors each vertex i with distinct colors so that adjacent vertices also receives distinct colors Can be represented as a (proper) graph coloring Replace vertex i with a fully connected graph with nodes Connect each new node to the previous connected nodes Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring (cont.) How about setting S to: = Max Clique number This may give us a too small number, since there are cases where, Chromatic number > Max Clique number Example: Chromatic # = 3 Max Clique # = 2 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic THE Min S for FCA using Weighted Coloring (cont.) Algorithms for weighted coloring problem C. McDiarmid and B. Reed, “Channel Assignment and Weighted Coloring”, Networks, Vol. 36, No. 2, pp. 114-117, 2000 A polynomial time algorithm that colors the graph with 4/3 times the minimum number M. Caramia and P. Dell’Olmo, “Solving the Minimum Weighted Coloring Problem”, Technical Report, University of Rome, Italy. A branch and bound algorithm capable of solving 90 vertices Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Finding The Min S for FCA using Weighted Coloring Algorithm 4.2: Weighted-Coloring-Based, Cell- Indep., Traffic-Dependent FCA (WCB-CI-TD FCA) Take Step 1 above to calculate the required , Take Step 2 above to form the Reuse Constraints Use a polynomial heuristic or an optimal algorithm (NP-Hard) to find the color sets (channels) for each cell i For cells 1 to n, allocate the channels found by the algorithm Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Benefit of Detailed Traffic-Dependent FCA Example 4.7: 5-tier hexagonal cells with reuse distance = 2 Question: By how much can we increase the network utilization with frequency allocation based on the detailed loads , , rather than based on the average load ? Network Data 49 cells in 5-tier hexagonal layout Reuse distance = 2 Total offered load = 490 Erlangs Blocking prob. = 0.05 4/7 of the cells have 3/7 of the cells have Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Benefit of Detailed Traffic-Dependent FCA Solution 1: The Uniform Traffic Suppose we use FCA based only on the average load The no. of channels required per each cell is resolved from C=15 The reuse constraints graph with C=1 is colored with 7 colors. Thus, a total of 7x15 = 105 channels is required The average utilization per cell is then Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Benefit of Detailed Traffic-Dependent FCA Solution 1: The Uniform Traffic - Summary Cell Utilization Frequency Allocation 15 15 15 15 15 x 15 15 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Benefit of Detailed Traffic-Dependent FCA Solution 2: The Non-uniform Traffic Solution Suppose we use FCA based only on the detailed loads and The no. of channels required for a cell with C=22 The no. of channels required for a cell with C=9 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Benefit of Detailed Traffic-Dependent FCA Solution 2: The Non-uniform Traffic Solution (cont.) The reuse constraints graph with the following mixture of 9 and 22 channel requirements is colored with (4x9)+(3x22) = 102 colors (channels) The cell utilization is then 22 9 9 9 22 x 22 9 Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Benefit of Detailed Traffic-Dependent FCA Solution 1 Vs. Solution 2 Solution 1 Solution 2 No. of Channels 105 102 Avg. Cell Util. 8.328 9.589 Conclusion Detailed traffic-dependent FCA improves the system utilization by more than 15% with 3 less channels Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic How To Best Use The Extra Channels ? Suppose we want to maximize the utilization The Erlang-B, as function C for any given If is the same – we gain more for lower C If C is the same - we gain more for lower 1 - blocking prob. c no. of channels 1 - blocking prob. offered load (in Erlangs) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic How To Best Use The Extra Channels ? To maximize utilization Gradually allocate each additional channel to a cell with the largest marginal gain To “equalize” blocking probabilities Gradually allocate each additional channel to cell with the largest blocking probability A systematic methodology to allocate available channels is presented below Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Cell Occupancy with FCA Suppose channels are allocated to cell i and the reuse constraints in (4.7) are not violated. If the mobiles stay put in their cells, the blocking prob. In each is cell i is , By cell independence, the stationary pdf of the cell occupancy vector is: Q: Find K (4.8) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA: Further Considerations – Limited Channels Usually, an operator is given a spectrum span (bandwidth) that determines the no. of available channels, S, to allocate On the other hand, traffic requirements and operator goal for blocking prob. determine the , Thus, for a given reuse constraints we may not get a feasible coloring with S colors Two resolution paths: Increase blocking prob. (less revenue for the operator) Relax the reuse constraints (lower quality service ) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA: Further Considerations – Adjacent Channel Interference Reuse constraints reflect only co-channel interference A lesser degree of interference to frequency f occurs by adjacent channels using a near frequency g typically: low |f - g| high interference Clearly, adj. chan. interfer. Corresponds only the edges of the reuse constraints graph Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA: Further Considerations – Adjacent Channel Interference Example The values denote the minimum frequency band separation distance with respect to Intra-cell channels are separated by 4 Cells at 1-tier away - by 2 Cells at > 1-tier away - by 0 Leads into a new set of constraints For any edge (v,w) in the reuse constr. 2 2 2 4 2 2 2 = the set of forbidden distances Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA: Further Considerations – Adjacent Channel Interference Up & down links used in the same cell may interfere Thus, must be further separated Assigned in two different bands sufficiently separated by up down Channels used to handle handover must also be separated In GSM, BCCH (broadcast channel) are used by two BTS to transmit to a mobile during its handover process Thus, BCCH channels in adjacent cells should be separated too Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA: Further Considerations – Adjacent Channel Interference The T - Weighted Graph Coloring Problem If two channels f and g, interfer or not depending on |f - g| It lends itself into the T- weighted coloring problem See the paper: F. S. Roberts, T-Colorings of graphs: Recent results and open problems, Discrete Mathematics 93 (1991), pp. 229-245. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA: Further Considerations – Restricted Channels European countries and competing operators within a country, often use the same system, e.g., GSM A further restriction exist on their boundary cells: They must be separated administratively Further restrictions may be imposed by the army & military Leads into a new set of constraints: For any vertex (cell) v = the set of feasible channels for cell v Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA – An Integer Linear Program Solution (ILP) Taking into account the following considerations of: Pre-assigned channels Intra and inter cell - channel separation distance Cell-dependent list of feasible channels Non-dichotomy of Interference Leads into a penalty function approach: For every edge (v,w) in the reuse constraints graph and a pair of allocation frequencies and , we penalize the choice by that depends on the interference level Bang-bang type penalty: if > ; 0 otherwise Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA – An Integer Linear Program Solution (ILP) Determining the penalty function User are mobiles not practical to use FCA for uplinks Downlinks are allocated first, then symmetric alloc. from the downlink band is used for the downlinks with a constant separation The penalty function is taken as a function of the SIRs (one/several locations & theoretical/measured) v w g f f g SIR BTS BTS SIR Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Maximum Service FCA using ILP - 1 if channel f is assigned to cell v; 0 otherwise - the no. of channels assigned to cell v - the no. of channels required by cell v Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Maximum Service FCA using ILP To avoid “unfair” allocation by the ILP, we may add the constraints Alternatively, we may replace the linear objective function by the convex function (and minimize) - the offered load weight of cell v Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Other FCA Optimization programs See the survey paper: K. I. Aardal, S. P.M. Van Hoesel, A. M.C.A. Koster,C. Mannino and A. Sassano, Models and Solution Techniques for Frequency Assignment Problems, ZIB-Report 01–40 (December 2001), Konrad-Zuse-Zentrum fur Informationstechnik, Berlin. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Dynamic Channel Allocation (DCA) The next question we address is: How much can be gained by DCA compared with FCA? With FCA, channels are not shared among the cells With DCA, a cell can borrow channels from other cells An architecture that supports handover can support DCA Channels are shared within a Base Station Controller (BSC) domain BSC BTS BTS Telephone Network MSC Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Dynamic Channel Allocation (DCA) The following “idealized” Maximum Packing (MP) DCA alg. is the most one may hope for Based on the paper D. Everitt and D. Manfield, “Performance analysis of cellular mobile communication systems with dynamic channel assignment”, IEEE JSAC, Vol. 7, no. 8, pp. 1172-1180, Oct. 1989. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Maximum Packing DCA Consider a network layout of M cells and S channels Define the Reuse Constraints graph MP Algorithm Upon every new call arrival to cell i at state Accept the new call if the reuse constraint graph can be weight-colored with S colors, where the weights are given by If accepted, re-arrange the channels accordingly Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP Example 4.8: 5-tier hexagonal cells with reuse distance = 2 Network Data 49 homogeneous cells in a 5-tier hexagonal layout Reuse distance = 2 Total of 350 channels Blocking prob. either 0.02 or 0.05 To model spatial traffic we take i.i.d with known and ,where is calibrated to give the required blocking prob. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Optimal FCA for Uniform Traffic Since cells are homogeneous the 7-color pattern is optimal provides 50 channels per cell Calibrating for the blocking prob. yields We use various to get different degrees of spatial traffic Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Experiment For every iteration draw the actual offered loads using For FCA, calculate the cell blocking prob. by the average , where Average over many iterations Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Experiment (cont.) Weighted coloring is computational complex so we use the following approx. for MP DCA Approximated MP Construct the set of Maximal Cliques (done only once!) When a call arrives to cell i in clique K at state : accept the new call iff Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Experiment (cont.) Remarks on the approximated MP For where the weighted-chromatic # < S, it is OK ! Otherwise, a call is mistakenly accepted ! (rear events!) The feasible state space of the Approximated MP are: (4.9) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Experiment (cont.) E.g., for the single cluster case the states are the vectors s.t. Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Experiment (cont.) For Poisson call arrivals and Exponential holding times the Markov chain is time reversible The stationary state distribution is therefore solved from the local balance equations: Truncated Poisson Distributions (4.10) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Experiment (cont.) The dist. Is not very practical for direct computation since finding the normalization constant is NP-complete However, for cases where the non-truncated distribution is know (as here), we can use an efficient technique called the “Monte-Carlo Simulation” Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic The Blocking Probability with MP The Experiment (cont.) Monte-Carlo simul. to compute the blocking prob. Of MP Every iteration, generate M independent values If one of the reuse constraints in (4.9) is violated – ignore the sample Otherwise, add to a counter, BC, the no. of cells in the cliques with Repeat for a large number of iterations, T MP_Block_Prob = BC / (49 x T) Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA vs. DCA (ref point of blocking prob. 0.02 ) 49 hexagonal cells with N=7 and S =350 channels. Blocking prob. produced with 200 independent spatial traffic distribution. Top curve: FCA Bottom curve: MP DCA operating with the same offered loads Middle curve: MP DCA dimensioned to higher load as to meet the same blocking prob. as in the uniform traffic Q: Why the top and bottom curves don’t coincide at variance 0? Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA vs. DCA (ref point of blocking prob. 0.05 ) 49 hexagonal cells with N=7 and S =350 channels. Blocking prob. produced with 200 independent spatial traffic distribution. Top curve: FCA Bottom curve: MP DCA operating with the same offered loads Middle curve: MP DCA dimensioned to higher load as to meet the same blocking prob. as in the uniform traffic Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic FCA vs. DCA – Experiment Conclusion DCA may significantly reduce the blocking prob. From 0.05 down to less than 0.005 and from 0.02 down to less than 0.002 DCA is significantly less sensitive to traffic variability compared with FCA The main benefit of DCA is that it is adaptive to the traffic Important since the offered loads are not known! Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -

Channel Allocation for Non-Uniform Traffic Other DCA Algorithm Families Maximal Packing is an instance of Traffic Adaptive DCA algorithms Other families of DCA algorithms are those that are: Adaptive to the received signal power Adaptive to interference power Lecture 4&5: Chan. alloc. for non-uniform traffic Adv. Wireless Comm. Systems - Cellular Networks -