Capacity and Scheduling in Small-Cell HetNets

Transcription

1 Capacity and Scheduling in Small-Cell HetNets Stephen Hanly Macquarie University North Ryde, NSW 2109 Joint Work with Sem Borst, Chunshan Liu and Phil Whiting Tuesday, January 13, 2015 Hanly Capacity and Scheduling 1 / 41

2 Talk Summary 1 Small Cells and Research Challenges Hanly Capacity and Scheduling 2 / 41

3 Talk Summary 1 Small Cells and Research Challenges 2 Model Hanly Capacity and Scheduling 2 / 41

4 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results Hanly Capacity and Scheduling 2 / 41

5 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program Hanly Capacity and Scheduling 2 / 41

6 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program 5 Cell Association and Scheduling Algorithms Hanly Capacity and Scheduling 2 / 41

7 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program 5 Cell Association and Scheduling Algorithms 6 Understanding the Converse Hanly Capacity and Scheduling 2 / 41

8 Small Cells and Data Offloading Re-use spectrum: make cells smaller Offload traffic from macro-cells onto pico and femto-cells Hanly Capacity and Scheduling 3 / 41

9 Small Cells: Theoretical Challenges What is the benefit? Base station densification gain? Hanly Capacity and Scheduling 4 / 41

10 Small Cells: Theoretical Challenges What is the benefit? Base station densification gain? Characterizing capacity Hanly Capacity and Scheduling 4 / 41

11 Small Cells: Theoretical Challenges What is the benefit? Base station densification gain? Characterizing capacity Optimizing resource allocation and cell association Hanly Capacity and Scheduling 4 / 41

12 System Model One macro Base Station (BS) L pico BSs C 4 C 1 All users in coverage of macro BS C l coverage area of pico BS l Power levels are fixed C 3 L=4 C 2 Macro BS uses higher power than pico BSs Hanly Capacity and Scheduling 5 / 41

13 Time Sharing and Cell Association Time Share Spectrum Macro Cell versus Pico Cells Almost Blanking SubFrames Hanly Capacity and Scheduling 6 / 41

14 Time Sharing and Cell Association Time Share Spectrum Macro Cell versus Pico Cells Almost Blanking SubFrames Cell Range Expansion for Picos Expand pico-cells to cover more mobiles Contract pico-cells and send at Higher Rate Hanly Capacity and Scheduling 6 / 41

15 Research Problems 1 How to split the time between macro and picos? 2 How to decide the cell association? Hanly Capacity and Scheduling 7 / 41

16 Research Problems 1 How to split the time between macro and picos? 2 How to decide the cell association? For cell association, one way is via biasing: add a bias to the measured power level to encourage offloading. Hanly Capacity and Scheduling 7 / 41

17 Research Problems 1 How to split the time between macro and picos? 2 How to decide the cell association? For cell association, one way is via biasing: add a bias to the measured power level to encourage offloading. We will address both 1. and 2. in a joint approach Hanly Capacity and Scheduling 7 / 41

18 Research Problems 1 How to split the time between macro and picos? 2 How to decide the cell association? For cell association, one way is via biasing: add a bias to the measured power level to encourage offloading. We will address both 1. and 2. in a joint approach We will discover biasing based on rate ratios, not power levels Hanly Capacity and Scheduling 7 / 41

19 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program 5 Cell Association and Scheduling Algorithms 6 Understanding the Converse Hanly Capacity and Scheduling 8 / 41

20 Arrivals in Space and Time λ a Α Files arrive at rate λ a files/slot (Poisson) η( ) gives a probability measure on the macrocell area Spatial arrival intensity is λ(dξ) = λ a η(dξ) η(α) λ(d ξ)=λ a η(d ξ) Hanly Capacity and Scheduling 9 / 41

21 Arrivals in Space and Time λ a Α η(α) λ(d ξ)=λ a η(d ξ) Files arrive at rate λ a files/slot (Poisson) η( ) gives a probability measure on the macrocell area Spatial arrival intensity is λ(dξ) = λ a η(dξ) nth arrival has length D n bits; E[D n ] = D Hanly Capacity and Scheduling 9 / 41

22 Arrivals in Space and Time λ a Α η(α) λ(d ξ)=λ a η(d ξ) Files arrive at rate λ a files/slot (Poisson) η( ) gives a probability measure on the macrocell area Spatial arrival intensity is λ(dξ) = λ a η(dξ) nth arrival has length D n bits; E[D n ] = D How large can λ a be? Hanly Capacity and Scheduling 9 / 41

23 Pico versus Macro time arrival initial wait slots Macro BS schedules files in macro-time All pico BSs can be scheduled simultaneously in pico-time t t+1 slots f t pico time 1 f t macro time Hanly Capacity and Scheduling 10 / 41

24 Pico versus Macro time arrival initial wait t t+1 slots slots Macro BS schedules files in macro-time All pico BSs can be scheduled simultaneously in pico-time A pico BS schedules files in its coverage area Not all files need be scheduled f t pico time 1 f t macro time Hanly Capacity and Scheduling 10 / 41

25 Pico versus Macro time arrival initial wait t t+1 slots slots Macro BS schedules files in macro-time All pico BSs can be scheduled simultaneously in pico-time A pico BS schedules files in its coverage area Not all files need be scheduled f t pico time 1 f t macro time We will consider only clearing schedules Clearing schedules include FCFS (one at a time) and PS (parallel processing) Hanly Capacity and Scheduling 10 / 41

26 Location based policies y π l (ξ, F ) bits x l π (ξ, F ) bits ξ(location) C l F bits Schedules determined via only the location ξ and the size F (bits) of the file Hanly Capacity and Scheduling 11 / 41

27 Location based policies y π l (ξ, F ) bits x l π (ξ, F ) bits ξ(location) C l F bits Schedules determined via only the location ξ and the size F (bits) of the file If π is such a scheduler, ξ C l, and F is the file size then: x π l (ξ, F ) bits are from pico BS l y π l (ξ, F ) are from macro BS Hanly Capacity and Scheduling 11 / 41

28 Location based policies y π l (ξ, F ) bits x l π (ξ, F ) bits ξ(location) C l F bits Schedules determined via only the location ξ and the size F (bits) of the file If π is such a scheduler, ξ C l, and F is the file size then: x π l (ξ, F ) bits are from pico BS l y π l (ξ, F ) are from macro BS If nth arrival is of size D n bits and located at ξ n then x π l (ξ n, F n ) + y π l (ξ n, F n ) = D n Hanly Capacity and Scheduling 11 / 41

29 Data rates R l (ξ) bits/slot ξ S (ξ) bits/slot C l Assume no interference between picocells (will relax later) So there is no interference in the system! Hanly Capacity and Scheduling 12 / 41

30 Data rates R l (ξ) bits/slot ξ S (ξ) bits/slot C l Assume no interference between picocells (will relax later) So there is no interference in the system! At any point ξ there is a macro-cell rate of S(ξ) bits/slot At any point ξ C l there is a pico-cell rate of R l (ξ) bits/slot Hanly Capacity and Scheduling 12 / 41

31 Data rates R l (ξ) bits/slot ξ S (ξ) bits/slot C l Assume no interference between picocells (will relax later) So there is no interference in the system! At any point ξ there is a macro-cell rate of S(ξ) bits/slot At any point ξ C l there is a pico-cell rate of R l (ξ) bits/slot A file can be served by macro and pico BSs in the same slot Hanly Capacity and Scheduling 12 / 41

32 Buildup of work arrivals slots V π 20 (ω) Fix a location-based policy π and outcome ω Ω. Let VT π (ω) be time needed to clear all files that arrive in [0, T ] Hanly Capacity and Scheduling 13 / 41

33 Buildup of work arrivals slots V π 20 (ω) Fix a location-based policy π and outcome ω Ω. Let VT π (ω) be time needed to clear all files that arrive in [0, T ] Clearly π is NOT stable if lim inf T on an event of nonzero probability. V π T T > 1 Hanly Capacity and Scheduling 13 / 41

34 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program 5 Cell Association and Scheduling Algorithms 6 Understanding the Converse Hanly Capacity and Scheduling 14 / 41

35 Recall main parameters λ a η (dξ) = λ (dξ), η (dξ) spatial intensity of arrivals R l (ξ), S l (ξ) Rates for pico and macro at location ξ x l (ξ), y l (ξ) bit assignments at location ξ D mean download file size Hanly Capacity and Scheduling 15 / 41

36 Continuous Linear Program Consider the following continuous LP: min τ = f + sub L l=1 xl (ξ) R l (ξ) λ (dξ) f x l (ξ) + y l (ξ) D, where f represents pico-time. Let τ be the optimal value of the program. yl (ξ) S l (ξ) λ (dξ) l Hanly Capacity and Scheduling 16 / 41

37 Sufficiency Theorem (Hanly, Whiting) Let τ be optimal solution to the LP. If τ < 1, a clearing schedule π with ergodic properties. Also define S π n (ω) := sojourn time nth job, then π satisfies, E [S π n (ω)] < S < (1) Hanly Capacity and Scheduling 17 / 41

38 Converse Theorem (Hanly, Whiting) Let τ be the solution to the continuous LP. Suppose that τ > 1 then there is a fixed constant η > 0, such that for any clearing schedule π lim inf T V π T (ω) T = 1 + η almost surely. Hanly Capacity and Scheduling 18 / 41

39 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program 5 Cell Association and Scheduling Algorithms 6 Understanding the Converse Hanly Capacity and Scheduling 19 / 41

40 Discrete Model 3 4 R 3, n S 3,n 3, n Let instantaneous rate of nth user provided by pico BS be denoted by R n Instantaneous rate of user provided by macro BS be denoted by S n ρ 3, n = R 3, n S 3, n Hanly Capacity and Scheduling 20 / 41

41 Discrete Model 3 4 R 3, n S 3,n 3, n 0 ρ 3, n = R 3, n S 3, n 1 2 Let instantaneous rate of nth user provided by pico BS be denoted by R n Instantaneous rate of user provided by macro BS be denoted by S n The rate ratio is defined by ρ n = Rn S n Hanly Capacity and Scheduling 20 / 41

42 Problem Formulation y 3, n bits Let D n denote the amount of bits of data required by a user. D n can be split into x n bits of data from pico BS and y n bits of data from the macro BS. x 3,n bits R 3,n S 3,n 3, n 0 D l, n =x 3, n + y 3,n bits Hanly Capacity and Scheduling 21 / 41

43 Problem Formulation x 3,n bits R 3,n y 3, n bits S 3,n 3, n 0 D l, n =x 3, n + y 3,n bits Let D n denote the amount of bits of data required by a user. D n can be split into x n bits of data from pico BS and y n bits of data from the macro BS. It therefore requires xn R n secs from pico BS and yn S n secs from macro BS The problem is to minimize the total time to satisfy all the data demands in the network. Hanly Capacity and Scheduling 21 / 41

44 Problem Formulation: General The problem to be solved is the following linear program: min f + sub N l n=1 L N l l=0 n=1 x l,n R l,n f y l,n S l,n l x l,n + y l,n D l,n l, n = 1, 2,... N l f 0, x l,n 0, y l,n 0 l, n = 1, 2,... N l where f is the time allocated to the picocells. Hanly Capacity and Scheduling 22 / 41

45 A One Dimensional Formulation decreasing rate ratio m j =4 Pico cell j N j =6 Linear Programming theory tells us that the rate ratios ρ := R S are the key to the optimal cell association Order the users in pico cell in decreasing order of the rate ratio Hanly Capacity and Scheduling 23 / 41

46 A One Dimensional Formulation decreasing rate ratio m j =4 Pico cell j N j =6 Linear Programming theory tells us that the rate ratios ρ := R S are the key to the optimal cell association Order the users in pico cell in decreasing order of the rate ratio Then there will be a user m j in pico cell j such that users 1, 2,..., m j 1 will be 100 % served by the pico cell BS (y j,n = 0 for these users) users m j + 1, 2,..., N j will be 100 % served by the macro cell BS (x j,n = 0 for these users) user m j may get its service from both base stations (pico and macro) Hanly Capacity and Scheduling 23 / 41

47 A One Dimensional Formulation (time allocated to pico BSs) n D j, n R j, n f D j,1 R j, 1 + D j, 2 R j, 2 D j,1 served by pico j p 1cm j (f) served by macro R j, n j (f) m j (f) =5 6 n j Hanly Capacity and Scheduling 24 / 41

48 A one dimensional formulation (time allocated to pico BSs) n D j, n R j, n f D j,1 R j, 1 + D j, 2 R j, 2 D j,1 served by pico j p 1cm j (f) served by macro R j, n j (f) m j (f) =5 The macro-cell time required to service users users near pico cell j is g j (f ) = p j (f ) D N j,m j j (f ) D j,n + S j,mj (f ) S j,n n=m j (f )+1 6 n j Hanly Capacity and Scheduling 25 / 41

49 A one dimensional formulation The problem is therefore to minimize the following function of the scalar parameter f : L f + g l (f ) l=1 Hanly Capacity and Scheduling 26 / 41

50 A one dimensional formulation The problem is therefore to minimize the following function of the scalar parameter f : L f + g l (f ) The range for the optimization is 0 f max L Nl D l,n l=1 n=1 S l,n. l=1 Hanly Capacity and Scheduling 26 / 41

51 A one dimensional formulation g j ( f ) slope = R j,1 S j, 1 = ρ j, 1 N j D j, n n=1 S j, n N j D j, n n= 2 S j,n slope = ρ j,2 slope = ρ j, N j 1 D j, N j 1 + D j, N j S j, N j 1 S j, N j D j, N j slope = ρ j, N j S j, N j D D j,1 + D N N j,1 j, 2 j 1 D j j, n R R j, 1 R n=1 n=1 j, 1 j, 2 R j, n f D j, n R j, n Hanly Capacity and Scheduling 27 / 41

52 Edge rate condition Recall that the problem is to minimize the function f + L l=1 g l(f ) Hanly Capacity and Scheduling 28 / 41

53 Edge rate condition Recall that the problem is to minimize the function f + L l=1 g l(f ) The derivative at any non-break-point f is therefore 1 L l=1 ρ l(f ). Hanly Capacity and Scheduling 28 / 41

54 Edge rate condition Recall that the problem is to minimize the function f + L l=1 g l(f ) The derivative at any non-break-point f is therefore 1 L l=1 ρ l(f ). But the optimum will occur at one of the break points, where the derivative changes. Hanly Capacity and Scheduling 28 / 41

55 Edge rate condition Recall that the problem is to minimize the function f + L l=1 g l(f ) The derivative at any non-break-point f is therefore 1 L l=1 ρ l(f ). But the optimum will occur at one of the break points, where the derivative changes. edge rate condition: The time allocation f to pico-cells is optimal if and only if L L ρ l,f 1 l=1 l=1 ρ l,f + Hanly Capacity and Scheduling 28 / 41

56 Edge rate condition Recall that the problem is to minimize the function f + L l=1 g l(f ) The derivative at any non-break-point f is therefore 1 L l=1 ρ l(f ). But the optimum will occur at one of the break points, where the derivative changes. edge rate condition: The time allocation f to pico-cells is optimal if and only if L L ρ l,f 1 l=1 l=1 ρ l,f + We only need to check the 1 + L l=1 N l break points, where the derivative changes, for the edge-rate condition. Hanly Capacity and Scheduling 28 / 41

57 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program 5 Cell Association and Scheduling Algorithms 6 Understanding the Converse Hanly Capacity and Scheduling 29 / 41

58 Recall theorem for existence of schedule Theorem (Hanly, Whiting) Let τ be optimal solution to the LP. If τ < 1, a clearing schedule π with ergodic properties. Hanly Capacity and Scheduling 30 / 41

59 Recall theorem for existence of schedule Theorem (Hanly, Whiting) Let τ be optimal solution to the LP. If τ < 1, a clearing schedule π with ergodic properties. The optimal solution of the continuous LP is characterized by rate ratio thresholds ρ l, l = 1, 2,..., L: x l (ξ) = { D ρl (ξ) > ρ l 0 o.w. (2) We will now show that these thresholds can be used to construct an optimal schedule. Hanly Capacity and Scheduling 30 / 41

60 Construction of a stabilizing schedule t t+1 slots f * τ * (τ * f * ) pico time τ * macro time Suppose τ < 1 and let f be the optimal value of f from continuous LP. Allocate f τ of each slot to picos. Allocate τ f τ of each slot to the macro. Assign each file to pico or macro based on rate-ratio threshold ρ l. Hanly Capacity and Scheduling 31 / 41

61 Construction of a stabilizing schedule merged arrivals slots All files that arrive during a slot are merged into one job Service time of each job can be computed (location-based policy) Each BS serves jobs in FCFS order D/G/1 queue at each base station. Hanly Capacity and Scheduling 32 / 41

62 Construction of a stabilizing schedule The workload arriving at the macro BS queue can be shown to be τ f slots/slot The service rate of the macro BS is τ f τ slots/slot So the utilization of the macro BS is τ Hanly Capacity and Scheduling 33 / 41

63 Construction of a stabilizing schedule The workload arriving at the macro BS queue can be shown to be τ f slots/slot The service rate of the macro BS is τ f τ slots/slot So the utilization of the macro BS is τ The workload arriving at a pico BS can be shown to be at most f slots/slot The service rate of each pico BS is f slots/slot So the utilization of each pico BS is at most τ τ Hanly Capacity and Scheduling 33 / 41

64 Construction of a stabilizing schedule The workload arriving at the macro BS queue can be shown to be τ f slots/slot The service rate of the macro BS is τ f τ slots/slot So the utilization of the macro BS is τ The workload arriving at a pico BS can be shown to be at most f slots/slot The service rate of each pico BS is f slots/slot So the utilization of each pico BS is at most τ τ If τ < 1 then each D/G/1 queue is stable. Hanly Capacity and Scheduling 33 / 41

65 Talk Summary 1 Small Cells and Research Challenges 2 Model 3 Main Stability Results 4 Discrete Linear Program 5 Cell Association and Scheduling Algorithms 6 Understanding the Converse Hanly Capacity and Scheduling 34 / 41

66 Understanding the converse N 4 =10 arrivals slots n=5 th arrival (x n,y n ) bits Suppose τ > 1, and let π be a clearing schedule. Let N T be the number of arrivals during [0, T ]. Let x n be number of pico-cell bits for the nth arrival Let y n be number of macro-cell bits for the nth arrival Hanly Capacity and Scheduling 35 / 41

67 Understanding the converse Imagine all the N T arrivals being present at time zero 4 1 S 3,n 3 R 3, n 3, n 0 2 ρ 3, n = R 3, n S 3, n Hanly Capacity and Scheduling 36 / 41

68 Understanding the converse 4 1 Imagine all the N T arrivals being present at time zero Let VT LP be the minimum time in the corresponding discrete LP S 3,n 3 R 3, n 3, n 0 2 ρ 3, n = R 3, n S 3, n Hanly Capacity and Scheduling 36 / 41

69 Understanding the converse 4 1 S 3,n R 3, n 3, n Imagine all the N T arrivals being present at time zero Let VT LP be the minimum time in the corresponding discrete LP We can analyze this and show V that lim inf LP T T T > 1 + η ρ 3, n = R 3, n S 3, n Hanly Capacity and Scheduling 36 / 41

70 Understanding the converse N 20 arrivals slots f π 20 (ω) = time when a pico is sending to at least one of N 20 arrivals Now consider the actual system under policy π Let ft π be the total time in which at least one of the N T files is getting pico-service Discrete Linear Program min f + l sub. n x l,n n R l,n f y l, n S l,n x l,n + y l,n D l,n l l, n Hanly Capacity and Scheduling 37 / 41

71 Understanding the converse N 20 arrivals slots f π 20 (ω) = time when a pico is sending to at least one of N 20 arrivals Discrete Linear Program min f + l sub. n x l,n n R l,n f y l, n S l,n x l,n + y l,n D l,n l l, n Now consider the actual system under policy π Let ft π be the total time in which at least one of the N T files is getting pico-service Then N T VT π = ft π y n +, S n n C l n=1 x n R n f π T Hanly Capacity and Scheduling 37 / 41

72 Understanding the converse N 20 arrivals slots f π 20 (ω) = time when a pico is sending to at least one of N 20 arrivals Discrete Linear Program min f + l sub. n x l,n n R l,n f y l, n S l,n x l,n + y l,n D l,n l l, n Now consider the actual system under policy π Let ft π be the total time in which at least one of the N T files is getting pico-service Then N T VT π = ft π y n +, S n n C l n=1 x n R n f π T So (f π T, x n, y n,...) is feasible for the earlier discrete LP Hanly Capacity and Scheduling 37 / 41

73 Understanding the converse Hence lim inf T V π T T which implies that π is unstable. V T LP T > 1 + η Hanly Capacity and Scheduling 38 / 41

74 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved Hanly Capacity and Scheduling 39 / 41

75 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Hanly Capacity and Scheduling 39 / 41

76 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Assumption that statistics are known can be relaxed Hanly Capacity and Scheduling 39 / 41

77 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Assumption that statistics are known can be relaxed The assumption that there is no pico-cell interference can be relaxed: Assume fixed power levels when pico scheduled to be on Hanly Capacity and Scheduling 39 / 41

78 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Assumption that statistics are known can be relaxed The assumption that there is no pico-cell interference can be relaxed: Assume fixed power levels when pico scheduled to be on Many different modes: subsets of pico BSs that are simultaneously activated Hanly Capacity and Scheduling 39 / 41

79 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Assumption that statistics are known can be relaxed The assumption that there is no pico-cell interference can be relaxed: Assume fixed power levels when pico scheduled to be on Many different modes: subsets of pico BSs that are simultaneously activated The pico rates at a location are mode-dependent Hanly Capacity and Scheduling 39 / 41

80 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Assumption that statistics are known can be relaxed The assumption that there is no pico-cell interference can be relaxed: Assume fixed power levels when pico scheduled to be on Many different modes: subsets of pico BSs that are simultaneously activated The pico rates at a location are mode-dependent Problem remains a continuous LP, but with a huge number of variables Hanly Capacity and Scheduling 39 / 41

81 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Assumption that statistics are known can be relaxed The assumption that there is no pico-cell interference can be relaxed: Assume fixed power levels when pico scheduled to be on Many different modes: subsets of pico BSs that are simultaneously activated The pico rates at a location are mode-dependent Problem remains a continuous LP, but with a huge number of variables Good but suboptimal schemes can be found focusing on the most important modes Hanly Capacity and Scheduling 39 / 41

82 Extensions Given stabilizing schedule is NOT dynamic and can greatly be improved We can analyze Processor Sharing at each BS which gets much better delay performance Assumption that statistics are known can be relaxed The assumption that there is no pico-cell interference can be relaxed: Assume fixed power levels when pico scheduled to be on Many different modes: subsets of pico BSs that are simultaneously activated The pico rates at a location are mode-dependent Problem remains a continuous LP, but with a huge number of variables Good but suboptimal schemes can be found focusing on the most important modes Extensions to multiple macro-cells Hanly Capacity and Scheduling 39 / 41

83 Applications Theory can be used as a basis for the search for adaptive algorithms Can be used as a cell planning tool since capacity is a function of base station locations Hanly Capacity and Scheduling 40 / 41

84 Concluding Remarks Formulated a notion of capacity for a dynamic HetNet consisting of one macrocell, multiple picocells Hanly Capacity and Scheduling 41 / 41

85 Concluding Remarks Formulated a notion of capacity for a dynamic HetNet consisting of one macrocell, multiple picocells Characterized capacity in terms of the solution of a deterministic, continuous Linear Program Hanly Capacity and Scheduling 41 / 41

86 Concluding Remarks Formulated a notion of capacity for a dynamic HetNet consisting of one macrocell, multiple picocells Characterized capacity in terms of the solution of a deterministic, continuous Linear Program Showed how ABS slot and cell association can be solved jointly Hanly Capacity and Scheduling 41 / 41

87 Concluding Remarks Formulated a notion of capacity for a dynamic HetNet consisting of one macrocell, multiple picocells Characterized capacity in terms of the solution of a deterministic, continuous Linear Program Showed how ABS slot and cell association can be solved jointly Showed how rate ratio thresholds provide the right way to bias pico-cells Hanly Capacity and Scheduling 41 / 41

88 Concluding Remarks Formulated a notion of capacity for a dynamic HetNet consisting of one macrocell, multiple picocells Characterized capacity in terms of the solution of a deterministic, continuous Linear Program Showed how ABS slot and cell association can be solved jointly Showed how rate ratio thresholds provide the right way to bias pico-cells Questions? Hanly Capacity and Scheduling 41 / 41