Design Aspects of Satellite-Cellular Hybrid Wireless Systems Tamer ElBatt Information Sciences Lab HRL Laboratories, LLC Malibu, CA 90265, USA

Anthony Ephremides * ECE Dept. University of Maryland College Park, MD 20742, USA

[email protected]

[email protected]

Key words: Satellite-Cellular Hybrid Systems, Channel Partitioning, Call Assignment, Frequency Reuse, Cell Size, Dynamic Call Re-Assignment, Optimization.

Abstract In this paper, we investigate various issues related to the design of satellite-cellular hybrid systems. First, we review the fundamental problems of channel partitioning and call admission/assignment. Second, we study the impact of dierent frequency reuse constraints, in both layers, on the optimum channel partitioning. Third, we investigate, analytically and via simulation, the eect of reducing the cell size. We emphasize the blocking-forced termination probabilities trade-o for pure cellular and satellite-cellular hybrid systems. Accordingly, an optimization problem with respect to the cell size is formulated. Finally, we search for the optimum dynamic call re-assignment policy that "This work was supported by the Center for Satellite & Hybrid Communication Networks, a NASA Commercial Space Center (CSC) at the University of Maryland, under NASA Cooperative Agreement NCC3-528." 

improves the system capacity at the expense of the complexity associated with tearing down a connection in one system and setting-up an alternative one in the other system. For a small hybrid system, we characterized the optimum re-assignment policies that minimize the blocking probability, dropping probability, and a weighted cost function of these probabilities.

1 Introduction In this paper, we investigate several issues related to the design of satellite-cellular hybrid wireless systems. Such networks are expected to enhance performance without additional wireless resources because: 1) The space segment provides a shared pool of resources and 2) The frequency reuse patterns achievable in the space segment can be much denser. The reason for this is two fold; rst, the interference conditions in mobile satellite systems are dierent from its terrestrial counterpart due to the presence of the on-board antenna that acts as a spatial lter [1]. Thus, in satellite systems, co-channel interference is a consequence of the presence of sidelobes in the on-board antenna radiation pattern. Second, the development in antennae technology and careful evaluation of the propagation eects support the above hypothesis [2]. In this paper, we address various problems related to the design of such networks when the trac carried is connection-oriented, typically voice calls. First, we study the fundamental problems of channel partitioning and call admission/assignment. In addition, we investigate the impact of dierent frequency reuse constraints in the two layers. Afterwards, we investigate the performance gains and tradeos associated with reducing the cell size. Due to the shared capacity advantage in the space segment, we may allow dynamic re-assignment of on-going calls between the two segments of the hybrid system. Therefore, the last part of this paper is dedicated to searching for the optimal re-assignment policy. Various techniques have been introduced in the literature to improve the capacity of terrestrial cellular networks. Commonly used methods include: 1) Reducing the frequency reuse factor via cell sectorization; 2) Reducing the cell size and 3) Hierarchical cell layout 2

[3], with larger macro-cells overlaying a collection of small micro-cells. Satellite-cellular hybrid networks can be considered a generalization of hierarchical cellular networks where the highest layer in the hierarchy is a satellite system. This, in turn, imposes a new set of design constraints due to the vastly dierent propagation characteristics, power requirements, and delay constraints in both systems. The problem of partitioning the bandwidth between micro-cells and macro-cells was previously addressed in [4]. However, the call assignment policy was assumed to be speed-dependent. Performance analysis of a satellite- cellular hybrid system has also been studied [5]. However, the reuse pro le for the satellite system was assumed to be the same as the terrestrial system. Moreover, the call assignment policy employed therein is a special case of the class policies considered in this paper. The work in section 3 builds upon earlier work [6] in which users' mobility, hand-os, and call assignment policies were not considered but, rather, only a static split of the total bandwidth into a terrestrial and a satellite component. In [7, 8] the authors emphasized that smaller cells reduce the new call blocking probability. However, nothing was mentioned about its impact on hand-o failures. It is evident that the hand-o rate increases as the cell size is reduced [9], while it is not quite clear how the dropping probability behaves. Therefore, we study analytically and via simulation the impact of reducing the cell size on hand-o failures in satellite-cellular hybrid systems. Dynamic call re-assignment is, to some extent, related to the classical channel reassignment problem that received considerable attention in the literature[10, 11, 12, 13]. The common aspect among the two problems is the re-assignment of on-going calls to dierent types of channels. On the other hand, the context and motivation of the two problems are quite dierent. Dynamic call re-assignment in satellite- cellular hybrid systems is relatively a new problem that needs further attention. In [6], a heuristic call re-assignment policy was proposed, where it was shown to improve the system capacity compared to the reference system (with no re-assignment capability). In this paper, we search for the optimum call re-assignment policy that balances a trade-o between the blocking and dropping probabilities. 3

The paper is organized as follows: In section 2, the assumptions are introduced and the system model is described. The joint channel partitioning-call assignment optimization problem is addressed in section 3. Extending the study to incorporate the eect of dierent frequency reuse factors in both layers is done in section 4. The cell size optimization problem is investigated in section 5. Section 6 is devoted to searching for the optimal call re-assignment policy for small hybrid systems. Finally, the conclusions are drawn in section 7.

2 Assumptions The network under consideration consists of N cells overlaid by m spot-beams covering the same area as shown in Figure 1. We assume that the hybrid system supports connection-oriented services (e.g. voice calls). The stochastic processes governing the system dynamics are assumed to be memoryless. Thus, new calls are assumed to arrive at each cell, Ci, according to a Poisson arrival process with rate  calls/min. The duration of each call is exponentially distributed with mean 1/ min. The inter-hando time for a mobile in cell Ci is exponentially distributed with mean 1/h min. This is done in order to facilitate modeling the system as a multi-dimensional birth-death Markov process. In this study, we consider "channelized" hybrid systems, i.e. either FDMA or TDMA with orthogonal (non-overlapping) components. Throughout this paper, we use the generic term "resources" to indicate either FDMA channels or TDMA slots. We de ne Ks as the satellite reuse factor, that is the number of spot-beams per cluster, where all the spotbeams in a cluster use distinct resource sets. Likewise, de ne Kc as the cellular reuse factor. Accordingly, the total number of \pre-design" resources available to the system, denoted M, is fully allocated to any cluster as shown below, M = Ki=1c Mci + Kj =1s Msj where, Mci = number of resources dedicated to cell Ci. Msj = number of resources dedicated to spot-beam Sj . 4

The base stations could communicate via either a terrestrial wireline connection or a wireless gateway located on the border between the cells. Using either assumption does not have any impact on the validity of the mathematical model adopted, it only aects the shape of the state space boundaries. Thus, in section 3, we adopt the wireless gateway Satellite

Spot Beam Out−of−Area Coverage

Footprints Base Stations

Terrestrial Network

Figure 1: A Hybrid Wireless System assumption to accommodate areas that do not support xed infrastructure. Later in section 4, a wireline interconnection of base stations is assumed in order to model multi5

cell/multi-spot beam hybrid systems. A mobile user can access the satellite directly (not through its Base Station (BS)) using a dual mode satellite-cellular mobile terminal[14]. All call types have the same priority and all calls considered in the model are mobile-tomobile calls. It is straightforward to extend the model to take mobile-to- xed calls into consideration. However, we believe that it does not aect in a substantial way the tradeos addressed in this paper. Base stations and spot-beams are assumed to be stationary. We de ne f as the fraction of calls originating in one cell and destined to another cell. When mobiles reach cell/spot-beam boundaries, hand-os could be either intra-layer or inter-layer. In this study, we restrict call hand-os to be intra-layer. This assumption is motivated by the technical diculties associated with tearing down a connection in one system and setting up an alternative one in the other system in a timely manner. Finally, we assume that blocked calls are cleared.

3 Joint Channel Partitioning and Call Assignment Optimization In this section, we focus on a simple 2 cell/1 spot-beam system as shown in Figure 2. We intentionally limit our attention to this simple context in order to gain insights and better understand the trade-o between the satellite and the terrestrial resources. This system is modeled as a 4-dimensional continuous time Markov chain, where the state x is de ned as x = (n11 , n12 , n22, ns), and nij is the number of calls of type ij; that is, calls served by BSi and BSj , where one of the parties is in Ci and the other is in Cj , and ns is the number of calls served by the satellite S. As mentioned earlier, we assume the two base stations, BS1 and BS2, communicate via a wireless gateway located on the borders between the cells. Accordingly, a mobile-tomobile call originating in a cell and destined to the other cell needs 4 duplex channels if served by the cellular network. Moreover, we assume that Kc = 2. Therefore, the set of 6

S

Gateway

BS1

BS2

C1

C2

Figure 2: A Hybrid Mobile Network of 2 Cells and 1 Spot-beam feasible states should satisfy the following constraints: 2n11 + 2n12  Mc1

(1)

2n22 + 2n12  Mc2 2ns  Ms

3.1 Call Admission/Assignment Policies In this section, we address the fundamental problem of assigning an incoming call to the cellular network or to the satellite. As mentioned earlier, all call types considered in this study have the same priority. Thus, according to [15], the class of Complete Sharing admission policies turns out to be the optimum. Accordingly, we limit our search here within this class of policies. In addition, we limit the search process to the class of "randomized" call assignment rules due to the computational complexity associated with searching for the optimum decision at each state, specially for large state spaces. The class of randomized rules is characterized as follows: any call is assigned, according to the outcome of a coin ipped, either to the cellular network or to the satellite provided 7

that they both have available resources. The call is blocked if and only if no resources are available in both layers. The call assignment probabilities can be chosen to depend on the call types, since dierent call types generally have dierent channel requirements and hand-o rates. Accordingly, a call of type ij is assigned to the cellular network with probability Pij and to the overlaying satellite with probability (1 ; Pij ). This allows more degrees of freedom in the optimization problem. The question that arises next, is how the call assignment is performed at the state space boundaries where one of the two layers is lled-up while the other still has resources. Clearly, the assignment rule has to be switched to a deterministic one to make full use of the available resources. Two switching boundaries are considered; switching at the state space boundaries, and switching at threshold boundaries [16].

3.2 Problem De nition and Solution Approach Our main objective is to determine the channel partitioning policy (Mc1 ; Mc2 ; Ms) and call assignment probabilities (P11 ; P12; P22) that solve the following multi-objective optimization problem [17]: min

P11 ;P12 ;P22 ;Mc1 ;Mc2 ;Ms

s.t.

(Pb + 1 :Pd + 2 : )

M = Mc1 + Mc2 + Ms 0  P11  1 0  P12  1 0  P22  1

(2)

where, Pb = average new call blocking probability, Pd = average hand-o dropping probability, = average percentage of satellite calls in the system, (this represents the consideration of the satellite propagation delay in the optimization problem), Pij = probability of assigning a call of type ij to the cellular system, 8

1 and 2 are weighting factors that are chosen to re ect the desired relative importance of the dierent components in the cost function. In the above formulation, the choice of the parameters 1 and 2, is rather unguided since there is no well-de ned procedure for choosing them. We employ, instead, an equivalent formulation which consists of minimizing one component of the composite cost function subject to the other components staying below pre-determined acceptable thresholds. Due to the sheer complexity of the overall problem, we consider simpler sub-problems each of which addresses one aspect of the original problem. Thus we attempt to solve the following sub-problems:

 Determine Mc , Mc , Ms (channel partitioning) for a given assignment rule. 1

2

 Determine P11 , P12 , P22 (assignment rule) for given Mc , Mc , Ms. 1

2

 Re ne the assignment rule by optimizing over threshold boundaries. The formulation of the aforementioned sub-problems can be solved via discrete exhaustive search [18]. First, we attempted solving the global balance equations numerically, but it was found to be extremely complex for even modest sizes of the problem. This is specially aggravated by the lack of a product-form solution [19]. Therefore, a simulation process was developed using C++ and run on Sun-Ultra architectures. To increase the eciency of the simulation we employed the so-called Standard Clock (SC) simulation[20]. This is a simulation procedure that permits simultaneous measurement of performance of multiple dierent control policies with a single simulation run. The bene ts of SC simulation are two fold, simulation time is signi cantly reduced and fair comparison is guaranteed due to the correlation among dierent sample paths generated using the same random seed. Moreover, as we are more interested in the relative ranking of policies, rather than their actual performance values, and to further speed up the simulations, Ordinal Optimization [21] was employed. This is an optimization methodology that determines the control policies that perform relatively well compared to other candidate policies without necessarily obtaining accurate estimates of their performance values. In this work, 9

ordinal optimization based on short simulation runs was implemented. Of course, there was no apriori assurance that ordinal optimization would be applicable to our problem. Fortunately, our results showed that it applies very satisfactorily, since the ordinal ranking of the dierent candidate policies under fast and ecient simulations was, to a great extent, the same as the exact (accurate) ranking (determined via long simulations).

3.3 Numerical and Simulation Results The network shown in Figure 2 was analyzed assuming the numerical parameters given in Table 1. Considering the rst sub-problem, the optimum channel partitioning for a given Total Pre-design Bandwidth (M) Call Arrival Rate per Cell () Call Service Rate () Call Hand-o Rate (h) Fraction of calls originating in a cell and destined to the other cell (f)

8 channels 0.33 calls/min 0.33 calls/min 0.5 calls/min 0.5

Table 1: System Parameters call assignment policy was determined for the following call assignment probabilities: P11 = P12 = P22 = 0.5 It is worth mentioning that the results given throughout this section are not typical values, due to the small state space generated by the small number of resources assumed. Nevertheless, these results are very useful in comparing the relative performance of various policies under investigation. First, it can be noticed in [16] that the call blocking and dropping probabilities obtained numerically and via SC simulation have good agreement for all channel partitioning policies. Moreover, the optimum channel partitioning policy turns out to be the "All-Channels-to-Satellite" policy if no constraints are enforced on the dropping probabilities or the propagation delays. This is intuitively clear since the 10

satellite provides maximum exibility in sharing its resources to the users of all cells. In Figure 3, ordinal rankings based on call blocking probabilities were plotted for simulation runs of various lengths versus the exact ranking obtained numerically. It was noticed that short simulation runs, while giving inaccurate call blocking probabilities, retain the correct ordinal ranking of the best policies. In Figure 4, the channel partitioning policies are compared for their average blocking probability, average dropping probability and average percentage of satellite calls (representing the consideration of satellite propagation delays). As mentioned earlier, the weighted cost function can be minimized via enforcing upper bounds on Pd and while minimizing Pb as shown below, Pd  1  0.6 It can be obtained from Figure 4 that the minimum value of Pb is 0.29 and is achieved by policy #4 (Mc1 = 2; Mc2 = 4; Ms = 2). Accordingly, it can be concluded that splitting the available number of resources into terrestrial and satellite components minimizes the multi-faceted cost function as compared to the two extremes, namely pure cellular and pure satellite systems. This is gained at the expense of longer propagation delays and more complexity to integrate and inter-operate the two systems with minimum changes to both of them. For the second sub-problem, the optimum call assignment probabilities were determined for the following channel partitioning: Mc1 = 2, Mc2 = 2, Ms = 4 We have chosen a nite subset of 64 policies that cover the whole [0,1] range. The ranking of the policies based on blocking and dropping probabilities generated by various simulation lengths are shown in Figures 5 and 6. It can be noticed that the blocking trajectory for 10K arrival events has a wider performance range between best and worst policies than the one associated with 1M arrival events. Furthermore, by increasing the simulation length, this range shrinks monotonically. Therefore, given the small performance range between extreme policies, two simple call assignment policies, introduced 11

10 Ranking based on 10 K Arrival Event SC Simulation Ranking based on 100 K Arrival Event SC Simulation Ranking based on 1 M Arrival Event SC Simulation

9

8

7

6

5

4

3

2

1

0

1

2

3 4 5 6 7 BW splits ranked based on Exact Blocking Probabilities

8

9

Figure 3: Ordinal Rankings based on Blocking Probabilities for Channel Allocation Policies

12

1 0.9 Avg. % of Satellite Calls 0.8 0.7 0.6 0.5

Avg. Blocking Probability

0.4 0.3 Avg. Dropping Probability 0.2 0.1 0 1

2 3 4 5 6 7 8 BW splits ranked based on 1M Event SC Simulation (Blocking Probabilities)

9

Figure 4: Simulation-based Call Blocking, Dropping, and Propagation Delay Performance for the Channel Allocation Policies

13

earlier in [6], were examined. It was noticed that the Cellular First (CF) policy outperforms the Satellite First (SF) policy (with respect to Pb and Pd). Second, the CF policy gave blocking and dropping rates (Pb = 0.1854, Pd = 0.0366) that are not much inferior to those achieved by the optimum policy (Pb = 0.1841, Pd = 0.0359). Thus, the CF policy might be considered a simple near-optimal call assignment rule. 0.2

10K Arrival Events 40K Arrival Events 80K Arrival Events 200K Arrival Events 1M Arrival Events

Average Call Blocking Probability

0.195

0.19

0.185

0.18

0.175

0.17

10 20 30 40 50 60 Assignment Policies ranked based on Blocking Probabilities (calculated by SC simulation)

Figure 5: Call Blocking Probabilities for Call Assignment Policies based on SC Simulations of dierent lengths The call assignment policy was further modi ed to improve the system's dropping performance via optimizing over switching thresholds as presented in [16] in more details. The results therein emphasize that switching from the randomized assignment rule to a deterministic rule at appropriate thresholds improves the call-dropping probability as compared to the case of switching at the original state space boundaries. This is achieved 14

0.06

10K Arrival Events 40K Arrival Events 80K Arrival Events 400K Arrival Events 1M Arrival Events

Average Call Dropping Probability

0.055

0.05

0.045

0.04

0.035

0.03

0.025

10 20 30 40 50 60 Assignment Policies ranked based on Dropping Probabilities (calculated by SC simulation)

Figure 6: Hand-o Dropping Probabilities for Call Assignment Policies Ranked based on SC Simulations of dierent lengths

15

at the expense of tolerable degradation in the blocking probability. Therefore, based on the fact that dropping an on-going call is less desired than blocking a new one, this assignment strategy controls the relative importance of the blocking and dropping rates.

4 Frequency Reuse Impact on the Optimum Channel Partitioning In this section, we focus on one of the fundamental dierences between hybrid wireless systems and hierarchical cellular networks, that is the frequency reuse constraints. The dierent satellite propagation characteristics in addition to the on-board antenna that acts as a spatial lter permit a much denser frequency reuse pattern in the space segment. In section 3, we focused on showing the trade-o and solving the problem for a system of just two cells overlaid by one spot-beam. Here, we extend the model to a more realistic case of multiple cells and spot-beams. Our prime concern is to show how the optimum channel partitioning is aected by the frequency reuse constraints. We assume that the network under consideration consists of N = 8 cells and a satellite emitting m = 4 spot-beams covering the same area, and supported by on-board switching. It is worth mentioning that the additional resources provided by the overlapping spotbeams are not considered in this model. In this section, base stations are assumed to be interconnected via a wireline infrastructure. This emphasizes the applicability of the developed model to wireline as well as wireless interconnection of base stations. The only dierence in this case is that any mobile-to-mobile call needs only 2 duplex channels. The inter-hando time of a mobile from spot-beam Sl to spot-beam Sk is exponentially distributed with mean 1/hs min, where the hand-o rate is assumed to be inversely proportional to the spot-beam radius and l,k=1,...,4. The state of the system is de ned by the vector x = (n11 ; n12 ; n13; :::; nij ; :::; n88; ns11 ; ns12 ; :::; nskl ; :::; ns44 ), where i,j = 1,2,3,...,8, i  j and l,k = 1,...,4, l  k. nij is the number of active calls of type ij; that is, calls served by BSi and BSj , where one of the parties is in Ci and the other is in Cj . On the other 16

hand, nskl is the number of active calls of type skl ; that is, calls served by spot-beams Sk and Sl , where one of the parties is within foot-print Sk and the other is within footprint Sl . Along the same lines of section 3, the system is modeled as a multi-dimensional continuous-time Markov chain.

4.1 Problem De nition and Solution Approach The objective is to solve the following minimization problem for dierent frequency reuse constraints in both layers, min

Mc1 ;Mc2 ;:::;Mc8 ;Ms1 ;Ms2 ;:::;Ms4

s.t.

Pb

Pd  M = Ki=1 Mc + Kj =1 Ms c

i

s

(3)

j

where, Msj = number of channels dedicated to spot-beam Sj , and, = pre-speci ed upper bound on the dropping probability. As indicated earlier, the search space for this problem is very large. Hence, it is infeasible to search for the optimum in one phase, and a tree-search type of random search algorithms [22] is needed in conjunction with SC simulation and Ordinal Optimization. According to the \Adaptive Partitioned Random Search" (APRS) global optimization technique [23], the search region of the objective function is to be partitioned into a pre-speci ed number of sub-regions. Then, using the sampled function values from each sub-region, determine how promising each sub-region is. The "most promising" subregion is then further partitioned. The simulation results show that, while APRS does not necessarily reach the global optimum, it is guaranteed to reach a near-optimal solution quite fast. This is achieved at a computational cost much less than discrete exhaustive search.

17

4.2 Simulation Results The hybrid wireless system of interest was analyzed assuming the numerical parameters given in Table 2. It should be pointed out that the following results were obtained with no constraint enforced on Pd while minimizing Pb, i.e. was assumed to be 1 in (3). The optimum channel partitioning policy was determined for various frequency reuse constraints and the following call assignment probabilities: Pij = 0.5, i,j = 1,2,...,8, i  j For the rst frequency reuse pair, Kc = 4, Ks = 1, the frequency reuse in the satellite layer is optimistic in the sense that neighboring, or even overlapping foot-prints, may use the same resources. This assumption is supported by the satellite propagation characteristics and the on-board antenna which may permit a much denser frequency reuse pattern in the space segment. On the other hand, the frequency reuse in the cellular layer was, relatively, conservative by assuming that each cell cluster has 4 cells. For this set of frequency reuse factors, the shared satellite resources assisted by the denser frequency reuse pattern in the space segment, give the superiority to the satellite layer. For the numerical parameters given in Table 2, the spatial symmetry of the call arrival rates, service rates, and hando rates among the cells and spot-beams can be noticed. Therefore, the search space is restricted to those policies having equal shares among cells and equal shares among spotbeams, i.e. Mci = Mc, i=1,2,...,8, Msj = Ms, j=1,...,4. The simulation results shown in Table 3 indicate that the optimum policy (shown in bold font) is to assign all the resources to the satellite system. Consider next the hybrid system having Kc = 4, Ks = 2. In this case, both layers have good, but not extensive reuse patterns. Again, the shared capacity advantage of the space segment wins and the "All-Channels-to-Satellite" allocation policy achieves the minimum blocking probability as given in [24]. For the third frequency reuse set, Kc = 3; Ks = 2, we assume an optimistic reuse pattern for the terrestrial layer. We expect that the dense frequency reuse pattern in the terrestrial layer might overcome the shared capacity advantage of the satellite, and 18

Total System Bandwidth (M) Call Arrival Rate per Cell () Call Service Rate () Call Hand-o Rate (h) Fraction of calls originated in a cell and destined to any other cell (f)

40 channels 0.6 calls/min 0.6 calls/min 0.5 calls/min 0.125

Table 2: System Parameters (Mc1 ; Mc2 ; Mc5 ; Mc6 ; Ms1 ) Pb

Pd

(0,0,0,0,40)

0.000007

0.000009

(1,1,1,1,36) (2,2,2,2,32) (3,3,3,3,28) (4,4,4,4,24) (5,5,5,5,20) (6,6,6,6,16) (7,7,7,7,12) (8,8,8,8,8) (9,9,9,9,4) (10,10,10,10,0)

0.000031 0.000046 0.000052 0.000079 0.000090 0.000174 0.000279 0.001485 0.019534 0.080307

0.000020 0.000027 0.000036 0.000057 0.000070 0.000080 0.000183 0.003517 0.030689 0.125289

Table 3: Blocking and Dropping Performance of Channel Allocation Policies (Kc = 4, Ks = 1)

19

this turns out to be the case. The simulation results, see Table 4, show the optimum allocation policy, in terms of minimizing the blocking probability, to be Mc1 = 2, Mc2 = 2, Mc3 = 2, Ms1 = 17, Ms2 = 17. Finally, the last set of frequency reuse factors, (Mc1 ; Mc2 ; Mc3 ; Ms1 ; Ms2 ) Pb (0,0,0,20,20) 0.000029 (1,1,1,18,19) 0.000026 (2,2,2,17,17)

0.000023

(3,3,3,15,16) (4,4,4,14,14) (5,5,5,13,12) (6,6,6,11,11) (7,7,7,10,9) (8,8,8,8,8) (9,9,9,6,7) (10,10,10,5,5) (11,11,11,4,3) (12,12,12,2,2) (13,13,13,1,0)

0.000034 0.000068 0.000080 0.000104 0.000537 0.001812 0.005413 0.008635 0.017908 0.036162 0.051285

Pd 0.000016 0.000011

0.000020 0.000027 0.000037 0.000051 0.000091 0.000310 0.004610 0.008814 0.016808 0.0042381 0.073978 0.125041

Table 4: Blocking and Dropping Performance of Channel Allocation Policies (Kc = 3, Ks = 2)

Kc = 3; Ks = 4, indicates that bandwidth partitioning is the optimum allocation policy since the terrestrial network has the best achievable frequency reuse factor, while the satellite layer has the worst one. The simulation results for this case are given in Table 5. It can be noticed that the optimum channel allocation policy in this case is Mc1 = 4, Mc2 = 4, Mc3 = 4, Ms1 = 7, Ms2 = 7, Ms3 = 7, Ms4 = 7. In order to reach the previous results, we made use of the spatial symmetry of the 20

(Mc1 ; Mc2 ; Mc3 ; Ms1 ; Ms2 ; Ms3 ; Ms4 ) (0,0,0,10,10,10,10) (1,1,1,9,9,9,10) (2,2,2,8,9,9,8) (3,3,3,7,8,8,8)

Pb 0.017 0.009 0.008 0.006

(4,4,4,7,7,7,7)

0.0058

(5,5,5,6,7,6,6) (6,6,6,5,6,6,5) (7,7,7,5,4,5,5) (8,8,8,4,4,4,4) (9,9,9,3,3,4,3) (10,10,10,2,3,3,2) (11,11,11,2,1,2,2) (12,12,12,1,1,1,1) (13,13,13,1,0,0,0)

0.009 0.011 0.015 0.019 0.031 0.038 0.044 0.055 0.063

Pd 0.011 0.012 0.005

0.0053 0.006 0.011 0.016 0.023 0.027 0.051 0.067 0.086 0.11 0.136

Table 5: Blocking and Dropping Performance of Channel Allocation Policies (Kc = 3, Ks = 4)

21

parameters of the stochastic processes governing the system dynamics in limiting the search space. We restricted the search process to those policies having equal shares among cells and equal shares among spot-beams. For the general case, the search space is expected to be extremely large and it would be impossible to search for the optimum in one phase. Therefore, we recommend employing a tree search type of algorithms, like the APRS global optimization technique. We applied this optimization technique to our system with Kc = 4; Ks = 2, and the same numerical parameters given in Table 2. To verify our earlier results, given in [24], we solved the optimization problem again without taking into account the spatial symmetry. Instead, we searched for the optimum, blindly, in the whole policy space. In this case, the space of channel allocation policies was 6dimensional. The search space was partitioned to 12 regions in each phase and a sample policy was picked randomly from each partition according to a uniform distribution. The partitioning was performed using hyperplanes parallel to the space axes. In each search phase, we marked the partition having the policy that gave the minimum bloking rate as the "most promising" partition, and it was partitioned further in the next phase. It is noticed in [24] that the partitioning process is approaching the "All-Channels-to-Satellite" policy as obtained earlier. Therefore, we conclude that the APRS algorithm reaches a near-optimal solution quite fast as compared to exhaustive search. Finally, we found out that our results carry for large hybrid systems re ecting practical network setups [24].

5 Optimum Cell Size In this section, we investigate, analytically and via simulation, the impact of reducing the cell size on the system capacity and hand-o failure rate of hybrid wireless systems. Earlier studies indicate that smaller cells permit denser frequency reuse patterns and hence improve the system capacity. In this section, the price paid is found to be two fold, the increase in the forced-termination probability and the signaling and processing overhead necessary for handling frequent hand-o attempts. This trade-o motivates the introduced optimization problem. 22

The chosen QoS parameters should re ect accurately the impact of reducing the cell size on both, system capacity and hand-o failure rate. For hand-o failures, we have two candidate QoS parameters, namely the dropping probability and the forced-termination probability[25]. The dropping probability is de ned as the fraction of hand-o attempts that are denied access due to lack of resources. On the other hand, the forced-termination probability is de ned as the probability that a call accepted to the system is interrupted, during its lifetime, due to hand-o failures. In this study, we consider both criteria and show the fundamental dierence between their behavior when the cell size is reduced.

5.1 Pure Cellular Systems First, a cellular system consisting of N cells, serving mobile-to- xed calls, was modeled as an N-dimensional Markov chain with diagonal state transitions caused by hand-os. This, in turn, did not allow a product-form solution and hence hindered further analytical studies. For small systems, we found, numerically, that the blocking and dropping probabilities decrease while the forced-termination probability increases as the cell size shrinks. Furthermore, we support these results by analysis using the following approximate model. We illustrate it using a simple 2-cell system. Each hand-o event is approximated by arrival and departure events from respective cells. Under this approximation, the 2-dimensional Markov chain becomes pure birth-death process with state-dependent arrival rates as given by,

8> <  + (n2 + 1)h 1(n2) = > :  8> <  + (n1 + 1)h 2(n1) = > : 

: 0  n2  Mc2 ; 1 : n2 = Mc2 : 0  n1  Mc1 ; 1 : n1 = Mc1

where, i = aggregate call arrival rate (new + hand-o) to Ci, i=1,2. ni = number of calls in cell Ci, i=1,2. 23

In addition, the aggregate call departure rate is given by, t =  + h Nevertheless, the detailed balance equations do not hold and thus the product-form solution remains inapplicable even under this approximation. To circumvent the statedependent arrival rates hurdle, we approximate the number of active calls by their expected values. Accordingly, we get the following expressions for 1 and 2 , 1 =  + E [n2] h 2 =  + E [n1] h Based on the aforementioned approximations, the detailed balance equations hold and the product-form solution becomes applicable. Accordingly, the system steady-state probabilities can be written in the following form, n1 n2 P (n1; n2) = G1 : n1 ! : n2 !

1

2

; 0  ni  Mc ; i = 1; 2 i

where G is the normalization constant and, 1 = 1t , 2 = 2t Using this model, we were able to develop closed form expressions for the blocking, dropping, and forced-termination probabilities, for a system of N cells, in terms of the simple Erlang-B formula[19]. The analysis in [26] and the approximate results therein emphasize the same trends obtained using the exact model. Furthermore, we veri ed the analytical results with simulations that enabled us to study large systems consisting of hundreds of cells. In the simulation study, we employ a more realistic mobility model that assumes the speed of each mobile user to be uniformly distributed between maximum and minimum values. Moreover, the direction of motion is uniformly distributed between 0 and 2. From the simulation results, shown in Figure 7, it can be noticed that the blocking and dropping probabilities decrease as the number of cells in the given area increases. On the other hand, the forced-termination probability tends to increase as the number of cells increases. Again, this emphasizes the blocking/ forced-termination trade-o and thus, motivates the following multi-objective 24

Total System Bandwidth (M) # Cells in the covered area (N) Total Call Arrival Rate Call Service Rate () Cellular Reuse Factor (Kc) Minimum Speed Maximum Speed

100 channels 50,...,400 cells 0:33 calls/min 0.33 calls/min 7 12 mph 58 mph

Table 6: Numerical Parameters for the Simulation Model optimization problem:

min Pb N

(4)

s.t.

Pf  where Pf = forced-termination probability. It is evident that as the number of cells in the covered area increases, the dimensionality of the Markov chain increases. Since the numerical solution becomes prohibitive, we resort to simulation to solve the optimization problem using parameters re ecting practical network setups as given in Table 6. Assume that the parameter in (4) is set to 0.01. Based on the simulation results in Figure 7, it can be easily deduced that the minimum achievable blocking probability under this constraint is 0.087. This is achievable via deploying 100 hexagonal cells in the area of interest.

5.2 Satellite-Cellular Hybrid Systems This section is devoted to discussing the eect of the cell size, or equivalently the number of cells per spot-beam, on the performance of a hybrid wireless system. Towards this end, we go through two phases. The rst phase focuses on showing the blocking probability/forced-termination probability trade-o. We believe that the trends of the 25

0

10

Blocking Prob. Dropping Prob. Forced Termination Prob.

−1

10

−2

10

−3

10

50

100

150

200 250 Number of Cells

300

350

400

Figure 7: QoS Parameters vs. Number of Cells for a Pure Cellular System: Simulation Results

26

QoS parameters would be similar to those obtained earlier for pure cellular systems. The only dierence we expect is the relative improvement achieved by the hybrid system. Once we have the trade-o emphasized, the second phase of the study is devoted to optimizing the system design. In order to study a hybrid system, assumptions similar to those in section 2 are made. We consider a hybrid system having the same numerical parameters in Table 6. In addition, the network under consideration consists of N cells overlaid by one satellite emitting 4 spot-beams covering the same area. We assume the satellite reuse factor Ks = 2. For satellite-cellular hybrid systems, there are more degrees of freedom to optimize the system design. In this section, we focus on two design parameters, namely the number of cells per spot-beam and the bandwidth partitioning. Our main objective is to show how jointly optimizing these two parameters could improve the system capacity and forcedtermination probability as compared to the pure cellular alternative. The optimization is carried out for given satellite and cellular reuse factors, call assignment policy, and new call arrival and hand-o processes. Accordingly, the optimization problem is formulated as follows, min Pb (5) N;Mc ;Ms i

j

s.t.

Pf  M = Ki=1Mc + Kj =1Ms It is worth mentioning that we restrict the search space to bandwidth partitioning policies having equal shares among all cells and among all spot-beams. This is due to the spatial symmetry of the call arrival rates, service rates, and hand-o rates assumed in Table 6. This, in turn, explains why the bandwidth partitioning degrees of freedom reduce to the two parameters Mc and Ms as given in (5). Due to the complexity of the joint optimization problem, we consider here the following sub-problem: Given Mc and Ms , solve the following problem: c

i

i

s

j

j

i

j

min Pb N 27

(6)

s.t.

Pf  The channel partitioning policy was assumed to be: Mc = 6 , Ms = 29 Figure 8 shows the trends of the QoS parameters versus N. Again, it can be noticed that the blocking and dropping probabilities decrease with N. On the other hand, the forcedtermination probability increases with N. By comparing Figures 7 and 8, we conclude that the hybrid system has better capacity and forced-termination performance for the same total number of resources. To solve the optimization problem, we assume the value of to be 0.0024. Based on the simulation results in Figure 8, it can be easily noticed that the minimum blocking probability achievable under this constraint is 0.0095. It is achievable via deploying 36 hexagonal cells underlying each spot-beam. i

j

5.3 Network Design Approaches The previous discussion brings up a new approach for increasing the capacity of pure cellular systems. In order to illustrate this approach, consider a pure cellular system having N cells. To increase the capacity of such system, we have two design alternatives. First, we can reduce the cell size as explained earlier. According to the results obtained, we concluded that reducing the cell size in pure cellular systems increases the system capacity. This is gained at the expense of degradation in the forced-termination probability and increase in the hand-o signaling and processing overhead. The second alternative is to keep the cell size xed and deploy an overlaying satellite layer. This causes the blocking and dropping probabilities to drop due to the shared capacity advantage of the space segment. This alternative is attractive in the case of co-existing satellite and cellular systems operating independently and providing coverage to the same geographical area. The only burden in this case is managing to inter-operate the two independent systems. The following example further illustrates the two alternatives: Consider a pure cellular system having N = 36 cells and the numerical parameters in 28

0

10

Blocking Prob. Dropping Prob. Forced Termination Prob.

−1

10

−2

10

−3

10

50

100

150

200 250 Number of Cells

300

350

400

Figure 8: QoS Parameters vs. Number of Cells for a Hybrid System: Simulation Results

29

Table 6. This system achieves blocking and dropping probabilities of 0.1205 and 0.0692 respectively. If we reduce the cell size such that N becomes 100, then the blocking and dropping probabilities drop to 0.079 and 0.0386 respectively. According to the second alternative, deploy a satellite emitting 4 spot-beams covering the same area covered by the N cells. If the total bandwidth M is partitioned between the two layers, such that Mci = 6 and Msj = 29, then the blocking and dropping probabilities become 0.0119 and 0.0138 respectively. By comparing the previous results, we conclude that deploying a hybrid wireless system is strongly recommended, especially if the satellite and cellular systems already co-exist.

6 Dynamic Call Re-assignment The class of assignment rules introduced in section 3 makes the decision only once, that is, at the time of admitting a new call or upon hand-os. Once a mobile user has been assigned a particular channel (either satellite or cellular), it continues to use this channel until it is completed or requires hand-o. Such policies yield poor performance as can be seen from the following scenario (shown in Figure 9): consider a hybrid wireless system that is designed to serve mobile-to-mobile calls. It consists of one satellite spot-beam S that covers the same area covered by the two cells C1 and C2. Assume that C1 has 4 channels, two of which are busy serving a local call in C1, while C2 has 2 channels which are busy. On the other hand, the satellite beam has 2 channels that are busy serving a local call in C1. If a new call request originates in C2, it is blocked. However, if the call served by the satellite had been re-assigned to C1, then the new call request in C2 would have been served by the satellite. This section explores dynamic call re-assignment that can be exploited to increase the capacity of hybrid wireless systems. More speci cally, we search for the optimum re-assignment policy that strikes a balance between call blocking and hand-o dropping probabilities. We develop a Markov decision based formulation which suits this kind of sequential decision problems. Its main drawback is that the computational complexity 30

increases with the size of the system. Therefore, we limit our investigation to a small (2 cell/1 spot-beam) hybrid system with small number of channels.

S

C1

M =4 C 1

C2

M C 2

=2

M

=2

S

Figure 9: Satellite-to-Cellular Re-Assignment Scenario

6.1 Problem De nition and Solution Approach Motivated by practical considerations, we assume hand-os to be intra-layer only, i.e. no hand-os are allowed between the cellular system and the overlaying satellite or vice versa. In this restricted hand-o environment, the dropping probability would be high and searching for ecient call re-assignment schemes becomes essential. In the next section, we characterize the optimal re-assignment policy that minimizes the dropping probability. Unfortunately, it is infeasible due to implementation constraints and searching for good heuristic policies turns out to be essential. 31

The total number of resources, M = 6, is equally divided among C1, C2 and S. The dynamics of this system is modeled as a multi-dimensional Markov chain of 6 dimensions. It is important, for the operation of the re-assignment algorithm, to distinguish between dierent types of satellite calls depending on the location of participating mobiles. Accordingly, the state of the system is de ned by the vector x = (n11 ; n12; n22 ; ns11 ; ns12 ; ns22 ) where nsij is the number of active calls of type sij ; that is, calls served by S and one of the parties is in Ci and the other is in Cj . We approach this problem by focusing on one QoS parameter at a time. Accordingly, the study goes through the following three phases. In the rst phase we assume no users' mobility and the objective would be to nd the optimum re-assignment policy that minimizes the blocking probability. The second phase incorporates users' mobility and hand-os into the problem with the objective shifted to minimizing the hand-o dropping probability. Finally, we focus on minimizing a weighted cost function of the blocking and dropping probabilities. First, we consider the case of no users' mobility. We present the Markov decision process (MDP) formulation of the problem and the adopted iterative solution approach. Due to the memoryless nature of the governing processes, the time elapsed between various decision epochs (state transitions) turns out to be exponentially distributed. Therefore, the formulation developed is that of a Semi Markov Decision Process (SMDP) that can be converted to a discrete MDP via a data transformation operation. The corner stones of the formulation are the state space, action space, event space, transition probabilities and the one-step cost function that is to be minimized. It is evident that the one-step cost is the average number of calls blocked since users' mobility is not considered. According to the de ned system state and capacity constraints, the state space, denoted , turns out to consist of 24 states. The feasible events at each state are arrivals and departures. For the system under investigation, the event space is given by E = faij , asij , dij , dsij g where aij and asij represent arrivals of new calls of type ij that are assigned to the cellular system or to the satellite respectively. On the other hand, dij and dsij represent the corresponding call termination events. It is worth mentioning that the underlying Markov chain is 32

stationary, hence the optimal re-assignment rule is expected to be stationary as well. To be able to specify the action space, we need to investigate the characteristics of call re-assignment policies. These characteristics involve:

 Direction of Re-Assignment: either "downwards", from the satellite to the cellular system, or "upwards", from the cellular system to the overlaying satellite.

 Re-Assignment Decision: depends on the current state and the current event.  Number of calls to re-assign at each decision epoch: one or multiple. According to the no mobility assumption, it can be shown that the optimal direction of re-assignment is "downwards". The intuitive explanation is that the optimal policy tends to free the shared pool of resources in the space segment at the expense of lling up the dedicated resources in the cellular network. Typical Scenarios can be given to show that "upwards" re-assignments degrade the blocking probability. In addition, we limit our attention to policies that attempt re-assignment only once at each decision epoch. This is dictated by the limited state space where the states are expected to have at most one call to re-assign. In larger state spaces, multiple call re-assignments at each decision epoch should be taken into consideration. According to the aforementioned characteristics of the optimal re-assignment policy for the 2-cell/1-spot beam hybrid system, the action space, denoted A, is given by:

8 >> 0 >> <1 A=> >> 2 >: 3

: No ReAssignment : ReAssign a 0110 call from S to C1 : ReAssign a 0120 call from S to C1 and C2 : ReAssign a 0220 call from S to C2 If we denote the re-assignment policy space as R, and jj = !, jE j = e, and jAj = a, then, jRj = a!e which gives an idea of the exponential growth of the policy space with the sizes of the state and event spaces. The re-assignment policy at state x is de ned, for all possible 33

events, by the vector r(x) = (ra11 (x); :::; ras11 (x); :::; rds22 (x)). The scalability problem of MDPs is unavoidable, it is one of their inherent drawbacks. Consequently, we focus here on small systems in order to gain insights about the characteristics of the optimal policy as a step towards characterizing optimal re-assignment policies for more realistic hybrid systems. The stochastic equations governing the system dynamics are characterized by the transition probabilities P (x; y; r) that is; the probability of making a transition from state x to state y if the action taken is ri (x). The construction of the transition probabilities is quite an elaborate process, yet straightforward. Thus, we demonstrate it via a simple example here and eliminate further details. In the following, we construct P for a sample of arrival events. Consider an arrival of a new call of type '11' that is assigned to cell C1:

8> >> 11  (x; r)(ra11 ) ><   (x; r)(r ; 1) a11 P (x; y; r) = > 11 >> 11 (x; r)(ra11 ; 2) >: 11 (x; r)(ra ; 3) 11

where,

: : : :

y = (n11 + 1; n12 ; n22; ns11 ; ns12 ; ns22 ) y = (n11 + 2; n12 ; n22; ns11 ; 1; ns12 ; ns22 ) y = (n11 + 1; n12 + 1; n22; ns11 ; ns12 ; 1; ns22 ) y = (n11 + 1; n12 ; n22 + 1; ns11 ; ns12 ; ns22 ; 1)

8> < 1 : n=0 (n) = > : 0 : n 6= 0

11 =  (1 ; f )P11 and  (x; r) is the expected time to next state if you are at state x and the action vector is r. It is straightforward to notice that  (x; r) is given by:  (x; r) = l(x; r) +  (P 1 n + P n ) (7) i;j ij i;j s where l(x; r) is the rate of transition out of state x due to call arrival events depending on the action taken. Various solution techniques for MDPs have been introduced in the literature. Among those are policy iteration and value iteration (VI). Value iteration is computationally ecient as it performs normal computations in each iteration step. As pointed out in [27], the VI algorithm is applied only to discrete MDPs where the decision epochs are separated ij

34

by constant time periods. In order to apply VI to SMDP, it should be transformed rst to a discrete MDP via a process called uniformization. The details of this operation can be reviewed in [27]. The VI algorithm aims at minimizing the following value function with respect to the re-assignment decision taken at each state. It is worth mentioning that minimizing this value function is equivalent to minimizing the long-run blocking probability. The value function for discrete MDPs is given by[27],

VT (x) = rmin fb (r) + (x)2R x

X y2

P (x; y; r)VT ;1(y)g

(8)

where, VT (x) : minimum value function for state x at iteration T. bx(r) : one-step average blocking probability at state x when the action vector is r. For a SMDP, the value function is slightly modi ed from (8) due to the uniformization process. The other two phases of the study, which take users' mobility into consideration, are formulated and solved in a similar fashion [28].

6.2 Numerical Results Total Bandwidth (M) Call Arrival Rate per Cell () Call Service Rate () Call Hand-o Rate (h) Fraction of calls originating in a cell and destined to the other cell (f)

6 channels 0.4, 0.6, ... 2 calls/min 0.33 calls/min 0.0833 calls/min 0.5

Table 7: System Parameters First, we show the characteristics of the re-assignment policy that minimizes the blocking probability in case of no users' mobility. It is characterized as "Re-assign a call from 35

the satellite system to the cellular system upon arrival of a new call i the call is blocked in case of no re-assignment". Figure 10 shows the performance of the optimal re-assignment policy compared to the reference system (no re-assignment) for dierent call arrival rates. It is worth mentioning that the given values of blocking probabilities are not typical, due to the small number of resources assumed. We are more concerned here with the trends and the relative performance of various policies. From Figure 10, the improvement achieved by the optimal policy can be noticed, it reduces the blocking probability by 3% on the average. The improvement is relatively small due to the limited capacity and state space. We expect the percentage improvement to be more signi cant for larger systems as shown later in this section. Next, we examine two sub-optimal re-assignment policies, 0.8 No Re−Assign. Optimal Re−Assign. 0.75

0.7

Call Blocking Probability

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3 0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

Figure 10: Optimal Re-Assignment Policy for min. Pb 36

1.8

2

that were previously introduced in the literature, and compare their performance to the optimum. The rst policy suggests attempting call re-assignments periodically. Figure 11 shows the performance of this policy when the period is taken to be ve times the mean inter-event time. This policy is inferior to the optimum by a factor of 2% on the average. The second re-assignment policy attempts re-assignments as soon as there is room in the cellular system. This is motivated by the intuition that freeing the shared pool of resources in the space segment, as soon as possible, improves performance. It can be noticed from Figure 11 that this policy is approximately 2% inferior to the optimum as well. Thus, there is room for developing heuristic policies with performance closer to the optimum. 0.8

0.75

No Re−Assign. Optimal Re−Assign. Periodic Re−Assign. Re−Assign. at Termination

0.7

Call Blocking Probability

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3 0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

1.8

2

Figure 11: Sub-optimal Re-Assignment Policies In the second phase of the study, we obtained the characteristics of the re-assignment 37

policy that minimizes the dropping probability. Figure 12 shows the improvement achieved by the optimal re-assignment policy compared to the reference system. It is worth mentioning that this policy is characterized as: "Re-assign a call from the cellular system to the satellite system i a call is dropped upon hand-o in case of no re-assignment". This implies that, in order to minimize the dropping probability, inter-layer hand-os must be enabled. Unfortunately, inter-layer hand-os are practically infeasible due to the aforementioned reasons. Therefore, heuristic policies need to be introduced. A candidate class of policies would take the re-assignment decision and direction based on the utilization of resources in the two segments. Accordingly, the objective would be to maintain a balanced amount of residual capacity in each layer to serve restricted hand-os. Again, the dropping probabilities given here are not typical values, yet they are useful for comparison purposes. From Figure 12, the relative improvement of the optimal policy compared to the reference system was found to be 16%, which is quite signi cant. On the other hand, the blocking probability is slightly degraded as can be noticed from Figure 13. In the following, we examine a heuristic re-assignment policy which attempts call reassignment from the cellular system to the satellite periodically. Figure 14 shows the relative performance of this policy compared to the optimum. This policy is inferior to the optimum by a factor of 10% on the average. Finally, we show the performance achieved by various re-assignment policies when the objective is to minimize a weighted cost function of the blocking and dropping probabilities. It can be noticed from Figure 16 that the two extreme points represent the two special cases considered earlier, while intermediate points on the graph represent various optimal re-assignment policies that minimize Pb while preserving Pd below a pre-speci ed threshold denoted . For example, if is set to 0.5, then there is a re-assignment policy that can achieve a blocking probability of 0.513 and a dropping probability of 0.497 simultaneously. From the previous results, it can be noticed that the performance improvement due to dynamic call re-assignment is limited due to the limited number of resources and hence the small state space. In the following, we investigate the performance gains of dynamic 38

0.7

No Re−Assign. Optimal Re−Assign.

Handoff Dropping Probability

0.6

0.5

0.4

0.3

0.2 0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

1.8

2

Figure 12: Hand-o Dropping Performance of the Optimal Re-Assignment Policy for min. Pd

39

0.8 No Re−Assign. Optimal Re−Assign. 0.75

0.7

Call Blocking Probability

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3 0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

1.8

2

Figure 13: Call Blocking Performance of the Optimal Re-Assignment Policy for min. Pd

40

0.7

No Re−Assign. Optimal Re−Assign. Periodic Re−Assign.

Handoff Dropping Probability

0.6

0.5

0.4

0.3

0.2 0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

1.8

2

Figure 14: Hand-o Dropping Performance of the "Periodic" Re-Assignment Policy

41

0.8

0.75

No Re−Assign. Optimal Re−Assign. Periodic Re−Assign.

0.7

Call Blocking Probability

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3 0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

1.8

Figure 15: Call Blocking Performance of the "Periodic" Re-Assignment Policy

42

2

0.6 Blocking Probability Dropping Probability 0.58

0.56

0.54

0.52

0.5

0.48

0.46

0.44

0.42

0.4

0.44

0.46

0.48

0.5 β

0.52

0.54

Figure 16: Balancing the Blocking-Dropping Trade-o

43

0.56

re-assignment for a 2 cell/1 spot-beam with more resources. We consider the case where the total number of resources M = 30. We focus here on the re-assignment policies that improve the dropping probability due to their practical signi cance. It can be noticed from Figure 17, that the optimal policy reduces the hand-o dropping probability by more than one order of magnitude for relatively light loads and this improvement decreases monotonically to about half an order of magnitude at higher loads. On the other hand, the degradation in the new call blocking probability is acceptable as can be noticed from Figure 18. Thus, it can be concluded that the performance gains due to call re-assignment are signi cant in large hybrid systems. 0

10

No Re−Assign. Optimal Re−Assign.

−1

Handoff Dropping Probability

10

−2

10

−3

10

−4

10

−5

10

0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

1.8

2

Figure 17: Hand-o Dropping Performance of the Optimal Policy for M = 30 channels

44

0

10

No Re−Assign. Optimal Re−Assign. −1

10

Call Blocking Probability

−2

10

−3

10

−4

10

−5

10

−6

10

0.4

0.6

0.8

1

1.2 λ calls/min

1.4

1.6

1.8

2

Figure 18: Call Blocking Performance of the Optimal Policy for M = 30 channels

45

7 Conclusions In this paper, we investigated various issues pertaining to the design of satellite-cellular hybrid systems. First, we addressed the fundamental problems of channel partitioning and call admission/assignment. We characterized the optimal policies for a simple 2 cell/1 spot-beam hybrid system. Second, we found out that dierent frequency reuse constraints, in both systems, play a major role in the design of satellite-cellular hybrid wireless systems. Third, we investigated, analytically and via simulation, the eect of reducing the cell size on the blocking, dropping and forced-termination probabilities. We showed the blockingforced termination probabilities trade-o for pure cellular and satellite-cellular hybrid systems. Accordingly, an optimization problem with respect to the cell size was formulated and solved for practical network setups. Finally, we searched for the optimum dynamic call re-assignment policy that improves the hybrid system performance. For a small hybrid system, we characterized the optimum re-assignment policies that minimize the blocking probability, dropping probability, and a weighted cost function of these probabilities. In addition, the optimal policies were examined in the context of large hybrid systems and were found to increase the capacity signi cantly.

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[25] S. Rappaport, "The Multiple-Call Hand-o Problem in High-Capacity Cellular Communication Systems," IEEE Transactions on Vehicular Technology, vol. 40, no. 3, pp. 546-557, August 1991. [26] T. ElBatt and A. Ephremides, "Cell Size in Hybrid Wireless Systems," IEEE Vehicular Technology Conference VTC'99, Houston, Texas, May 1999. [27] H. Tijms, Stochastic Modelling and Analysis: A Computational Approach. New York:Wiley, 1986. [28] T. ElBatt and A. Ephremides, "Optimal Call Re-Assignment in Hybrid Wireless Systems," 19th AIAA International Communications Satellite Systems Conference, April 2001, Toulouse, France.

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