A Time-scale Decomposition Approach to Optimize Wireless Packet Resource Allocation and Scheduling

Transcription

1 A Time-scale Decomposition Approach to Optimize Wireless Packet Resource Allocation and Scheduling 1 Abstract Wireless channels are error-prone and susceptible to several kinds of interference from different time scales. This paper investigates the resource optimization and scheduling of wireless networks through a time-scale decomposition approach. We decompose the dynamics of time-varying wireless channel conditions into random processes in two different time scales: a slow time-varying process in a larger time scale, called frame-scale, and a stationary random process with high variation in a smaller time scale, called slot-scale. This results in two different algorithms dealing with each time scale: the resource optimizer optimize the sum of utilities in the slowly changed time scale, and the slot scheduler exploits the efficiency in the highly variable time scale. Our scheme can obtain a high utilization while providing service guarantee. Simulation results show our scheme can improve the performance and efficiency substantially. Keywords Wireless Scheduling, Pricing, Radio Resource Allocation. I. Introduction It is a challenging task for the wireless networks to allocate the wireless resource to meet the requirement of certain quality of service (QoS) objectives as well as to gain high radio frequency utilization over the shared, scarce and time-varying wireless channel. Resource allocation schemes and scheduling policies are critical to achieve these goals. There have been numerous works which develop elaborate scheduling algorithms in wireline networks (see [?], [?] and therein) in order to provide certain service performance guarantees. Unfortunately, those schemes can t be deployed directly in wireless environment because wireless channel has its unique characteristics: first, the wireless channel is error-prone compared with wired link and furthermore the errors are location-dependant and timevarying; second, wireless links are susceptible to air interference and the capacity loss is unavoidable. There have been several pieces of works (see [?] and therein) to deal with such problems. In [?], [?] and [?], authors provide their modified wireless version of fair queueing algorithms based on the idealized Generalized Processor Sharing (GPS) policy. These algorithms inherit the advantage of the ones in wireline networks, such as short term and long term fairness, as well as short term and long term throughput bounds. However, they oversimplify the channel conditions by modelling the wireless channel as either good or bad, so that the channel capacity might be under-utilized, especially when some adaptive techniques [?] can be used. Another drawback of those works is that they pay little attention to the efficiency and utilization. All authors are with the Complexity System Engineering Laboratory, Department of Electronics Engineering, Tsinghua University, Beijing, China. wuw99@mails.tsinghua.edu.cn There is also some research in the area of wireless scheduling and resource allocation. They ([?][?] and more) focus on the efficiency and throughput of the wireless networks, but neglect the service guarantee in a short term. In [?][?], authors present a scheduling scheme for the Qualcomm CDMA/HDR system. The HDR algorithm exploits time-varying channel conditions while maintaining wellknown proportional fairness in a long term. Reference [?] proposes an opportunistic scheduling policy to optimize the efficiency of wireless transmission stochastically while guaranteeing the long-term fairness among users. However, the development of the algorithm assumes the channel conditions are stationary, which may not be the case in practical system. Furthermore, both HDR and opportunistic schemes are designed for high rate data. They can not provide any service guarantees to the real-time applicaions. In this paper, we investigate the resource allocation and scheduling algorithms to support a wide range of service. Because the wireless channel conditions variate in different time scales, we propose a time-scale decomposition framework to exploit the optimal resource allocation of wireless links and maintain service guarantee to various applications. In a large time scale of our framework, we allocate time slots in a frame by optimizing the sum of upper application utilities through a pricing-based approach. The optimization problem can be formulated as a dual problem: the applications transmit with their optimal rate according to current price while the base station (BS) coordinate the global price according to user aggregate rate. Such an approach has gained success in the area of flow control [?]. The difference is taking channel conditions into account in the optimization. In a smaller time scale, we arrange slot sequence allocated to users in a frame to gain a higher utilization by exploiting fast channel fading. The rest of the paper is organized as follows. In section 2, we first present the system model, the wireless channel model and the structure of our system. In section 3, we reformulate the slot allocation in a frame as a dual problem, maximize the total utilities of all upper applications and discuss the algorithm and its implementation. We subsequently propose our scheduling algorithm in the small time scale in section 4. In section 5, we describe some numerical experiments to illustrate the performance of our approach. Conclusions and future work are presented in section 6. II. Model Description We consider a base station (BS) serving a set of users, M = {1,..., m}. BS transmits in slots of some fixed duration while C slots build up a frame. In each slot, BS

2 2 transmits to or receives from exactly one of users. We assume signal to noise ratio (SNR) of each user in a slot can be measured or estimated before scheduling next time slot. For an uplink channel, they can be measured or estimated directly by BS, while for a down-link channel, the information can be measured by mobile users and feed back to BS through pilot channel or other mechanisms. Because each user may have multiple connections of different service types with BS, we assume each user has a connection set N i, N i = {1,..., n i } and i M n i = n, where N = {1,..., n} is the set of all connections to the base station. The connections of a user usually transmit independently and belong to different service classes, so they may have different traffic profile and different QoS objectives. There are several metrics to characterize end-to-end QoS of an application, such as available bandwidth, delay, jitter and packet loss rate. In this paper we use bandwidth and packet loss rate as our QoS metric and delay bound can be derived from the bandwidth, just like GPS and its derivatives. Let U i (x i )(i N) quantify the level of performance that the application gains when it transmits with rate x i. The goal is to distribute the limited time slot to the applications so that the maximum of overall utilities i N U i(x i ) can be achieved. Note that the transmission rate of application we discuss here is not the time slot allocated to the application because the wireless channel is error-prone and need the protection by forward error correction (FEC) and retransmission. We consider different service classes in our system model, which have different requirements on packet error rate (PER). Traditional voice service can typically tolerate PER up to 1 3, while realtime data service might need PER guarantee even higher than 1 6. To achieve respective PER under certain SNR, applications can select different coding schemes, thus result in different coding overhead. So we define a function f c (SNR, P ER) to quantify the efficiency, f c = bits transmitted from upper application. (1) overall bits transmitted It is straightforward to derive f c (SNR, P ER) when modulation and coding have been selected. Fig.1 shows the curves of f c (SNR, 1 3 ), f c (SNR, 1 6 ) when BCH codes with a block size of 255 bits are used adaptively to achieve the target PER in a Rayley-fading channel. Also in fig.1, we plot f c when a static coding scheme is used: low PER deteriorates the transmission efficiency to upper applications. This description is quite general and can be extended to a broad range. A wireless channel is roughly susceptible to four independent phenomena: path-loss variation with distance, slow log-normal shadowing, flat-fading and fast multipathfading. They can be divided into three time scales. The power-law path-loss propagation is due to that the average received signal power decreases logarithmically with distance and the log-normal shadowing causes quite small shifts of received power variations. The time scale of these Efficiency (%) Pm<1e 3 Pm<1e 6 local error correct SNR (db) Fig. 1. The efficiency function versus SNR two is about one-tenth second and is comparable to the frame-scale in wireless system. The time scale of flat-fading depends on the speed of users: in a bad case, a user with the speed 8 miles/hour suffers a flat-fading duration in a millisecond scale, which is comparable to slot-scale in wireless networks. Multipath-fading fluctuates symbol by symbol and is in the smallest time scale, which is left to be recovered by FEC at a lower layer and out of scope of this paper. Fig. 2. Applying time-scale decomposition to the SNR of a user Wireless scheduling algorithms allocate slots as the smallest time unit and mainly suffers two larger time scale interference. Our time decomposition approach decouples these two time scales and deals with them respectively in a different way: we neglect flat-fading in the large time scale, and optimize the slowly changing system through service pricing and utility optimization; in the small time scale, we needn t consider the slow variation of channel in the long term and view SNR as a random process dominated by flat-fading with constant expectation. This idea can be demonstrated by fig.??. So in our system framework of BS, we have two layer: the upper layer, Frame Optimizer, which is working in the large time scale, is responsible for fairness and service guarantee; while the lower layer, Slot Scheduler, which is working in the small time-scale, mainly

3 3 deals with link efficiency. The architecture of our system is outlined in fig.??. The channel predictor can predict SNR and averaged SNR using the measure channel condition. The efficiency calculator output the current efficiency after inputting the predicted averaged SNR in the large time scale. The admission control is responsible for blocking the incoming connection request after considering the load of the traffic in the cell. The details of Frame Optimizer and Slot Scheduler are demonstrated in section 3,4. i, f c, so that x i = f c,i s i according to (??). In our framework, the role of utility functions is to link the performance metrics of the lower physical layer to the satisfactory level of the upper application layer. Our objective (the primal problem) is to choose the resource allocation vector s = (s i, i N) so as to P : max i N U i (x i ) (2) subject to i N s i C, x i = f c,i s i (3) Define the Lagrangian L(s, p) = U i (f c,i s i ) p( s i C) i N i N = (U i (f c,i s i ) ps i ) + pc (4) i N Notice that the first term (??) is separable in s i and the primal problem P can be derived into its dual problem: D : min p max s i L(s, p) (5) The solution to problem D is largely dependent on the expected range of the transmission rate from upper applications, x i I i. If all the U i (x i ) is concave in I i, the problem can be decomposed into N separable subproblems Fig. 3. The system structure of our time-scale decomposition algorithm at the base station III. Upper Layer: Utility-based Resource Allocation In this section, we will propose and discuss a resource allocation optimizer, which is working in the large time scale of our time-decomposition approach. We first formulate resource allocation into an optimization problem and discuss its solution in subsection 3.1. In subsection 3.2, we elaborate on the details of the design and implementation of the optimizer. A. Utility Optimization There have been numerous works based on the notion of utility in the networking research while the definition of it is quite different. In this paper, utility is introduced to quantify the satisfactory of upper applications for bandwidth: U i (x i ) denotes the reward from application i when rate x i is obtained. Note that the approach of the utility function is very general so as to deal with both rigid applications which need hard-qos assurances and adaptive ones which can coordinate their transmission according to available resources and service quality. Here x i is the rate obtained by upper application i, which is different from the actual wireless resource allocated in slots, s i. In wireless networks, the wireless resource is limited thus i N s i C, where C is the capacity of a frame in slots. x i is related to s i by the efficiency function of application max (U i (f i s i ) ps i ) (6) f i s i I i and has a unique optimal solution, which can be easily achieved by the gradient projection method in a distributed way (e.g., [?]). Note that p can be interpreted as the price per bandwidth thus (??) denotes the net reward of application i. Define x i,min = min{x i, U i ( ) is concave in [x i, + )} and we come to a more complex case: application i transmits packets if x i > x i,min, otherwise it transmits nothing; and furthermore U i is concave if x i [x i,min, + ). Thus I i = {} [x i,min, + ) for i N. Such a form of utility function is general enough to accommodate almost all the current applications such as multimedia with hardor soft-qos objectives and elastic data transfer. Define p i,cutoff = max U i(f i s i ) s i, a critical price that will determine whether application i should transmit taking the economics into account, shown as the following theorem. Lemma 1: Suppose p is fixed, all f i (i N) are known and x i = s i f i I i = {} [x i,min, + ). Set Z = {i, p i,cutoff < p}. For a resource allocation vector s = {s i, i N}, we consider s, s i = { si, i N\Z, i Z Then vector s is more optimal than s, L(s, p) > L(s, p). Proof: i Z, U i(f i s i ) s i p i,cutoff < p, so U i (f i s i ) ps i <. L(s, p) L(s, p) = i Z U i(f i s i ) ps i <, thus L(s, p) > L(s, p).

4 4 Theorem 1: Suppose f i (i N) are known and s i f i I i = {} [x i,min, + ). The optimal solution to problem D can be achieved by s i (n + 1) = { U 1 i ( p(n) f i )/f i, i N\Z, i Z p(n + 1) = [p(n) + γ(c i N (7) s i ) (8) where γ is the adaptation step of the price and sufficiently small, U i (x) is the differential function of U i(x i ) and U 1 i (x i ) is its inverse function. Proof: For a certain p, according to Lemma 1, i Z, s i =. Then the problem becomes to a concave optimization problem. Its unique solution can be obtained by L(s, p) s i = f i U i p =. This can achieve the inner maximum of (??). The outer minimum can be achieved asymptotically by gradient projection method: p(n + 1) = p(n) γ This can get (??) directly. L(s, p). p B. The Implementation of the Optimizer Algorithm Above we can see that theorem 1 has provided an adaptation algorithm to gain the optimal utility of the wireless system. Next we will present the implementation details of the algorithm. The algorithm of the optimizer works periodically at the beginning of each time frame and can be formulated by following steps: 1. Channel Prediction. (a) Channel Measurement. The channel predictor collects and averages the channel condition of each user in frame n, and get SNR j (n), j M as a result. (b) SNR Prediction. We use a one-step predictor to approximate SNR j (n + 1), thus SNR j (n + 1) = SNR j (n). 2. Efficiency Calculation: According to the estimated average SNR j (j M) in the next frame and the QoS objective of application i, i N j, we can estimate the efficiency of application i, f c,i (SNR j ). 3. Price Adaptation. The price is updated according to (??). 4. Application Adaptation. The number of slots allocated to application i is coordinated as following way s i (n + 1) = { U 1 i ( p(n+1) f i )/f i, p(n + 1) p i,cutoff, p(n + 1) > p i,cutoff 5. Repeat step 3 and 4 for K times. K=5. 6. Slot Assignment. s i s i si C to ensure s i = C. Note that in this paper we assume SNR at each slot can be acquired by measuring it directly at BS for uplink transmission or by measuring it at each mobile user and feeding it back to BS through some pilot channel. We use step 5 to speed up the convergence of the optimizer towards the optimal allocation vector. According to our simulations, K = 5 is good enough to track the channel dynamics in the large time scale. IV. Lower Layer: Exploiting Channel Variation Through Scheduling After the optimization in the large time scale, SNR in a short term can be viewed as a random process fluctuated around the averaged SNR, SNR j. This fluctuation is mainly contributed by flat-fading, which is varied in a range larger than 2dB. Such a high variation enables us to achieve a further performance gain by arranging the slots in a frame intelligently. The key idea here is that the probability all users suffer bad channel conditions is far less than that of a single user so that each user can obtain a higher SNR than SNR j actually. The scheduler can be formulated by following steps: At the beginning of each frame: b i = s i, where b i denotes the number of left slots in the rest part of the frame. 1. Channel Prediction. We use the measured SNR j to approximate the one at the next slot directly. 2. Allocation Selection. (a) User Selection. W = {j, j M and b j > } is the set that the application has slots unallocated, then choose user j, SNR j j = arc max j W SNR j (b) Application Selection. Choose application i, i N j b and i = arc max i i Nj s i. 3. Slot allocation. Allot current slot to application i of user j, b i b i Repeat step 1, 2, 3 till the end of current frame. In step 2.a, we choose the user who has the highest relative gain and still has slots to send so as to achieve better SNR while maintaining the resource allocated in the frame is fixed. It is obvious that a tradeoff exists between efficiency and service guarantee. Step 2.b can make the resource allotted to user j distribute fairly among applications N j. The computation complexity of the algorithm is approximately O(m). V. Simulation Results In this section, we present numerical results from computer simulations of our time-scale decomposition scheme. Our simulation environment is a single cell which is a round area with radius 1km. Tens of mobile users move in the cell with the velocity distributed uniformly between the minimum (3.6km/h) and the maximum (18km/h). The directions of mobile users are independent random variables uniformly distributed between and 2π. The location of a user is randomly generated at the beginning of the simulation in the cell. A mobile user choose its velocity periodically. If a user moves out of the border of the cell, we assume it reappears at a point that is symmetric to the existing point about the center base station.

5 5 In this paper, we simulate three independent phenomena in the wireless networks: path-loss variation, slow lognormal shadowing and flat-fading. We adopt the path-loss model (Lee s model)[?], the slow log-normal shadowing model in [?] and Clarke s model [?]for flat-fading. Specifically, the channel gain g(k) (in db) at time slot k between an arbitrary user at a distance d from a base station is given by: g(k) = l p (k) + s(k) + R(k) (9) where l p (k), s(k) and R(k) are terms representing pathloss, shadowing and flat-fading respectively. The path-loss l p (k) (db) is obtained as Utility M 2 M 1 D V l p (k) = K log 1 (d(k)) α (1) where α is a correction factor used to account for different mobile station antenna heights, transmission powers and antenna gains, and K = is a constant during the simulation. The shadowing term s(k) is usually modelled as a zero-mean stationary Gaussian process with autocorrelation function given by E(s(k)s(k + m)) = σ 2 oξ vt/d d where ξ d is the correlation between two points separated by a spatial distance D (meters), and v is velocity of the mobile user. We use a value of σ o = 4.3dB, corresponding to a correlation of.3 at a distance of 1 meters, as reported by Gudmundson [?]. The flat-fading R(k) is assumed to be Rayleigh-distributed random process and details can be found in [?]. Though our scheme can be applied to a broad spectrum of utility functions, we mainly focus our discussion on four type, which is shown in fig.??. Curve V can be viewed as the utility function of voice services, which need a requirement of hard-qos, low bandwidth and P ER < 1 3 ; concave curve D can represent that of data services with low loss rate (P ER < 1 6 ), zero minimum bandwidth and elastic rate range; Curve M1 and M2 denote two classes of multimedia services: M1 can bear a higher loss rate (P ER < 1 3 ) and has lower bandwidth requirement comparing with M2 (P ER < 1 6 ). In our simulations, we mainly compare four algorithms to show the performance gain of out time decomposition scheme: weighted round robin (WRR), only optimizer, only scheduler and our time decomposition approach (optimizer+scheduler). In WRR both the number and the sequence of slots are allocated statically. Only Optimizer allot the number of slots in a frame optimally while the slot sequence in a frame is fixed. Only Scheduler has the slot number fixed and change the sequence of slots in a frame. The simulation settings are given in Tab.??. Different simulation scenarios correspond to different traffic mixture, and the traffic ratio of each type can be adjusted by controlling the user number of each type. The simulation results are drawn in fig.??, in which y-axis denotes the overall revenue of the wireless link, t Ui (t). From the results in fig.??, we can observe that: 2 Fig. 4. Revenue Rate (kbps) The utility function of four typical application classes WRR Only Scheduler Only Optimizer Optimizer & Scheduler Scenario ID Fig. 5. Comparison of four algorithms: WRR, Only Optimizer, Only Scheduler, Optimizer+Scheduler TABLE I Simulation Scenario Settings. Scenario ID Type V Type D Type M1 Type M2 (PER) % 67% 2 28% 42% 14% 16% 3 13% 37% 5% 4 13% 37% 25% 25% 5 13% 37% 33% 34%

6 WRR Only Scheduler Only Optimizer Optimizer & Scheduler 5 Revenue Fig. 6. The adaptation of price p and i N s i/c V V V D D D M M User ID 1. Our time-scale decomposition scheme can obtain nearly 1% performance gain over WRR, which does not take the characteristics of wireless channel into account. 2. Both the optimizer and the scheduler can achieve a substantial overall revenue improvement in their own time scale respectively and the time-scale decomposition approach can inherit these two improvements. 3. Only Optimizer achieves a higher revenue than Only Scheduler in all the scenarios in this simulation. Note Only Scheduler is very like the scheduler in [?], IWFQ. Our timescale decomposition scheme has more than 25% revenue increase than the scheme using scheduler only. Now we take one simulation scenario, Scenario 2, to see how our approach works. Firstly let us come to a look on the dynamical behavior of price p and i N s i/c before unified in step 6 in section III.B, which is drawn in fig.??. i N s i/c demonstrates the convergence of the optimizer to the optimal resource allocation. It also shows how the optimizer adapts to time-varying channel conditions. The curve of price p embodies the economics in the cellular wireless networks: when the channel conditions are comparably good, the price drops because less upper applications are willing to pay for excessive bandwidth; when the wireless link capacity is deteriorated, price increases due to the increased requirement for bandwidth. The revenue of some typical users are shown in fig.??. There are three voice, three data and two multimedia applications, which is denoted as V, D, M respectively. Observing the comparison of four algorithms in all applications, we can understand our algorithm more clearly: 1. From fig.?? Only Scheduler usually gain a higher revenue than WRR, because its rearrangement of slots can guarantee its averaged SNR of each user is better than WRR. 2. Only Scheduler can t guarantee that each user gains a better performance than WRR. When a user suffers strong interference, the optimizer can shut down its transmission while WRR can t. The objective of the optimizer is to maximize the overall utilities even through sacrificing some users. In fig.??, data application D4 and D6 are sacrificed. Fig. 7. The revenue gained by some users 3. Considering fairness issues among the applications in the wireless environment, we can see that WRR can be viewed as input-fairness (or slot-fairness), in which the resource is allocated proportionally to s i. The optimizer can be viewed as output-fairness (or revenue-fairness), in which the resource is allotted according to U i. 4. Observing voice services in fig.??, the revenue is far more improved by our time decomposition algorithm, especially when the voice service suffers bad channel conditions, such as V1, V2. VI. Conclusions and Future Works In this paper, we have studies a time-scale decomposition approach to achieve both QoS objectives and system efficiency. By decomposing the random process of each user s SNR into two different time scales, we can deal with each time scale with specific methods: in the large time scale, we propose an adjustment algorithm to achieve the optimal revenue of all the application utilities. We prove the algorithm can achieve the optimal solution asymptotically. Simulations show that the convergence of adjustment algorithms is very fast and it can adapt itself fast to the averaged channel conditions, which is slowly changed. In the small time scale, we use a scheduler to exploit the variation of the channel conditions in a short term by rearranging the slots allocated. Simulations show that the superiority of our time-scale decomposition approach over the traditional scheduler, WRR, is more than 1% in revenue. The contribution of this paper can be concluded as follows: 1. Introducing the idea of time decomposition to exploit the characteristics of wireless links and achieve high efficiency and QoS guarantee. 2. Proposing a pricing-based adaptation algorithm to achieve the optimal solution by maximizing the system revenue. 3. Using a efficiency function f i to quantify the relation-

7 7 ship between the channel conditions and the application throughput. In this paper we mainly focus our attention to QoSobjectives and system revenue (or efficiency narrowly speaking). We leave the fairness problem to the network pricing or billing and call such a fairness as utility-fairness. The utility function decides the performance of the upper applications so we will study it in the future work. References [1] Hui Zhang, Service Displines For Guaranteed Performance Service in Packet-Switching Networks, Proceeding of IEEE, Oct. 1995, 83: [2] Vijay Sivaraman, Fabio M. Chiussi and Mario Gerla, End-to- End Statistical Delay Service under GPS and EDF Scheduling: A Comparison Study, Proceeding of IEEE INFOCOM, Apr. 21. [3] Yaxin Cao and Victor O. K. Li, Scheduling Algorithms in Braod-Band Wireless Networks, Proceedings of IEEE, vol.89, no.1, Jan. 21. [4] Songwu Lu, Vaduvur Bharghavan and R. Srikant, Fair Scheduling in Wireless Packet Networks, IEEE/ACM Transactions on Networking, Vol.7, no.4, Aug [5] T. Nandagopal, S. Lu and V. Bharghavan, A Unified Architecture for the Design and Evaluation of Wireless Fair Queueing Algorithms, in ACM Mobicom 99, Aug [6] T. S. Eugene Ng, Ion Stoica and Hui Zhang, Packet Fair Queueing Algorithms for Wireless Networks with Location-Dependant Errors, Proceedings of IEEE INFOCOM, [7] Charles Chien et al, Adaptive Radio for Multimedia Wireless Links, IEEE Journal on Selected Areas in Communications, vol.17, no.5, May [8] A. Sampath, P. S. Kumar and J. M. Holtzman, Power Control and Resource Management for a Multimedia CDMA Wireless System, PIMRC 1995, vol.1, pp21-25, [9] Majid Soleimanipour, Weihua Zhuang and George H. Freeman, Optimal Resource Management in Multimedia WCDMA Systems, Proceedings of IEEE GlebeCom 2, 2. [1] Paul Bender et al, CDMA/HDR: A Bandwidth-Efficient High- Speed Wireless Data Service for Nomadic Users, IEEE Communication Magazine, July 2. [11] A. Jalali, R. Padovani and R. Pankaj, Data Throughput of CDMA-HDR a High Efficiency-High Data Rate Personal Communication Wireless Systems, IEEE VTC, 2. [12] X. Liu, E. K. P. Chong, and N. B. Shroff, Efficient Scheduling for Efficient Wireless Utilization, Proceedings of IEEE INFO- COM 21, April 21. [13] Steven H. Low and David E. Lapsley, Optimization Flow Control I: Basic Algorithm and Convergence, IEEE/ACM Transactions on Networking, vol.7, no.6, Dec [14] William C. Y. Lee, Mobile Communications Engineering, by McGra-Hill Book Company, [15] Theodore S. Rappaport, Wireless Communications: Principles and Practice, by Prentice Hall, [16] Clarke, R. H., A Statistical Theory of Mobile-radio Reception, Bell Systems Technical Journal, Vol. 47 pp , [17] M. Gudmundson, Correlation model for shadow fading in mobile radio systems, Electronics Letters, vol. 27, pp , November 1991.