Throughput Analysis of the Proportional Fair Scheduler in HSDPA

Transcription

1 Throughput Analysis of the Proportional Fair Scheduler in SDPA Gábor orváth Csaba Vulkán Abstract igh Speed Downlink Packet Access SDPA provides increased user data rate by introducing additional functionalities at the Node B, such as fast packet scheduling based on the instantaneous radio link quality. One of the algorithms that is able to provide this capability is the proportional fair P-FR scheduler. In this paper we describe an approximate analysis method for the mean throughput of the proportional fair scheduler with two traffic classes. We extend the existing results that are based on the assumption of continuous rate distributions to the more realistic discrete distributions. The analysis with discrete distributions entails two problems that do not appear in the continuous case, namely that the priority of several users can be equal with a non-zero probability and that the sample path of throughputs does not always converge to a mean throughput. The proposed approximate method is evaluated with NS simulations. Index Terms Proportional Fair Scheduling, SDPA, 3.5G I. Introduction SDPA igh Speed Downlink Packet Access [] is an evolutionary step of the 3GPP 3rd Generation Partnership Project technology that enables the deployment of packet based services over cellular networks. The aim of SDPA is to decrease the radio layer latency, increase the system capacity, peak data rate and throughput achievable by the end users the maximum SDPA data rate is 4.4 Mbps and to improve the overall performance of the packet based services web browsing, file transfer, etc. over the radio access system. These improvements are achieved by introducing additional functionalities at the Node B, that are enabling Node B controlled fast adaptation of the modulation and coding scheme, fast packet scheduling and retransmission handling with the ybrid ARQ ARQ functionality. The fast, fixed Transmission Time Interval TTI based packet scheduling at the Node B is responsible of granting access to the radio interface resources. The SDPA packet scheduler should provide reasonable level of fairness to the users and at the same time it should keep the radio interface under optimal load. It can follow a round robin policy that ensures equivalent head-of-line delay for each SDPA user, or other, more sophisticated algorithms that are able to consider the traffic type and the instantaneous channel quality as well upon scheduling decisions. A large The research work of Gábor orváth is partially supported by the NAPA-WINE FP7 project. The content of this paper has been developed in cooperation with Nokia Siemens Networks. G. orváth is with the Department of Communications, Budapest University of Technology and Economics, Budapest. ghorvath@hit.bme.hu Cs. Vulkán is with the Nokia Siemens Networks, Budapest. csaba.vulkan@nsn.com number of scheduling algorithms that are addressing S- DPA have been published [], some of these are trying to increase the level of the radio interface usage as much as possible, while others intend to provide some level of QoS to the users based on requirements like maximum delay. One of the most investigated algorithms that is considered a good solution for SDPA fast packet scheduling is the proportional fair scheduling algorithm P-FR. Several papers appeared that are providing analytical results for the mean throughput of the P-FR scheduler with various restrictions. In [3], [4] an iterative solution is given assuming that there are two traffic classes and the instantaneous rate is a continuous random variable. In [5] the authors provide a closed formula for the mean throughput for multiple traffic classes, but the instantaneous rate is assumed to be exponentially distributed. In [6] a deep analysis of P-FR schedulers is discussed, but the mean throughput is provided only if the differential equation describing the throughput change has a solution. In this paper we show with examples that if the instantaneous rate is a discrete variable as it is in case of SDPA i.e. the number of packets that can be transmitted during one TTI then the mean throughput can oscillate in some cases. We provide a solution that approximates the mean throughput in these cases. The rest of the paper is organized as follows. Section II gives a brief introduction of the proportional scheduler and introduces notations. The way how this scheduler can be used in case of SDPA is also described in this section. Section III summarizes the results of the published papers regarding to the P-FR scheduler with continuous instantaneous rate. Section IV provides the main contribution of the paper with the approximate analysis of the P-FR scheduler in the discrete case. In Section V we evaluate our algorithm and check how well it can be used for SDPA throughput approximation. concludes the paper. Finally, Section VI II. The Proportional Fair Scheduler A. The Basic Algorithm The proportional fair algorithm is a packet scheduling method that provides reasonable level of fairness to the users and in the same time it is able to achieve high overall throughput. At each scheduling instance, the priority of user i, R i, is calculated based on the user s instantaneous and measured average throughput: R i = N i, i=...m T i In the formula N i is the instantaneous rate and T i is the measured mean throughput of user i. The cdf of N i is de-

2 noted by F i x, F i x =P N i <x. The corresponding probability density function is denoted by f i x. At each scheduling instance the P-FR secheduler selects for transmission the user having the highest priority. B. P-FR Scheduler in SDPA systems In case of SDPA systems, the packet scheduler is located at the Node B. In this way, fast scheduling decisions can be made based on the instantaneous radio link conditions of the users and resources can be granted to the user with the best channel quality. The scheduling is done at each ms TTI. In this work it is assumed that a separate buffer is allocated to each user at the Node B. Furthermore, it is assumed that the SDPA flow control algorithm [7] is able to achieve its purpose by keeping the buffer of each user at optimal level so that the air interface capacity is not wasted and the delay on Node B buffer is not too high. The air interface capacity is not wasted if there is enough pending data in the scheduled user s buffer thus the user is able to benefit from having favorable channel conditions. The scheduler uses the reports received from the user equipment about the observed channel quality. Based on this instantaneous channel quality information CQI, the available transmission power for the user s power controlled downlink physical channel, and the amount of available codes the modulation and coding scheeme is selected. Eventually this defines the amount of data number of packets N i n user i is able to send over the next TTI. The scheduler selects the user that maximizes the relative CQI defined as follows [8]: RCQI i n= N in T i n = min{n in,b i n}, T i n T i n is the measured average throughput of user i in the past, n is the scheduling interval and B i n is the number of packets waiting in the buffer allocated to user i. In this paper we assume fixed packet size, that is the size of the MAC-d PDUs referred to as packets scheduled by the P-FR algorithm is considered fixed. Furthermore, it is assumed that in case of active users B i n >N i n, i.e. there are always enough packets waiting in the scheduled user s buffer. The measured average throughput T i n is calculated by a moving average with a supporting window, R, with the following formula: T i n= I {Bin>} T i n + R R N in, 3 where N i n is the actual number of packets transmitted by user i in the nth scheduling interval. The multiplication by the indicator I {Bin>} ensures that the priority of user i does not increase during the silent periods. III. Continuous Instantaneous Rate Distribution As the P-FR is scheduling packets, the instantaneous rate N i is a discrete quantity, that depends on the number of packets transmitted over one TTI. The research results published so far are all investigating the case when instantaneous rate is continuous. Before we describe our discrete analysis results, we summarize the common analysis techniques of the continuous case, because our method is based on similar ideas. In [6] it is proven that if the instantaneous rate follows a continuous distribution, the measured mean throughput T i converges to a unique point ET i that equals the mean throughput of user i. Note that the convergence does not necessary apply when N i has a discrete distribution, see the next section. The mean throughput ET i can be calculated as follows. User i can send x amount of data in a given time slot with density f i x, if its priority is the highest in that time slot. Thus, assuming the users are independent, we have: ET i = = x x f i xp ET i > N ET,..., x ET i > N m ET m dx F j x ET j dt ET i x f i x j i In this way we have m equations for i =,...,mto calculate the m unknowns ET i. This set of equations has a closed form solution when F i x is an exponential distribution, see [5], but obtaining a closed form solution in more general cases is not possible. In this section we recall the numerical method published in [4] to find the solution, if there are two user classes in the system. The instantaneous rate of class, its cdf and pdf are denoted by N N,F xf x,f x, f x, respectively. The number of users in class is m m. By specializing equation 4 and introducing the ratio of throughputs = ET /ET wehave: ET = 4 xf xf x m F x m dx 5 If ET is expressed similarly, and divided by ET, the following equation can be created for : where: g= = g, 6 xf xf x m F x/ m dx xf xf x m F x m dx It is easy to see that the only unknown in eq. 6 is. If we could find its solution, the mean throughputs of the users could be computed using eq. 5. In [4] this equation is solved by a fixed point iteration. Starting with =, the following iteration computes n from n : 7 n = g n 8 But contrary to the statement of the author in [4], this method does not always find the solution. An important

3 criterion for such fixed point methods is that the derivative of the function around the solution has to be greater than, criterion that does not hold in several cases. For example when N is uniformly distributed over, thus f x =.,F x =.x, x and N is exponentially distributed with parameter λ =. m = m =. Figure depicts g for this example. The solution of the equation 6 is at the intersection of the two plots, but the fixed point iteration method diverges and oscillates between and. Fig.. g = g Ratio of the throughputs g As a conclusion, we recommend to use an interval bisectioning-based numerical method instead of the fixed point iteration to compute the ratio of the mean throughputs. aving computed, the mean throughput of class users is calculated by eq. 5. IV. Discrete Instantaneous Rate Distribution As mentioned before, there are no published results available when the instantaneous rate distribution is discrete, although it is a typical situation in several practical applications. Such a case is the SDPA scheduler, where N i is the number of packets the Node B can transmit in a TTI. This section describes the problems appearing during the analysis of the discrete case and provides an approximation method for the mean throughput if there are two traffic classes. If N i is a discrete random variable, we denote its distribution function by p i k=pn i = k, and its cumulative distribution function by P i k=pn i <k. In SDPA, N i depends on the reported channel quality based on which the amount of bytes frame size that can be sent to the user is defined. Thus if the size of packets is given, N i equals to number of packets that can be fitted to this frame. Contrary to the continuous case, in the discrete case T i does not necessarily converge to ET i asymptotically see [6]. According to our experience with the simulation of the system, we found that in some cases the measured throughput T i converged to ET i, but in some cases it showed a kind of oscillation as simulation time passed instead of the convergence. This oscillation is the feature of the system the reason of this behaviour will be clear later in this section. The window size in which the throughput average is measured determines the amplitude of the oscillation. With larger window size the amplitude of the oscillation decreases but the frequency decreases as well. Unfortunately we could not derive explicit formulas for the mean throughput for the discrete case. We present an approximation for the two class case in this section. The approximation is based on the assumption that the averaging window size is sufficiently large, such that we can approximate the possibly oscillating T i variables by a constant ET i with a reasonable accuracy. The idea of the calculation is similar to the continuous case. We introduce the ratio of the mean measured throughputs = ET /ET, and compute the mean throughputs ET andet byassumingafixed value. The computation of the mean throughput is more complex compared to the continuous case. Since ET and ET are assumed to be fixed, it is possible that several users belonging to the same class have the same priority value R i this is the case if their instantaneous rates are equal. If the number of users having the same priority is a, the user that is selected to be scheduled at the time slot is selected with probability /a if this priority value is the largest among the users. If or / is an integer, it is even possible that several class and class users have the same priority. In this case the scheduled user selection is done by a random choice. Let us express the throughput of a tagged class i customer in the simpler case, thus if the ratio of measured throughputs is not an integer. In this case the tagged customer can transmit k amount of data with probability p i k, if its priority is higher than all class j, j i customers second term of eq. 9, cf. eq. 4 and if its priority is higher than the other class i customers first term of eq. 9. If there are a other class i customers having the same priority, the tagged one is selected with probability / + a. Thus, for i, j {, ;, } we have: ET i = = kp i k k= m i mi p i k a P i k mi a a a= }{{} the tagged is selected from class i P j k i mj }{{} the class i user has priority over the other class +a 9 If is an integer, we have to take into consideration that some class j customers can have the same priority, too. In this case we have: ET i = kp i k k= m j mj b= b m i mi p i k a P i k mi a a a= p j k i b P j k i mj b +a + b,

4 where i is defined as follows: { = ET /ET, if i =, i = / = ET /ET, if i =. If we would like to follow the same idea as in the continuous case and define g=et /ET, the ratio of the mean throughputs could be computed by solving the equation of = g. The problem is that it is possible that no value satisfies this equation. This is because P i k is a discrete cumulative distribution function, thus P kandp k/ take the same value at several settings. Figure depicts g as a function of. Itcan be seen from the figure that curves and g haveno common points. When = g has no solution, the trajectory of T i does not converge to ET i, but it necessarily oscillates around the equilibrium denoted by in the figure. The reason of this interesting phenomenon is the following. When the measured mean throughputs are such that <, then the scheduler begins to prefer class customers over class customers, that increases the ratio of the measured mean throughputs. At a given point will be larger than, and the behaviour will be the opposite: the scheduler will prefer class customers over class, that decreases. Since the instantaneous rate is discrete, both the increase and the decrease of is performed in discrete steps. When the equation g = has no solution, the ratio of throughputs can never take the value needed for equilibrium =. In this case has an oscillating behaviour around as time passes. Although there is no solution of equation = g, we can find the value at which the jump in g occurs that crosses denoted by. The mean throughput of the users is approximated as follows. The two values of g that are closest to the solution are denoted by and + see figure. We calculate the throughput of class i users by assuming a mean throughput ratio of and + by using eq. 9,, and denote the results by ET i and ET i +, respectively. The mean throughput ET i is then the average of these two conditional throughputs weighted by the distance between + and : ET i = + ET i ET i + V. Numerical Results In this section we evaluate the applicability of the presented analytical method to the throughput analysis of the P-FR scheduler in SDPA. A. Assumptions In the approximate analytical method described in this paper we have introduced simplifications by not considering aspects of SDPA systems that might affect the throughput. Such an aspect is that in practice the instantaneous rate that depends on reported CQI, available power and code resources is not an i.i.d. variable, but has Fig.. g g g a Ratio of throughputs + g Ratio of the throughputs b Magnified a correlation structure due for example to the shadow fading. Another aspect that affects the throughput is that the sources are not saturated. In reality, SDPA has been designed for data connections that are mostly using TCP as transport protocol. The TCP flow control adapts its rate according to the available transport bandwidth. The transport network capacity connecting the RNC to the Node B is a limited resource that is shaping the SDPA traffic. The SDPA flow control is not perfect in the sense that it is not able to continuously feed the air interface with packets. There are idle times when there is no data to transmit. In our analysis we consider only the active users that are having packects waiting at the Node B. These aspects have not been taken into account in the mathematical model, thus we assume saturated traffic sources and perfect flow control algorithm. In our idealized analytical model it is assumed that there is always at least one available -ARQ entity. According to our best knowledge, these assumptions are applied in all papers about analytical P-FR modelling. B. Parameters of the Numerical Example The analytical results have been evaluated with simulations. Two user profiles have been considered: Pedestrian and Vehicular.

5 Pedestrian Vehicular Profile Ped-A Veh-A Speed 3kmh 5 kmh Distance 3 m 5 m Trace length 9 s 9 s Avg. # of packets/ ms The radio channel condition has been modeled separately for each user profile. For each profile a radio channel trace file has been generated with MATLAB scripts of the Eurane project [9] based on the assumptions below. These trace files have been used by both analytical and simulation models. The average number of scheduled packets per TTI shown in the table has been calculated assuming only one user at the Node B, and that the scheduler is able to schedule this user at each TTI. The CQI estimation error is modeled with a constant delay of 6 ms []. Users are modeled with ITU-T Pedestrian A and Vehicular A model, assuming that chase combining is implemented in the user equipments. The Signal to Noise Ratio SNR has been calculated considering the following: distance loss according to Okumara-ata model for urban cell with base station antenna height of 3 m, mobile antenna height of.5 m and carrier frequency of 95 Mhz []; multi-path fast fading; Rake receiver assuming that channel estimation is ideal and the power levels of all paths are known; shadow slow fading log-normal distribution correlated in time []; constant Node B antenna gain 7 dbi; intercell interference -7 dbm and intra-cell interference 3dBm. The number of S-DSC igh Speed Dedicated Shared Channel codes was set to five. The simulation assumes user equipment category 5-6 that means that the maximum rate the user can achieve is 3.6 Mbps. As mentioned above, the analytical results have been evaluated with two sets of simulations. These two sets of simulations correspond to different level of detail regarding the radio access system with SDPA: Simulation I. In the first set of simulations the effect of shadow fading and the real, non-saturated traffic behaviour was neglected. From the trace files the distribution of the number of packets that could be transmitted for a user over the ms TTI has been created. This distribution was used in both the analytical and the simulation models based on the assumption that the air interface behaves in an i.i.d. manner. Thus, the number of transmittable packets has been defined by a random choice at each TTI according to the distribution extracted from the trace. Simulation II. In the second set of simulations the effect of the shadow fading has been included, but the traffic still was considered saturated. In this case the simulator used the trace files directly. To avoid the synchronization between the traffic sources, the connections have been started randomly. Twenty simulations have been executed with different random shifts of the trace files, and the results were averaged. C. Evaluation with the Simulation Models The analytical model and the Simulation I. are using the distribution of the number of packets that a user could transmit at each ms TTI. The distribution can be calculated based on the CQI, the available Node B power and amount of S-DSC codes as in addition to the scheduling decision at each TTI the Node B is also defining the amount of data that can be sent. The distributions corresponding to the two user classes are shown in Figure 3. Fixed sized packets MAC-d PDU size = 336 bits including Radio Link Control Protocol header have been assumed. The transport block size the amount of the data that can be transmitted depends on the coding and modulation scheeme used at the transmission and on the available power and codes. The number of 336 bit packets that can be transmitted should fit into the transport block size. There are packet numbers that do not correspond to any coding modulation schemes, in this case the probabilities corresponding to these packet numbers are zeros. As the plot reflects, the parameters of the air interface profiles has been selected such that these distributions correspond to a definitely better pedestrian and worse vehicular air interface conditions. Fig. 3. Probability Probability # of MAC-d PDUs a Pedestrian user 5 5 # of MAC-d PDUs b Vehicular user Distribution of the number of MAC-d PDUs/TTI During the analysis, the sample path of the throughput was not always converging, thus in some cases = g had no solution, Figure 4 shows that with one user from both classes the solution thus, the equilibrium exists, while with three users it does not exist, meaning that the approximation given by eq. has to be applied.

6 g g Mean throughput [# MAC-hs PDUs/TTI] class Analysis class Analysis class Simulation II. class Simulation II. Fig. 4. g g vs. a One user g b Three users from both classes Figure 5 compares the results of the analysis and Simulation I. The two curves match almost perfectly, the difference is less than two percent. Mean throughput [# MAC-hs PDUs/TTI] class Analysis class Analysis class Simulation I. class Simulation I Number of vehicular users Fig. 5. Mean throughput, analysis vs. simulation I. Figure 6 compares the results of the analysis and Simulation II. It is clear that the shadow fading thus, when the instantaneous rates are correlated does indeed affect the throughput, that the analytical method can not capture. The accuracy of the approximation is still reasonable, the error of the analytical method is eight percent in the worst case. VI. Conclusion In this paper we described an approximate analysis method for the mean throughput of the proportional fair Fig Number of vehicular users Mean throughput, analysis vs. simulation II. scheduler with two traffic classes. In the provided method we propose a simple iterative method to find the throughput around which the sample path of the measured mean throughput oscillates. The paper contains a detailed study in order to evaluate how the different modeling assumptions are affecting the accuracy of the approximation when it is used in the context of SDPA. As a possible future work, further research has to be done to develop a method that takes the effect of shadow fading into consideration. References []. olma and A. Toscala, SDPA/SUPA for UMTS, John Wiley & Sons, 6. [] A.R. Braga, E.B. Rodrigues, and F.R.P. Cavalcanti, Packet scheduling for VOIP over SDPA in mixed traffic scenarios, in Personal, Indoor and Mobile Radio Communications, 6 IEEE 7th International Symposium on, Sept 6, pp. 5. [3] J.M. oltzman, CDMA forward link waterfilling power control, in Vehicular Technology Conference Proceedings,, pp [4] J.M. oltzman, Asymptotic analysis of proportional fair algorithm, in th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications,, pp [5] Jin-Ghoo Choi and Saewoong Bahk, Cell throughput analysis of the proportional fair scheduling policy, in NETWORKING 4, Nikolas Mitrou, Kimon Kontovasilis, George N. Rouskas, Ilias Iliadis, and Lazaros Merakos, Eds., Athens, Greece, 4, vol. 34 of LNCS, pp [6].J. Kushner and P.A. Whiting, Convergence of proportionalfair sharing algorithms under general conditions, Wireless Communications, IEEE Transactions on, 4. [7] M. C. Necker and A. Weber, Parameter selection for SDPA Iub flow control, in Proc. nd International Symposium on Wireless Communication Systems ISWCS 5, Siena, Italy, 5. [8] T. E. Kolding, Link and system performance aspects of Proportional Fair scheduling in WCDMA/SDPA, in Proceedings of 58th IEEE Vehicular Technology Conference VTC, Florida, USA, 3, pp [9] Eurane, The eurane project, 4, []. van den Berg, R. Litjens, and J. Laverman, SDPA flow level performance: The impact of key system and traffic aspects, in Proceedings of the 7th ACM International Symposium on Modeling, Analysis and Simulation of Wireless and Mobile Systems, Venice, Italy, 4. [] F. Brouwer, I. de Bruin, J. C. Silva, N. Suoto, F. Cercas, and A. Correia, Usage of link-level performance indicators for S- DPA network-level simulations in E-UMTS, in Proceedings of IEEE ISSSTA 4, Sydney, Australia, 4.