A General Distribution Approximation Method for Mobile Network Traffic Modeling

Transcription

1 Applied Mathematical Sciences, Vol. 6, 2012, no. 23, A General Distribution Approximation Method for Mobile Network Traffic Modeling Spiros Louvros Department of Telecommunication Systems and Networks Technological Educational Institute of Messologhi Nafpaktos, Greece slouvros@teimes.gr Athanassios C. Iossifides Department of Electronics Alexander Technological Educational Institute of Thessaloniki Sindos, Thessaloniki, Greece aiosifidis@el.teithe.gr Abstract This paper addresses the problem of calculating the number of active users in a cell on time t, given the total number of users entering the system from the beginning of the counting process. This calculation is especially useful for estimating the active users after a certain period of time, in order to optimize scheduling decisions. In applied wireless networks, the service time most usually follows a general distribution which for some simplified models can be considered as negative exponential. This paper proposes a technique to approximate a general distribution by a Gaussian like distribution based on Hermitian polynomial expansion. Keywords: Network traffic models, general distribution, Hermitian polynomial expansion 1 Introduction Nowadays Long Term Evolution (LTE is becoming the first choice of operators when designing new mobile networks, because of its high throughput, low latency and high bandwidth efficiency that can be exploited using proper Medium Access Control (MAC scheduling. Therefore, a lot of papers in the

2 1106 S. Louvros and A. C. Iossifides literature propose scheduling algorithms and test them through simulations. For instance, in [1], a specific study on the conventional time domain Proportional Fair (PF algorithm is proposed in order to maximize proportional fair criteria in the frequency-domain setting. A performance comparison of two distinct scheduling schemes for LTE uplink (fair fixed assignment and fair work-conserving has been presented in [2], trying to consider packet level characteristics and flow level dynamics due to the random user behavior at the same time. Moreover, it is widely accepted that MAC scheduling optimization is service dependent. Based on this fact, a complete, practical and low complexity algorithm for multiclass traffic in LTE is presented in [3], while [4] proposes two novel schedulers that assume a joint and a separate implementation of scheduling and Adaptive Modulation and Coding (AMC schemes. Simulation modeling is the standard tool to test network performance and the vast majority of the work during the last years avoid a general theoretical approach of scheduling. Since throughput depends mostly on MAC scheduling and since scheduling depends mostly on the expected traffic in the system under minimum throughput constraints per user, simulation approaches undercover the basic mathematical principles of scheduling under long term traffic estimations. The aim of this paper is twofold. First, a pure mathematical approach is proposed in order to estimate analytically the expected number of users existing in the cell (system in a certain period of time for a given number of call arrivals. Having this estimation tool, an optimum scheduling algorithm could be designed provided that the MAC scheduler could estimate the expected capacity in the cell and the average user throughput leading to a more sophisticated performance over longer periods. However, it is obvious that the expected number of users depends also on the service time distribution. All previously mentioned papers suppose a general negative exponential service time distribution (see also [5]-[7]. Though, this is not always the case, since in LTE and other high speed networks the service time depends on the user behavior, cell sizes (due to user mobility and the type (class of service. In this paper, a general distribution is used in the corresponding analysis to provide a general tool for any kind of service behavior. In real networks this general service time distribution may be deduced from statistical analysis, e.g. [7]-[8] and may not follow any already known distribution. Based on Hermitian Polynomial expansion [9], a new method is proposed in order to fit the general service time distribution into a modified Gaussian distribution which is well known and thoroughly analyzed. 2 General traffic theory model Suppose that there is a cellular system with a stochastic process of call arrivals where N(t represents a counting procedure that defines the total number

3 A general distribution approximation method 1107 of calls (events which have occurred up to time period t. This stochastic process is also considered to be Poisson process, thus fulfilling by definition the following properties: (i lim t 0 N(t 0, (ii the process has stationary and independent increments, (iii the number of calls in any interval (s, t + s of length t has a Poisson distribution with mean λt, that is for any arbitrary time s: λt (λtm P {N(t + s N(s m} e, m! where m is the number of calls in the system. As a consequence the call arrivals at the system server will follow Poisson distribution with rate λ. We moreover consider that the cellular system behaves like an infinite server Poisson Queue system, where each call is immediately served by one of the infinite number of servers. Thus we study the system behavior under normal traffic conditions. Service time in the system is considered to follow a general distribution G(t. Assume that N(t n calls have arrived into the system up to time t and let T i denote the arrival time of the ith call, where i 1, 2,...,n. We partition the observation time interval [0,t]into(n+1 partitions 0 <t 1 <t 2 <...<t n <t, so that t i <T i <t i +Δτ i, where Δτ i is an infinitesimal time interval. Keeping in mind that only one call is arriving in each interval Δτ i, due to the Poisson nature of the process, the probability that n calls will arrive in the observation time interval can be expressed as: P {t 1 T 1 t 1 +Δτ 1,...,t n T n t n +Δτ n N(t n} P {t 1 T 1 t 1 +Δτ 1,...,t n T n t n +Δτ n, no calls in Ū}, P {N(t n} where U is the union of the intervals [t i,t i +Δτ i ],i1, 2,...,n and Ū is the union of the intervals remaining when subtracting U from [0,t]. Since the arrival events from Poisson process are independent -property (ii-, it follows that P {t 1 T 1 t 1 +Δτ 1,...,t n T n t n +Δτ n N(t n} P {t 1 T 1 t 1 +Δτ 1 } P{t n T n t n +Δτ n } P{no calls in Ū}. P {N(t n} (1 Using Poisson property (iii yields P {t i T i t i +Δτ i } λδτ i e λδτ i,i1, 2,...,n (2 P {no calls in Ū} e λ(t Δτ 1 Δτ 2... Δτ n, (3 and substituting for (2 and (3 in (1, results P {t 1 T 1 t 1 +Δτ 1,...,t n T n t n +Δτ n N(t n}

4 1108 S. Louvros and A. C. Iossifides λδτ 1e λδτ1 λδτ 2 e λδτ2 λδτ n e λδτn e λ(t Δτ 1 Δτ 2... Δτ n e λt (λtn t Δτ n 1 Δτ 2 Δτ n or P {t i T i t i +Δτ i,i1, 2,...,n N(t n} Δτ 1 Δτ 2 Δτ n t. n Taking the limits of the infinitesimal intervals Δτ i to zero yields P {t i T i t i +Δτ i,i1, 2,...,n N(t n} lim (Δτ 1,Δτ 2,...,Δτ n 0 Δτ 1 Δτ 2 Δτ n n P Δτ 1 Δτ 2 Δτ n t. n Therefore, the probability density function of the joint n call arrivals in the cellular system is f T1,T 2,...T n (t 1,t 2,...t n t n, 0 <t 1 <t 2 <...<t n <t However, following the service distribution G(t, some of these calls would have been already terminated. For a single call arrival the probability density function becomes f T1 (t 1 1, 0 < t t 1 < t and the probability π that an arbitrary call arrival out of total n is still active in the cellular system at time t is π t Setting u t τ, yields π t 0 0 (1 G(t τf T1 (t 1 dτ 1 G(t τ t t 0 1 G(t τ t dτ. 0 1 G(u t 1 G(τ dτ du dτ. (4 t t 0 t The number of active calls in the cellular system at time t is a stochastic process A(t. The sample space of call arrivals S {N n N(t :n 1, 2,...} is a finite countable space with infinite partitions, as a result S {N 1 N 2 N 3...} and the probability that k active calls exist in the cellular system is calculated by the total probability law: P {A(t k} P {A(t k N(t n}p {N(t n} n0 λt (λtn P {A(t k N(t n}e. (5 n0

5 A general distribution approximation method 1109 For any possible combination of getting exactly A(t k active call arrivals in total time t with total success probability π k out of total n call arrivals and n k terminated calls with probability (1 π n k into the cellular system, we engage the Binomial distribution: { ( n P {A(t k N(t n} k π k (1 π n k k 0, 1, 2,...,n 0 k>n (6 Substituting for (5 and (6 in (4 yields or P {A(t k} n0 nk λt (λtn P {A(t k N(t n}e e λt π k nk e λt (λtπ k ( n π k (1 π n k λt (λtn e k (n k! [λt(1 π] n k (n k! nk P {A(t k} e λt (λtπ k which, due to e x x i i0, results to i! (1 πn k (λtn [λt(1 π] i, i (i! λtπ (λtπk P {A(t k} e t (λt 1 G(τ e λt t 1 G(τ dτ 0 t 0 t dτ k (7 3 General distribution approximation In most applications, distribution functions are not well known or well defined, and they do follow from specific statistical characteristics that could not be defined into the well know pdf functions. Such a case is definitely the service time in a cellular system, where for most of the applications we do consider to be either exponential or Poisson defined. However this is not the case in modern nowadays wireless data networks where multi-services are considered to be the dominant parallel traffic. In such networks cumulative distribution functions G(t and the corresponding pdf g(t d G(t of service time is not following dt any specific distribution, rather it is better described by general distributions. However, we could still use some interesting results from functional analysis and statistical theory and approximate such general distributions with the well

6 1110 S. Louvros and A. C. Iossifides known and studied Gaussian distribution with mean value μ and variance 2, with a pdf given by 1 (t μ2 ϕ(t e 2 2. (8 2π 2 The two important characteristics of Gaussian distribution are the single maximum value and that both sides of distribution extend to infinity as time reaches infinity. Considering that a general distribution g(t is a converging function with strict property g(t 2 dt <, given a Gaussian distribution ϕ(t with mean μ and variance 2 and the orthogonal Hermitian polynomials H v (t, we can use following result from functional analysis to approximate the unknown or general described distribution by a modified Gaussian distribution as lim g(t ϕ(t N ( β ν t μ N ν0 ν! H 2 ν ν dt 0, (9 which implies that g(t lim ϕ(t N β ν N ν0 ν! H ν ν, (10 thus providing an approximation of any pdf function with a modified Gaussian distribution based on orthogonal Hermitian polynomials. Hermitian polynomials for a pdf normal distribution function ϕ(z are defined as H n (zϕ(z ( d dz n ϕ(z, where H n (z ( 1 n e z2 /2 dn /2 dz n e z2, n 0, 1, 1,...,. (11 The orthogonality of Hermitian polynomials for a cumulative distribution normal function Φ(x states that: H n (zh m (z dφ(z δ mn H n (zh m (zϕ(z dz δ mn (12 which then implies that the weights on expansion (9 are calculated as a time average on the general pdf g(t, that is, ( t μ β ν ν g(th ν dt (13 The definition of Hermitian polynomials (11, it is proven that H 0 (z 1, H 1 (z z, H 2 (z z 2 1, H 3 (z z 3 3z, etc., implies that H 0 ( t μ 1, H 1 ( t μ t μ, H 2( t μ (t μ 2 1( t2 2tμ+μ 2 etc. Then, following the 2

7 A general distribution approximation method 1111 definition of (13, the weight coefficients of the expansion could be calculated as: β 0 0 β 1 1 β 2 2 ( t μ g(th 0 g(th 1 g(th 1 dt dt g(t dt 1 dt 2 g(t dt 0 ( t 2 2tμ + μ g(t dt 0 Thus, using (10, we can approximate the unknown pdf with a finite number N of terms, instead of an infinite number of terms, as g(t d dt G(t ϕ(t {1+ N ν3 [ 1 ν! ( ] t μ g(th ν dt H ν }, introducing a small error. Therefore, the generalized distribution can be approximated by G(τ τ 0 { ϕ(x 1+ N ν3 [ 1 ν! ( ] z μ g(zh ν dz H ν ( x μ } dx (14 Eventually, the probability that k active calls exist in the cellular system could be calculated for any general service distribution G(t by substituting (14 into (7. References [1] S-B. Lee, I. Pefkianakis, A. Meyerson, S. Xu and S. Lu, Proportional Fair Frequency-Domain Packet Scheduling for 3GPP LTE Uplink, Proc. of IEEE INFOCOM, Rio de Janeiro, Brazil, (2009, [2] D.C. Dimitrova, H. van den Berg, R. Litjens, and G. Heijenk, Scheduling Strategies for LTE Uplink with Flow Behaviour Analysis, Proc. of 4th ERCIM Workshop on emobility, Lulea, Sweden, (2010, [3] B. Sadiq, R. Madan, and A. Sampath, Downlink Scheduling for Multiclass Traffic in LTE, EURASIP Journal on Wireless Communications and Networking, (2009, doi: /2009/ [4] M. Assaad and A. Mourad, New Frequency-Time Scheduling Algorithms for 3GPP/LTE-like OFDMA Air Interface in the Downlink, Proc. of IEEE Vehicular Technology Conference (VTC, Singapore, (2008,

8 1112 S. Louvros and A. C. Iossifides [5] F. Barcelo, J. Jordan, Channel Holding Time Distribution in Public Cellular Telephony, Elsevier Science B.V., (1999, [6] T.K. Christensen, B.F. Nielsen, and V.B. Iversen, Phase-Type Models of Channel-Holding Times in Cellular Communication Systems, IEEE Transactions on Vehicular Technology, 53, 3, (2004, [7] E.A. Yavuz and V.C.M. Leung, Modeling Channel Occupancy Times for Voice Traffic in Cellular Networks, Proc. of IEEE International Conference on Communications (ICC, Glascow, Scotland, (2007, [8] A. Jayasuriya, D. Green, and J. Asenstorfer, Modelling Service Time Distribution in Cellular Networks Using Phase-Type Service Distributions, Proc. of IEEE international Conference on Communications (ICC, Beijing, China, (2001, [9] J.L. Lopez and N.M Temme, Hermite Polynomials in Asymptotic Representations of Generalized Bernoulli, Euler, Bessel, and Buchholf Polynomials, Journal of Mathematical Analysis and Applications, 239, 2, (1999, Received: June, 2011