On the Effectiveness of Loss-conserving Disciplines under a Long-range Dependent Process

In this paper, we investigate the effectiveness of selective discard in a multiplexer subject to a long-range dependent process. We consider loss conserving disciplines, and we evaluate the impact of the input process traffic characteristics such as the Hurst parameter and variance into the per class loss rate and the loss gap. We found out that the Complete Sharing discipline is clearly worth adopting whereas Complete Sharing with Guaranteed Queue Minimum may be not. Furthermore, we show that the choice of push out policy may impact significantly the perceived QoS. 3 I) Introduction Several studies [1]-[4] have claimed that different types of network traffic, e.g. local area network traffic (LAN), can be accurately modeled by a self-similar process. Even though the selfsimilar nature of network traffic has not been fully understood, a self-similar process is able to capture the long-range dependence (LRD) phenomenon exhibited by this traffic. Moreover, a series of simulation and analytical studies [5]-[10] demonstrated that this phenomenon might have a pervasive effect on queueing performance, i.e., there is clear evidence that it can potentially cause massive cell losses in ATM networks. In fact, Norros [8] and Duffield [11] showed that the buffer overflow probability for an ATM queueing system with fractional Brownian arrivals follows a Weibull distribution. Furthermore, this queueing system suffers from the buffer inefficacy phenomenon [7], [12], i.e., by just increasing the buffer size we are not able to decrease the buffer overflow probability considerably. Different multimedia applications have diverse loss requirements. For instance, a telephone conversation my afford a loss rate of 10-3, whereas an MPEG video transmission may tolerate a loss rate of the order of 10-9. Coping with different loss requirements is a challenging task. Selective discard is a congestion control mechanism aimed at enabling the network to deal with diverse loss requirements [13], [14]. In a selective discard mechanism cells are discarded in an overflow situation according to their priority level. Selective discard has been studied in the past few years. However, most of these studies considered short-range dependent processes. In this paper, we investigate the effectiveness of selective discard mechanism in a multiplexer subject to a long-range dependent process. We consider loss conserving disciplines, and we evaluate the impact of the input process traffic characteristics such as the Hurst parameter and variance into the per class loss rate and the loss gap length [15]. We found out that the Complete Sharing discipline is clearly worth adopting, whereas Complete Sharing with Guaranteed Queue Minimum [16] may be not. Furthermore, we show that the choice of push out policy may significantly impact the perceived QoS. Our study is a first step in understanding end-to-end loss phenomenon in networks driven by realistic LRD processes [17], [18]. This paper is organized as follows: Section II introduces the Fractional Brownian Motion process. Section III explains the buffer inefficacy phenomenon. Section IV briefly introduces selective discard mechanisms. Section V describes the traffic generator used in this paper. In Section VI the main findings are discussed. Finally, conclusions are drawn in section VII. 4 II) Fractional Brownian Motion The ordinary Brownian motion, B(t), describes the movement of a particle in a liquid subjected to collisions and other forces [19]. It is a real random function with independent Gaussian increments such that Mandelbrot defines fractional Brownian motion (fBm) as being the moving average of dB(t) in which past increments of B(t) are weighted by the kernel (t-s)h 1/2 [20]. Definition: Let H be such that 0 < H < 1. The fBm is defined as the Weyl’s fractional integral of B(t). This equation leads to the ordinary Brownian motion if H = 1/2. Its self-similar property is based on the fact that BH(ρs) is identical in distribution to ρh * BH(s). The increments of the fBm, Yj form a stationary sequence called fractional Gaussian noise (fGn): We should note that these increments are not independent unless you have pure Brownian motion, i.e., H = 1/2. Moreover, Hurst law states that Var[BH(t+s) BH(t)]=σs, i.e., a fBm arrival model is also able to capture the inherent high variability exhibited by real network traffic. III) The Buffer Inefficacy Phenomenon The buffer inefficacy phenomenon is the queueing phenomenon in which by just increasing the buffer size, we are not able to decrease the buffer overflow probability considerably. This phenomenon has been reported earlier by several other studies [7],[12], [21]. In this section, we present a very intuitive explanation for it and show that it is of particular importance when the traffic source exhibits long-range dependence. We model an ATM node as a deterministic queueing system with constant departure rate given by c and finite buffer size given by b. The input traffic is given by the stochastic process A(t) with mean input rate . It defines the aggregate number of cell arrivals up to time . E B t s + ( ) B t ( ) – [ ] 0 = Var B t s + ( ) B t ( ) – [ ] σ s = BH t ( ) 1 Γ H 1 2 ⁄ + ( ) -----------------------------t s – ( ) H 1 2 ⁄ – s – ( ) H 1 2 ⁄ – –     B s ( ) d ∞ – 0 ∫ t s – ( ) H 1 2 ⁄ – B s ( ) d 0 t ∫ + = Yj BH j 1 + ( ) BH j ( ) j , – ... 1 0 1 ... , , , – , = =