CIPP — a versatile analytical model for VBR traffic

Abstract

Correlated interarrival time Poisson process (CIPP) has been proposed in Proceedings of the Fifth Biennial Conference on Signal Processing and Communications (SPCOM’99), IISc, Bangalore, July 1999, pp. 43–50; J. Indian Inst. Sci. 79 (3) (1999) 233–249] for modeling both the composite arrival process of packets in broadband networks and the individual video source modeling. The CIPP — a generalization of the Poisson process — is a stationary counting process and is parameterized by correlation parameter ‘ρ’, the degree of correlation in adjacent interarrivals and ‘λ’, the intensity of the process. In this paper, we develop the theory for CIPP/M/1 queue and undertake the performance modeling of statistical multiplexer with VBR video traffic in broadband networks using the CIPP/M/1 queue. We first derive the expressions for stationary distributions for queue length and waiting time in a CIPP/M/1 queue. Then, we derive the queuing performance measures of interest. For reasons of feasibility of theoretical performance modeling and realistic compulsions, we propose a deterministic smoothing with random (geometrically distributed) packet sizes. We simulate a queue with (thus smoothed) VBR video trace data as input to compare with the theoretical performance measures derived above. Experimental results show that the CIPP/M/1 queue models well the statistical multiplexer performance with the real-world MPEG-1 VBR video traffic input.

Introduction

Digital video communications (like video phone, video conferencing, television distribution, etc.) form a major class of services provided by broadband networks. Accordingly, the video transport over broadband networks is of significant interest. Most of the video encoding is done using the MPEG standards. The output of the encoder in MPEG systems can be either constant bit rate (CBR) or variable bit rate (VBR). A constant quality video signal is delivered at the receiver by VBR, but at the cost of complex transmission. Broadband network like ATM takes advantage of the bit rate variations of individual sources by statistical multiplexing. In this way, the individual (video) sources share a link of capacity less than the sum of individual peak rates, achieving significant multiplexing gain. This multiplexing gain is restricted by the quality of service (QoS) requirements of the sources. Thus statistical multiplexing could lead to more efficient use of network resources, but may require new kinds of bandwidth management, QoS provisioning and traffic control.

Statistical multiplexing involves N input streams (which are asynchronous and statistical) and N output streams, so that traffic from any of the N input streams can be switched over to any of the N output streams. Now each of the N output streams can be modeled as a single server queue, with the arrival streams forming the superposition of N input streams in general. Accordingly, this single server (with such arrival streams) has been of interest to telecommunication researchers over the years. Hence, understanding the nature of traffic in such heterogeneous multiservice networks is essential for engineering, operations and performance evaluation of these networks. Moreover, to facilitate the mechanization of multiplexing, routing and control, it is imperative to appropriately model and characterize the input traffic and to map the input source parameters (could be mean bit rate, peak bit rate or it could be even Hurst parameter (H) which measures the degree of self-similarity) into network parameters (could be mean cell delay, cell loss rate or delay jitter). In light of the above, an important question often raised by network planners and designers is: “Is there an accurate and useful traffic model in the form of a simple stochastic process with minimal parameters and when fed into the single server queue, gives the same queuing performance as a real traffic stream?” Such a traffic model will be very useful in network design tools or in tools supporting real-time traffic management. We intend here to use the fact that the interarrival correlation plays a major role in imitating the self-similar behavior in increments of the corresponding counting process and to make a contribution towards this key aspect of video traffic modeling.

Two classes of traffic models have been proposed in literature, for modeling the bit rate of a VBR video source, namely, short range dependent (SRD) models [3], [4], [5], [6] and long range dependent (LRD) (or self-similar) models [7], [8]. SRD models addressed in the literature, in turn, broadly fall into two categories: single source models [3], [4], [5], [9] and multiplexed source models [3], [6]. For low motion video telephony and medium-motion video conferencing, there exist the autoregressive (AR) model [3], [5], the discrete autoregressive (DAR(1)) model [4] and the discrete state continuous time Markov model [3], [5]. For full motion MPEG video source, a three class AR model with time-varying coefficients has been developed in [9]. Xu et al. [10] proposed a GAR(1) model for VBR traffic. In [11], a model based on arrival rate histograms of VBR traffic has been proposed and a quasi-static approximation is made for buffer occupancy distributions and cell loss rates. Most of the SRD models proposed for VBR traffic are analytically intractable [4], [6], [9], [11]. Moreover, those SRD models which are analytically tractable [3], [5] predict the queuing performance measures over optimistically. Note that the LRD (or self-similar) models also lack analytical tractability, except that few bounds are obtained for some queuing measures [12], [13].

The above models are intended to capture the correlation in the counts sequence (cells per frame) by a DAR(1) model as in [4] or in the rate process as in ARMA models [3], [4] or by the modulating Markov chain (MC) as in SRD models. In models based on self-similarity (or LRD) or AR models, correlations are in rate process (equivalently cells per unit time). In [1], [2], we proposed a model called correlated interarrival time Poisson process (CIPP) for composite traffic modeling and individual video source modeling. We argued that it is more appropriate to consider a model which captures the correlation in the interarrival sequence rather than the correlation in counts sequence as in the above models. In CIPP, the interarrivals form a first order Markov sequence. In [1], [2], it is shown that the CIPP does exhibit self-similarity over a range of time scales of interest. It is also analytically tractable (see, for instance, [14], [15]). In this paper, after giving a brief account of CIPP, we derive the stationary distributions in a CIPP/M/1 queue and then undertake the performance modeling of a statistical multiplexer with VBR video traffic input using the CIPP/M/1 queue. The rest of the paper is organized as follows. In Section 2, we give a short mathematical account of the CIPP process. Section 3 gives analytical results for the CIPP/M/1 queue. In Section 4, we present the performance modeling of VBR traffic using CIPP/M/1 queue. In Section 5, simulation results corresponding to MPEG-1 real-world VBR traffic traces are given. Section 6 concludes the paper.