Abstract - We study the performance of a statistical multiplexer whose inputs consist of a superposition of packetized voice sources and data. The performance analysis predicts voice packet delay distributions, which usually have a stringent requirement, as well as data packet delay distributions. The superposition is approximated by a correlated Markov modulated Poisson process (MMPP), which is chosen such that several of its statistical characteristics identically match those of the superposition. Matrix analytic methods are then used to evaluate system performance measures. In particular, we obtain moments of voice and data delay distributions and queue length distributions. We also obtain Laplace-Stieltjes transforms of the voice and data packet delay distributions, which are numerically inverted to evaluate tails of delay distributions. It is shown how the matrix analytic methodology can incorporate practical system considerations such as finite buffers and a class of overload control mechanisms discussed in the literature. Comparisons with simulation show the methods to be accurate. The numerical results for the tails of the voice packet delay distribution show the dramatic effect of traffic variability and correlations on performance.
While the problem was motivated by the desire to analytically gain insight into the performance of an integrated voice/data network, the techniques developed are more broadly applicable. The methodology presented here provides new insights and can also be viewed as a step in the direction of reducing the dependence on simulations, which can be expensive and are typically used in studying performance issues for this class of problems. In addition, the analysis enables one to obtain low probability tails and high percentiles' of distributions, which cannot be obtained with simulations.
The input process to the statistical multiplexer is a fairly complex process and can possess correlations, in the number of arrivals in adjacent time intervals, which can significantly affect queueing performance. These correlations result from the fact that the aggregate voice packet arrival rate is a modulated process obtained by the individual voice source packet rate by the number of voice sources in their talk spurt, which is itself a correlated process. Even if a component voice process is approximated as a renewal process, with detenninistically spaced packets during a talk spurt followed by an exponentially distributed silence period, the superposition process is a complex nonrenewal process. Exact analysis of systems to which this superposition process is offered is intractable, especially when such systems contain finite buffers and overload control mechanisms.
The approach we take in this paper is to approximate the aggregate arrival process by a simpler, correlated, nonrenewal stream, which is modulated in a Markovian manner. The approximating stream is chosen such that several of its statistical characteristics identically match those of the original superposition. In choosing the approximating stream we are driven by the need for analytic simplicity as well as the desire for versatility. A natural choice is the Markov modulated Poisson process (MMPP), which is a doubly stochastic Poisson process where the rate process is detennined by the state of a continuous-time Markov chain. This process has previously been used to accurately approximate a superposition of packet arrival processes for a related problem . One advantage of our characterization of the superposition of voice sources and data as an MMPP is that once we obtain the parameters of the process we can feed it into any system we like.
In this paper the packet voice/data multiplexer is modeled by feeding the MMPP into a single-server queue, served first-in-first-out (FIFO), with general service time distribution where the service distribution is the appropriate mixture of the voice and data packet service time distributions. A detailed analysis of this queueing system is presented where matrix analytic-algorithmic procedures - are used to compute, for example,
These measures are obtained for both voice and data traffic. The first quantity is of importance, for example, in systems that have a performance criterion on the tail of the voice packet delay distribution.
We also show how the powerful matrix-analytic methodology can be applied to handle finite buffers and a particular type of over1oad control discussed in the literature. The control mechanism here is to use a variable bit rate on voice packets during congestion (see, e.g., -). This would provide a graceful degradation of system performance during overload. The variable bit rate overload control is incorporated into the model by using state-dependent service times in the matrix-analytic methodology.
There has been a considerable amount of related work on this problem, and we refer the reader to - for details and further references. In a companion paper in this issue , and in , an approach based on approximating the superposition process by a renewal process with an inflated coefficient of variation for the interarrival time distribution is presented. The inflation factor for the arrival process depends on the system to which the process is offered, and simple closed-form formulas for the first two moments of delay, which capture the qualitative behavior, are given. The multiplexer is modeled as a FIFO queue with infinite buffers and state-independent service times. In  it is observed that although successive interarrival times can be nearly independent and exponentially distributed, these low correlations can have a cumulative effect over long time periods and can result in behavior significantly different than that of a Poisson process. In  a similar approach is used to analyze a multiplexer serving only packetized voice. For the purposes of comparison, the numerical examples presented here correspond to those in [l7].
In , approximate queue length distributions are obtained for the case where all sources are identical and have the same deterministic service time. This precludes mixing voice and data sources with different packet lengths.
We finally note that the methodology presented is fairly general, and the application to the voice/data problem may be viewed as an illustration.
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