On the blocking probability in batch Markovian arrival queues

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Abstract

In this paper we investigate the blocking probability in a finite–buffer queue whose arrival process is given by the batch Markovian arrival process (BMAP). BMAP generalizes a wide set of Markovian processes and is especially useful as a precise model of aggregated IP traffic. We first give a detailed description of the BMAP, next we prove a formula for the transform of the blocking probability and show how time-dependent and stationary characteristics can be obtained by means of this formula. Then we discuss the computational complexity and other computational issues. Finally, we present a set of numerical results for two different BMAP parameterizations. In particular, we show sample transient and stationary blocking probabilities and the impact of the auto-correlated structure of the arrival process on the blocking probability.

Introduction

Among several performance parameters of buffering processes in telecommunication networks, the following two are of special interest: the queueing delay and the blocking probability. The queueing delay (also: virtual waiting time or workload) is the length of time a packet waits in a buffer before transmission. The blocking probability represents the probability that the buffer in a network element is full and the arriving packets will be lost. High values of these parameters, for obvious reasons, have a negative impact on the quality of service perceived by network users. Unfortunately, by enlarging the buffer we decrease the blocking probability but increase the delay and vice versa. In order to reach a reasonable trade-off between these parameters we have to be able to compute both of them.

In this paper we deal with the blocking probability in a finite–buffer queue fed by a batch Markovian arrival process. Complementary results, devoted to the delay in a BMAP queue, can be found by the reader in [1].

The first reason we use the BMAP is that it can accurately model aggregated IP traffic. IP traffic often possesses negative statistical properties (correlated arrivals, burstiness, self-similarity [2], [3]) and its model must be able to imitate such behaviour. BMAP is not only able to imitate these properties on the level of packet arrival moments but it can also capture the correlation between the arriving packet size and the current packet arrival intensity.

The second good reason for dealing with the BMAP is that this process is a generalization of a very wide class of Markovian processes. In this class we can find the Poisson process, the batch (compound) Poisson process, the Markov-modulated Poisson process (MMPP), the Markovian arrival process (MAP), the phase-type renewal process and others. Therefore, all results obtained for BMAP traffic can be directly applied to all of the aforementioned processes. Some of them are very popular in teletraffic modeling (for instance, MMPP [4], [5], [6], [7]).

Another good reason for dealing with the BMAP is that the process, despite its flexibility, is very suitable for simulation purposes (see also Appendix A).

The practical usefulness of the theoretical results based on the BMAP depends on how good the model is to the original trace file, so a good parameter fitting algorithm has to be applied. Once the BMAP is well parameterized in terms of its marginal distribution, batch size distribution and autocorrelation structure, its queueing performance characteristics become very close to those based on the original data – this effect was shown, for instance, in [8].

Therefore, to take advantage of BMAPs, some efficient and numerically stable algorithms for its parameter estimation have been developed [8], [9]. Moreover, an adequate software package has been written1 and we may now easily obtain parameterizations based on IP traces and use them in the analysis of particular performance issues connected with IP networks. For instance, in [10] the down link delay of IP traffic between the Mobile Gateway Server and the End User in a UMTS mobile network was evaluated using the BMAP.

Since the classic works by Ramaswami and Lucantoni on BMAP queues [11], [12], [13], [14], a considerable number of papers have been published on this subject. Unfortunately, very few of them deal with the finite–buffer queueing scheme (for instance [15], [16]). In real devices all buffers are finite. The overflows and losses connected with this fact are crucial to the performance of the network, thus the applicability of the infinite–buffer model is limited.

The main result of this paper is the Laplace transform of the time-dependent blocking probability in a finite–buffer BMAP queue. To the best of the authors’ knowledge, there have been no reported results of this type yet. The closest we found is a simple approximate formula for the stationary loss probability in a BMAP queue [16]. The bounds for an error of this formula are not known and the error may be large in some cases. The results presented herein are exact and by using them, we can easily calculate both the transient and stationary characteristics.

The remaining part of the paper is organized in the following way. In Section 2, the definition and basic properties of the batch Markovian arrival process are recalled. In Section 3, a description of the queueing model and the notation used throughout the paper are presented. The definition of the blocking probability and the main results devoted to the blocking probability are shown in Section 4. The computational complexity and other computational issues are discussed in Section 5. Examples of numerical results are presented in Section 6. In Section 7, concluding remarks are given.

Section snippets

Batch Markovian Arrival Process

The Batch Markovian Arrival Process was invented by Neuts [17] and then reformulated by Lucantoni [12], who introduced a more elegant parameterization of it. It is also called the N-process or the versatile Markovian point process.

Following [12], the Batch Markovian Arrival Process is constructed by considering a 2-dimensional Markov process (N(t), J(t)) on the state space {(i, j):i  0, 1  j  m} with an infinitesimal generator Q in the form:Q=D0D1D2D3··D0D1D2··D0D1·····,where Dk, k  0 are m × m

Queueing model and notation

We consider herein a single server queueing system whose arrival process is given by a BMAP. The service time is distributed according to a distribution function F( · ), which is not further specified, and the standard independence assumptions are made. The capacity of the system (buffer size) is finite and equal to b (including service position). In Kendall’s notation, the system described is denoted by BMAP/G/1/b. We assume also that the time origin corresponds to a departure epoch.

The

Blocking probability

By the blocking probability at the moment t we mean the probability that the buffer is full at this moment. Its value depends on the initial queue size, X(0), and the initial state of the modulating process, J(0). It will be denoted herein by p¯n,i(t),p¯n,i(t)=P(X(t)=b|X(0)=n,J(0)=i).

As t  ∞, the dependence on X(0) and J(0) vanishes and we may denote the stationary blocking probability by:p¯=limtp¯n,i(t).

The main result of this paper will be presented in terms of the Laplace transform of p¯n,i(

Computational issues

In order to make Theorem 1 and Corollary 1 usable, we need two additional tools.

Firstly, we have to be able to compute all the coefficients (vectors and matrices) appearing in these theorems. They are either trivial (like Yk(s)) or based on integral matrices Ak(s) and D¯k(s). The numerical methods for computing integral matrices of this type are very well developed and based on the uniformization technique [12]. Applying this technique we obtain the following representations for Ak(s) and D¯k(s)

Example 1

In this example we use a simple BMAP parameterization to demonstrate sample transient and stationary blocking probabilities and the impact of the auto-correlated arrival process on it. Let us consider the following BMAP parameterization:D0=-32314-54411-127,D1=1412132235,D3=1211321412,D10=23214130105.

Thus we have three modulating states (m = 3) and the size of the batch can be 1, 3, 10. The fundamental arrival rate is Λ = 441.4. The considered BMAP was constructed in such a way that the modulating

Conclusions

In the article we proved the formula for the Laplace transform of the time-dependent blocking probability in a BMAP queue. Using this formula, one can calculate the blocking probability in the transient and stationary regime. To demonstrate this, a set of numerical illustrations was presented. There are at least two important observations based on the numerical examples.

Firstly, the correlation structure of the BMAP is important in the evaluation of the blocking probability. In one of the

Acknowledgements

This is an extended version of the paper presented during 11th IEEE Symposium on Computers and Communications, Pula, Italy, July 2006. The material is based upon work supported by the Polish Ministry of Scientific Research and Information Technology under Grant No. 3 T11C 014 26. The authors wishes to thank the referees for their valuable remarks on this work.

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