Analysis of a discrete‐time queue with integrated bursty inputs in ATM networks

Abstract

In this paper, we carry out an exact analysis of a discrete-time queue system with a number of independent Markov modulated inputs in ATM networks, using a generating function approach. We assume that the queueing system has an infinite buffer with M servers. The cell arrival process is characterized by a number of independent Markov modulated geometrical batch arrival processes. We first obtain the generating function of the queue-size distribution at steady-state in vector form, then derive an expression for the average queue-size in terms of the unknown boundary probabilities. To obtain those unknown probabilities, we use the technique proposed in Reference 1. This involves decomposing the system characteristic function to evaluate the roots and solving a set of linear equations. One of the contributions of this paper is presented in Lemma 1, which characterizes the property of the underlying eigenvalues. For one special case of at least M-1 cell arrivals during one slot at one Markov state and of at least M arrivals at all other states, the determination of the unknowns is straightforward. If every Markov modulated arrival process can be further decomposed into a number of i.i.d. two-state, or three-state, or even four-state Markov modulated arrival processes, then each root can be obtained separately using an iterative algorithm. Numerical results are presented to validate the proposed traffic models against actual traffic measurements.

Analysis of a versatile batch-service queueing model with correlation in the arrival process

“…The condition that A(z) is diagonalizable is standard in order to be able to apply the spectral decomposition approach ( [35], [49], [50], [67], [71]) and is not a specific restriction on the generality of the model. Contribution [35] contains a detailed and extensive analysis on the conditions under which such a solution exists.…”

“…Several variants of MAP exist: in case of a BMAP, customers arrive in batches instead of individually, whereas D-MAP and D-BMAP represent the discrete-time analogues of MAP and BMAP. Queueing models with MAP (or variants) have been studied extensively in the past, for instance the MAP is considered in [4], [5], [8], [14], [44], [48] and [52], the D-MAP is covered in [21], [25], [41], [68], [69], the BMAP is studied in [1], [10], [33], [53]- [55], [58], [59], [64] and [15], [21], [32], [34], [42], [47], [49], [63], [71] deal with D-BMAP.…”

Finite buffer discrete-time queues with multiple Markovian arrivals and services in ATM networks

“…Queueing systems with integrated traffic have been analyzed extensively through various approaches [l-61. In [7], the generating function of the queue size distribution is obtained for the case when the buffer capacity is assumed to be infinite. Later on that result is extended to the case in which the number of arrivals is a random variable governed by the state of the underlying Markov chain [8]. The model studied in [7] is considered again in a recent paper by Li [9] using the generating function approach.…”

A Batch-Service Queueing Model with a Discrete Batch Markovian Arrival Process

“…For instance, in telecommunications, a traffic source which is inactive in a given time slot is very likely to remain inactive for a long time (or during a large number of time slots). In order to capture most traffic characteristics up to any desired precision, one often adopts the discrete batch Markovian arrival process (D-BMAP) [4], [13], [18], [20], [22], [27], [30]. In [30], the queue length is studied in a multiserver system with finite buffer space and deterministic service times.…”

“…In order to capture most traffic characteristics up to any desired precision, one often adopts the discrete batch Markovian arrival process (D-BMAP) [4], [13], [18], [20], [22], [27], [30]. In [30], the queue length is studied in a multiserver system with finite buffer space and deterministic service times. [4] analyses the queue length at departure instants in a singleserver, finite-buffer system with general service times, whereas [18] investigates the queue length at arbitrary instants and the waiting time for this system.…”