Elsevier

Computer Communications

Volume 21, Issue 11, 10 August 1998, Pages 1010-1019
Computer Communications

Correlation properties of the token leaky bucket departure process

https://doi.org/10.1016/S0140-3664(98)00157-1Get rights and content

Abstract

In this paper we focus on the behavior of the leaky bucket mechanism with respect to second-order statistics of the departure process under various correlated cell arrival and deterministic token arrival distributions. We derive expressions for the lag-k auto-correlation of the inter-departure times for the leaky bucket cell departure stream and we produce numerical examples for the lag-k auto-correlation of the leaky bucket inter-departure times for several different cell arrival distributions exhibiting both positive and negative auto-correlations.

Introduction

Buffer size requirements at a particular node or switch in an ATM network is an important aspect of design. It has been shown that analytical techniques using the assumption of renewal arrivals result in buffer designs which frequently exhibit high blocking under actual traffic conditions, which is often highly correlated. Short and long term dependencies have been observed in LAN traffic [1]as well as TCP applications over wide-area networks [2]and both are supported by ATM implementations. The leaky bucket has been proposed as a traffic shaping and traffic policing mechanism in ATM networks. Analyzing correlations in the departure process of a leaky bucket is thus important in developing more accurate analytical models of ATM networks.

Window control mechanisms are not feasible in high speed networks because of the impact of propagation delay on cell transmission time. Large propagation delay renders message acknowledgment infeasible. A proactive mechanism for flow control is the leaky bucket. The leaky bucket has been proposed as a rate based traffic shaping mechanism by Turner [3]and Woodruff et al. [4]. A credit based mechanism known as the buffered or token leaky bucket proposed by Sidi [5]is the focus of this study.

Fig. 1 presents a model of the token leaky bucket traffic shaping mechanism. Cells arrive at the system and if a token is present in the token buffer (bucket), the cell is transmitted without further delay and a token is removed from the token buffer. If no token is available (the token bucket is empty), the cell is queued in the customer buffer. Similarly, tokens arrive at the system and if one or more cells are queued, the first cell in the buffer is transmitted. If no cells are waiting in the cell buffer, tokens are queued in the token buffer (bucket).

Both the cell buffer and the token buffer have finite waiting room. K tokens can be stored in the token buffer and N cells can be stored in the cell buffer. When a cell arrives and finds the cell buffer filled, the cell is discarded, and lost. When a token arrives and finds the token buffer full, the token is discarded.

The departure process of the leaky bucket has been studied extensively in the literature. Gun et al. [6]quantified the impact of leaky bucket parameters on the queueing behavior of the departure process. Anantharam and Konstantopoulos [7]studied the leaky bucket based on infinite cell buffer space and a token buffer of size K. They examined the burstiness of the departure process as a function of K. Wittevrongel and Bruneel [8]included lag-1 correlations in their analysis of the departure process. Hughes discovered through simulation that the leaky bucket can introduce positive correlations into the departure stream 9, 10.

Studies by Pruneski and Li [11]show that long term dependencies pass through a finite queue unchanged. Further, it has been shown that the effect of the correlated traffic on the system is related to the size of the buffers in the network 12, 13. Correlations in queues in general have been studied by 14, 15, 16, 17, 18, 19, 20. The effect of the leaky bucket on the correlation structures of non-renewal arrival processes and its subsequent effect on cell delay and loss statistics is generally unknown.

In this paper we use Linear Algebra Queueing Theory (LAQT) to model the behavior of the leaky bucket as a traffic shaper. Using techniques presented by Lipsky et al. [18], we model the leaky bucket departure process as a semi-Markov process from which we derive the joint inter-departure interval distributions and lag-k correlations.

In Section 2of this paper, we present our linear algebra queueing theory (LAQT) representation of the cell arrival processes and leaky bucket departure process as semi-Markov processes. In Section 3, we present the derivation and solution for any lag-k correlations in the departures of a semi-Markov process. In Section 4, we produce numerical results for the auto-correlation of the inter-departure times for any lag-k for several cell arrival distributions. Section 5concludes this paper.

Section snippets

The semi-Markov process

Semi-Markov arrival (departure) processes are represented as the joint density function of the first n+k-successive arrival (departure) intervalsfn+k(x1,x2,…,xn+k)=π(0)exp(−x1B)Lexp(−x2B)Lexp(−xn+kB)Le

B is the process-rate operator which must be non-singular, L is the event-rate operator which must be non-zero, and π(0) is the starting vector for the process. e′ is a column vector of 1,s and is used as a summing vector.

The leaky bucket departure process and all of the arrival processes into

Auto-correlations of the semi-Markov process

We review here the derivations for the auto-correlation structure of the semi-Markov process. First consider the joint density function of the first n+k-successive departure intervals from any semi-Markov process see [18].fn+k(x1,x2,…,xn+k)=π(0)exp(−x1B)Lexp(−x2B)Lexp(−xn+kB)Le

Since only the covariance of the two variables Xn and Xn+k are of interest, we integrate over all the other variables. We denote the marginal density functions by:fn+k(xn+k)=000fn+k(x1,…,xn+k)(dx1,…,dxn+k−1)=π(0)(

Numerical results

In this section we produce some numerical results for the auto-correlation structure of the leaky bucket inter-departure times for several different cell arrival processes which exhibit either positive, negative, or zero auto-correlations. Specifically, we show the arrival and departure auto-correlations for the Poisson, Erlang-2, IPP, alternating exponential, and several MMPP processes which exhibit short and long term positive and negative auto-correlations. We study the leaky bucket

Conclusion

In this paper we analyzed the second-order behavior of the leaky bucket based on the auto-correlation lag-k of the inter-departure process. We derived equations to analyze correlations in the departure process of the leaky bucket under several correlated and non-correlated cell arrival processes for varying cell buffer and token bucket sizes, and for varying lag-k. The ability to analyze the correlation structures in the output process of the leaky bucket should be helpful in the analytical

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