Analysis of DAR(1)/D/s Queue with Quasi-Negative Binomial-II as Marginal Distribution

Abstract

In this paper we consider the arrival process of a multiserver queue governed by a discrete autoregressive process of order 1 [DAR(1)] with Quasi-Negative Binomial Distribution-II as the marginal distribution. This discrete time multiserver queueing system with autoregressive arrivals is more suitable for modeling the Asynchronous Transfer Mode(ATM) multiplexer queue with Variable Bit Rate (VBR) coded teleconference traffic. DAR(1) is described by a few parameters and it is easy to match the probability distribution and the decay rate of the autocorrelation function with those of measured real traffic. For this queueing system we obtained the stationary distribution of the system size and the waiting time distribution of an arbitrary packet with the help of matrix analytic methods and the theory of Markov regenerative processes. Also we consider negative binomial distribution, generalized Poisson distribution, Borel-Tanner distribution defined by Frank and Melvin(1960) and zero truncated generalized Poisson distribution as the special cases of Quasi-Negative Binomial Distribution-II. Finally, we developed computer programmes for the simulation and empirical study of the effect of autocorrelation function of input traffic on the stationary distribution of the system size as well as waiting time of an arbitrary packet. The model is applied to a real data of number of customers waiting for checkout in an airport and it is established that the model well suits this data.

In this paper we consider the arrival process of a multiserver queue governed by a discrete autoregressive process of order 1 [DAR(1)] with Quasi-Negative Binomial Distribution-II as the marginal distribution. This discrete time multiserver queueing system with autoregressive arrivals is more suitable for modeling the Asynchronous Transfer Mode(ATM) multiplexer queue with Variable Bit Rate (VBR) coded teleconference traffic. DAR(1) is described by a few parameters and it is easy to match the probability distribution and the decay rate of the autocorrelation function with those of measured real traffic. For this queueing system we obtained the stationary distribution of the system size and the waiting time distribution of an arbitrary packet with the help of matrix analytic methods and the theory of Markov regenerative processes. Also we consider negative binomial distribution, generalized Poisson distribution, Borel-Tanner distribution defined by Frank and Melvin(1960) and zero truncated generalized Poisson distribution as the special cases of Quasi-Negative Binomial Distribution-II. Finally, we developed computer programmes for the simulation and empirical study of the effect of autocorrelation function of input traffic on the stationary distribution of the system size as well as waiting time of an arbitrary packet. The model is applied to a real data of number of customers waiting for checkout in an airport and it is established that the model well suits this data.

Keywords

Discrete Autoregressive Process of Order [DAR(1)], Multiserver ATM Multiplexer, Matrix Analytic Methods, Markov Renewal Process, Markov Regenerative Theory, Teleconference Traffic, Quasi-Negative Binomial Distribution-II, Generalized Poisson Distribution, Borel-Tanner Distribution

Discrete Autoregressive Process of Order [DAR(1)], Multiserver ATM Multiplexer, Matrix Analytic Methods, Markov Renewal Process, Markov Regenerative Theory, Teleconference Traffic, Quasi-Negative Binomial Distribution-II, Generalized Poisson Distribution, Borel-Tanner Distribution

Cite this paper

nullK. Jose and B. Abraham, "Analysis of DAR(1)/D/s Queue with Quasi-Negative Binomial-II as Marginal Distribution,"*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1159-1169. doi: 10.4236/am.2011.29161.

nullK. Jose and B. Abraham, "Analysis of DAR(1)/D/s Queue with Quasi-Negative Binomial-II as Marginal Distribution,"

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