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,"

References

[1] P. A. Jacobs and P. A.W. Lewis, “Discrete Time Series generated by Mixtures III: Autoregressive Processes (DAR(p)),” Naval Postgraduate School, Monterey, 1978.

[2] A. Elwalid, D. Heyman, T. V. Laksman, D. Mitra and A. Weiss, “Fundamental Bounds and Approximations for ATM Multiplexers with Applications to Video Teleconferencing,” IEEE Journal of Selected Areas in Communications, Vol. 13, No. 6, 1995, pp. 1004-1016. doi:10.1109/49.400656

[3] F. Kamoun and M. M. Ali, “A New Theortical Approach for the Transient and Steady—State Analysis of Multiserver ATM Multiplexers with Correlated Arrivals,” 1995 IEEE International Conference on Communications, Vol. 2, 1995, pp. 1127-1131. doi:10.1109/ICC.1995.524276

[4] G. U. Hwang and K. Sohraby, “On the Exact Analysis of a Discrete Time Queueing System with Autoregressive Inputs,” Queueing Systems, Vol. 43, No. 1-2, 2003, pp. 29-41. doi:10.1023/A:1021848330183

[5] G. U. Hwang, B. D. Choi and J. K. Kim, “The Waiting Time Analysis of a Discrete Time Queue with Arrivals as an Autoregressive Process of Order 1,” Journal of Applied Probability, Vol. 39, No. 3, 2003, pp. 619-629.

[6] B. D. Choi, B. Kim, G. U. Hwang and J. K. Kim, “The Analysis of a Multiserver Queue Fed by a Discrete Autoregressissive Process of Order 1,” Operations Research Letters, Vol. 32, No. 1, 2004, pp. 85-93. doi:10.1016/S0167-6377(03)00068-3

[7] J. Kim, B. Kim and K. Sohraby, “Mean Queue Size in a Queue with Discrete Autoregressive Arrivals of Order p,” Annals of Operations Research, Vol. 162, No. 1, 2008, pp. 69-68. doi:10.1007/s10479-008-0318-1

[8] M. F. Neuts, “Structured Stochastic Matrices of the M/G/1 Type and Their Applications,” Dekker, New York. 1989.

[9] K. G. Janardhan, “Markov-Polya urn Model with Pre- Determined Strategies,” Gujarat Statistical Review, Vol. 2, No. 1, 1975, pp. 17-32.

[10] K. Sen and R. Jain “Generalized Markov-Polya urn Model Withpre-Determined Strategies,” Journal Statistical Planning and Inference, Vol. 54, 1996, pp. 119-133. doi:10.1016/0378-3758(95)00161-1

[11] G. C. Jain and P. C. Consul, “A Generalized Negative Bi- nomial Distribution,” Society for Industrial Mathematics, Vol. 21, No. 4, 1971, pp. 501-513. doi:10.1137/0121056

[12] S. B. Ahmad, A. Hassan and M. J. Iqbal, “On a Quasi Negative Binomial Distribution–II and Its Applications,” Preprint, 2010.

[13] A. H. Frank and A. B. Melvin, “The Borel-Tannerdistri- bution,” Biometrica, Vol. 47, No. 1-2, 1960, pp. 143-150.

[14] G. Latouche and V. Ramaswamy, “Introduction to Matrix Analytical Method in Stochastic Modelling,” Society for Industrial Mathematics, Pennsylvania, 1991.

[15] L. Breuer and D. Baum “An Introduction to Queueing Theory and Matrix Analytic Method,” Springer, Berlin, 2004.

[16] V. G. Kulkarni “Modelling and Analysis of Stochastic Systems,” Chapman & Hall, London, 1995.

[17] Customer Check Out. xlsx www.westminstercollege.edu.

[1] P. A. Jacobs and P. A.W. Lewis, “Discrete Time Series generated by Mixtures III: Autoregressive Processes (DAR(p)),” Naval Postgraduate School, Monterey, 1978.

[2] A. Elwalid, D. Heyman, T. V. Laksman, D. Mitra and A. Weiss, “Fundamental Bounds and Approximations for ATM Multiplexers with Applications to Video Teleconferencing,” IEEE Journal of Selected Areas in Communications, Vol. 13, No. 6, 1995, pp. 1004-1016. doi:10.1109/49.400656

[3] F. Kamoun and M. M. Ali, “A New Theortical Approach for the Transient and Steady—State Analysis of Multiserver ATM Multiplexers with Correlated Arrivals,” 1995 IEEE International Conference on Communications, Vol. 2, 1995, pp. 1127-1131. doi:10.1109/ICC.1995.524276

[4] G. U. Hwang and K. Sohraby, “On the Exact Analysis of a Discrete Time Queueing System with Autoregressive Inputs,” Queueing Systems, Vol. 43, No. 1-2, 2003, pp. 29-41. doi:10.1023/A:1021848330183

[5] G. U. Hwang, B. D. Choi and J. K. Kim, “The Waiting Time Analysis of a Discrete Time Queue with Arrivals as an Autoregressive Process of Order 1,” Journal of Applied Probability, Vol. 39, No. 3, 2003, pp. 619-629.

[6] B. D. Choi, B. Kim, G. U. Hwang and J. K. Kim, “The Analysis of a Multiserver Queue Fed by a Discrete Autoregressissive Process of Order 1,” Operations Research Letters, Vol. 32, No. 1, 2004, pp. 85-93. doi:10.1016/S0167-6377(03)00068-3

[7] J. Kim, B. Kim and K. Sohraby, “Mean Queue Size in a Queue with Discrete Autoregressive Arrivals of Order p,” Annals of Operations Research, Vol. 162, No. 1, 2008, pp. 69-68. doi:10.1007/s10479-008-0318-1

[8] M. F. Neuts, “Structured Stochastic Matrices of the M/G/1 Type and Their Applications,” Dekker, New York. 1989.

[9] K. G. Janardhan, “Markov-Polya urn Model with Pre- Determined Strategies,” Gujarat Statistical Review, Vol. 2, No. 1, 1975, pp. 17-32.

[10] K. Sen and R. Jain “Generalized Markov-Polya urn Model Withpre-Determined Strategies,” Journal Statistical Planning and Inference, Vol. 54, 1996, pp. 119-133. doi:10.1016/0378-3758(95)00161-1

[11] G. C. Jain and P. C. Consul, “A Generalized Negative Bi- nomial Distribution,” Society for Industrial Mathematics, Vol. 21, No. 4, 1971, pp. 501-513. doi:10.1137/0121056

[12] S. B. Ahmad, A. Hassan and M. J. Iqbal, “On a Quasi Negative Binomial Distribution–II and Its Applications,” Preprint, 2010.

[13] A. H. Frank and A. B. Melvin, “The Borel-Tannerdistri- bution,” Biometrica, Vol. 47, No. 1-2, 1960, pp. 143-150.

[14] G. Latouche and V. Ramaswamy, “Introduction to Matrix Analytical Method in Stochastic Modelling,” Society for Industrial Mathematics, Pennsylvania, 1991.

[15] L. Breuer and D. Baum “An Introduction to Queueing Theory and Matrix Analytic Method,” Springer, Berlin, 2004.

[16] V. G. Kulkarni “Modelling and Analysis of Stochastic Systems,” Chapman & Hall, London, 1995.

[17] Customer Check Out. xlsx www.westminstercollege.edu.