January  2014, 10(1): 167-192. doi: 10.3934/jimo.2014.10.167

A dual tandem queueing system with GI service time at the first queue

1. 

Department of Telecommunications, Budapest University of Technology and Economics, Budapest

2. 

Department of Intelligence and Informatics, Konan University, 8-9-1 Okamoto, Kobe 658-8501

Received  September 2012 Revised  June 2013 Published  October 2013

In this paper we consider the analysis of a tandem queueing model $M/G/1 -> ./M/1$. In contrast to the vast majority of the previous literature on tandem queuing models we consider the case with GI service time at the first queue and with infinite buffers. The system can be described by an M/G/1-type Markov process at the departure epochs of the first queue. The main result of the paper is the steady-state vector generating function at the embedded epochs, which characterizes the joint distribution of the number of customers at both queues. The steady-state Laplace-Stieljes transform and the mean of the sojourn time of the customers in the system are also obtained.
    We provide numerical examples and discuss the dependency of the steady-state mean of the sojourn time of the customers on several basic system parameters. Utilizing the structural characteristics of the model we discuss the interpretation of the results. This gives an insight into the behavior of this tandem queuing model and can be a base for developing approximations for it.
Citation: Zsolt Saffer, Wuyi Yue. A dual tandem queueing system with GI service time at the first queue. Journal of Industrial and Management Optimization, 2014, 10 (1) : 167-192. doi: 10.3934/jimo.2014.10.167
References:
[1]

D. Bertsekas and R. Gallager, "Data Networks," 2nd Edition, Prentice Hall, 1991.

[2]

O. J. Boxma and J. A. C. Resing, Tandem queues with deterministic service times, Annals of Operations Research, 49 (1994), 221-239. doi: 10.1007/BF02031599.

[3]

G. Casale, P. G. Harrison and M. Vigliotti, Product-Form Approximation of Tandem Queues via Matrix Geometric Methods, in "6th International Workshop on the Numerical Solution of Markov Chains (NSMC 2010)," (2010).

[4]

A. N. Dudin, C. S. Kim, V. I. Klimenok and O. S. Taramin, A dual tandem queueing system with a finite intermediate buffer and cross traffic, in "5th International Conference on Queueing Theory and Network Applications (QTNA 2010)," Beijing, (2010), 102-109. doi: 10.1145/1837856.1837872.

[5]

A. Heindl, Decomposition of general tandem queueing networks with mmpp input, Performance Evaluation, 44 (2001), 5-23.

[6]

B. Van Houdt and A. S. Alfa, Response time in a tandem queue with blocking, Markovian arrivals and phase-type services, Operations Research Letters, 33 (2005), 373-381. doi: 10.1016/j.orl.2004.08.004.

[7]

L. Kleinrock, "Queuing Systems. Vol I: Theory," John Wiley, 1975.

[8]

V. Klimenok, A. Dudin and V. Vishnevsky, On the stationary distribution of tandem queue consisting of a finite number of stations, in "Computer Networks, Communications in Computer and Information Science," 291 (2012), 383-392. doi: 10.1007/978-3-642-31217-5_40.

[9]

G. Latouche and V. Ramaswami, "Introduction to Matrix Geometric Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; American Statistical Association, Alexandria, VA, 1999. doi: 10.1137/1.9780898719734.

[10]

L. Le and E. Hossain, Tandem queue models with applications to QoS routing in multihop wireless networks, IEEE Trans. on Mobile Computing, 7 (2008), 1025-1040.

[11]

D. L. Lucantoni, New results on the single server queue with a batch markovian arrival process, Comm. Statist. Stochastic Models, 7 (1991), 1-46. doi: 10.1080/15326349108807174.

[12]

R. A. Marie, Calculating equilibrium probabilities for $\lambda(n)$/$C_k$/1/N queue, in "Proceedings Performance' 80," Toronto, (1980), 117-125.

[13]

D. G. Pandelis, Optimal control of flexible servers in two tandem queues with operating costs, Probability in the Engineering and Informational Sciences, 22 (2008), 107-131. doi: 10.1017/S0269964808000077.

[14]

Y. E. Sagduyu and A. Ephremides, Network Coding in Wireless Queueing Networks: Tandem Network Case, in "Proc. IEEE International Symposium on Information Theory," Seattle, WA, (2006).

[15]

M. van Vuuren, "Performance Analysis of Manufacturing Systems: Queueing Approximations and Algorithms," Ph.D thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 2007.

[16]

M. van Vuuren, I. J. B. F. Adan and A. E. Resing-Sassenb, Performance analysis of multi-server tandem queues with finite buffers and blocking, OR Spektrum, 27 (2005), 315-338. doi: 10.1007/s00291-004-0189-z.

show all references

References:
[1]

D. Bertsekas and R. Gallager, "Data Networks," 2nd Edition, Prentice Hall, 1991.

[2]

O. J. Boxma and J. A. C. Resing, Tandem queues with deterministic service times, Annals of Operations Research, 49 (1994), 221-239. doi: 10.1007/BF02031599.

[3]

G. Casale, P. G. Harrison and M. Vigliotti, Product-Form Approximation of Tandem Queues via Matrix Geometric Methods, in "6th International Workshop on the Numerical Solution of Markov Chains (NSMC 2010)," (2010).

[4]

A. N. Dudin, C. S. Kim, V. I. Klimenok and O. S. Taramin, A dual tandem queueing system with a finite intermediate buffer and cross traffic, in "5th International Conference on Queueing Theory and Network Applications (QTNA 2010)," Beijing, (2010), 102-109. doi: 10.1145/1837856.1837872.

[5]

A. Heindl, Decomposition of general tandem queueing networks with mmpp input, Performance Evaluation, 44 (2001), 5-23.

[6]

B. Van Houdt and A. S. Alfa, Response time in a tandem queue with blocking, Markovian arrivals and phase-type services, Operations Research Letters, 33 (2005), 373-381. doi: 10.1016/j.orl.2004.08.004.

[7]

L. Kleinrock, "Queuing Systems. Vol I: Theory," John Wiley, 1975.

[8]

V. Klimenok, A. Dudin and V. Vishnevsky, On the stationary distribution of tandem queue consisting of a finite number of stations, in "Computer Networks, Communications in Computer and Information Science," 291 (2012), 383-392. doi: 10.1007/978-3-642-31217-5_40.

[9]

G. Latouche and V. Ramaswami, "Introduction to Matrix Geometric Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; American Statistical Association, Alexandria, VA, 1999. doi: 10.1137/1.9780898719734.

[10]

L. Le and E. Hossain, Tandem queue models with applications to QoS routing in multihop wireless networks, IEEE Trans. on Mobile Computing, 7 (2008), 1025-1040.

[11]

D. L. Lucantoni, New results on the single server queue with a batch markovian arrival process, Comm. Statist. Stochastic Models, 7 (1991), 1-46. doi: 10.1080/15326349108807174.

[12]

R. A. Marie, Calculating equilibrium probabilities for $\lambda(n)$/$C_k$/1/N queue, in "Proceedings Performance' 80," Toronto, (1980), 117-125.

[13]

D. G. Pandelis, Optimal control of flexible servers in two tandem queues with operating costs, Probability in the Engineering and Informational Sciences, 22 (2008), 107-131. doi: 10.1017/S0269964808000077.

[14]

Y. E. Sagduyu and A. Ephremides, Network Coding in Wireless Queueing Networks: Tandem Network Case, in "Proc. IEEE International Symposium on Information Theory," Seattle, WA, (2006).

[15]

M. van Vuuren, "Performance Analysis of Manufacturing Systems: Queueing Approximations and Algorithms," Ph.D thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 2007.

[16]

M. van Vuuren, I. J. B. F. Adan and A. E. Resing-Sassenb, Performance analysis of multi-server tandem queues with finite buffers and blocking, OR Spektrum, 27 (2005), 315-338. doi: 10.1007/s00291-004-0189-z.

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