Analytical Solution for Time-Dependent Queue-Size Behavior in the Manufacturing Line with Finite Buffer Capacity and Machine Setup and Closedown Times

Abstract:

A single reliable-machine manufacturing line with finite buffer capacity is considered, in which each idle period is preceded by a random closedown time, and each busy period is preceded by a random setup time. The stream of arriving jobs is described by a simple Poisson process, while processing (service), closedown and setup times are generally distributed random variables. A system of Volterra-type integral equations for the transient queue-size distribution conditioned by the initial level of buffer saturation is found by using the Markov property of successive service completion epochs and the continuous version of total probability law. A parallel system written for Laplace transforms is obtained and written in a specific form. The solution of the latter system is derived in a compact-form applying the linear algebraic approach. The final representations are found in terms of “input” system parameters (transforms of processing, closedown and setup times’ distributions) and certain sequence defined recursively.

• Periodical: Applied Mechanics and Materials (Volumes 809-810)
• Pages: 1360-1365
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.809-810.1360
• Online since: November 2015
• Authors: Wojciech M. Kempa*, Iwona Paprocka
• Keywords: Closedown Time, Queue-Size Distribution, Setup Time, Single-Machine Manufacturing Line, Transient State

References

[1] B. Krishna Kumar, R. Rukmani, S.R. Anantha Lakshmi, Performance analysis of an M/G/1 queueing system under Bernoulli vacation schedules with server setup and close down periods, Comput. Ind. Eng. 66, 1 (2013) 1-9.

DOI: 10.1016/j.cie.2013.06.005

[2] P. Moreno, A discrete-time single-server queueing system under multiple vacations and setup-closedown times, Stoch. Anal. Appl. 27, 2 (2009) 221-239.

DOI: 10.1080/07362990802558303

[3] Z. Niu, T. Shu, Y. Takahashi, A vacation queue with setup and close-down times and batch Markovian arrival processes, Perform. Evaluation. 64 (2003) 225-248.

DOI: 10.1016/s0166-5316(03)00058-0

[4] W.M. Kempa, On transient queue-size distribution in the batch-arrivals system with a single vacation policy, Kybernetika. 50, 1 (2014) 126-141.

DOI: 10.14736/kyb-2014-1-0126

[5] W.M. Kempa, I. Paprocka, K. Kalinowski, C. Grabowik, On transient queue-size distribution in a single-machine production system with breakdowns, in: C. Carausu, A. Wróbel (Eds. ), Modern technologies in industrial engineering II, Trans Tech Publications, Staffa-Zurich, 2014, pp.505-510.

DOI: 10.4028/www.scientific.net/amr.1036.505

[6] W.M. Kempa, A direct approach to transient queue-size distribution in a finite-buffer queue with AQM, Appl. Math. Inf. Sci. 7, 3 (2013) 909-515.

DOI: 10.12785/amis/070308

[7] W.M. Kempa, On main characteristics of the M/M/1/N queue with single and batch arrivals and the queue size controlled by AQM algorithms, Kybernetika. 47, 6 (2011) 930-943.

DOI: 10.14736/kyb-2014-1-0126

[8] W.M. Kempa, The transient analysis of the queue-length distribution in the batch arrival system with N-policy. Multiple vacations and setup times, in: Applications of mathematics in engineering and economics, American Institute of Physics, Melville, 2010, pp.235-242.

DOI: 10.1063/1.3515592

[9] V.S. Korolyuk, Boundary-value problems for compound Poisson process, Naukova Dumka, Kiev, (1975).