On Departure Process in a Production Model with Cyclic Working and Repair Periods

Abstract:

A finite-buffer queueing system of the M/M/1/N type is used for modeling the operation of a single-machine production line with cyclic failure-free and repair periods. The arriving jobs enter randomly according to a Poisson process and are being processed individually with service times having the common exponential distribution. After an exponentially distributed working period a breakdown of the machine occurs, starting an exponentially distributed repair time during which the service process is stopped. At the completion epoch of the repair time a new working period begins and so on. A system of integral equations for conditional probability distributions of the number of jobs completely processed before the fixed time t (departure process) is built, using the concept of embedded Markov chain and the total probability law. Applying linear-algebraic approach the compact-form solution of the corresponding system written for double transforms of departure process is found.

• Periodical: Advanced Materials Research (Volume 1036)
• Pages: 846-851
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.1036.846
• Online since: October 2014
• Authors: Wojciech M. Kempa*, Iwona Paprocka, Krzysztof Kalinowski, Cezary Grabowik
• Keywords: Departure Process, Finite-Buffer Queue, Machine Breakdown, Total Probability Law, Transient State

References

[1] Dalley Y.; Gershwin S.B.: Manufacturing flow line systems: a review of models and analytical results. Queueing Systems 12 (1992), 3-94.

DOI: 10.1007/bf01158636

[2] Kempa W.M.: Some new results for departure process in the MX/G/1 queueing system with a single vacation and exhaustive service. Stochastic Analysis and Applications 28 (1) (2010), 26-43.

DOI: 10.1080/07362990903417920

[3] Kempa W.M.: Analysis of departure process in batch arrival queue with multiple vacations and exhaustive service. Communications in Statistics - Theory and Methods 40 (16) (2011), 2856-2865.

DOI: 10.1080/03610926.2011.562767

[4] Kempa W.M.: Transient analysis of the output process in the GI/M/1-type queue with finite buffer. AIP Conference Proceedings 1487 (2012), 193-200.

DOI: 10.1063/1.4758958

[5] Kempa W.M., Paprocka I.: Estimation of reliability characteristics in a production scheduling model with the renewal theory application. First part. Information systems architecture and technology. Wroclaw University of Technology Press, Wroclaw (2012).

[6] Kempa W. M.: Output process in batch-arrival queue with N-policy and multiple vacations. Lecture Notes in Computer Science 7314 (2013), 247-261.

DOI: 10.1007/978-3-642-39408-9_18

[7] Li W.; Shi D.; Shao X.: Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations. Journal of Applied Probability 34 (1997), 546-555.

DOI: 10.1017/s0021900200101172

[8] Paprocka I., Kempa W.M.: Estimation of reliability characteristics in a production scheduling model with the renewal theory application. Second part: numerical example. Information systems architecture and technology. Wroclaw University of Technology Press, Wroclaw (2012).

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