Distribution of time to buffer overflow in a finite-buffer manufacturing model with unreliable machine

Abstract

Abstract.One of the most important characteristic of each queueing model is time to buffer overflow. In the paper the distribution of that time in a single-channel manufacturing system, modelled by a Markovian finitebuffer queue with machine breakdowns, is studied. By using the analytical approach based on the idea of embedded Markov chain, total probability law and linear algebra, the formula for Laplace transform of the time to buffer overflow conditional distribution is found. Numerical illustration is presented. PreliminariesThe problem of finite buffer (magazine) capacity is a typical one in manufacturing engineering. Jobs arriving to the system being saturated are being lost or must be redirected to another destination. The in-depth investigation of the loss process cannot be based only on the well-known coefficient as the loss ratio, defined as the part of the total number of jobs being directed to the system (manufacturing line), which are lost due to the buffer overflow. To analyze the phenomenon more precisely from the probabilistic point of view, it is necessary to know, e.g. the probability distributions of buffer overflow durations and times of reaching them. The problem becomes more essential in the case of unreliable machine, where successive failure-free times are followed by repair periods during which the processing of jobs is suspended.In [1] and [2] the studies on the application of different-type queueing models in manufacturing engineering can be found. As one may notice, the analytical results for distributions of the time to the buffer overflow in one-channel queues modelling singlemachine manufacturing lines, in fact, are restricted mainly to systems without machine breakdowns. In [3][4] the formulae for the time-to-buffer overflow period were obtained for

On Time-to-Buffer Overflow Distribution in a Single-Machine Discrete-Time System With Finite Capacity

“…in [13] and [14], where the multiple vacation policy and the control mechanism based on setup and closedown times are considered, respectively. The model of a production line with unreliable machine is investigated in [15]. In [12] the distribution of the time to buffer overflow is analyzed in the case of a model with server working vacations.…”

Probabilistic Analysis of a Buffer Overflow Duration in Data Transmission in Wireless Sensor Networks

“…The generalization of these results for the case of a multiple vacation policy and for the compound Poisson arrival stream can be found in [ 10 , 11 ], respectively [ 12 , 13 ]. Analytical results for the CDF of the time to the first buffer overflow are obtained, e.g., in [ 14 , 15 ], where the multiple vacation policy and an unreliable server subject to breakdowns are assumed, respectively, in the finite-buffer model with Poisson arrivals. As a tool for avoiding the risk of a buffer overflow, active queue management (AQM) is often applied.…”