The G/M/r machine interference model

Bunday, B.D.; Scraton, R. E.

doi:10.1016/0377-2217(80)90192-7

Cited by 56 publications

(7 citation statements)

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“…Note that the repair facility utilization does not change provided the repair time distribution is exponential. This is another statement of the result given, for example, in Bunday andScraton (1980) andCarmichael (1987).…”

Section: A Numerical Solutionmentioning

confidence: 57%

Machine interference with general repair and running times

Carmichael

1987

Zeitschrift f�r Operations Research

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Machine maintenance and repair operations, alternatively called machine interference, may be effectively represented as finite source queueing processes. This paper presents a general solution to such processes where both the repair times and running (operating) times are modelled as Erlang distributions (the (Eh/Em/1)/K model). Such a general solution is unavailable elsewhere.The solution is made possible by two devices. Firstly the finite source process is replaced by an equivalent two stage cyclic queueing process where the two stages are the repair stage and the operating stage. Secondly a one-dimensional state definition is used for each stage giving a total two-dimensional state definition for the whole process. This two-dimensional state permits the straightforward visualization and drawing of the rate diagram and the consequent extraction of the steady state balance equations from this rate diagram. A numerical solution procedure for the balance equations is also outlined as is an extension to the multiple repair facility case (the (Eh/Em/c)/K model).Notation: The notation used in this paper for finite source queueing models is (-/-/e)/K where the first slot refers to the probability distribution for the machine operating times, the second slot refers to the probability distribution for the repair times, e = 1, 2, ... refers to the number of repair facilities and K = 1, 2, ... refers to the number of machines allocated to the c repair facilities.

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Section: A Numerical Solutionmentioning

confidence: 57%

Machine interference with general repair and running times

Carmichael

1987

Zeitschrift f�r Operations Research

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“…We assume that the lifetimes of items are IID random variables with mean 1/ λ (independent of the priority type), the j th vendor employs s j servers to repair items, and the repair times are IID exponential random variables with parameter μ ij . We point out an invariance result for G / M / r interference model that the steady state probabilities depend only on the mean failure time 1/ λ and not the failure distribution G ( · ) (Bunday and Scranton, 1980). Therefore, since we are only concerned with long‐run average cost, we need not assume that the lifetimes of items are exponentially distributed.…”

Section: Problem Descriptionmentioning

confidence: 85%

“…exponential random variables with parameter µ j . We point out an invariance result for a G/M/r interference model that the steady state probabilities depend only on the mean failure time 1/λ and not the failure distribution G(·) (Bunday & Scranton 1980). Therefore, since we are only concerned with long-run average cost, we need not assume that the lifetimes of items are exponentially distributed.…”

Section: Problem Descriptionmentioning

confidence: 99%

Outsourcing prioritized warranty repairs

Buczkowski

Hartmann

Kulkarni

2005

Int J Qual & Reliability Mgmt

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We consider the problem of outsourcing warranty repairs to outside vendors when items have priorities in service. The manufacturer has a contract with a fixed number of repair vendors. The manufacturer pays a fixed fee for each repair done by a vendor which is independent of the repair type and priority class but depends on the vendor. There are a fixed number of items under warranty, and each item belongs to one of a fixed number of priority classes. The manufacturer also pays for holding costs incurred when the items are at the vendors, the holding cost being higher for the higher priority items. We focus on static allocation of the warranty repairs; that is, we assign all items to the vendors at the beginning of the warranty period. We give the known algorithm to optimally solve the one priority class problem and solve the multi-priority class problem by formulating it as a convex minimum cost network flow problem. Then, we give numerical examples to illustrate the cost benefits of a multi-priority structure.

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“…Note that in Table 5 , 1 -Po is very close to 1.0, thus justifying the assumption of heavy traffic approximation. The numerical results shown in Table 5 show that the probability of a nonempty repair stage, 1 -Po, is very close to unity for moderate to large values of (1) [4].…”

Section: Determination Offlx) Po S[xj and Var[xlmentioning

confidence: 86%

Diffusion approximation to theG/G/R machine repair problem with warm standby spares

Sivazlian

Wang

1990

Naval Research Logistics

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The G/G/R machine repair problem with M operating machines, S warm standby spares, and R repairmen is studied as a diffusion process. The steady‐state equations are formulated as diffusion equations subject to two reflecting barriers. The approximate diffusion parameters of the diffusion equations are obtained (1) under the assumption that the input characteristics of the problem are defined only by their first two moments rather than their probability distribution function, (2) under the assumption of heavy traffic approximation, that is, when queues of failed machines in the repair stage are almost always nonempty, and (3) using well‐known asymptotic results from renewal theory. Expressions for the probability density functions of the number of failed machines in the system are obtained. A study of the derived approximate results, compared to some of the exact results, suggests that the diffusion approach provides a useful method for solving complex machine‐repair problems.

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The G/M/r machine interference model

Cited by 56 publications

References 1 publication

Machine interference with general repair and running times

Machine interference with general repair and running times

Outsourcing prioritized warranty repairs

Diffusion approximation to theG/G/R machine repair problem with warm standby spares

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