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Abstract
Consider an M/G/1 production line in which a production item is failed with some probability and is then repaired. We consider three repair disciplines depending on whether the failed item is repaired immediately or first stockpiled and repaired after all customers in the main queue are served or the stockpile reaches a specified threshold. For each discipline, we find the probability generating function (p.g.f.) of the steady‐state size of the system at the moment of departure of the customer in the main queue, the mean busy period, and the probability of the idle period.
Highlights
Consider an M/G/1 queueing system in which some customers must be reserviced with probability p
We find the probability generating function (p.g.f.) of the steady-state size of the system at the moment of departure of the customer in the main queue, the mean busy period, and the probability of the idle period
The server has to switch from main queue (MQ) to failed queue (F Q) if the store is full or if there are no more items in MQ, and returns to MQ after reservicing all the failed items in F Q, if there are any items in MQ
Summary
Consider an M/G/1 queueing system in which some customers must be reserviced with probability p. We find the probability generating function (p.g.f.) of the steady-state size of the system at the moment of departure of the customer in the main queue, the mean busy period, and the probability of the idle period (the proportion of time that the server is idle). In this case, the server has to switch from MQ to F Q if the store is full (threshold N) or if there are no more items in MQ, and returns to MQ after reservicing all the failed items in F Q, if there are any items in MQ. For an M/G/1 queue in which the server starts with one customer, we know that the mean of the number of the departures in MQ is 1/(1 − ρ1) (see Salehi-Rad and Mengersen [8]). By (5.9) and using (a), (b), (c), and (d), we can find πI∗II
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