Maintenance in Single‐Server Queues: A Game‐Theoretic Approach

Abstract

We use antagonistic stochastic games and fluctuation analysis to examine a single-server queue with bulk input and secondary work during server's multiple vacations. When the buffer contents become exhausted the server leaves the system to perform some diagnostic service of a minimum ofLjobs clustered in packets of random sizes (event A). The server is not supposed to stay longer thanTunits of time (event B). The server returns to the system when A or B occurs, whichever comes first. On the other hand, he may not break service of a packet in a middle even if A or B occurs. Furthermore, the server waits for batches of customers to arrive if upon his return the queue is still empty. We obtain a compact and explicit form functional for the queueing process in equilibrium.

Time Sensitive Functionals in a Queue with Sequential Maintenance

“…[4,5] , with the phase I being different from the present one. We assumed there that the server sojourns at the MF no longer than time T and at the same time, he is limited by the number of jobs he can process (threshold L).…”

“…[2][3][4][5] , but they are still essentially different from ours. Furthermore, we pursue the goal to offer an explicit solution to our model without algorithms and at the same time preserving some generality which Refs.…”

“…Furthermore, we pursue the goal to offer an explicit solution to our model without algorithms and at the same time preserving some generality which Refs. [2][3][4][5] did not. New fluctuations techniques [2] , sequential games [9] , and time sensitive analysis embellished in Ref.…”

Time Sensitive Analysis of Antagonistic Stochastic Processes and Applications to Finance and Queueing

“…However, we need to know about the contents of the buffer (or waiting room) upon the end of the vacation and during the vacation for various reasons. One of them is to construct the semi-regenerative kernel of the queueing process in case of M/G/1-type or GI/M/1-type queues and a subsequent analysis of the continuous time parameter queueing process, cf., [1,[4][5][6]. Consequently, the time sensitive analysis rendered on phase 1, during server's vacations and concluded in Theorem 3.1 will have to be merged with a time sensitive functional on phase 2, i.e.…”

“…Relation to the Existing Literature. Our work topically falls into fluctuations of stochastic processes, which is a widely referred to area of random walk, renewal processes, random measures, recurrent processes, Markov processes, and semi-Markov processes, to name a few [7-9, 26, 33, 36] and their applications to astronomy and biology [24], queueing [4,10,11], stochastic games [14], finance and economics [12,13,16,20,27,30,31,38], physics [24,32], and earthquakes [33]. The main focus of our work is on time sensitive fluctuations, meaning that at least some of the component processes are with continuous time parameter.…”