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Abstract In this paper, we consider a multi-server, multiclass queue with reneging operating under the random order of service discipline. Interarrival times, service times, and patience times are assumed to be generally distributed. Under mild conditions, we establish a fluid limit theorem for a measure-valued process that keeps track of the remaining patience time for each job in the queue, when the number of servers and classes is held fixed. We prove uniqueness for fluid model solutions in all but one case. We characterize the unique invariant state for the fluid model and prove that fluid model solutions converge to the invariant state as time goes to infinity, uniformly for suitable initial conditions.
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Existence of stationary ballistic deposition on the infinite lattice
Abstract Ballistic deposition is one of the many models of interface growth that are believed to be in the KPZ universality class, but have so far proved to be largely intractable mathematically. In this model, blocks of size one fall independently as Poisson processes at each site on the ‐dimensional lattice, and either attach themselves to the column growing at that site, or to the side of an adjacent column, whichever comes first. It is not hard to see that if we subtract off the height of the column at the origin from the heights of the other columns, the resulting interface process is Markovian. The main result of this article is that this Markov process has at least one invariant probability measure. We conjecture that the invariant measure is not unique, and provide some partial evidence.
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- Award ID(s):
- 2113242
- PAR ID:
- 10419436
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 62
- Issue:
- 3
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 600-622
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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