Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues
In the Hammersley-Aldous-Diaconis process infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x$ whose nearest neighbor to the right is at y, jumps at rate y-x to a position uniformly distributed in the interval (x,y). The basic coupling between tra...
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| Format: | Journal article |
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2007
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| _version_ | 1797080279051927552 |
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| author | Ferrari, P Martin, J |
| author_facet | Ferrari, P Martin, J |
| author_sort | Ferrari, P |
| collection | OXFORD |
| description | In the Hammersley-Aldous-Diaconis process infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x$ whose nearest neighbor to the right is at y, jumps at rate y-x to a position uniformly distributed in the interval (x,y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rate \lambda and the (attempted) services with rate \rho>\lambda. Then put the first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for the n-class case by using n-1 queues in tandem with n-1 priority-types of customers. A multi-line process is introduced; it consists of a coupling (different from Liggett's basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves the dual points of the space-time Poisson process used in the graphical construction of the system. The coupled process is a transformation of the multi-line process and its invariant measure the transformation described above of the product measure. |
| first_indexed | 2024-03-07T00:57:42Z |
| format | Journal article |
| id | oxford-uuid:88a638a3-9e07-406c-b25f-fd785943762e |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:57:42Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:88a638a3-9e07-406c-b25f-fd785943762e2022-03-26T22:18:47ZMulticlass Hammersley-Aldous-Diaconis process and multiclass-customer queuesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:88a638a3-9e07-406c-b25f-fd785943762eSymplectic Elements at Oxford2007Ferrari, PMartin, JIn the Hammersley-Aldous-Diaconis process infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x$ whose nearest neighbor to the right is at y, jumps at rate y-x to a position uniformly distributed in the interval (x,y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rate \lambda and the (attempted) services with rate \rho>\lambda. Then put the first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for the n-class case by using n-1 queues in tandem with n-1 priority-types of customers. A multi-line process is introduced; it consists of a coupling (different from Liggett's basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves the dual points of the space-time Poisson process used in the graphical construction of the system. The coupled process is a transformation of the multi-line process and its invariant measure the transformation described above of the product measure. |
| spellingShingle | Ferrari, P Martin, J Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues |
| title | Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer
queues |
| title_full | Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer
queues |
| title_fullStr | Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer
queues |
| title_full_unstemmed | Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer
queues |
| title_short | Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer
queues |
| title_sort | multiclass hammersley aldous diaconis process and multiclass customer queues |
| work_keys_str_mv | AT ferrarip multiclasshammersleyaldousdiaconisprocessandmulticlasscustomerqueues AT martinj multiclasshammersleyaldousdiaconisprocessandmulticlasscustomerqueues |