Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime
A large-scale flexible service system with two large server pools and two types of customers is considered. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible – they can serve both types 1 and 2. (This is a so-called “N-system.”) The customer arrival processes are...
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| Format: | Article |
| Language: | English |
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Institute for Operations Research and the Management Sciences (INFORMS)
2015-12-01
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| Series: | Stochastic Systems |
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| Online Access: | http://www.i-journals.org/ssy/viewarticle.php?id=139&layout=abstract |
| _version_ | 1811340394981490688 |
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| author | Alexander L. Stolyar |
| author_facet | Alexander L. Stolyar |
| author_sort | Alexander L. Stolyar |
| collection | DOAJ |
| description | A large-scale flexible service system with two large server pools and two types of customers is considered. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible – they can serve both types 1 and 2. (This is a so-called “N-system.”) The customer arrival processes are Poisson and customer service requirements are independent exponentially distributed. The service rate of a customer depends both on its type and the pool where it is served. A priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1, is considered. We consider the Halfin-Whitt asymptotic regime, where the arrival rate of customers and the number of servers in each pool increase to infinity in proportion to a scaling parameter n, while the overall system capacity exceeds its load by O(√n). <br/>
For this system we prove tightness of diffusion-scaled stationary distributions. Our approach relies on a single common Lyapunov function <i>G</i><sup>(<i>n</i>)</sup>(<i>x</i>), depending on parameter <i>n</i> and defined on the entire state space as a functional of the drift-based fluid limits (DFL). Specifically, <i>G</i><sup>(<i>n</i>)</sup>(<i>x</i>)=∫<sup>∞</sup><sub>0</sub><i>g</i>(<i>y</i><sup>(<i>n</i>)</sup>(<i>t</i>))<i>dt</i>, where <i>y</i><sup>(<i>n</i>)</sup>(⋅) is the DFL starting at <i>x</i>, and <i>g</i>(⋅) is a “distance” to the origin. (<i>g</i>(⋅) is same for all <i>n</i>). The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function <i>G</i><sup>(<i>n</i>)</sup>(<i>x</i>). The approach, as well as many parts of the analysis, are quite generic and may be of independent interest. |
| first_indexed | 2024-04-13T18:41:20Z |
| format | Article |
| id | doaj.art-0715a11b9b084e569739c3c3242870ab |
| institution | Directory Open Access Journal |
| issn | 1946-5238 1946-5238 |
| language | English |
| last_indexed | 2024-04-13T18:41:20Z |
| publishDate | 2015-12-01 |
| publisher | Institute for Operations Research and the Management Sciences (INFORMS) |
| record_format | Article |
| series | Stochastic Systems |
| spelling | doaj.art-0715a11b9b084e569739c3c3242870ab2022-12-22T02:34:42ZengInstitute for Operations Research and the Management Sciences (INFORMS)Stochastic Systems1946-52381946-52382015-12-015223926710.1214/14-SSY139Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regimeAlexander L. Stolyar0Lehigh UniversityA large-scale flexible service system with two large server pools and two types of customers is considered. Servers in pool 1 can only serve type 1 customers, while server in pool 2 are flexible – they can serve both types 1 and 2. (This is a so-called “N-system.”) The customer arrival processes are Poisson and customer service requirements are independent exponentially distributed. The service rate of a customer depends both on its type and the pool where it is served. A priority service discipline, where type 2 has priority in pool 2, and type 1 prefers pool 1, is considered. We consider the Halfin-Whitt asymptotic regime, where the arrival rate of customers and the number of servers in each pool increase to infinity in proportion to a scaling parameter n, while the overall system capacity exceeds its load by O(√n). <br/> For this system we prove tightness of diffusion-scaled stationary distributions. Our approach relies on a single common Lyapunov function <i>G</i><sup>(<i>n</i>)</sup>(<i>x</i>), depending on parameter <i>n</i> and defined on the entire state space as a functional of the drift-based fluid limits (DFL). Specifically, <i>G</i><sup>(<i>n</i>)</sup>(<i>x</i>)=∫<sup>∞</sup><sub>0</sub><i>g</i>(<i>y</i><sup>(<i>n</i>)</sup>(<i>t</i>))<i>dt</i>, where <i>y</i><sup>(<i>n</i>)</sup>(⋅) is the DFL starting at <i>x</i>, and <i>g</i>(⋅) is a “distance” to the origin. (<i>g</i>(⋅) is same for all <i>n</i>). The key part of the analysis is the study of the (first and second) derivatives of the DFLs and function <i>G</i><sup>(<i>n</i>)</sup>(<i>x</i>). The approach, as well as many parts of the analysis, are quite generic and may be of independent interest.http://www.i-journals.org/ssy/viewarticle.php?id=139&layout=abstractMany server modelsdrift-based fluid limitdiffusion limittightness of stationary distributionslimit interchangecommon Lyapunov function |
| spellingShingle | Alexander L. Stolyar Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime Stochastic Systems Many server models drift-based fluid limit diffusion limit tightness of stationary distributions limit interchange common Lyapunov function |
| title | Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime |
| title_full | Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime |
| title_fullStr | Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime |
| title_full_unstemmed | Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime |
| title_short | Tightness of stationary distributions of a flexible-server system in the Halfin-Whitt asymptotic regime |
| title_sort | tightness of stationary distributions of a flexible server system in the halfin whitt asymptotic regime |
| topic | Many server models drift-based fluid limit diffusion limit tightness of stationary distributions limit interchange common Lyapunov function |
| url | http://www.i-journals.org/ssy/viewarticle.php?id=139&layout=abstract |
| work_keys_str_mv | AT alexanderlstolyar tightnessofstationarydistributionsofaflexibleserversysteminthehalfinwhittasymptoticregime |