scholarly journals Maximal stability region of a perturbed nonnegative matrix

2008 ◽  
Vol 19 (3) ◽  
pp. 364-376 ◽  
Author(s):  
Bertrand Haut ◽  
Georges Bastin ◽  
Paul Van Dooren
Keyword(s):  
Stability Region ◽  
Nonnegative Matrix ◽  
Maximal Stability

scholarly journals Stability of the Bipartite Matching Model

2013 ◽  
Vol 45 (2) ◽  
pp. 351-378 ◽  
Author(s):  
Ana Bušić ◽  
Varun Gupta ◽  
Jean Mairesse
Keyword(s):  
Bipartite Graph ◽  
Stability Region ◽  
Joint Probability ◽  
Matching Model ◽  
Time Step ◽  
Random Match ◽  
Bipartite Matching ◽  
Discrete Time Markov Chain ◽  
The Stability ◽  
Maximal Stability

We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles. There are finite sets C and S of customer and server classes, respectively. Time is discrete and at each time step one customer and one server arrive in the system according to a joint probability measure μ on C× S, independently of the past. Also, at each time step, pairs of matched customers and servers, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph (C, S, E⊂ C × S). A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete-time Markov chain. We study its stability under various admissible matching policies, including ML (match the longest), MS (match the shortest), FIFO (match the oldest), RANDOM (match uniformly), and PRIORITY. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and PRIORITY policies, we exhibit a bipartite graph with a non-maximal stability region.

scholarly journals Stability of the Bipartite Matching Model

2013 ◽  
Vol 45 (02) ◽  
pp. 351-378 ◽  
Author(s):  
Ana Bušić ◽  
Varun Gupta ◽  
Jean Mairesse
Keyword(s):  
Bipartite Graph ◽  
Stability Region ◽  
Joint Probability ◽  
Matching Model ◽  
Time Step ◽  
Random Match ◽  
Bipartite Matching ◽  
Discrete Time Markov Chain ◽  
The Stability ◽  
Maximal Stability

We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles. There are finite sets C and S of customer and server classes, respectively. Time is discrete and at each time step one customer and one server arrive in the system according to a joint probability measure μ on C× S, independently of the past. Also, at each time step, pairs of matched customers and servers, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph (C, S, E⊂ C × S). A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete-time Markov chain. We study its stability under various admissible matching policies, including ML (match the longest), MS (match the shortest), FIFO (match the oldest), RANDOM (match uniformly), and PRIORITY. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and PRIORITY policies, we exhibit a bipartite graph with a non-maximal stability region.

A scheduling policy with maximal stability region for ring networks with spatial reuse

1995 ◽  
Vol 19 (1-2) ◽  
pp. 131-148 ◽  
Author(s):  
Leonidas Georgiadis ◽  
Wojciech Szpankowski ◽  
Leandros Tassiulas
Keyword(s):  
Stability Region ◽  
Spatial Reuse ◽  
Ring Networks ◽  
Scheduling Policy ◽  
Maximal Stability
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