scholarly journals Markov processes with product-form stationary distribution

2008 ◽  
Vol 13 (0) ◽  
pp. 614-627 ◽  
Author(s):  
Krzysztof Burdzy ◽  
David White
Keyword(s):  
Stationary Distribution ◽  
Markov Processes ◽  
Product Form

scholarly journals A product form for the general stochastic matching model

2021 ◽  
Vol 58 (2) ◽  
pp. 449-468
Author(s):  
Pascal Moyal ◽  
Ana Bušić ◽  
Jean Mairesse
Keyword(s):  
Stationary Distribution ◽  
Product Form ◽  
Necessary Condition ◽  
Matching Model ◽  
Bipartite Matching ◽  
Compatibility Graph ◽  
Stochastic Matching ◽  
Dynamic Reversibility ◽  
Original Dynamic ◽  
Do So

AbstractWe consider a stochastic matching model with a general compatibility graph, as introduced by Mairesse and Moyal (2016). We show that the natural necessary condition of stability of the system is also sufficient for the natural ‘first-come, first-matched’ matching policy. To do so, we derive the stationary distribution under a remarkable product form, by using an original dynamic reversibility property related to that of Adan, Bušić, Mairesse, and Weiss (2018) for the bipartite matching model.

scholarly journals Entropic Equilibria Selection of Stationary Extrema in Finite Populations

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 631
Author(s):  
Marc Harper ◽  
Dashiell Fryer
Keyword(s):  
Markov Process ◽  
Population Size ◽  
Mutation Rate ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Local Maxima ◽  
Moran Process ◽  
The Stability ◽  
Definition Of ◽  
Selection Of

We propose the entropy of random Markov trajectories originating and terminating at the same state as a measure of the stability of a state of a Markov process. These entropies can be computed in terms of the entropy rates and stationary distributions of Markov processes. We apply this definition of stability to local maxima and minima of the stationary distribution of the Moran process with mutation and show that variations in population size, mutation rate, and strength of selection all affect the stability of the stationary extrema.

Online Network Optimization Using Product-Form Markov Processes

2016 ◽  
Vol 61 (7) ◽  
pp. 1838-1853 ◽  
Author(s):  
Jaron Sanders ◽  
Sem C. Borst ◽  
Johan S. H. van Leeuwaarden
Keyword(s):  
Markov Processes ◽  
Network Optimization ◽  
Product Form

Markov-renewal fluid queues

2004 ◽  
Vol 41 (3) ◽  
pp. 746-757 ◽  
Author(s):  
Guy Latouche ◽  
Tetsuya Takine
Keyword(s):  
Markov Process ◽  
Exponential Distribution ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Input Rate ◽  
Fluid Queue ◽  
Birth And Death Processes ◽  
Fluid Queues ◽  
Semi Markov Process ◽  
Semi Markov Processes

We consider a fluid queue controlled by a semi-Markov process and we apply the Markov-renewal approach developed earlier in the context of quasi-birth-and-death processes and of Markovian fluid queues. We analyze two subfamilies of semi-Markov processes. In the first family, we assume that the intervals during which the input rate is negative have an exponential distribution. In the second family, we take the complementary case and assume that the intervals during which the input rate is positive have an exponential distribution. We thoroughly characterize the structure of the stationary distribution in both cases.

scholarly journals STATIONARY DISTRIBUTION OF THE VOLUME AT THE BEST QUOTE IN A POISSON ORDER BOOK MODEL

2017 ◽  
Vol 20 (06) ◽  
pp. 1750039
Author(s):  
IOANE MUNI TOKE
Keyword(s):  
Stochastic Process ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Applied Probability ◽  
Order Book ◽  
Markovian Model ◽  
Limit Order ◽  
Market Order ◽  
Electronic Order

We develop a Markovian model that deals with the volume offered at the best quote of an electronic order book. The volume of the first limit is a stochastic process whose paths are periodically interrupted and reset to a new value, either by a new limit order submitted inside the spread or by a market order that removes the first limit. Using applied probability results on killing and resurrecting Markov processes, we derive the stationary distribution of the volume offered at the best quote. All proposed models are empirically fitted and compared, stressing the importance of the proposed mechanisms.

scholarly journals Product-Form Solutions for a Class of Structured Multidimensional Markov Processes

2014 ◽  
Vol 74 (3) ◽  
pp. 844-863 ◽  
Author(s):  
Jori Selen ◽  
Ivo J.B.F. Adan ◽  
Johan S.H. van Leeuwaarden
Keyword(s):  
Markov Processes ◽  
Product Form

Insensitivity and product-form decomposability of reallocatable GSMP

1993 ◽  
Vol 25 (2) ◽  
pp. 415-437 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Stochastic Process ◽  
Stationary Distribution ◽  
Jump Process ◽  
Product Form ◽  
Balance Equations ◽  
Independent Components ◽  
Extended Sense ◽  
Wide Range ◽  
Standard Models

A stochastic process, called reallocatable GSMP (RGSMP for short), is introduced in order to study insensitivity of its stationary distribution. RGSMP extends GSMP with interruptions, and is applicable to a wide range of queues, from the standard models such as BCMP and Kelly's network queues to new ones such as their modifications with interruptions and Serfozo's (1989) non-product form network queues, and can be used to study their insensitivity in a unified way. We prove that RGSMP supplemented by the remaining lifetimes is product-form decomposable, i.e. its stationary distribution splits into independent components if and only if a version of the local balance equations hold, which implies insensitivity of the RGSMP scheme in a certain extended sense. Various examples of insensitive queues are given, which include new results. Our proofs are based on the characterization of a stationary distribution for SCJP (self-clocking jump process) of Miyazawa (1991).

ON THE TIME-DEPENDENT BEHAVIOR OF A MARKOVIAN REENTRANT-LINE MODEL

2018 ◽  
Vol 33 (3) ◽  
pp. 367-386
Author(s):  
Brian Fralix
Keyword(s):  
Stationary Distribution ◽  
Time Dependent ◽  
Product Form ◽  
Line Model ◽  
Transition Functions ◽  
Geometric Product ◽  
Dependent Behavior ◽  
Product Technique ◽  
Form Representation ◽  
Similar Product

We use the random-product technique from [5] to study both the steady-state and time-dependent behavior of a Markovian reentrant-line model, which is a generalization of the preemptive reentrant-line model studied in the work of Adan and Weiss [2]. Our results/observations yield additional insight into why the stationary distribution of the reentrant-line model from [2] exhibits an almost-geometric product-form structure: indeed, our generalized reentrant-line model, when stable, admits a stationary distribution with a similar product-form representation as well. Not only that, the Laplace transforms of the transition functions of our reentrant-line model also have a product-form structure if it is further assumed that both Buffers 2 and 3 are empty at time zero.

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