The solution of certain two-dimensional Markov models

1982 ◽  
Vol 14 (2) ◽  
pp. 295-308 ◽  
Author(s):  
G. Fayolle ◽  
P. J. B. King ◽  
I. Mitrani
Keyword(s):  
Steady State ◽  
Solution Method ◽  
Markov Models ◽  
Transition Rates ◽  
Two Dimensional ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Dependent ◽  
Modelling Problems

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

The solution of certain two-dimensional Markov models

1982 ◽  
Vol 14 (02) ◽  
pp. 295-308 ◽  
Author(s):  
G. Fayolle ◽  
P. J. B. King ◽  
I. Mitrani
Keyword(s):  
Steady State ◽  
Solution Method ◽  
Markov Models ◽  
Transition Rates ◽  
Two Dimensional ◽  
Steady State Distribution ◽  
State Distribution ◽  
Birth And Death Processes ◽  
State Dependent ◽  
Modelling Problems

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

A Result on Networks of Queues with Customer Coalescence and State-Dependent Signaling

1998 ◽  
Vol 35 (01) ◽  
pp. 151-164
Author(s):  
Xiuli Chao ◽  
Shaohui Zheng
Keyword(s):  
Steady State ◽  
Single Unit ◽  
Form Solution ◽  
Product Form ◽  
Transition Rates ◽  
Steady State Distribution ◽  
State Distribution ◽  
State Dependent ◽  
Batch Services ◽  
Networks Of Queues

In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Hendersonet al. (1994), as well as those of Chaoet al. (1996).

A Result on Networks of Queues with Customer Coalescence and State-Dependent Signaling

1998 ◽  
Vol 35 (1) ◽  
pp. 151-164 ◽  
Author(s):  
Xiuli Chao ◽  
Shaohui Zheng
Keyword(s):  
Steady State ◽  
Single Unit ◽  
Form Solution ◽  
Product Form ◽  
Transition Rates ◽  
Steady State Distribution ◽  
State Distribution ◽  
State Dependent ◽  
Batch Services ◽  
Networks Of Queues

In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Henderson et al. (1994), as well as those of Chao et al. (1996).

scholarly journals Infinity Learning: Learning Markov Chains from Aggregate Steady-State Observations

2020 ◽  
Vol 34 (04) ◽  
pp. 3922-3929
Author(s):  
Jianfei Gao ◽  
Mohamed A. Zahran ◽  
Amit Sheoran ◽  
Sonia Fahmy ◽  
Bruno Ribeiro
Keyword(s):  
Steady State ◽  
Descent Method ◽  
Training Data ◽  
Stochastic Gradient Descent ◽  
Gradient Descent Method ◽  
Transition Rates ◽  
Steady State Distribution ◽  
State Distribution ◽  
Heavy Loads ◽  
Infinite Sums

We consider the task of learning a parametric Continuous Time Markov Chain (CTMC) sequence model without examples of sequences, where the training data consists entirely of aggregate steady-state statistics. Making the problem harder, we assume that the states we wish to predict are unobserved in the training data. Specifically, given a parametric model over the transition rates of a CTMC and some known transition rates, we wish to extrapolate its steady state distribution to states that are unobserved. A technical roadblock to learn a CTMC from its steady state has been that the chain rule to compute gradients will not work over the arbitrarily long sequences necessary to reach steady state —from where the aggregate statistics are sampled. To overcome this optimization challenge, we propose ∞-SGD, a principled stochastic gradient descent method that uses randomly-stopped estimators to avoid infinite sums required by the steady state computation, while learning even when only a subset of the CTMC states can be observed. We apply ∞-SGD to a real-world testbed and synthetic experiments showcasing its accuracy, ability to extrapolate the steady state distribution to unobserved states under unobserved conditions (heavy loads, when training under light loads), and succeeding in difficult scenarios where even a tailor-made extension of existing methods fails.

Kinetic analysis of mechanism of intestinal Na+-dependent sugar transport

1985 ◽  
Vol 248 (5) ◽  
pp. C498-C509 ◽  
Author(s):  
D. Restrepo ◽  
G. A. Kimmich
Keyword(s):  
Experimental Data ◽  
Steady State ◽  
Sugar Transport ◽  
Enzyme Kinetic ◽  
Steady State Distribution ◽  
State Distribution ◽  
Least Squares Fit ◽  
Coupling Stoichiometry ◽  
Rate Limiting ◽  
Kinetics Of

Zero-trans kinetics of Na+-sugar cotransport were investigated. Sugar influx was measured at various sodium and sugar concentrations in K+-loaded cells treated with rotenone and valinomycin. Sugar influx follows Michaelis-Menten kinetics as a function of sugar concentration but not as a function of Na+ concentration. Nine models with 1:1 or 2:1 sodium:sugar stoichiometry were considered. The flux equations for these models were solved assuming steady-state distribution of carrier forms and that translocation across the membrane is rate limiting. Classical enzyme kinetic methods and a least-squares fit of flux equations to the experimental data were used to assess the fit of the different models. Four models can be discarded on this basis. Of the remaining models, we discard two on the basis of the trans sodium dependence and the coupling stoichiometry [G. A. Kimmich and J. Randles, Am. J. Physiol. 247 (Cell Physiol. 16): C74-C82, 1984]. The remaining models are terter ordered mechanisms with sodium debinding first at the trans side. If transfer across the membrane is rate limiting, the binding order can be determined to be sodium:sugar:sodium.

Insensitivity of steady-state distributions of generalized semi-Markov processes by speeds

1978 ◽  
Vol 10 (04) ◽  
pp. 836-851 ◽  
Author(s):  
R. Schassberger
Keyword(s):  
Steady State ◽  
General System ◽  
State Distribution ◽  
Lifetime Distributions ◽  
State Dependent ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Infinite State ◽  
State Case ◽  
Residual Lifetimes

A generalized semi-Markov process with speeds describes the fluctuation, in time, of the state of a certain general system involving, at any given time, one or more living components, whose residual lifetimes are being reduced at state-dependent speeds. Conditions are given for the stationary state distribution, when it exists, to depend only on the means of some of the lifetime distributions, not their exact shapes. This generalizes results of König and Jansen, particularly to the infinite-state case.

PROCESSOR SHARING G-QUEUES WITH INERT CUSTOMERS AND CATASTROPHES: A MODEL FOR SERVER AGING AND REJUVENATION

2017 ◽  
Vol 31 (4) ◽  
pp. 420-435 ◽  
Author(s):  
J.-M. Fourneau ◽  
Y. Ait El Majhoub
Keyword(s):  
Steady State ◽  
Product Form ◽  
Processor Sharing ◽  
Service Capacity ◽  
Steady State Distribution ◽  
State Distribution ◽  
Signal Arrival ◽  
Open Networks ◽  
Networks Of Queues

We consider open networks of queues with Processor-Sharing discipline and signals. The signals deletes all the customers present in the queues and vanish instantaneously. The customers may be usual customers or inert customers. Inert customers do not receive service but the servers still try to share the service capacity between all the customers (inert or usual). Thus a part of the service capacity is wasted. We prove that such a model has a product-form steady-state distribution when the signal arrival rates are positive.

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