The basic equations for a supplemented GSMP and its applications to queues

1988 ◽  
Vol 25 (03) ◽  
pp. 565-578 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Stochastic Process ◽  
Steady State ◽  
Queueing Networks ◽  
General Setting ◽  
Main Concern ◽  
Stationary Condition ◽  
State Equations ◽  
Semi Markov Process ◽  
Basic Equations ◽  
Stieltjes Transforms

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

The basic equations for a supplemented GSMP and its applications to queues

1988 ◽  
Vol 25 (3) ◽  
pp. 565-578 ◽  
Author(s):  
Masakiyo Miyazawa ◽  
Genji Yamazaki
Keyword(s):  
Stochastic Process ◽  
Steady State ◽  
Queueing Networks ◽  
General Setting ◽  
Main Concern ◽  
Stationary Condition ◽  
State Equations ◽  
Semi Markov Process ◽  
Basic Equations ◽  
Stieltjes Transforms

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

Loss tandem networks with blocking - a semi-Markov approach

2010 ◽  
Vol 58 (4) ◽  
pp. 673-681
Author(s):  
W. Oniszczuk
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Markov Model ◽  
Markov Models ◽  
State Graph ◽  
State Equations ◽  
Semi Markov Process ◽  
Tandem Networks ◽  
Measures Of Effectiveness

Loss tandem networks with blocking - a semi-Markov approachBased on the semi-Markov process theory, this paper describes an analytical study of a loss multiple-server two-station network model with blocking. Tasks arrive to the tandem in a Poisson fashion at a rate λ, and the service times at the first and second stations are non-exponentially distributed with means sAand sB, respectively. Between these two stations there is a buffer with finite capacity. In this type of network, if the buffer is full, the accumulation of new tasks (jobs) by the second station is temporarily suspended (blocking factor) and tasks must wait on the first station until the transmission process is resumed. Any new task that finds all service lines at the first station occupied is turned away and is lost (loss factor). Initially, in this document, a Markov model of the loss tandem with blocking is investigated. Here, a two-dimensional state graph is constructed and a set of steady-state equations is created. These equations allow the calculation of state probabilities for each graph state. A special algorithm for transforming the Markov model into a semi-Markov process is presented. This approach allows calculating steady-state probabilities in the semi-Markov model. In the next part of the paper, the algorithms for calculation of the main measures of effectiveness in the semi-Markov model are presented. Finally, the numerical part of this paper contains an investigation of some special semi-Markov models, where the results are presented of the calculation of the quality of service (QoS) parameters and the main measures of effectiveness.

Approximation of the queue-length distribution of an M/GI/s queue by the basic equations

1986 ◽  
Vol 23 (2) ◽  
pp. 443-458 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Length Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
State Equations ◽  
Main Interest ◽  
Queue Length Distribution ◽  
Approximation Formulas ◽  
Basic Equations

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in the M/GI/s queue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived for M/GI/s/k with finite and infinite k. Similar approximations are possible for M/GI/s/k (k < +∞).

scholarly journals Semi-Markov-Based Approach for the Analysis of Open Tandem Networks with Blocking and Truncation

2009 ◽  
Vol 19 (1) ◽  
pp. 151-164 ◽  
Author(s):  
Walenty Oniszczuk
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Markov Model ◽  
Markov Models ◽  
Network Models ◽  
State Graph ◽  
State Equations ◽  
Semi Markov Process ◽  
Tandem Networks ◽  
Measures Of Effectiveness

Semi-Markov-Based Approach for the Analysis of Open Tandem Networks with Blocking and TruncationThis paper describes an analytical study of open two-node (tandem) network models with blocking and truncation. The study is based on semi-Markov process theory, and network models assume that multiple servers serve each queue. Tasks arrive at the tandem in a Poisson fashion at the rate λ, and the service times at the first and the second node are non-exponentially distributed with meanssAandsB, respectively. Both nodes have buffers with finite capacities. In this type of network, if the second buffer is full, the accumulation of new tasks by the second node is temporarily suspended (a blocking factor) and tasks must wait on the first node until the transmission process is resumed. All new tasks that find the first buffer full are turned away and are lost (a truncation factor). First, a Markov model of the tandem is investigated. Here, a two-dimensional state graph is constructed and a set of steady-state equations is created. These equations allow calculating state probabilities for each graph state. A special algorithm for transforming the Markov model into a semi-Markov process is presented. This approach allows calculating steady-state probabilities in the semi-Markov model. Next, the algorithms for calculating the main measures of effectiveness in the semi-Markov model are presented. In the numerical part of this paper, the author investigates examples of several semi-Markov models. Finally, the results of calculating both the main measures of effectiveness and quality of service (QoS) parameters are presented.

Approximation of the queue-length distribution of an M/GI/s queue by the basic equations

1986 ◽  
Vol 23 (02) ◽  
pp. 443-458 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Length Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
State Equations ◽  
Main Interest ◽  
Queue Length Distribution ◽  
Approximation Formulas ◽  
Basic Equations

We give a unified way of obtaining approximation formulas for the steady-state distribution of the queue length in theM/GI/squeue. The approximations of Hokstad (1978) and Case A of Tijms et al. (1981) are derived again. The main interest of this paper is in considering the theoretical meaning of the assumptions given in the literature. Having done this, we derive new approximation formulas. Our discussion is based on one version of the steady-state equations, called the basic equations in this paper. The basic equations are derived forM/GI/s/kwith finite and infinitek.Similar approximations are possible forM/GI/s/k(k&lt; +∞).

FURTHER CONSIDERATIONS ON THE CALCULATIONS OF SECRETION RATES. A CORRECTION

1961 ◽  
Vol 38 (3) ◽  
pp. 469-472 ◽  
Author(s):  
K. R. Laumas ◽  
J. F. Tait ◽  
S. A. S. Tait
Keyword(s):  
Steady State ◽  
Compartmental Model ◽  
Single Injection ◽  
Previous Treatment ◽  
Urinary Metabolite ◽  
Theoretical Treatment ◽  
Basic Equations ◽  
Special Circumstances ◽  
Secretion Rates

ABSTRACT Reconsideration of the question of the validity of the calculations of the secretion rates from the specificity activity of a urinary metabolite after the single injection of a radioactive hormone has led us to conclude that the basic equations used in a previous theoretical treatment are not generally applicable to the nonisotopic steady state if the radioactive steroid and hormone are introduced into the same compartment. If this is so, in a two compartmental model with metabolism occurring in both pools, it is now shown that the calculation (S = R — τ) is rigorously valid if certain precautions are taken. This is in contrast to the previous treatment which concluded (in certain special circumstances) that the calculation might not be correct. However, if the hormone is secreted in both compartments and the radioactive steroid is injected into only one, then the calculation (S = R — τ) may not be correct in certain circumstances as was previously concluded (Laumas et al. 1961).

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.

Validation of a new method for determination of free fatty acid turnover

1987 ◽  
Vol 252 (3) ◽  
pp. E431-E438 ◽  
Author(s):  
J. M. Miles ◽  
M. G. Ellman ◽  
K. L. McClean ◽  
M. D. Jensen
Keyword(s):  
Steady State ◽  
Fatty Acid ◽  
Free Fatty Acid ◽  
Clearance Rate ◽  
Experimental Period ◽  
Metabolic Clearance ◽  
Metabolic Clearance Rate ◽  
State Equations ◽  
Tracer Methods

The accuracy of tracer methods for estimating free fatty acid (FFA) rate of appearance (Ra), either under steady-state conditions or under non-steady-state conditions, has not been previously investigated. In the present study, endogenous lipolysis (traced with 14C palmitate) was suppressed in six mongrel dogs with a high-carbohydrate meal 10 h before the experiment, together with infusions of glucose, propranolol, and nicotinic acid during the experimental period. Both steady-state and non-steady-state equations were used to determine oleate Ra ([3H]oleate) before, during, and after a stepwise infusion of an oleic acid emulsion. Palmitate Ra did not change during the experiment. Steady-state equations gave the best estimates of oleate inflow approximately 93% of the known oleate infusion rate overall, while errors in tracer estimates of inflow were obtained when non-steady-state equations were used. The metabolic clearance rate of oleate was inversely related to plasma concentration (P less than 0.01). In conclusion, accurate estimates of FFA inflow were obtained when steady-state equations were used, even under conditions of abrupt and recent changes in Ra. Non-steady-state equations, in contrast, may provide erroneous estimates of inflow. The decrease in metabolic clearance rate during exogenous infusion of oleate suggests that FFA transport may follow second-order kinetics.

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