Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue

1989 ◽  
Vol 21 (01) ◽  
pp. 159-180 ◽  
Author(s):  
Bhaskar Sengupta
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Supply And Demand ◽  
Joint Probability ◽  
Matrix Exponential ◽  
Linear Matrix ◽  
Joint Probability Distribution ◽  
Rate Matrix ◽  
Steady State Distribution

This paper is concerned with a bivariate Markov process {Xt, Nt ; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt ; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue

1989 ◽  
Vol 21 (1) ◽  
pp. 159-180 ◽  
Author(s):  
Bhaskar Sengupta
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Supply And Demand ◽  
Joint Probability ◽  
Matrix Exponential ◽  
Linear Matrix ◽  
Joint Probability Distribution ◽  
Rate Matrix ◽  
Steady State Distribution

This paper is concerned with a bivariate Markov process {Xt, Nt; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

scholarly journals Perishable Inventory System with Postponed Demands and Negative Customers

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Paul Manuel ◽  
B. Sivakumar ◽  
G. Arivarignan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Inventory System ◽  
Arrival Process ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Time Interval ◽  
Negative Customers ◽  
Perishable Inventory System ◽  
Number Of Customers

This article considers a continuous review perishable (s,S) inventory system in which the demands arrive according to a Markovian arrival process (MAP). The lifetime of items in the stock and the lead time of reorder are assumed to be independently distributed as exponential. Demands that occur during the stock-out periods either enter a pool which has capacity N(<∞) or are lost. Any demand that takes place when the pool is full and the inventory level is zero is assumed to be lost. The demands in the pool are selected one by one, if the replenished stock is above s, with time interval between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The waiting demands in the pool independently may renege the system after an exponentially distributed amount of time. In addition to the regular demands, a second flow of negative demands following MAP is also considered which will remove one of the demands waiting in the pool. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. The measures of system performance in the steady state are calculated and the total expected cost per unit time is also considered. The results are illustrated numerically.

scholarly journals A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
R. Jayaraman ◽  
B. Sivakumar ◽  
G. Arivarignan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Life Time ◽  
Inventory System ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Cost Rate ◽  
Stochastic Inventory ◽  
Number Of Customers

A mathematical modelling of a continuous review stochastic inventory system with a single server is carried out in this work. We assume that demand time points form a Poisson process. The life time of each item is assumed to have exponential distribution. We assume(s,S)ordering policy to replenish stock with random lead time. The server goes for a vacation of an exponentially distributed duration at the time of stock depletion and may take subsequent vacation depending on the stock position. The customer who arrives during the stock-out period or during the server vacation is offered a choice of joining a pool which is of finite capacity or leaving the system. The demands in the pool are selected one by one by the server only when the inventory level is aboves, with interval time between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the inventory level and the number of customers in the pool is obtained in the steady-state case. Various system performance measures in the steady state are derived, and the long-run total expected cost rate is calculated.

Perishable Inventory System with Server Interruptions, Multiple Server Vacations, and N Policy

2015 ◽  
Vol 6 (2) ◽  
pp. 32-52 ◽  
Author(s):  
K. Jeganathan ◽  
N. Anbazhagan ◽  
B. Vigneshwaran
Keyword(s):  
Steady State ◽  
Poisson Process ◽  
Finite Size ◽  
Joint Probability ◽  
Inventory System ◽  
Joint Probability Distribution ◽  
Perishable Inventory ◽  
Waiting Area ◽  
Perishable Inventory System ◽  
Negative Exponential

This article presents a perishable inventory system under continuous review at a service facility in which a waiting area for customers is of finite size . The authors assume that the replenishment of inventory is instantaneous. The items of inventory have exponential life times. It is assumed that demand for the commodity is of unit size. The service starts only when the customer level reaches a prefixed level , starting from the epoch at which no customer is left behind in the system. The arrivals of customers to the service station form a Poisson process. The server goes for a vacation of an exponentially distributed duration whenever the waiting area is zero. If the server finds the customer level is less than when he returns to the system, he immediately takes another vacation. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The service process is subject to interruptions, which occurs according to a Poisson process. The interrupted server is repaired at an exponential rate. Also the waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the number customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived and the total expected cost is also considered. The results are illustrated with numerical examples.

scholarly journals Markovian inventory model with two parallel queues, jockeying and impatient customers

2016 ◽  
Vol 26 (4) ◽  
pp. 467-506 ◽  
Author(s):  
K. Jeganathan ◽  
J. Sumathi ◽  
G. Mahalakshmi
Keyword(s):  
Steady State ◽  
Finite Size ◽  
Joint Probability ◽  
Life Time ◽  
The Other ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Cost Rate ◽  
Parallel Queues ◽  
Number Of Customers

This article presents a perishable stochastic inventory system under continuous review at a service facility consisting of two parallel queues with jockeying. Each server has its own queue, and jockeying among the queues is permitted. The capacity of each queue is of finite size L. The inventory is replenished according to an (s; S) inventory policy and the replenishing times are assumed to be exponentially distributed. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The life time of each item is assumed to be exponential. Customers arrive according to a Poisson process and on arrival; they join the shortest feasible queue. Moreover, if the inventory level is more than one and one queue is empty while in the other queue, more than one customer are waiting, then the customer who has to be received after the customer being served in that queue is transferred to the empty queue. This will prevent one server from being idle while the customers are waiting in the other queue. The waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the inventory level, the number of customers in both queues, and the status of the server are obtained in the steady state. Some important system performance measures in the steady state are derived, so as the long-run total expected cost rate.

scholarly journals Network of interacting synthetic molecules in steady state

2008 ◽  
Vol 464 (2096) ◽  
pp. 2219-2228 ◽  
Author(s):  
Erol Gelenbe
Keyword(s):  
Steady State ◽  
Closed Form Solution ◽  
Joint Probability ◽  
Reaction Network ◽  
Mass Conservation ◽  
Reaction Rates ◽  
Form Solution ◽  
Joint Probability Distribution ◽  
State Behaviour ◽  
Steady State Behaviour

In this paper, we study the steady-state behaviour of a reaction network of interacting molecules using the chemical master equation (CME). The model considers a set of base species from which further compounds are created via binary reactions, as well as by monomolecular and dissipation reactions. The model includes external arrivals of molecules into the reaction volume and assumes that the reaction rates are proportional to the number of molecules of the reactants that are present. We obtain an explicit expression for the solution of the CME in equilibrium under the assumption that the system obeys a mass conservation law for the overall rate of incoming and outgoing molecules. This closed-form solution shows that the joint probability distribution of the number of molecules of each species is in ‘product form’, i.e. it is the product of the marginal distributions for the number of molecules of each species. We also show that the steady-state distribution of the number of molecules of each base and synthesized species follows a Poisson distribution.

scholarly journals Copula-based joint probability distribution of water supply and demand in Luhun irrigation district

2018 ◽  
Vol 19 (3) ◽  
pp. 932-943
Author(s):  
Jinping Zhang ◽  
Xixi Shi ◽  
Jian Li ◽  
Fawen Li
Keyword(s):  
Irrigation Water ◽  
Supply And Demand ◽  
Three Dimensional ◽  
Joint Probability ◽  
Irrigation District ◽  
Joint Probability Distribution ◽  
Joint Distributions ◽  
Best Fitting ◽  
T Copula ◽  
Student’S T

Abstract The potential of the copula method to construct the joint probability distribution of three hydrological variables characterizing water supply and demand (WSD) is explored for the Luhun irrigation district of China. The marginal distributions of rainfall, reference crop evapotranspiration (ET0) and irrigation water are simulated by the corresponding best-fitting cumulative distribution functions. Furthermore, the correlations between every pair of variables are quantified. On this basis, the two-dimensional joint distributions of rainfall and (ET0) (representing natural WSD), and irrigation water and (ET0) (representing man-made WSD), and the three-dimensional joint distribution of rainfall, irrigation water, and (ET0) (representing natural–man-made WSD) are established. The results reveal that the best-fitting marginal distributions for rainfall and (ET0) and irrigation water are the normal distribution and the Weibull distribution. Moreover, for rainfall and (ET0), the Student's t copula is applied to obtain the joint distribution, while the corresponding copula for (ET0) and irrigation water is the Clayton copula. Finally, the three-dimensional Student's t copula is selected to explore the dependence structure among rainfall, irrigation water, and (ET0). Therefore, these joint distributions provide an efficient approach to assess water shortage risks in the irrigation district.

scholarly journals Rate Equation Theory for Island Sizes and Capture Zone Areas in Submonolayer Deposition: Realistic Treatment of Spatial Aspects of Nucleation

2002 ◽  
Vol 749 ◽  
Author(s):  
J.W. Evans ◽  
Maozhi Li ◽  
M.C. Bartelt
Keyword(s):  
Rate Equation ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Capture Zone ◽  
Nucleation Event ◽  
Island Size ◽  
Extensive Information ◽  
The Impact ◽  
Island Density

ABSTRACTExtensive information on the distribution of islands formed during submonolayer deposition is provided by the joint probability distribution (JPD) for island sizes, s, and capture zone areas, A. A key ingredient determining the form of the JPD is the impact of each nucleation event on existing capture zone areas. Combining a realistic characterization of such spatial aspects of nucleation with a factorization ansatz for the JPD, we provide a concise rate equation formulation for the variation with island size of both the capture zone area and the island density.

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