Stationary Distribution in the One Server Queuing System $$M|M|1|\infty $$ with the Stopping Intensity of the Input Flow

2021 ◽  
pp. 444-449
Author(s):  
Gurami Tsitsiashvili
Keyword(s):  
Stationary Distribution ◽  
Queuing System ◽  
Input Flow ◽  
The One

scholarly journals Stochastic models for interference between searching insect parasites

1989 ◽  
Vol 30 (3) ◽  
pp. 268-277 ◽  
Author(s):  
Phil Diamond
Keyword(s):  
Stationary Distribution ◽  
Intraspecific Competition ◽  
Death Process ◽  
Confluent Hypergeometric Functions ◽  
Large Populations ◽  
The Mean ◽  
The One ◽  
Globally Stable ◽  
Insect Parasites ◽  
Birth Death

AbstractCompetition between a finite number of searching insect parasites is modelled by differential equations and birth-death processes. In the one species case of intraspecific competition, the deterministic equilibrium is globally stable and, for large populations, approximates the mean of the stationary distribution of the process. For two species, both inter- and intraspecific competition occurs and the deterministic equilibrium is globally stable. When the birth-death process is reversible, it is shown that the mean of the stationary distribution is approximated by the equilibrium. Confluent hypergeometric functions of two variables are important to the theory.

scholarly journals Simulation of a Queuing System of a Post Office in Anylogic Software

2020 ◽  
Vol 22 (2) ◽  
pp. 15-22
Author(s):  
Silvia Dutkova ◽  
Karol Achimsky ◽  
Pawel Drozdziel
Keyword(s):  
Probability Distribution ◽  
Service Time ◽  
Simulation Software ◽  
Queuing System ◽  
System Model ◽  
Cost Calculation ◽  
Post Office ◽  
Input Flow ◽  
Starting Point ◽  
Exponential Probability

This paper displays the design and application of a model that simulates the queuing system of a fictional post office. Starting point for solving more complicated optimization tasks is to create a system model that consists of elements of reality and the relationships between these elements. The key part of the paper includes the model of a queuing system of a post office created in Anylogic simulation software. The model of the post office displays post office with 5 postal counters, a certain input flow and a certain service time with an exponential probability distribution. The model also includes statistics and cost calculation.

scholarly journals Analysis and optimization of retrial systems and queue with variable rate of input flow

2020 ◽  
pp. 130-134
Author(s):  
I. Ya. Usar ◽  
I. A. Makushenko
Keyword(s):  
Stationary Distribution ◽  
Stochastic Systems ◽  
Convergence Rates ◽  
Control Strategies ◽  
Stationary Regime ◽  
Variable Rate ◽  
Input Flow ◽  
Threshold Strategy ◽  
Threshold Control ◽  
Repeated Calls

The paper is focused on in-depth study of the promising area of the stochastic systems theory related with scrutiny of queuing systems with repeated calls. We research Markov`s models of retrial systems with queue and variable rate of input flow controlled by threshold strategy with no restriction on the capacity of the orbit. We defined stationary regime existence conditions and investigated probability characteristics of process for two-dimension Markov process with continuous time which we took as a main model of the specified system. In stationary regime for probability characteristics of the service process were found explicit formulas. Research methods which we used are based on the initial process approximation by the process with bounded state space. Results of the research allow us to evaluate convergence rates of stationary distribution of finite systems with repeated calls to stationary distribution of infinite systems. Method of probability flow equating is used for obtain explicit expressions for stationary system probabilities through the closed path which are defined in a special way. For threshold control strategies the optimization problem of the total income of the system was stated and solved.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (1) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

scholarly journals ANALYSIS OF COMMUTER LINE TICKET PURCHASE QUEUING SYSTEM IN BOGOR STATION

2017 ◽  
Vol 2 (02) ◽  
pp. 35
Author(s):  
Resista Vikaliana
Keyword(s):  
Queuing Theory ◽  
Service Use ◽  
Single Channel ◽  
Performance Measure ◽  
Queuing System ◽  
Queuing Model ◽  
The One ◽  
Multi Phase ◽  
Ticket Purchase ◽  
First In First Out

Queue is a situation that happen to people, goods, and components that need to wait to get a service. The good quality of service will satisfy the customers and decrease the queue line. Queue often happens in a station especially in weekdays. A long queue line happens in the station is one of the problems that need to be solved. Instead of manual ticket purchasing that served by the operator, today PT KAI Commuter Line also serve ticket purchasing using THB machine. The purpose of this study is to compare the performance of queuing model that happen in Bogor station locket and to determine if the queuing model is efficient by comparing the service standard, between the manual and the one that used THB machine. The method used in this research was descriptive method by using queuing theory calculation. The model of locket queuing using THB machine in Bogor Station is Multi Channel-Multi Phase, in ticket purchasing using THB machine. Besides, in the operator locket service, the queuing model is Single Channel-Single Phase. Both s ticket purchasing service use First In First Out (FIFO) disciple. The maximum amount of the queue line and the source of customers’ arrival are infinite. Based on the value of system performance can be concluded that queuing system and the service given already great and effective (based on the performance measure and probability or passengers’’ chances), passengers who are waiting to buy tickets, either manual or using machine less than 1, or assumed 1 person. From the observation, the use of THB machine decrease the queue line, but need to be socialized because passengers does not know how to use THB machine to buy ticket independently.Keywords: queue, queuing model, commuter line ticket purchasing, Bogor station

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (01) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

scholarly journals Large deviations and quasi-stationarity for density-dependent birth-death processes

1998 ◽  
Vol 40 (2) ◽  
pp. 238-256 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Large Deviations ◽  
Stationary Distribution ◽  
Death Process ◽  
Large Deviations Principle ◽  
Density Dependent ◽  
Exponential Decay Rate ◽  
Finite State ◽  
Degenerate Distribution ◽  
The One ◽  
Birth Death

AbstractConsider a density-dependent birth-death process XN on a finite state space of size N. Let PN be the law (on D([0, T]) where T > 0 is arbitrary) of the density process XN/N and let πN be the unique stationary distribution (on[0,1]) of XN/N, if it exists. Typically, these distributions converge weakly to a degenerate distribution as N → ∞ so the probability of sets not containing the degenerate point will tend to 0; large deviations is concerned with obtaining the exponential decay rate of these probabilities. Friedlin-Wentzel theory is used to establish the large deviations behaviour (as N → ∞) of PN. In the one-dimensional case, a large deviations principle for the stationary distribution πN is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so πN no longer exists), the same elementary computations are still applicable to the quasi-stationary distribution, and we show that the quasi-stationary distributions obey the same large deviations principle as in the recurrent case. In addition, we address some questions related to the estimated time to absorption and obtain a large deviations principle for the invariant distribution in higher dimensions by studying a quasi-potential.

Sign in / Sign up

Export Citation Format

Share Document