scholarly journals Asymptotic Diffusion Analysis of Multi-Server Retrial Queue with Hyper-Exponential Service

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 531 ◽  
Author(s):  
Alexander Moiseev ◽  
Anatoly Nazarov ◽  
Svetlana Paul
Keyword(s):  
Numerical Analysis ◽  
Diffusion Process ◽  
Service Time ◽  
Probability Distributions ◽  
Retrial Queue ◽  
Stationary Probability ◽  
Diffusion Analysis ◽  
Number Of Customers ◽  
Multi Server

A multi-server retrial queue with a hyper-exponential service time is considered in this paper. The study is performed by the method of asymptotic diffusion analysis under the condition of long delay in orbit. On the basis of the constructed diffusion process, we obtain approximations of stationary probability distributions of the number of customers in orbit and the number of busy servers. Using simulations and numerical analysis, we estimate the accuracy and applicability area of the obtained approximations.

scholarly journals Diffusion Limit of Multi-Server Retrial Queue with Setup Time

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2232
Author(s):  
Anatoly Nazarov ◽  
Alexander Moiseev ◽  
Tuan Phung-Duc ◽  
Svetlana Paul
Keyword(s):  
Asymptotic Methods ◽  
Retrial Queue ◽  
Power Saving ◽  
Setup Time ◽  
Diffusion Limit ◽  
Steady State Probability ◽  
Retrial Queueing System ◽  
Limiting Condition ◽  
Number Of Customers ◽  
Multi Server

In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer. Customers that find all the servers busy join the orbit and retry for service after an exponentially distributed time. For this model, we derive the stability condition which depends on the setup time and turns out to be more strict than that of the corresponding model with an infinite buffer which is independent of the setup time. We propose asymptotic methods to analyze the system under the condition that the delay in the orbit is extremely long. We show that the scaled-number of customers in the orbit converges to a diffusion process. Using this diffusion limit, we obtain approximations for the steady-state probability distribution of the number of busy servers and that of the number of customers in the orbit. We verify the accuracy of the approximations by simulations and numerical analysis. Numerical results show that the retrial system under the limiting condition consumes more energy than that with an infinite buffer in front of the servers.

A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (1) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation.Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

On the convergence to stationarity of the many-server Poisson queue

1999 ◽  
Vol 36 (2) ◽  
pp. 546-557 ◽  
Author(s):  
Wolfgang Stadje ◽  
P. R. Parthasarathy
Keyword(s):  
Service Time ◽  
Stationary Probability ◽  
Speed Of Convergence ◽  
The Many ◽  
Number Of Customers ◽  
Arrival Intensity

We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c < 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then pn(t) = P(X(t) = n) converges to the stationary probability πn = P(X = n). The integrals ∫0∞[E(X)-E(X(t))]dt and ∫0∞[P(X≤n) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.

A service model in which the server is required to search for customers

1984 ◽  
Vol 21 (01) ◽  
pp. 157-166 ◽  
Author(s):  
Marcel F. Neuts ◽  
M. F. Ramalhoto
Keyword(s):  
Poisson Process ◽  
Numerical Computation ◽  
Probability Distributions ◽  
General Distribution ◽  
Search Process ◽  
Single Server ◽  
Stationary Probability ◽  
Service Model ◽  
Service Times ◽  
Number Of Customers

Customers enter a pool according to a Poisson process and wait there to be found and processed by a single server. The service times of successive items are independent and have a common general distribution. Successive services are separated by seek phases during which the server searches for the next customer. The search process is Markovian and the probability of locating a customer in (t, t + dt) is proportional to the number of customers in the pool at time t. Various stationary probability distributions for this model are obtained in explicit forms well-suited for numerical computation. Under the assumption of exponential service times, corresponding results are obtained for the case where customers may escape from the pool.

On the convergence to stationarity of the many-server Poisson queue

1999 ◽  
Vol 36 (02) ◽  
pp. 546-557
Author(s):  
Wolfgang Stadje ◽  
P. R. Parthasarathy
Keyword(s):  
Service Time ◽  
Stationary Probability ◽  
Speed Of Convergence ◽  
The Many ◽  
Number Of Customers ◽  
Arrival Intensity

We consider the many-server Poisson queue M/M/c with arrival intensity λ, mean service time 1 and λ/c &lt; 1. Let X(t) be the number of customers in the system at time t and assume that the system is initially empty. Then p n (t) = P(X(t) = n) converges to the stationary probability π n = P(X = n). The integrals ∫0 ∞[E(X)-E(X(t))]dt and ∫0 ∞[P(X≤n) − P(X(t)≤n)]dt are a measure of the speed of convergence towards stationarity. We express these integrals in terms of λ and c.

scholarly journals Output process of the M|GI|1 is an asymptotical renewal process

2021 ◽  
Vol 21 (1) ◽  
pp. 100-110
Author(s):  
Ivan L. Lapatin ◽  
◽  
Anatoly A. Nazarov ◽  
Keyword(s):  
Distribution Function ◽  
Service Time ◽  
Renewal Process ◽  
Retrial Queue ◽  
Arbitrary Distribution ◽  
Single Server ◽  
Output Process ◽  
Limit Condition ◽  
Repeated Calls ◽  
Number Of Customers

Most of the studies on models with retrials are devoted to the research of the number of applications in the system or in the source of repeated calls using asymptotic and numerical approaches or simulation. Although one of the main characteristics that determines the quality of the communication system is the number of applications served by the system per unit of time. Information on the characteristics of the output processes is of great practical interest, since the output process of one system may be incoming to another. The results of the study of the outgoing flows of queuing networks are widely used in the modeling of computer systems, in the design of data transmission networks and in the analysis of complex multi-stage production processes. In this paper, we have considered a single server system with redial, the input of which receives a stationary Poisson process. The service time in considered system is a random value with an arbitrary distribution function B(x). If the customer enters the system and finds the server busy, it instantly joins the orbit and carries out a random delay there during an exponentially distributed time. The object of study is the output process of this system. The output is characterized by the probability distribution of the number of customers that have completed service for time t. We have provided the study using asymptotic analysis method under low rate of retrials limit condition. We have shown in the paper that the output of retrial queue M|GI|1 is an asymptotical renewal process. Moreover, the lengths of the intervals in output process are the sum of an exponential random value with the parameter lambda + kappa and a random variable with the distribution function B(x). The results of a numerical experiment show that the probability distributions of the number of served customers in the system are practically the same for significantly different distribution laws B(x) of service time if the service times have the same first two moments.

scholarly journals Analysis of a Multiserver Queueing-Inventory System

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
A. Krishnamoorthy ◽  
R. Manikandan ◽  
Dhanya Shajin
Keyword(s):  
Steady State ◽  
Service Time ◽  
Conditional Distribution ◽  
Retrial Queue ◽  
Form Solution ◽  
Inventory System ◽  
Inventory Level ◽  
Steady State Distribution ◽  
State Distribution ◽  
Number Of Customers

We attempt to derive the steady-state distribution of theM/M/cqueueing-inventory system with positive service time. First we analyze the case ofc=2servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the(s,Q)policy. We obtain a product form solution of the steady-state distribution under the assumption that customers do not join the system when the inventory level is zero. An average system cost function is constructed and the optimal pair(s,Q)and the corresponding expected minimum cost are computed. As in the case ofM/M/cretrial queue withc≥3, we conjecture thatM/M/cforc≥3, queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach. We derive an explicit expression for the stability condition of the system. Conditional distribution of the inventory level, conditioned on the number of customers in the system, and conditional distribution of the number of customers, conditioned on the inventory level, are derived. The distribution of two consecutivestostransitions of the inventory level (i.e., the first return time tos) is computed. We also obtain several system performance measures.

scholarly journals Stochastic Approximations and Monotonicity of a Single Server Feedback Retrial Queue

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Mohamed Boualem ◽  
Natalia Djellab ◽  
Djamil Aïssani
Keyword(s):  
Markov Chains ◽  
Service Time ◽  
Retrial Queue ◽  
Stochastic Ordering ◽  
Stochastic Comparison ◽  
Single Server ◽  
Stochastic Approximations ◽  
Bernoulli Feedback ◽  
Monotonicity Results

This paper focuses on stochastic comparison of the Markov chains to derive some qualitative approximations for anM/G/1retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters.

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