scholarly journals Analysis of MAP/PH/1 Queueing System with Degrading Service Rate and Phase Type Vacation

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2387
Author(s):  
Alka Choudhary ◽  
Srinivas R. Chakravarthy ◽  
Dinesh C. Sharma
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Service System ◽  
Fixed Number ◽  
Service Rate ◽  
Queueing Models ◽  
Matrix Analytic Methods ◽  
Numerical Examples ◽  
Phase Type ◽  
Wear And Tear

Degradation of services arises in practice due to a variety of reasons including wear-and-tear of machinery and fatigue. In this paper, we look at MAP/PH/1-type queueing models in which degradation is introduced. There are several ways to incorporate degradation into a service system. Here, we model the degradation in the form of the service rate declining (i.e., the service rate decreases with the number of services offered) until the degradation is addressed. The service rate is reset to the original rate either after a fixed number of services is offered or when the server becomes idle. We look at two models. In the first, we assume that the degradation is instantaneously fixed, and in the second model, there is a random time that is needed to address the degradation issue. These models are analyzed in steady state using the classical matrix-analytic methods. Illustrative numerical examples are provided. Comparisons of both the models are drawn.

scholarly journals Two service units with interference in the access to servers

1999 ◽  
Vol 12 (4) ◽  
pp. 357-370
Author(s):  
Rosa E. Lillo ◽  
Marcel F. Neuts
Keyword(s):  
Steady State ◽  
Manufacturing Systems ◽  
Service System ◽  
Queueing Models ◽  
Steady State Regime ◽  
Primary Interest ◽  
Phase Type ◽  
Algorithmic Problems ◽  
State Regime ◽  
General Service

We examine the service mechanism of two queueing models with two units in tandem. In the first model, customers who complete service in Unit 1 must wait in an intermediate buffer until the ongoing service in Unit II ends. In the second model, jobs can be pre-positioned in an intermediate buffer to await service in Unit II. Under the assumption of phase-type service times, the steady-state regime of the service system is studied in detail.The models are inspired by the gas pump model of A.B. Clarke and by phenomena observed in cafeteria lines and certain manufacturing systems. However, their primary interest may lie in the methodology of their exceptionally tractable analysis. We derive formulas for the throughput and other quantities by using the familiar PH-formalism. These formulas turn out to be unusually transparent and have probabilistic interpretations that do not depend on the PH assumptions. These interpretations therefore also hold for general service time distributions. The methodology is general and can be applied to other systems with interactions between servers. The models also present interesting algorithmic problems of didactic interest.

scholarly journals A discrete single server queue with Markovian arrivals and phase type group services

1995 ◽  
Vol 8 (2) ◽  
pp. 151-176 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
K. Laurie Dolhun ◽  
S. Chakravarthy
Keyword(s):  
Steady State ◽  
Queueing System ◽  
State Probability ◽  
Arrival Process ◽  
Single Server ◽  
Probability Vector ◽  
Phase Type Distribution ◽  
Steady State Probability ◽  
Phase Type ◽  
Number Of Customers

We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.

scholarly journals Finite M/M/1 retrial model with changeable service rate

2021 ◽  
Vol 56 (1) ◽  
pp. 96-102
Author(s):  
M.S. Bratiichuk ◽  
A.A. Chechelnitsky ◽  
I.Ya. Usar
Keyword(s):  
Queueing System ◽  
Service Rate ◽  
Numerical Examples ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Explicit Formulae ◽  
Number Of Customers ◽  
Theoretical Results

The article deals with M/M/1 -type retrial queueing system with finite orbit. It is supposedthat service rate depends on the loading of the system. The explicit formulae for ergodicdistribution of the number of customers in the system are obtained. The theoretical results areillustrated by numerical examples.

ANALYSIS OF A MULTI-SERVER QUEUE WITH MARKOVIAN ARRIVALS AND SYNCHRONOUS PHASE TYPE VACATIONS

2009 ◽  
Vol 26 (01) ◽  
pp. 85-113 ◽  
Author(s):  
SRINIVAS R. CHAKRAVARTHY
Keyword(s):  
Steady State ◽  
Time Distribution ◽  
Queueing System ◽  
Queueing Model ◽  
Waiting Time Distribution ◽  
Steady State Analysis ◽  
Phase Type ◽  
Server Queue ◽  
Qbd Process ◽  
Multi Server

We study a MAP/M/c queueing system in which a group of servers take a simultaneous phase type vacation. The queueing model is studied as a QBD process. The steady-state analysis of the model including the waiting time distribution is presented. Interesting numerical results are discussed.

scholarly journals A k-out-of-n reliability system with an unreliable server and phase type repairs and services: the (N,T) policy

2001 ◽  
Vol 14 (4) ◽  
pp. 361-380 ◽  
Author(s):  
Srinivas R. Chakravarthy ◽  
A. Krishnamoorthy ◽  
P. V. Ushakumari
Keyword(s):  
Performance Measures ◽  
Matrix Analytic Methods ◽  
Steady State Analysis ◽  
Numerical Examples ◽  
Unreliable Server ◽  
Failure Times ◽  
Vacation Time ◽  
Phase Type ◽  
Repair Facility ◽  
Random Amount

In this paper we study a k-out-of-n reliability system in which a single unreliable server maintains n identical components. The reliability system is studied under the (N,T) policy. An idle server takes a vacation for a random amount of time T and then attends to any failed component waiting in line upon completion of the vacation. The vacationing server is recalled instantaneously upon the failure of the Nth component. The failure times of the components are assumed to follow an exponential distribution. The server is subject to failure with failure times exponentially distributed. Repair times of the component, fixing times of the server, and vacationing times of the server are assumed to be of phase type. Using matrix-analytic methods we perform steady state analysis of this model. Time spent by a failed component in service, total time in the repair facility, vacation time of the server, non-vacation time of the server, and time until failure of the system are all shown to be of phase type. Several performance measures are evaluated. Illustrative numerical examples are presented.

Study of a network of serial and non-serial servers with phase type service and finite queueing space

1972 ◽  
Vol 9 (01) ◽  
pp. 198-201 ◽  
Author(s):  
Krishan Lall Arya
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Steady State Solution ◽  
State Solution ◽  
System Service ◽  
Phase Type ◽  
In Series ◽  
Poisson Arrivals ◽  
Finite Space ◽  
Special Case

The paper develops the steady-state solution of a finite space queueing system wherein each of the two non-serial servers is separately in series with two non-serial servers. It is assumed that the arriving units of the same type may demand a different number of service phases. Poisson arrivals and exponential service times are assumed at all the four channels of the system. Service of units is completed on a first-come, first-served basis at each channel. The steady-state solution for infinite queueing space is obtained as a special case of finite queueing space.

An Unreliable Quorum Queueing System with Single and Multiple Vacations

2017 ◽  
Vol 34 (06) ◽  
pp. 1750036
Author(s):  
Lotfi Tadj
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Queueing System ◽  
System Size ◽  
Embedded Markov Chain ◽  
Queueing Models ◽  
Single Server ◽  
Multiple Vacations ◽  
Service Completion

This paper contributes to the literature of single server queueing models with a vacationing server. We have incorporated many features for a better control over the system. The server implements the N-policy, takes both single and multiple vacations, and is subject to breakdowns. The embedded Markov chain technique is used to obtain the pgf of the system size at a service completion epoch in the steady-state. The semi-regenerative technique is used to obtain the pgf of the system size at an arbitrary instant of time in the steady-state.

Study of a network of serial and non-serial servers with phase type service and finite queueing space

1972 ◽  
Vol 9 (1) ◽  
pp. 198-201 ◽  
Author(s):  
Krishan Lall Arya
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Steady State Solution ◽  
State Solution ◽  
System Service ◽  
Phase Type ◽  
In Series ◽  
Poisson Arrivals ◽  
Finite Space ◽  
Special Case

The paper develops the steady-state solution of a finite space queueing system wherein each of the two non-serial servers is separately in series with two non-serial servers. It is assumed that the arriving units of the same type may demand a different number of service phases. Poisson arrivals and exponential service times are assumed at all the four channels of the system. Service of units is completed on a first-come, first-served basis at each channel. The steady-state solution for infinite queueing space is obtained as a special case of finite queueing space.

scholarly journals Stability and moment bounds under utility-maximising service allocations: Finite and infinite networks

2020 ◽  
Vol 52 (2) ◽  
pp. 463-490
Author(s):  
Seva Shneer ◽  
Alexander Stolyar
Keyword(s):  
Steady State ◽  
Wide Class ◽  
Continuous Time ◽  
Service Rate ◽  
Fluid Limit ◽  
Natural Conditions ◽  
Infinite Systems ◽  
Moment Bounds ◽  
Infinite Networks ◽  
State Moment

AbstractWe study networks of interacting queues governed by utility-maximising service-rate allocations in both discrete and continuous time. For finite networks we establish stability and some steady-state moment bounds under natural conditions and rather weak assumptions on utility functions. These results are obtained using direct applications of Lyapunov–Foster-type criteria, and apply to a wide class of systems, including those for which fluid-limit-based approaches are not applicable. We then establish stability and some steady-state moment bounds for two classes of infinite networks, with single-hop and multi-hop message routes. These results are proved by considering the infinite systems as limits of their truncated finite versions. The uniform moment bounds for the finite networks play a key role in these limit transitions.

Analysis of R-out-of-N repairable systems: the case of phase-type distributions

2004 ◽  
Vol 36 (1) ◽  
pp. 116-138 ◽  
Author(s):  
Yonit Barron ◽  
Esther Frostig ◽  
Benny Levikson
Keyword(s):  
Repairable System ◽  
Repairable Systems ◽  
Regenerative Processes ◽  
Numerical Examples ◽  
Independent Components ◽  
Duality Results ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Two Systems ◽  
Repair Facility

An R-out-of-N repairable system, consisting of N independent components, is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R-1. A failed component is sent to a repair facility. After a failed component has been repaired it is as good as new. Formulae for the availability of the system using Markov renewal and semi-regenerative processes are derived. We assume that either the repair times of the components are generally distributed and the components' lifetimes are phase-type distributed or vice versa. Some duality results between the two systems are obtained. Numerical examples are given for several distributions of lifetimes and of repair times.

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