scholarly journals Bounding steady-state availability models with phase type repair distributions

2002 ◽  
Author(s):  
J.A. Carrasco
Keyword(s):  
Steady State ◽  
Phase Type

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 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 On some tractable growth-collapse processes with renewal collapse epochs

2011 ◽  
Vol 48 (A) ◽  
pp. 217-234 ◽  
Author(s):  
Onno Boxma ◽  
Offer Kella ◽  
David Perry
Keyword(s):  
Steady State ◽  
Stationary Distribution ◽  
Unit Interval ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Positive Real ◽  
Ratio Distribution ◽  
Phase Type

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

Algorithms for a continuous-review (s, S) inventory system

1981 ◽  
Vol 18 (02) ◽  
pp. 461-472
Author(s):  
V. Ramaswami
Keyword(s):  
Steady State ◽  
Inventory System ◽  
Steady State Distribution ◽  
State Distribution ◽  
Continuous Review ◽  
Markov Modulated Poisson Process ◽  
Special Cases ◽  
Phase Type ◽  
Markov Modulated ◽  
Stimulation Of

The steady-state distribution of the inventory position for a continuous-review (s, S) inventory system is derived in a computationally tractable form. Demands for items in inventory are assumed to form an N-process which is the ‘versatile Markovian point process' introduced by Neuts (1979). The N-process includes the phase-type renewal process, Markov-modulated Poisson process etc., as special cases and is especially useful in modelling a wide variety of qualitative phenomena such as peaked arrivals, interruptions, inhibition or stimulation of arrivals by certain events etc.

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 On some tractable growth-collapse processes with renewal collapse epochs

2011 ◽  
Vol 48 (A) ◽  
pp. 217-234 ◽  
Author(s):  
Onno Boxma ◽  
Offer Kella ◽  
David Perry
Keyword(s):  
Steady State ◽  
Stationary Distribution ◽  
Unit Interval ◽  
General Distribution ◽  
Phase Type Distribution ◽  
Steady State Distribution ◽  
State Distribution ◽  
Positive Real ◽  
Ratio Distribution ◽  
Phase Type

In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

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.

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