The analysis of queues by state-dependent parameters by Markov renewal processes

1971 ◽  
Vol 3 (01) ◽  
pp. 155-175
Author(s):  
Manfred Schäl
Keyword(s):  
Limiting Distribution ◽  
Embedded Markov Chain ◽  
Renewal Processes ◽  
Original Process ◽  
Queueing Process ◽  
State Dependent ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
The Relationship

In this paper, some results on the asymptotic behavior of Markov renewal processes with auxiliary paths (MRPAP's) proved in other papers ([28], [29]) are applied to queueing theory. This approach to queueing problems may be regarded as an improvement of the method of Fabens [7] based on the theory of semi-Markov processes. The method of Fabens was also illustrated by Lambotte in [18], [32]. In the present paper the ordinary M/G/1 queue is generalized to allow service times to depend on the queue length immediately after the previous departure. Such models preserve the MRPAP-structure of the ordinary M/G/1 system. Recently, the asymptotic behaviour of the embedded Markov chain (MC) of this queueing model was studied by several authors. One aim of this paper is to answer the question of the relationship between the limiting distribution of the embedded MC and the limiting distribution of the original process with continuous time parameter. It turns out that these two limiting distributions coincide. Moreover some properties of the embedded MC and the embedded semi-Markov process are established. The discussion of the M/G/1 queue closes with a study of the rate-of-convergence at which the queueing process attains equilibrium.

The analysis of queues by state-dependent parameters by Markov renewal processes

1971 ◽  
Vol 3 (1) ◽  
pp. 155-175 ◽  
Author(s):  
Manfred Schäl
Keyword(s):  
Limiting Distribution ◽  
Embedded Markov Chain ◽  
Renewal Processes ◽  
Original Process ◽  
Queueing Process ◽  
State Dependent ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
The Relationship

In this paper, some results on the asymptotic behavior of Markov renewal processes with auxiliary paths (MRPAP's) proved in other papers ([28], [29]) are applied to queueing theory. This approach to queueing problems may be regarded as an improvement of the method of Fabens [7] based on the theory of semi-Markov processes. The method of Fabens was also illustrated by Lambotte in [18], [32]. In the present paper the ordinary M/G/1 queue is generalized to allow service times to depend on the queue length immediately after the previous departure. Such models preserve the MRPAP-structure of the ordinary M/G/1 system. Recently, the asymptotic behaviour of the embedded Markov chain (MC) of this queueing model was studied by several authors. One aim of this paper is to answer the question of the relationship between the limiting distribution of the embedded MC and the limiting distribution of the original process with continuous time parameter. It turns out that these two limiting distributions coincide. Moreover some properties of the embedded MC and the embedded semi-Markov process are established. The discussion of the M/G/1 queue closes with a study of the rate-of-convergence at which the queueing process attains equilibrium.

On the multiserver queue with finite waiting room and controlled input

1985 ◽  
Vol 17 (2) ◽  
pp. 408-423 ◽  
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Markov Chain ◽  
Queue Length ◽  
Limiting Distribution ◽  
Embedded Markov Chain ◽  
Single Server ◽  
Simple Relationship ◽  
Waiting Room ◽  
Original Process ◽  
State Dependent ◽  
Server System

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (03) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij (t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

Limit theorems for α-recurrent semi-Markov processes

1976 ◽  
Vol 8 (3) ◽  
pp. 531-547 ◽  
Author(s):  
Esa Nummelin
Keyword(s):  
Markov Processes ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transition Probabilities ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Process ◽  
Limit Behaviour ◽  
Semi Markov Processes ◽  
Special Case

In this paper the limit behaviour of α-recurrent Markov renewal processes and semi-Markov processes is studied by using the recent results on the concept of α-recurrence for Markov renewal processes. Section 1 contains the preliminary results, which are needed later in the paper. In Section 2 we consider the limit behaviour of the transition probabilities Pij(t) of an α-recurrent semi-Markov process. Section 4 deals with quasi-stationarity. Our results extend the results of Cheong (1968), (1970) and of Flaspohler and Holmes (1972) to the case in which the functions to be considered are directly Riemann integrable. We also try to correct the errors we have found in these papers. As a special case from our results we consider continuous-time Markov processes in Sections 3 and 5.

On the multiserver queue with finite waiting room and controlled input

1985 ◽  
Vol 17 (02) ◽  
pp. 408-423 ◽  
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Markov Chain ◽  
Queue Length ◽  
Limiting Distribution ◽  
Embedded Markov Chain ◽  
Single Server ◽  
Simple Relationship ◽  
Waiting Room ◽  
Original Process ◽  
State Dependent ◽  
Server System

In this paper we study a multi-channel queueing model of type with N waiting places and a non-recurrent input flow dependent on queue length at the time of each arrival. The queue length is treated as a basic process. We first determine explicitly the limit distribution of the embedded Markov chain. Then, by introducing an auxiliary Markov process, we find a simple relationship between the limiting distribution of the Markov chain and the limiting distribution of the original process with continuous time parameter. Here we simultaneously combine two methods: solving the corresponding Kolmogorov system of the differential equations, and using an approach based on the theory of semi-regenerative processes. Among various applications of multi-channel queues with state-dependent input stream, we consider a closed single-server system with reserve replacement and state-dependent service, which turns out to be dual (in a certain sense) in relation to our model; an optimization problem is also solved, and an interpretation by means of tandem systems is discussed.

Insensitivity of steady-state distributions of generalized semi-Markov processes by speeds

1978 ◽  
Vol 10 (04) ◽  
pp. 836-851 ◽  
Author(s):  
R. Schassberger
Keyword(s):  
Steady State ◽  
General System ◽  
State Distribution ◽  
Lifetime Distributions ◽  
State Dependent ◽  
Semi Markov Process ◽  
Semi Markov Processes ◽  
Infinite State ◽  
State Case ◽  
Residual Lifetimes

A generalized semi-Markov process with speeds describes the fluctuation, in time, of the state of a certain general system involving, at any given time, one or more living components, whose residual lifetimes are being reduced at state-dependent speeds. Conditions are given for the stationary state distribution, when it exists, to depend only on the means of some of the lifetime distributions, not their exact shapes. This generalizes results of König and Jansen, particularly to the infinite-state case.

Analysis of an M/G/1 queue with two types of impatient units

1995 ◽  
Vol 27 (03) ◽  
pp. 840-861 ◽  
Author(s):  
M. Martin ◽  
J. R. Artalejo
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Mathematical Analysis ◽  
Service System ◽  
Probability Distributions ◽  
Embedded Markov Chain ◽  
Renewal Processes ◽  
Stationary Probability ◽  
Markov Renewal Processes ◽  
Specific Performance

This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later. We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/l type and in expressions of ‘Takács' equation' type.

Central limit theorems for multivariate semi-Markov sequences and processes, with applications

1999 ◽  
Vol 36 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Central Limit ◽  
Limit Theorems ◽  
Markov Models ◽  
Central Limit Theorems ◽  
Renewal Processes ◽  
Counting Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Processes

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

Analysis of an M/G/1 queue with two types of impatient units

1995 ◽  
Vol 27 (3) ◽  
pp. 840-861 ◽  
Author(s):  
M. Martin ◽  
J. R. Artalejo
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Mathematical Analysis ◽  
Service System ◽  
Probability Distributions ◽  
Embedded Markov Chain ◽  
Renewal Processes ◽  
Stationary Probability ◽  
Markov Renewal Processes ◽  
Specific Performance

This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later.We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/l type and in expressions of ‘Takács' equation' type.

Exponential ergodicity in Markov renewal processes

1968 ◽  
Vol 5 (2) ◽  
pp. 387-400 ◽  
Author(s):  
Jozef L. Teugels
Keyword(s):  
Markov Renewal Process ◽  
Renewal Processes ◽  
Continuous Time Markov Chain ◽  
Uniform Estimates ◽  
Exponential Ergodicity ◽  
Markov Renewal Processes ◽  
Recurrent Case ◽  
Transient Case ◽  
Positive Recurrent ◽  
Semi Markov Processes

In [3], Kendall proved a solidarity theorem for irreducible denumerable discrete time Markov chains. Vere-Jones refined Kendall's theorem by obtaining uniform estimates [14], while Kingman proved analogous results for an irreducible continuous time Markov chain [4], [5].We derive similar solidarity theorems for an irreducible Markov renewal process. The transient case is discussed in Section 3, and Section 4 deals with the positive recurrent case. Recently Cheong also proved solidarity theorems for Semi-Markov processes [1]. His theorems use the Markovian structure, while our emphasis is on the renewal aspects of Markov renewal processes.An application to the M/G/1 queue is included in the last section.

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