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

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.

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Non-Homogeneous Semi-Markov and Markov Renewal Processes and Change of Measure in Credit Risk

Mathematics ◽

10.3390/math9010055 ◽

2020 ◽

Vol 9 (1) ◽

pp. 55

Author(s):

P.-C.G. Vassiliou

Keyword(s):

Markov Chain ◽

Probability Measure ◽

Renewal Process ◽

Markov Chain Model ◽

Markov Renewal Process ◽

Renewal Processes ◽

Chain Model ◽

Markov Renewal Processes ◽

Risky Bonds ◽

Markov Structure

For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theorem that the forward probability measure does not alter the semi Markov structure. In addition, foundation of a G-inhohomogeneous Markov renewal process is done and a theorem is provided where it is proved that the Markov renewal process is maintained under the forward probability measure. We show that for an inhomogeneous semi-Markov there are martingales that characterize it. We show that the same is true for a Markov renewal processes. We discuss in depth the calibration of the G-inhomogeneous semi-Markov chain model and propose an algorithm for it. We conclude with an application for risky bonds.

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

Journal of Applied Probability ◽

10.1239/jap/1032374462 ◽

1999 ◽

Vol 36 (2) ◽

pp. 415-432 ◽

Cited By ~ 3

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.

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Superposition of Markov renewal processes and applications

Advances in Applied Probability ◽

10.2307/1427525 ◽

1993 ◽

Vol 25 (3) ◽

pp. 585-606 ◽

Cited By ~ 5

Author(s):

C. Teresa Lam

Keyword(s):

Queueing Networks ◽

Service System ◽

Queueing Systems ◽

Renewal Processes ◽

State Spaces ◽

System Availability ◽

Renewal Equations ◽

Markov Renewal Processes ◽

Countable State ◽

Bulk Service

In this paper, we study the superposition of finitely many Markov renewal processes with countable state spaces. We define the S-Markov renewal equations associated with the superposed process. The solutions of the S-Markov renewal equations are derived and the asymptotic behaviors of these solutions are studied. These results are applied to calculate various characteristics of queueing systems with superposition semi-Markovian arrivals, queueing networks with bulk service, system availability, and continuous superposition remaining and current life processes.

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

Advances in Applied Probability ◽

10.2307/1426332 ◽

1971 ◽

Vol 3 (1) ◽

pp. 155-175 ◽

Cited By ~ 16

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.

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

Advances in Applied Probability ◽

10.1017/s0001867800037617 ◽

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.

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Superposition of Markov renewal processes and applications

Advances in Applied Probability ◽

10.1017/s0001867800025568 ◽

1993 ◽

Vol 25 (03) ◽

pp. 585-606 ◽

Cited By ~ 1

Author(s):

C. Teresa Lam

Keyword(s):

Queueing Networks ◽

Service System ◽

Queueing Systems ◽

Renewal Processes ◽

State Spaces ◽

System Availability ◽

Renewal Equations ◽

Markov Renewal Processes ◽

Countable State ◽

Bulk Service

In this paper, we study the superposition of finitely many Markov renewal processes with countable state spaces. We define the S-Markov renewal equations associated with the superposed process. The solutions of the S-Markov renewal equations are derived and the asymptotic behaviors of these solutions are studied. These results are applied to calculate various characteristics of queueing systems with superposition semi-Markovian arrivals, queueing networks with bulk service, system availability, and continuous superposition remaining and current life processes.

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Markov renewal processes, counters and repeated sequences in Markov chains

Advances in Applied Probability ◽

10.1017/s000186780001675x ◽

1987 ◽

Vol 19 (03) ◽

pp. 521-545 ◽

Cited By ~ 7

Author(s):

J. D. Biggins ◽

C. Cannings

Keyword(s):

Markov Chain ◽

Markov Chains ◽

Restriction Enzymes ◽

Repeated Sequences ◽

Renewal Processes ◽

Basic Process ◽

Markov Renewal Processes ◽

Specific Sequences

The theory of Markov renewal processes is applied to study the occurrence of specific sequences of states in a Markov chain. Çinlar&s (1969) results are used to study both the basic process, and that obtained when the overlap of sequences is not permitted, as in the theory of counters. These results are applied to the fragments formed when DNA is digested using one, or more, restriction enzymes.

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Stationary probability distributions for a non-commutative Markov chain

Russian Mathematical Surveys ◽

10.1070/rm1995v050n06abeh002664 ◽

1995 ◽

Vol 50 (6) ◽

pp. 1284-1285

Author(s):

E A Morozova

Keyword(s):

Markov Chain ◽

Probability Distributions ◽

Stationary Probability

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Thinning of point processes—covariance analyses

Advances in Applied Probability ◽

10.1017/s0001867800014695 ◽

1985 ◽

Vol 17 (01) ◽

pp. 127-146

Author(s):

Jagadeesh Chandramohan ◽

Robert D. Foley ◽

Ralph L. Disney

Keyword(s):

Markov Chain ◽

Poisson Process ◽

Point Process ◽

Point Processes ◽

Arbitrary Point ◽

Renewal Processes ◽

Markov Renewal Processes ◽

Zero Correlation

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

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