Filtering of Markov renewal queues, II: Birth-death queues

In this part we extend and particularise results developed by the author in Part I (pp. 349–375) for a class of queueing systems which can be formulated as Markov renewal processes. We examine those models where the basic transition consists of only two types: ‘arrivals' and ‘departures'. The ‘arrival lobby' and ‘departure lobby' queue-length processes are shown, using the results of Part I to be Markov renewal. Whereas the initial study focused attention on the behaviour of the embedded discrete-time Markov chains, in this paper we examine, in detail, the embedded continuous-time semi-Markov processes. The limiting distributions of the queue-length processes in both continuous and discrete time are derived and interrelationships between them are examined in the case of continuous-time birth–death queues including the M/M/1/M and M/M/1 variants. Results for discrete-time birth–death queues are also derived.

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Limit theorems for α-recurrent semi-Markov processes

Advances in Applied Probability ◽

10.1017/s0001867800042397 ◽

1976 ◽

Vol 8 (03) ◽

pp. 531-547 ◽

Cited By ~ 2

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.

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Filtering of Markov renewal queues, I: Feedback queues

Advances in Applied Probability ◽

10.1017/s0001867800021212 ◽

1983 ◽

Vol 15 (02) ◽

pp. 349-375 ◽

Cited By ~ 3

Author(s):

Jeffrey J. Hunter

Keyword(s):

Queue Length ◽

Queueing Systems ◽

Renewal Processes ◽

Focus Attention ◽

Bernoulli Feedback ◽

State Dependent ◽

Markov Renewal Processes ◽

Special Cases ◽

Filtering Procedure ◽

Feedback Queues

Queueing systems which can be formulated as Markov renewal processes with basic transitions of three types, ‘arrivals', ‘departures' and ‘feedbacks' are examined. The filtering procedure developed for Markov renewal processes by Çinlar (1969) is applied to such queueing models to show that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also Markov renewal. In this part we focus attention on the derivation of stationary and limiting distributions (when they exist) for each of the embedded discrete-time processes, the embedded Markov chains. These results are applied to birth–death queues with instantaneous state-dependent feedback including the special cases of M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback.

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Limit theorems for α-recurrent semi-Markov processes

Advances in Applied Probability ◽

10.2307/1426143 ◽

1976 ◽

Vol 8 (3) ◽

pp. 531-547 ◽

Cited By ~ 8

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.

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Filtering of Markov renewal queues, I: Feedback queues

Advances in Applied Probability ◽

10.2307/1426440 ◽

1983 ◽

Vol 15 (2) ◽

pp. 349-375 ◽

Cited By ~ 11

Author(s):

Jeffrey J. Hunter

Keyword(s):

Queue Length ◽

Queueing Systems ◽

Renewal Processes ◽

Focus Attention ◽

Bernoulli Feedback ◽

State Dependent ◽

Markov Renewal Processes ◽

Special Cases ◽

Filtering Procedure ◽

Feedback Queues

Queueing systems which can be formulated as Markov renewal processes with basic transitions of three types, ‘arrivals', ‘departures' and ‘feedbacks' are examined. The filtering procedure developed for Markov renewal processes by Çinlar (1969) is applied to such queueing models to show that the queue-length processes embedded at any of the ‘arrival', ‘departure', ‘feedback', ‘input', ‘output' or ‘external' transition epochs are also Markov renewal. In this part we focus attention on the derivation of stationary and limiting distributions (when they exist) for each of the embedded discrete-time processes, the embedded Markov chains. These results are applied to birth–death queues with instantaneous state-dependent feedback including the special cases of M/M/1/N and M/M/1 queues with instantaneous Bernoulli feedback.

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Filtering of Markov renewal queues, IV: Flow processes in feedback queues

Advances in Applied Probability ◽

10.2307/1427147 ◽

1985 ◽

Vol 17 (2) ◽

pp. 386-407 ◽

Cited By ~ 3

Author(s):

Jeffrey J. Hunter

Keyword(s):

Queue Length ◽

Queueing Systems ◽

Arrival Process ◽

Markov Renewal Process ◽

Renewal Processes ◽

Flow Process ◽

Special Cases ◽

Flow Processes ◽

Transition Points ◽

The Times

This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

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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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On the parity of individuals in a branching process

Journal of Applied Probability ◽

10.1017/s0021900200094286 ◽

1976 ◽

Vol 13 (02) ◽

pp. 219-230 ◽

Cited By ~ 9

Author(s):

J. Gani ◽

I. W. Saunders

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Branching Process ◽

Yeast Cells ◽

Death Process ◽

Second Moments ◽

The Mean ◽

Watson Process ◽

Number Of Cells ◽

Birth Death

This paper is concerned with the parity of a population of yeast cells, each of which may bud, not bud or die. Two multitype models are considered: a Galton-Watson process in discrete time, and its analogous birth-death process in continuous time. The mean number of cells with parity 0, 1, 2, … is obtained in both cases; some simple results are also derived for the second moments of the two processes.

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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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