scholarly journals Superposition of renewal processes

1991 ◽  
Vol 23 (1) ◽  
pp. 64-85 ◽  
Author(s):  
C. Y. Teresalam ◽  
John P. Lehoczky
Keyword(s):  
Asymptotic Behavior ◽  
Queueing Systems ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Key Renewal Theorem

This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of queueing systems.

scholarly journals Superposition of renewal processes

1991 ◽  
Vol 23 (01) ◽  
pp. 64-85 ◽  
Author(s):  
C. Y. Teresalam ◽  
John P. Lehoczky
Keyword(s):  
Asymptotic Behavior ◽  
Queueing Systems ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Key Renewal Theorem

This paper extends the asymptotic results for ordinary renewal processes to the superposition of independent renewal processes. In particular, the ordinary renewal functions, renewal equations, and the key renewal theorem are extended to the superposition of independent renewal processes. We fix the number of renewal processes, p, and study the asymptotic behavior of the superposition process when time, t, is large. The key superposition renewal theorem is applied to the study of queueing systems.

Superposition of Markov renewal processes and applications

1993 ◽  
Vol 25 (3) ◽  
pp. 585-606 ◽  
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.

Uniform limit theorems for non-singular renewal and Markov renewal processes

1978 ◽  
Vol 15 (1) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie
Keyword(s):  
Recurrence Time ◽  
Renewal Processes ◽  
Convergence Properties ◽  
Renewal Theorem ◽  
Uniform Limit ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Positive Recurrent ◽  
Forward Recurrence Time ◽  
Initial Distributions

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

Superposition of Markov renewal processes and applications

1993 ◽  
Vol 25 (03) ◽  
pp. 585-606 ◽  
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.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

scholarly journals A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (2) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

A Diffusion Approximation for Markov Renewal Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 366-378
Author(s):  
Steven P. Clark ◽  
Peter C. Kiessler
Keyword(s):  
Markov Chain ◽  
Asymptotic Variance ◽  
Time Parameter ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Novel Method ◽  
Countably Infinite

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

Uniform limit theorems for non-singular renewal and Markov renewal processes

1978 ◽  
Vol 15 (01) ◽  
pp. 112-125 ◽  
Author(s):  
Elja Arjas ◽  
Esa Nummelin ◽  
Richard L. Tweedie
Keyword(s):  
Recurrence Time ◽  
Renewal Processes ◽  
Convergence Properties ◽  
Renewal Theorem ◽  
Uniform Limit ◽  
Markov Renewal Processes ◽  
Key Renewal Theorem ◽  
Positive Recurrent ◽  
Forward Recurrence Time ◽  
Initial Distributions

We show that if the increment distribution of a renewal process has some convolution non-singular with respect to Lebesgue measure, then the skeletons of the forward recurrence time process are φ-irreducible positive recurrent Markov chains. Known convergence properties of such chains give simple proofs of uniform versions of some old and new key renewal theorems; these show in particular that non-singularity assumptions on the increment and initial distributions enable the assumption of direct Riemann integrability to be dropped from the standard key renewal theorem. An application to Markov renewal processes is given.

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

1985 ◽  
Vol 17 (2) ◽  
pp. 386-407 ◽  
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.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

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