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.

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.

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.

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.

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.

Countable-state-space Markov chains with two time scales and applications to queueing systems

2002 ◽  
Vol 34 (3) ◽  
pp. 662-688 ◽  
Author(s):  
G. Yin ◽  
Hanqin Zhang
Keyword(s):  
Markov Chains ◽  
Time Scale ◽  
Transition Probability ◽  
Queueing Systems ◽  
Fast Time ◽  
State Spaces ◽  
Occupation Measures ◽  
Countable State ◽  
Probability Matrices ◽  
Markov Modulated

Motivated by various applications in queueing systems, this work is devoted to continuous-time Markov chains with countable state spaces that involve both fast-time scale and slow-time scale with the aim of approximating the time-varying queueing systems by their quasistationary counterparts. Under smoothness conditions on the generators, asymptotic expansions of probability vectors and transition probability matrices are constructed. Uniform error bounds are obtained, and then sequences of occupation measures and their functionals are examined. Mean square error estimates of a sequence of occupation measures are obtained; a scaled sequence of functionals of occupation measures is shown to converge to a Gaussian process with zero mean. The representation of the variance of the limit process is also explicitly given. The results obtained are then applied to treat Mt/Mt/1 queues and Markov-modulated fluid buffer models.

Filtering of Markov renewal queues, I: Feedback queues

1983 ◽  
Vol 15 (02) ◽  
pp. 349-375 ◽  
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.

Regularity of Stochastic Processes: A Theory Based on Directional Convexity

1993 ◽  
Vol 7 (3) ◽  
pp. 343-360 ◽  
Author(s):  
Ludolf E. Meester ◽  
J. George Shanthikumar
Keyword(s):  
Stochastic Processes ◽  
Large Class ◽  
Convex Functions ◽  
Queueing Systems ◽  
Single Stage ◽  
Renewal Processes ◽  
Doubly Stochastic ◽  
Directional Convexity ◽  
Stochastic Convexity ◽  
Markov Renewal Processes

We define a notion of regularity ordering among stochastic processes called directionally convex (dcx) ordering and give examples of doubly stochastic Poisson and Markov renewal processes where such ordering is prevalent. Further-more, we show that the class of segmented processes introduced by Chang, Chao, and Pinedo [3] provides a rich set of stochastic processes where the dcx ordering can be commonly encountered. When the input processes to a large class of queueing systems (single stage as well as networks) are dcx ordered, so are the processes associated with these queueing systems. For example, if the input processes to two tandem /M/c1→/M/c2→…→/M/cm queueing systems are dcx ordered, so are the numbers of customers in the systems. The concept of directionally convex functions (Shaked and Shanthikumar [15]) and the notion of multivariate stochastic convexity (Chang, Chao, Pinedo, and Shanthikumar [4]) are employed in our analysis.

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.

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

1983 ◽  
Vol 15 (2) ◽  
pp. 376-391 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Queue Length ◽  
Queueing Systems ◽  
Initial Study ◽  
Focused Attention ◽  
Renewal Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Processes ◽  
Birth Death

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.

Filtering of Markov renewal queues, I: Feedback queues

1983 ◽  
Vol 15 (2) ◽  
pp. 349-375 ◽  
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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