Recently discovered properties of generalised birth and death state-dependent queueing systems

1977 ◽  
Vol 9 (02) ◽  
pp. 216
Author(s):  
B. W. Conolly
Keyword(s):  
Queueing Systems ◽  
State Dependent ◽  
Death State

scholarly journals On intensities of modulated Cox measures

1998 ◽  
Vol 11 (3) ◽  
pp. 411-423 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Stochastic Process ◽  
Stochastic Models ◽  
Queueing Systems ◽  
Random Measure ◽  
Random Measures ◽  
Explicit Formulas ◽  
State Dependent

In this paper we introduce and study functionals of the intensities of random measures modulated by a stochastic process ξ, which occur in applications to stochastic models and telecommunications. Modulation of a random measure by ξ is specified for marked Cox measures. Particular cases of modulation by ξ as semi-Markov and semiregenerative processes enabled us to obtain explicit formulas for the named intensities. Examples in queueing (systems with state dependent parameters, Little's and Campbell's formulas) demonstrate the use of the results.

scholarly journals Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

2018 ◽  
Vol 28 (1) ◽  
pp. 141-154 ◽  
Author(s):  
Alexander Zeifman ◽  
Rostislav Razumchik ◽  
Yacov Satin ◽  
Ksenia Kiseleva ◽  
Anna Korotysheva ◽  
...  
Keyword(s):  
Rate Of Convergence ◽  
Queueing Systems ◽  
Approximation Error ◽  
Linear Operators ◽  
Upper And Lower Bounds ◽  
Logarithmic Norm ◽  
Batch Arrivals ◽  
State Dependent ◽  
The Mean ◽  
Number Of Customers

AbstractIn this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

scholarly journals A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

2013 ◽  
Vol 50 (3) ◽  
pp. 612-631 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Steady State ◽  
Storage Systems ◽  
Storage System ◽  
Queueing Systems ◽  
Local Maxima ◽  
State Dependent ◽  
State Densities ◽  
Duality Approach ◽  
And Storage ◽  
Level Process

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

scholarly journals Factorization Identities for Reflected Processes, with Applications

2013 ◽  
Vol 50 (03) ◽  
pp. 632-653
Author(s):  
Brian H. Fralix ◽  
Johan S. H. van Leeuwaarden ◽  
Onno J. Boxma
Keyword(s):  
Lévy Processes ◽  
Diffusion Processes ◽  
Queueing Systems ◽  
Levy Processes ◽  
Initial State ◽  
Batch Arrivals ◽  
Preemptive Resume ◽  
State Dependent ◽  
Arbitrary Initial State ◽  
And Diffusion

We derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Lévy processes, diffusion processes, and certain types of growth‒collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Lévy case, our identities simplify to both the well-known Wiener‒Hopf factorization, and another interesting factorization of reflected Lévy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth‒death processes and diffusion processes.

scholarly journals State-dependent M/G/1 queueing systems

2015 ◽  
Vol 82 (1-2) ◽  
pp. 121-148 ◽  
Author(s):  
Hossein Abouee-Mehrizi ◽  
Opher Baron
Keyword(s):  
Queueing Systems ◽  
State Dependent

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.

scholarly journals Factorization Identities for Reflected Processes, with Applications

2013 ◽  
Vol 50 (3) ◽  
pp. 632-653 ◽  
Author(s):  
Brian H. Fralix ◽  
Johan S. H. van Leeuwaarden ◽  
Onno J. Boxma
Keyword(s):  
Lévy Processes ◽  
Diffusion Processes ◽  
Queueing Systems ◽  
Levy Processes ◽  
Initial State ◽  
Batch Arrivals ◽  
Preemptive Resume ◽  
State Dependent ◽  
Arbitrary Initial State ◽  
And Diffusion

We derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Lévy processes, diffusion processes, and certain types of growth‒collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Lévy case, our identities simplify to both the well-known Wiener‒Hopf factorization, and another interesting factorization of reflected Lévy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth‒death processes and diffusion processes.

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