Conditional expected sojourn times in insensitive queueing systems and networks

1984 ◽  
Vol 16 (4) ◽  
pp. 906-919 ◽  
Author(s):  
Uwe Jansen
Keyword(s):  
Queueing Networks ◽  
Point Processes ◽  
Response Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Main Tool ◽  
Closed Queueing Networks ◽  
Conditional Mean ◽  
Sojourn Times ◽  
Special Cases

We consider queueing systems where the stationary state probabilities are insensitive with respect to the distribution of certain basic random variables such as service requirements, interarrival times, repair times, etc. The conditional expected sojourn times are stated as Radon–Nikodym densities of the stationary distribution at jump points of the queueing system. The conditions are the given values of such basic random variables for which the insensitivity is valid. We use stationary point processes as our main tool. This means that dependences between certain basic random variables are permitted. Conditional expected real service times, conditional mean response times in closed queueing networks, and similar conditional expected values, are dealt with as special cases.

Conditional expected sojourn times in insensitive queueing systems and networks

1984 ◽  
Vol 16 (04) ◽  
pp. 906-919 ◽  
Author(s):  
Uwe Jansen
Keyword(s):  
Queueing Networks ◽  
Point Processes ◽  
Response Times ◽  
Queueing Systems ◽  
Random Variables ◽  
Main Tool ◽  
Closed Queueing Networks ◽  
Conditional Mean ◽  
Sojourn Times ◽  
Special Cases

We consider queueing systems where the stationary state probabilities are insensitive with respect to the distribution of certain basic random variables such as service requirements, interarrival times, repair times, etc. The conditional expected sojourn times are stated as Radon–Nikodym densities of the stationary distribution at jump points of the queueing system. The conditions are the given values of such basic random variables for which the insensitivity is valid. We use stationary point processes as our main tool. This means that dependences between certain basic random variables are permitted. Conditional expected real service times, conditional mean response times in closed queueing networks, and similar conditional expected values, are dealt with as special cases.

scholarly journals On the saturation rule for the stability of queues

1995 ◽  
Vol 32 (02) ◽  
pp. 494-507 ◽  
Author(s):  
François Baccelli ◽  
Serguei Foss
Keyword(s):  
Queueing Networks ◽  
Stability Region ◽  
Queueing Systems ◽  
Main Tool ◽  
Stationary Regime ◽  
Arrival Process ◽  
State Variables ◽  
Arrival Times ◽  
Harris Recurrence ◽  
The Stability

This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule: ‘saturate' the queues which are fed by the external arrival stream; look at the ‘intensity' μ of the departure stream in this saturated system; then stability holds whenever the intensity of the arrival process, say λ satisfies the condition λ < μ, whereas the network is unstable if λ > μ. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes-type backward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is subadditive ergodic theory. It is shown that, for various problems, the proposed method provides an alternative to the methods based on Harris recurrence and regeneration; this is particularly true in the Markov case, where we show that the distributional assumptions commonly made on service or arrival times so as to ensure Harris recurrence can in fact be relaxed.

Conditional job-observer property for multitype closed queueing networks

2002 ◽  
Vol 39 (4) ◽  
pp. 865-881 ◽  
Author(s):  
Hans Daduna ◽  
Ryszard Szekli
Keyword(s):  
Cycle Time ◽  
Queueing Networks ◽  
Closed Queueing Networks ◽  
Basic Formula ◽  
Sojourn Times ◽  
Number Of Customers ◽  
Insight Into

For functionals of multitype closed queueing networks, a conditional job-observer property is shown which provides more insight into the classical job-observer property. Applications and examples are given, including the classical job-observer property for the number of customers in a network, a representation of cycle time distributions and a basic formula for sojourn times.

scholarly journals A new look at transient versions of Little's law, and M/G/1 preemptive last-come-first-served queues

2010 ◽  
Vol 47 (2) ◽  
pp. 459-473 ◽  
Author(s):  
Brian H. Fralix ◽  
Germán Riaño
Keyword(s):  
Point Processes ◽  
Queueing System ◽  
Queueing Systems ◽  
Time Dependent ◽  
Structural Form ◽  
Preemptive Resume ◽  
Special Cases ◽  
Number Of Customers ◽  
Regulated Brownian Motion ◽  
New Look

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

Closed queueing networks with blocking-a model for flexible manufacturing systems

1984 ◽  
Vol 16 (1) ◽  
pp. 9-9
Author(s):  
David D. W. Yao ◽  
J.A. Buzacott
Keyword(s):  
Flexible Manufacturing ◽  
Queueing Networks ◽  
Manufacturing Systems ◽  
Waiting Times ◽  
Queueing Systems ◽  
Priority Queue ◽  
Single Server ◽  
Closed Queueing Networks ◽  
State Behavior ◽  
Dynamic Priority

We consider a family of single-server queueing systems with two priority classes. The system operates under a dynamic priority queue discipline in which the relative priorities of customers increase with their waiting times, and which can be characterized by the urgency number. We investigate the transient as well as the steady-state behavior of the virtual waiting times of the two classes of customer as functions of the urgency number. Stochastic orderings, the joint distribution, and surprising limit results for these processes are obtained for the first time.

scholarly journals A new look at transient versions of Little's law, and M/G/1 preemptive last-come-first-served queues

2010 ◽  
Vol 47 (02) ◽  
pp. 459-473 ◽  
Author(s):  
Brian H. Fralix ◽  
Germán Riaño
Keyword(s):  
Point Processes ◽  
Queueing System ◽  
Queueing Systems ◽  
Time Dependent ◽  
Structural Form ◽  
Preemptive Resume ◽  
Special Cases ◽  
Number Of Customers ◽  
Regulated Brownian Motion ◽  
New Look

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

1989 ◽  
Vol 26 (03) ◽  
pp. 552-565 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Stochastic Process ◽  
Discrete Time ◽  
Queueing Systems ◽  
Random Variables ◽  
Decomposition Theorem ◽  
Asymptotic Decomposition ◽  
Ergodic Decomposition ◽  
Special Cases ◽  
Unified View ◽  
Lindley Process

We consider a discrete-time stochastic process {Wn , n≧0} governed by i.i.d random variables {ξ n } whose distribution has support on (–∞,∞) and replacement random variables {Rn } whose distributions have support on [0,∞). Given Wn, Wn + 1 takes the value Wn + ζ n + 1 if it is non-negative. Otherwise Wn + 1 takes the value Rn + 1 where the distribution of Rn + 1 depends only on the value of Wn + ζn + 1 . This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

1989 ◽  
Vol 26 (3) ◽  
pp. 552-565 ◽  
Author(s):  
J. George Shanthikumar ◽  
Ushio Sumita
Keyword(s):  
Stochastic Process ◽  
Discrete Time ◽  
Queueing Systems ◽  
Random Variables ◽  
Decomposition Theorem ◽  
Asymptotic Decomposition ◽  
Ergodic Decomposition ◽  
Special Cases ◽  
Unified View ◽  
Lindley Process

We consider a discrete-time stochastic process {Wn, n≧0} governed by i.i.d random variables {ξ n} whose distribution has support on (–∞,∞) and replacement random variables {Rn} whose distributions have support on [0,∞). Given Wn, Wn+ 1 takes the value Wn + ζ n+ 1 if it is non-negative. Otherwise Wn+ 1 takes the value Rn +1 where the distribution of Rn+ 1 depends only on the value of Wn + ζn +1. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Conditional job-observer property for multitype closed queueing networks

2002 ◽  
Vol 39 (04) ◽  
pp. 865-881 ◽  
Author(s):  
Hans Daduna ◽  
Ryszard Szekli
Keyword(s):  
Cycle Time ◽  
Queueing Networks ◽  
Closed Queueing Networks ◽  
Basic Formula ◽  
Sojourn Times ◽  
Number Of Customers ◽  
Insight Into

For functionals of multitype closed queueing networks, a conditional job-observer property is shown which provides more insight into the classical job-observer property. Applications and examples are given, including the classical job-observer property for the number of customers in a network, a representation of cycle time distributions and a basic formula for sojourn times.

Sojourn times in closed queueing networks

1983 ◽  
Vol 15 (03) ◽  
pp. 638-656 ◽  
Author(s):  
F. P. Kelly ◽  
P. K. Pollett
Keyword(s):  
Free Path ◽  
Queueing Networks ◽  
Joint Distribution ◽  
The State ◽  
Product Form ◽  
Closed Queueing Networks ◽  
Jackson Network ◽  
Simple Representation ◽  
Sojourn Times

This paper obtains the stationary joint distribution of a customer's sojourn times along an overtake-free path in a closed multiclass Jackson network. The distribution has a simple representation in terms of the product form distribution for the state of the network at an arrival instant.

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